[{"key":"Gibbons2006","type":"article","title":"Unbounded spigot algorithms for the digits of pi","author":"Gibbons, J.","abstract":"","comment":"There are several algorithms which produce digits of \u03c0 one at a time. This paper presents one which doesn't commit to a number of digits at the start, with Haskell code. The idea is that you write \u03c0 as the composition of infinitely many Mobius transforms, which are implemented as integer matrix multiplications.","date_added":"2010-01-05","date_published":"2006-08-06","urls":["http:\/\/www.cs.ox.ac.uk\/jeremy.gibbons\/publications\/spigot.pdf"],"collections":"Basically computer science,Fun maths facts","journal":"American Mathematical Monthly","number":"4","pages":"318--328","publisher":"Citeseer","url":"http:\/\/www.cs.ox.ac.uk\/jeremy.gibbons\/publications\/spigot.pdf","volume":"113","year":"2006","urldate":"2010-01-05"},{"key":"Quisquater","type":"article","title":"How to explain zero-knowledge protocols to your children","author":"Quisquater, JJ and Quisquater, M","abstract":"","comment":"A whimsical explanation of zero-knowledge protocols, by a story about Ali Baba's cave. The thieves, who know the secret phrase to open a door, can demonstrate that they can get out of the cave without telling Ali Bab how they do it.","date_added":"2010-03-24","date_published":"1989-08-06","urls":["https:\/\/link.springer.com\/chapter\/10.1007\/0-387-34805-0_60","http:\/\/pages.cs.wisc.edu\/~mkowalcz\/628.pdf"],"collections":"Attention-grabbing titles,Easily explained,Protocols and strategies","journal":"Advances in Cryptology\u2014 \\ldots","url":"https:\/\/link.springer.com\/chapter\/10.1007\/0-387-34805-0_60 http:\/\/pages.cs.wisc.edu\/~mkowalcz\/628.pdf","urldate":"2010-03-24","year":"1989"},{"key":"Socolar2010","type":"article","title":"An aperiodic hexagonal tile","author":"Socolar, Joshua E. S. and Taylor, Joan M.","abstract":"We show that a single tile can fill space uniformly but not admit a periodic tiling. The space--filling tiling that can be built from copies of the tile has the structure of a union of honeycombs with lattice constants of $2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all positive integer values. We present the tile, prove that the tilings it admits are not periodic, and discuss some of their remarkable properties, including their relation to a previously known tiling. We also clarify some subtleties in the definitions of the terms \"nonperiodic tiling\" and \"aperiodic tile\". For a reasonable interpretation of these terms, the tile presented here is the only known example of an aperiodic tile.","comment":"A single shape which tiles the plane, but not periodically. The catch is that it's not a single connected component.","date_added":"2010-03-26","date_published":"2010-03-01","urls":["http:\/\/arxiv.org\/abs\/1003.4279","http:\/\/arxiv.org\/pdf\/1003.4279v2"],"collections":"Geometry","keywords":"Combinatorics,Other Condensed Matter","month":"mar","pages":"21","url":"http:\/\/arxiv.org\/abs\/1003.4279 http:\/\/arxiv.org\/pdf\/1003.4279v2","year":"2010","archivePrefix":"arXiv","eprint":"1003.4279","primaryClass":"math.CO","urldate":"2010-03-26"},{"key":"Avigad2008","type":"article","title":"A formal system for Euclid's Elements","author":"Avigad, Jeremy and Dean, Edward and Mumma, John","abstract":"We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.","comment":"","date_added":"2010-04-04","date_published":"2008-10-01","urls":["http:\/\/arxiv.org\/abs\/0810.4315","http:\/\/arxiv.org\/pdf\/0810.4315v3"],"collections":"Geometry,About proof","keywords":"Logic","month":"oct","url":"http:\/\/arxiv.org\/abs\/0810.4315 http:\/\/arxiv.org\/pdf\/0810.4315v3","year":"2008","archivePrefix":"arXiv","eprint":"0810.4315","primaryClass":"math.LO","urldate":"2010-04-04"},{"key":"Hall2001","type":"article","title":"The Mathematics of Musical Instruments","author":"Hall, Rachel W. and Josic, Kresimir","abstract":"","comment":"","date_added":"2010-07-24","date_published":"2001-04-01","urls":["http:\/\/www.jstor.org\/stable\/2695241?origin=crossref"],"collections":"Music","journal":"The American Mathematical Monthly","month":"apr","number":"4","pages":"347","url":"http:\/\/www.jstor.org\/stable\/2695241?origin=crossref","volume":"108","year":"2001","urldate":"2010-07-24"},{"key":"Moritz2008","type":"book","title":"On Mathematics and Mathematicians","author":"Moritz, Robert Edoward","abstract":"ON MATHEMATICS AND MATHEMATICIANS Formerly titled Memorabilia Mathematica or the Philomathss Quotation-Book By Robert Edouard Moritz DOVER PUBLICATIONS INC., NEW YORK Copyright 1914 by Robert Edouard Moritz Copyright 1942 by Cassia K. Moritz This new Dover edition first published in 1958 is an unabridged and unaltered republication of the first edition which was originally en titled Memorabilia, Mathematical, or The Philo maths Quotation-Book. Manufactured in the United States of America Dover Publications, Inc. 920 Broadway New York 10, N. Y. PREFACE EVERY one knows that the fine phrase God geometrizes is attributed to Plato, but few know where this famous passage is found, or the exact words in which it was first expressed. Those who, like the author, have spent hours and even days in the search of the exact statements, or the exact references, of similar famous passages, will not question the timeliness and usefulness of a book whose distinct purpose it is to bring together into a single volume exact quotations, with their exact references, bearing on one of the most time-honored, and even today the most active and most fruitful of all the sciences, the queen mother of all the sciences, that is, mathematics. It is hoped that the present volume will prove indispensable to every teacher of mathematics, to every writer on mathe matics, and that the student of mathematics and the related sciences will find its perusal not only a source of pleasure but of encouragement and inspiration as well. The layman will find it a repository of useful information covering a field of knowledge which, owing to the unfamiliar and hence repellant character of the language employed by mathematicians, ispeculiarly in accessible to the general reader. No technical processes or technical facility is required to understand and appreciate the wealth of ideas here set forth in the words of the worlds great thinkers. No labor has been spared to make the present volume worthy of a place among collections of a like kind in other fields. Ten years have been devoted to its preparation, years, which if they could have been more profitably, could scarcely have been more pleasurably employed. As a result there have been brought together over one thousand more or less familiar passages pertaining to mathematics, by poets, philosophers, historians, statesmen, scientists, and mathematicians. These have been gathered from over three hundred authors, and have been vi PREFACE grouped under twenty heads, and cross indexed under nearly seven hundred topics. The authors original plan was to give foreign quotations both in the original and in translation, but with the growth of mate rial this plan was abandoned as infeasible. It was thought to serve the best interest of the greater number of English readers to give translations only, while preserving the references to the original sources, so that the student or critical reader may readily consult the original of any given extract. In cases where the translation is borrowed the translators name is inserted in brackets immediately after the authors name. Brackets are also used to indicate inserted words or phrases made necessary to bring out the context. The absence of similar English works has made the authors work largely that of the pioneer. Rebi res Math6matiques et Math naticiens and Ahrens Scherz und Ernst in der Mathematik have indeed been frequentlyconsulted but rather with a view to avoid overlapping than to receive aid. Thus certain topics as the correspondence of German and French mathematicians, so excellently treated by Ahrens, have pur posely been omitted. The repetitions are limited to a small number of famous utterances whose absence from a work of this kind could scarcely be defended on any grounds. No one can be more keenly aware of the shortcomings of a work than its author, for none can have so intimate an acquaint ance with it...","comment":"A collection of quotes about mathematics and mathematicians, from 1914.","date_added":"2010-08-24","date_published":"2008-08-06","urls":["http:\/\/books.google.com\/books?id=2sML4rpqucEC&pgis=1"],"collections":"The act of doing maths","pages":"448","publisher":"READ BOOKS","url":"http:\/\/books.google.com\/books?id=2sML4rpqucEC&pgis=1","year":"2008","urldate":"2010-08-24"},{"key":"Alpher1948","type":"article","title":"The Origin of Chemical Elements","author":"Alpher, R. and Bethe, H. and Gamow, G.","abstract":"","comment":"The \"Alpher-Bethe-Gamow\" paper.","date_added":"2010-09-03","date_published":"1948-04-01","urls":["http:\/\/en.wikipedia.org\/wiki\/Alpher%E2%80%93Bethe%E2%80%93Gamow_paper"],"collections":"Basically physics","journal":"Physical Review","month":"apr","number":"7","pages":"803--804","url":"http:\/\/en.wikipedia.org\/wiki\/Alpher%E2%80%93Bethe%E2%80%93Gamow_paper","volume":"73","year":"1948","urldate":"2010-09-03"},{"key":"Weber2002","type":"inproceedings","title":"The role of instrumental and relational understanding in proofs about group isomorphisms","author":"Weber, K.","abstract":"","comment":"\"Instrumental\" understanding is recalling an algorithm and being able to apply it. \"Relational\" understanding is knowing what the algorithm is for and why it works. The authors say you need relational understanding of group isomorphisms to be able to prove statements about them.","date_added":"2010-09-03","date_published":"2002-08-06","urls":["http:\/\/www.math.uoc.gr\/~ictm2\/Proceedings\/pap86.pdf"],"collections":"The act of doing maths,About proof,The groups group","booktitle":"Proceedings from the 2nd International Conference for the Teaching of Mathematics","url":"http:\/\/www.math.uoc.gr\/~ictm2\/Proceedings\/pap86.pdf","year":"2002","urldate":"2010-09-03"},{"key":"Grunbaum2006","type":"article","title":"What symmetry groups are present in the Alhambra?","author":"Gr\u00fcnbaum, Branko","abstract":"","comment":"Contrary to popular belief, not all the wallpaper groups are used in the decoration of the Alhambra. Some further musing on whether that's even a well-defined thing to say, and whether it's appropriate.","date_added":"2010-09-06","date_published":"2006-08-06","urls":["http:\/\/www.ams.org\/notices\/200606\/comm-grunbaum.pdf"],"collections":"History,Easily explained,Geometry","journal":"Notices of the American Mathematical Society","number":"ICM","pages":"2--5","url":"http:\/\/www.ams.org\/notices\/200606\/comm-grunbaum.pdf","year":"2006","urldate":"2010-09-06"},{"key":"Brunvand1963","type":"article","title":"A classification for shaggy dog stories","author":"Brunvand, J.H.","abstract":"","comment":"Over 300 shaggy dog stories, organised into six sections and over 200 types.","date_added":"2011-01-12","date_published":"1963-08-06","urls":["http:\/\/www.jstor.org\/stable\/538078"],"collections":"Lists and catalogues,Animals","journal":"Journal of American Folklore","number":"299","pages":"42--68","publisher":"JSTOR","url":"http:\/\/www.jstor.org\/stable\/538078","volume":"76","year":"1963","urldate":"2011-01-12"},{"key":"Ord2002","type":"article","title":"Hypercomputation: computing more than the Turing machine","author":"Ord, Toby","abstract":"Due to common misconceptions about the Church-Turing thesis, it has been\nwidely assumed that the Turing machine provides an upper bound on what is\ncomputable. This is not so. The new field of hypercomputation studies models of\ncomputation that can compute more than the Turing machine and addresses their\nimplications. In this report, I survey much of the work that has been done on\nhypercomputation, explaining how such non-classical models fit into the\nclassical theory of computation and comparing their relative powers. I also\nexamine the physical requirements for such machines to be constructible and the\nkinds of hypercomputation that may be possible within the universe. Finally, I\nshow how the possibility of hypercomputation weakens the impact of Godel's\nIncompleteness Theorem and Chaitin's discovery of 'randomness' within\narithmetic.","comment":"A probably overblown argument for a model of computation that does more than Turing machines.","date_added":"2011-02-09","date_published":"2002-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/0209332","http:\/\/arxiv.org\/pdf\/math\/0209332v1"],"collections":"Basically computer science,Unusual computers","journal":"Arxiv preprint math\/0209332","url":"http:\/\/arxiv.org\/abs\/math\/0209332 http:\/\/arxiv.org\/pdf\/math\/0209332v1","year":"2002","archivePrefix":"arXiv","eprint":"math\/0209332","primaryClass":"math.LO","urldate":"2011-02-09"},{"key":"Duarte2005","type":"article","title":"A discursive grammar for customizing mass housing: the case of Siza's houses at Malagueira","author":"Duarte, J","abstract":"","comment":"","date_added":"2011-02-15","date_published":"2005-03-01","urls":["http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0926580504000810"],"collections":"Basically computer science,Art","doi":"10.1016\/j.autcon.2004.07.013","issn":"09265805","journal":"Automation in Construction","keywords":"design automation,grammars,housing,mass customization,siza","month":"mar","number":"2","pages":"265--275","url":"http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0926580504000810","volume":"14","year":"2005","urldate":"2011-02-15"},{"key":"Blasjo","type":"article","title":"The isoperimetric problem","author":"Blasjo, Viktor","abstract":"","comment":"","date_added":"2011-03-14","date_published":"2001-08-06","urls":["http:\/\/books.google.com\/books?hl=en&lr=&id=HRr3jdBDXIgC&oi=fnd&pg=PA175&dq=The+Isoperimetric+Problem&ots=sh66HDGNF7&sig=98xmkIZ71_NHrpnKYs0WSQ0UFqc"],"collections":"Puzzles,Geometry","isbn":"0821835874","issn":"1534-6455","journal":"The Mathematical Association of America Montly","number":"112","pages":"526--566","publisher":"Amer Mathematical Society","url":"http:\/\/books.google.com\/books?hl=en&lr=&id=HRr3jdBDXIgC&oi=fnd&pg=PA175&dq=The+Isoperimetric+Problem&ots=sh66HDGNF7&sig=98xmkIZ71_NHrpnKYs0WSQ0UFqc","urldate":"2011-03-14","year":"2001"},{"key":"Comon1997","type":"article","title":"Tree automata techniques and applications","author":"Comon, Hubert and Dauchet, M and Gilleron, R","abstract":"","comment":"","date_added":"2011-03-25","date_published":"1997-08-06","urls":["http:\/\/en.scientificcommons.org\/42494218"],"collections":"Basically computer science","url":"http:\/\/en.scientificcommons.org\/42494218","year":"1997","urldate":"2011-03-25"},{"key":"Nollenburg2010","type":"article","title":"Drawing and Labeling High-Quality Metro Maps by Mixed-Integer Programming.","author":"N\u00f6llenburg, Martin and Wolff, Alexander","abstract":"Metro maps are schematic diagrams of public transport networks that serve as visual aids for route planning and navigation tasks. It is a challenging problem in network visualization to automatically draw appealing metro maps. There are two aspects to this problem that depend on each other: the layout problem of finding station and link coordinates and the labeling problem of placing non-overlapping station labels. In this paper we present a new integral approach that solves the combined layout and labeling problem (each of which, independently, is known to be NP-hard) using mixed-integer programming (MIP). We identify seven design rules used in most real-world metro maps. We split these rules into hard and soft constraints and translate them into a MIP model. Our MIP formulation finds a metro map that satisfies all hard constraints (if such a drawing exists) and minimizes a weighted sum of costs that correspond to the soft constraints. We have implemented the MIP model and present a case study and the results of an expert assessment to evaluate the performance of our approach in comparison to both manually designed official maps and results of previous layout methods.","comment":"","date_added":"2011-04-07","date_published":"2010-05-01","urls":["http:\/\/www.ncbi.nlm.nih.gov\/pubmed\/20498505"],"collections":"Basically computer science","doi":"10.1109\/TVCG.2010.81","issn":"1077-2626","journal":"IEEE transactions on visualization and computer graphics","month":"may","pages":"1--25","pmid":"20498505","url":"http:\/\/www.ncbi.nlm.nih.gov\/pubmed\/20498505","year":"2010","urldate":"2011-04-07"},{"key":"Khosla2002","type":"article","title":"Accurate estimation of forward path geometry using two-clothoid road model","author":"Khosla, D","abstract":"","comment":"","date_added":"2011-04-12","date_published":"2002-08-06","urls":["http:\/\/ieeexplore.ieee.org\/xpls\/abs_all.jsp?arnumber=1187944"],"collections":"Basically computer science,Geometry,Modelling","journal":"Intelligent Vehicle Symposium, 2002. IEEE","url":"http:\/\/ieeexplore.ieee.org\/xpls\/abs_all.jsp?arnumber=1187944","year":"2002","urldate":"2011-04-12"},{"key":"Boolos1994","type":"article","title":"G\u00f6del's Second Incompleteness Theorem Explained in Words of One Syllable","author":"Boolos, George","abstract":"","comment":"","date_added":"2011-09-24","date_published":"1994-08-06","urls":["http:\/\/mind.oxfordjournals.org\/cgi\/doi\/10.1093\/mind\/103.409.1"],"collections":"Easily explained,Fun maths facts","doi":"10.1093\/mind\/103.409.1","issn":"0026-4423","journal":"Mind","number":"409","pages":"1--3","url":"http:\/\/mind.oxfordjournals.org\/cgi\/doi\/10.1093\/mind\/103.409.1","volume":"103","year":"1994","urldate":"2011-09-24"},{"key":"Hardy1971","type":"article","title":"The elasto-plastic indentation of a half-space by a rigid sphere","author":"Hardy, C. and Baronet, C. N. and Tordion, G. V.","abstract":"","comment":"","date_added":"2011-10-02","date_published":"1971-10-01","urls":["http:\/\/doi.wiley.com\/10.1002\/nme.1620030402"],"collections":"","issn":"0029-5981","journal":"International Journal for Numerical Methods in Engineering","month":"oct","number":"4","pages":"451--462","url":"http:\/\/doi.wiley.com\/10.1002\/nme.1620030402","volume":"3","year":"1971","urldate":"2011-10-02"},{"key":"Calogero2003","type":"article","title":"Cool irrational numbers and their rather cool rational approximations","author":"Calogero, Francesco","abstract":"","comment":"","date_added":"2011-10-13","date_published":"2003-12-01","urls":["https:\/\/link.springer.com\/article\/10.1007%2FBF02984865"],"collections":"Attention-grabbing titles,Fun maths facts","issn":"0343-6993","journal":"The Mathematical Intelligencer","keywords":"Mathematics","month":"dec","number":"4","pages":"72--76","publisher":"Springer New York","url":"https:\/\/link.springer.com\/article\/10.1007%2FBF02984865","volume":"25","year":"2003","urldate":"2011-10-13"},{"key":"Ball2009","type":"article","title":"Scholarly communication in transition: The use of question marks in the titles of scientific articles in medicine, life sciences and physics 1966\u20132005","author":"Ball, Rafael","abstract":"The titles of scientific articles have a special significance. We examined nearly 20 million scientific articles and recorded the development of articles with a question mark at the end of their titles over the last 40 years. Our study was confined to the disciplines of physics, life sciences and medicine, where we found a significant increase from 50% to more than 200% in the number of articles with question-mark titles. We looked at the principle functions and structure of the titles of scientific papers, and we assume that marketing aspects are one of the decisive factors behind the growing usage of question-mark titles in scientific articles.","comment":"","date_added":"2011-10-14","date_published":"2009-01-01","urls":["https:\/\/link.springer.com\/article\/10.1007\/s11192-007-1984-5"],"collections":"History,The act of doing maths","issn":"0138-9130","journal":"Scientometrics","keywords":"Computer Science","month":"jan","number":"3","pages":"667--679","publisher":"Akad\u00e9miai Kiad\u00f3, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic Publishers B.V.","url":"https:\/\/link.springer.com\/article\/10.1007\/s11192-007-1984-5","volume":"79","year":"2009","urldate":"2011-10-14"},{"key":"Hilhorst2005","type":"article","title":"Asymptotic statistics of the n-sided planar Poisson\u2013Voronoi cell: I. Exact results","author":"Hilhorst, H.J.","abstract":"","comment":"","date_added":"2011-12-08","date_published":"2005-08-06","urls":["http:\/\/iopscience.iop.org\/1742-5468\/2005\/09\/P09005"],"collections":"Probability and statistics,Geometry","archivePrefix":"arXiv","arxivId":"arXiv:cond-mat\/0507567v1","eprint":"0507567v1","journal":"Journal of Statistical Mechanics: Theory and Experiment","keywords":"exact results,random graphs","pages":"P09005","primaryClass":"arXiv:cond-mat","publisher":"IOP Publishing","url":"http:\/\/iopscience.iop.org\/1742-5468\/2005\/09\/P09005","volume":"2005","year":"2005","urldate":"2011-12-08"},{"key":"Fink2000","type":"article","title":"Tie knots, random walks and topology","author":"Fink, T and Mao, Y","abstract":"","comment":"","date_added":"2011-12-09","date_published":"2000-02-01","urls":["http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0378437199002265"],"collections":"Probability and statistics","doi":"10.1016\/S0378-4371(99)00226-5","issn":"03784371","journal":"Physica A: Statistical Mechanics and its Applications","keywords":"knot theory,random walks,tie knots,topology","month":"feb","number":"1-2","pages":"109--121","url":"http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0378437199002265","volume":"276","year":"2000","urldate":"2011-12-09"},{"key":"Ekhad2011","type":"article","title":"How to Gamble If You're In a Hurry","author":"Ekhad, Shalosh B and Georgiadis, Evangelos and Zeilberger, Doron","abstract":"The beautiful theory of statistical gambling, started by Dubins and Savage (for subfair games) and continued by Kelly and Breiman (for superfair games) has mostly been studied under the unrealistic assumption that we live in a continuous world, that money is indefinitely divisible, and that our life is indefinitely long. Here we study these fascinating problems from a purely discrete, finitistic, and computational, viewpoint, using Both Symbol-Crunching and Number-Crunching (and simulation just for checking purposes).","comment":"","date_added":"2011-12-15","date_published":"2011-12-01","urls":["http:\/\/arxiv.org\/abs\/1112.1645","http:\/\/arxiv.org\/pdf\/1112.1645v2"],"collections":"Attention-grabbing titles,Games to play with friends,Probability and statistics","archivePrefix":"arXiv","arxivId":"1112.1645","eprint":"1112.1645","journal":"Strategy","keywords":"Computer Science and Game Theory,Probability","month":"dec","pages":"6","url":"http:\/\/arxiv.org\/abs\/1112.1645 http:\/\/arxiv.org\/pdf\/1112.1645v2","year":"2011","primaryClass":"math.PR","urldate":"2011-12-15"},{"key":"Petersen2010","type":"article","title":"Carrots for dessert","author":"Petersen, Carsten Lunde and Roesch, Pascale","abstract":"Carrots for dessert is the title of a section of the paper `On polynomial-like mappings' by Douady and Hubbard. In that section the authors define a notion of dyadic carrot fields of the Mandelbrot set M and more generally for Mandelbrot like families. They remark that such carrots are small when the dyadic denominator is large, but they do not even try to prove a precise such statement. In this paper we formulate and prove a precise statement of asymptotic shrinking of dyadic Carrot-fields around M. The same proof carries readily over to show that the dyadic decorations of copies M' of the Mandelbrot set M inside M and inside the parabolic Mandelbrot set shrink to points when the denominator diverge to infinity.","comment":"","date_added":"2012-01-02","date_published":"2010-03-01","urls":["http:\/\/arxiv.org\/abs\/1003.3947","http:\/\/arxiv.org\/pdf\/1003.3947v1"],"collections":"Attention-grabbing titles,Food","archivePrefix":"arXiv","arxivId":"1003.3947","eprint":"1003.3947","keywords":"Dynamical Systems","month":"mar","pages":"21","url":"http:\/\/arxiv.org\/abs\/1003.3947 http:\/\/arxiv.org\/pdf\/1003.3947v1","volume":"0","year":"2010","primaryClass":"math.DS","urldate":"2012-01-02"},{"key":"Gajardo2002","type":"article","title":"Complexity of Langton's ant","author":"Gajardo, A and Moreira, A and Goles, E","abstract":"","comment":"","date_added":"2012-01-04","date_published":"2002-08-06","urls":["http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0166218X00003346"],"collections":"Basically computer science,Easily explained,Animals","journal":"Discrete Applied Mathematics","keywords":"arti\u00ffcial life,complexity,universality,virtual ant","number":"1","pages":"41--50","publisher":"Elsevier","url":"http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0166218X00003346","volume":"117","year":"2002","urldate":"2012-01-04"},{"key":"Schmidt1966","type":"article","title":"On badly approximable numbers and certain games","author":"Schmidt, WM","abstract":"","comment":"","date_added":"2012-01-04","date_published":"1966-08-06","urls":["http:\/\/www.ams.org\/journals\/tran\/1966-123-01\/S0002-9947-1966-0195595-4\/S0002-9947-1966-0195595-4.pdf"],"collections":"Games to play with friends","journal":"Trans. Amer. Math. Soc","number":"C","pages":"178--199","url":"http:\/\/www.ams.org\/journals\/tran\/1966-123-01\/S0002-9947-1966-0195595-4\/S0002-9947-1966-0195595-4.pdf","year":"1966","urldate":"2012-01-04"},{"key":"Fisher2007","type":"article","title":"Three-dimensional finite point groups and the symmetry of beaded beads","author":"Fisher, GL and Mellor, B.","abstract":"","comment":"","date_added":"2012-01-04","date_published":"2007-06-01","urls":["http:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/17513470701416264","http:\/\/myweb.lmu.edu\/bmellor\/beadedbeads.pdf"],"collections":"Easily explained,Things to make and do,The groups group","doi":"10.1080\/17513470701416264","issn":"1751-3472","journal":"Journal of Mathematics and the Arts","keywords":"00a06,2000 mathematics subject classifications,20h15,51f25,a needle and thread,beaded bead,frieze group,introduction,polyhedron,section 1,symmetry,three-dimensional finite point group,to make decorative objects,to sew beads together,weavers of beads use","month":"jun","number":"2","pages":"85--96","publisher":"Taylor & Francis","url":"http:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/17513470701416264 http:\/\/myweb.lmu.edu\/bmellor\/beadedbeads.pdf","volume":"1","year":"2007","urldate":"2012-01-04"},{"key":"Funes1999","type":"article","title":"Computer evolution of buildable objects","author":"Funes, Pablo and Pollack, Jordan","abstract":"","comment":"","date_added":"2012-01-09","date_published":"1999-08-06","urls":["http:\/\/books.google.com\/books?hl=en&lr=&id=EgC6LBAH5r8C&oi=fnd&pg=PA387&dq=Computer+Evolution+of+Buildable+Objects&ots=DghBxZL9U8&sig=7N-w2EI4tyT7cZdbH9-F5SrHRk4"],"collections":"Basically computer science","journal":"Evolutionary design by computers","pages":"387--403","publisher":"Morgan Kaufmann, San Francisco","url":"http:\/\/books.google.com\/books?hl=en&lr=&id=EgC6LBAH5r8C&oi=fnd&pg=PA387&dq=Computer+Evolution+of+Buildable+Objects&ots=DghBxZL9U8&sig=7N-w2EI4tyT7cZdbH9-F5SrHRk4","year":"1999","urldate":"2012-01-09"},{"key":"Diaconis2011","type":"article","title":"Analysis of Casino Shelf Shuffling Machines","author":"Diaconis, Persi and Fulman, Jason and Holmes, Susan","abstract":"Many casinos routinely use mechanical card shuffling machines. We were asked to evaluate a new product, a shelf shuffler. This leads to new probability, new combinatorics, and to some practical advice which was adopted by the manufacturer. The interplay between theory, computing, and real-world application is developed.","comment":"","date_added":"2012-01-11","date_published":"2011-07-01","urls":["http:\/\/arxiv.org\/abs\/1107.2961","http:\/\/arxiv.org\/pdf\/1107.2961v2"],"collections":"Basically physics,Probability and statistics","archivePrefix":"arXiv","arxivId":"1107.2961","eprint":"1107.2961","journal":"Analysis","keywords":"Combinatorics,Probability","month":"jul","pages":"23","url":"http:\/\/arxiv.org\/abs\/1107.2961 http:\/\/arxiv.org\/pdf\/1107.2961v2","year":"2011","primaryClass":"math.CO","urldate":"2012-01-11"},{"key":"Viglietta2012","type":"article","title":"Gaming is a hard job, but someone has to do it!","author":"Viglietta, Giovanni","abstract":"We establish some general schemes relating the computational complexity of a video game to the presence of certain common elements or mechanics, such as destroyable paths, collecting items, doors activated by switches or pressure plates, etc.. Then we apply such \"metatheorems\" to several video games published between 1980 and 1998, including Pac-Man, Tron, Lode Runner, Boulder Dash, Deflektor, Mindbender, Pipe Mania, Skweek, Prince of Persia, Lemmings, Doom, Puzzle Bobble 3, and Starcraft. We obtain both new results, and improvements or alternative proofs of previously known results.","comment":"","date_added":"2012-01-27","date_published":"2012-01-01","urls":["http:\/\/arxiv.org\/abs\/1201.4995","http:\/\/arxiv.org\/pdf\/1201.4995v5"],"collections":"Basically computer science,Games to play with friends,Computational complexity of games","archivePrefix":"arXiv","arxivId":"1201.4995","eprint":"1201.4995","keywords":"Computational Complexity,Computer Science and Game Theory","month":"jan","pages":"12","url":"http:\/\/arxiv.org\/abs\/1201.4995 http:\/\/arxiv.org\/pdf\/1201.4995v5","year":"2012","primaryClass":"cs.CC","urldate":"2012-01-27"},{"key":"Moszkowski2011","type":"article","title":"Compositional Reasoning Using Intervals and Time Reversal","author":"Moszkowski, Ben","abstract":"","comment":"","date_added":"2012-01-27","date_published":"2011-09-01","urls":["http:\/\/ieeexplore.ieee.org\/lpdocs\/epic03\/wrapper.htm?arnumber=6065235"],"collections":"","doi":"10.1109\/TIME.2011.25","isbn":"978-1-4577-1242-5","journal":"2011 Eighteenth International Symposium on Temporal Representation and Reasoning","month":"sep","pages":"107--114","publisher":"Ieee","url":"http:\/\/ieeexplore.ieee.org\/lpdocs\/epic03\/wrapper.htm?arnumber=6065235","year":"2011","urldate":"2012-01-27"},{"key":"Wolf2010","type":"article","title":"Continued fractions constructed from prime numbers","author":"Wolf, Marek","abstract":"We give 50 digits values of the simple continued fractions whose denominators are formed from a) prime numbers, b) twin primes, c) generalized $d$-twins, d) primes of the form $m^2+n^4$, e)primes of the form $m^2+1$, f) Mersenne primes and g) primorial primes. All these continued fractions belong to the set of measure zero of exceptions to the theorems of Khinchin and Levy. We claim that all these continued fractions are transcendental numbers. Next we propose the conjecture which indicates the way to deduce the transcendence of some continued fractions from transcendence of another ones.","comment":"","date_added":"2012-02-01","date_published":"2010-03-01","urls":["http:\/\/arxiv.org\/abs\/1003.4015","http:\/\/arxiv.org\/pdf\/1003.4015v2"],"collections":"Easily explained","archivePrefix":"arXiv","arxivId":"1003.4015","eprint":"1003.4015","isbn":"9999637051","journal":"Arxiv preprint arXiv:1003.4015","keywords":"History and Overview,Number Theory","month":"mar","number":"1","pages":"35","url":"http:\/\/arxiv.org\/abs\/1003.4015 http:\/\/arxiv.org\/pdf\/1003.4015v2","volume":"1","year":"2010","primaryClass":"math.NT","urldate":"2012-02-01"},{"key":"Izhakian2005","type":"article","title":"Tropical Arithmetic and Tropical Matrix Algebra","author":"Izhakian, Zur","abstract":"This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.","comment":"","date_added":"2012-02-03","date_published":"2005-05-01","urls":["http:\/\/arxiv.org\/abs\/math\/0505458","http:\/\/arxiv.org\/pdf\/math\/0505458v3"],"collections":"Unusual arithmetic","archivePrefix":"arXiv","arxivId":"math\/0505458","eprint":"math\/0505458","journal":"ReCALL","keywords":"Algebraic Geometry,Combinatorics","month":"may","pages":"17","primaryClass":"math.AG","url":"http:\/\/arxiv.org\/abs\/math\/0505458 http:\/\/arxiv.org\/pdf\/math\/0505458v3","year":"2005","urldate":"2012-02-03"},{"key":"Hodge2010","type":"article","title":"Gerrymandering and Convexity","author":"Hodge, Jonathan K. and Marshall, Emily and Patterson, Geoff","abstract":"","comment":"","date_added":"2012-02-03","date_published":"2010-09-01","urls":["http:\/\/www.jstor.org\/stable\/10.4169\/074683410X510317"],"collections":"Easily explained,Geometry","doi":"10.4169\/074683410X510317","issn":"07468342","journal":"The College Mathematics Journal","month":"sep","number":"4","pages":"312--324","url":"http:\/\/www.jstor.org\/stable\/10.4169\/074683410X510317","volume":"41","year":"2010","urldate":"2012-02-03"},{"key":"Ball2011","type":"article","title":"In retrospect: On the Six-Cornered Snowflake","author":"Ball, Philip","abstract":"","comment":"","date_added":"2012-02-06","date_published":"2011-12-01","urls":["http:\/\/dx.doi.org\/10.1038\/480455a"],"collections":"","issn":"0028-0836","journal":"Nature","month":"dec","number":"7378","pages":"455--455","publisher":"Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.","shorttitle":"Nature","url":"http:\/\/dx.doi.org\/10.1038\/480455a","volume":"480","year":"2011","urldate":"2012-02-06"},{"key":"Meier1980","type":"article","title":"Benjamin Peirce and the Howland will","author":"Meier, Paul and Zabell, Sandy","abstract":"","comment":"","date_added":"2012-02-07","date_published":"1980-08-06","urls":["http:\/\/www.jstor.org\/stable\/10.2307\/2287637","http:\/\/ben-israel.rutgers.edu\/711\/Meier-Zabell.pdf"],"collections":"History","journal":"Journal of the American Statistical Association","keywords":"19th-century mathematical sued,charles s,distinguished,peirce,thomas blandel retained equally","number":"371","pages":"497--506","publisher":"JSTOR","url":"http:\/\/www.jstor.org\/stable\/10.2307\/2287637 http:\/\/ben-israel.rutgers.edu\/711\/Meier-Zabell.pdf","volume":"75","year":"1980","urldate":"2012-02-07"},{"key":"Speyer2004","type":"article","title":"Tropical Mathematics","author":"Speyer, David and Sturmfels, Bernd","abstract":"These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the ``tropical approach'' in mathematics, which has gotten a lot of attention recently in combinatorics, algebraic geometry and related fields. It offers an an elementary introduction to this subject, touching upon Arithmetic, Polynomials, Curves, Phylogenetics and Linear Spaces. Each section ends with a suggestion for further research. The bibliography contains numerousreferences for further reading in this field.","comment":"","date_added":"2012-02-12","date_published":"2004-08-01","urls":["http:\/\/arxiv.org\/abs\/math\/0408099","http:\/\/arxiv.org\/pdf\/math\/0408099v1"],"collections":"Unusual arithmetic,Easily explained","archivePrefix":"arXiv","arxivId":"math\/0408099","eprint":"math\/0408099","keywords":"Algebraic Geometry,Combinatorics","month":"aug","number":"July","pages":"15","primaryClass":"math.CO","url":"http:\/\/arxiv.org\/abs\/math\/0408099 http:\/\/arxiv.org\/pdf\/math\/0408099v1","year":"2004","urldate":"2012-02-12"},{"key":"Erlichson2003","type":"article","title":"Passage to the limit in Proposition I, Book I of Newton's Principia","author":"Erlichson, Herman","abstract":"","comment":"","date_added":"2012-02-15","date_published":"2003-11-01","urls":["http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0315086002000083"],"collections":"History","doi":"10.1016\/S0315-0860(02)00008-3","issn":"03150860","journal":"Historia Mathematica","month":"nov","number":"4","pages":"432--440","url":"http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0315086002000083","volume":"30","year":"2003","urldate":"2012-02-15"},{"key":"Bartholdi2012","type":"article","title":"Orange Peels and Fresnel Integrals","author":"Bartholdi, Laurent and Henriques, Andr\u00e9 G.","abstract":"There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral. The Euler spiral has applications in many fields of science. In optics, the illumination intensity at a point behind a slit is computed from the distance between two points on the Euler spiral. The Euler spiral also provides optimal curvature for train tracks between a straight run and an upcoming bend. It is striking that it can be also obtained with an orange and a kitchen knife.","comment":"","date_added":"2012-02-15","date_published":"2012-02-01","urls":["http:\/\/arxiv.org\/abs\/1202.3033","http:\/\/arxiv.org\/pdf\/1202.3033v1"],"collections":"Attention-grabbing titles,Easily explained,Things to make and do,Food,Geometry,Fun maths facts","archivePrefix":"arXiv","arxivId":"1202.3033","eprint":"1202.3033","journal":"Time","month":"feb","pages":"1--3","url":"http:\/\/arxiv.org\/abs\/1202.3033 http:\/\/arxiv.org\/pdf\/1202.3033v1","year":"2012","primaryClass":"math.HO","urldate":"2012-02-15"},{"key":"Pepperberg2012","type":"article","title":"Further evidence for addition and numerical competence by a Grey parrot (Psittacus erithacus)","author":"Pepperberg, Irene M.","abstract":"A Grey parrot ( Psittacus erithacus ), able to quantify sets of eight or fewer items (including heterogeneous subsets), to sum two sequentially presented sets of 0\u20136 items (up to 6), and to identify and serially order Arabic numerals (1\u20138), all by using English labels (Pepperberg in J Comp Psychol 108:36\u201344, 1994 ; J Comp Psychol 120:1\u201311, 2006a ; J Comp Psychol 120:205\u2013216, 2006b ; Pepperberg and Carey submitted), was tested on addition of two Arabic numerals or three sequentially presented collections (e.g., of variously sized jelly beans or nuts). He was, without explicit training and in the absence of the previously viewed addends, asked, \u201cHow many total?\u201d and required to answer with a vocal English number label. In a few trials on the Arabic numeral addition, he was also shown variously colored Arabic numerals while the addends were hidden and asked \u201cWhat color number (is the) total?\u201d Although his death precluded testing on all possible arrays, his accuracy was statistically significant and suggested addition abilities comparable with those of nonhuman primates.","comment":"","date_added":"2012-02-21","date_published":"2012-02-01","urls":["https:\/\/www.ncbi.nlm.nih.gov\/pubmed\/22402776"],"collections":"Animals","issn":"1435-9448","journal":"Animal Cognition","keywords":"Biomedical and Life Sciences","month":"feb","pages":"1--7","publisher":"Springer Berlin \/ Heidelberg","url":"https:\/\/www.ncbi.nlm.nih.gov\/pubmed\/22402776","year":"2012","urldate":"2012-02-21"},{"key":"Ghrist2008","type":"article","title":"Barcodes: the persistent topology of data","author":"Ghrist, Robert","abstract":"","comment":"","date_added":"2012-02-23","date_published":"2008-08-06","urls":["http:\/\/www.ams.org\/bull\/2008-45-01\/S0273-0979-07-01191-3\/S0273-0979-07-01191-3.pdf"],"collections":"Probability and statistics","journal":"Bulletin-American Mathematical Society","number":"3","pages":"1--15","url":"http:\/\/www.ams.org\/bull\/2008-45-01\/S0273-0979-07-01191-3\/S0273-0979-07-01191-3.pdf","year":"2008","urldate":"2012-02-23"},{"key":"Washington1986","type":"article","title":"Quotients Homophones des Groupes Libres Homophonic Quotients of Free Groups","author":"Washington, Lawrence and Zagier, Don","abstract":"","comment":"","date_added":"2012-02-27","date_published":"1986-08-06","urls":["http:\/\/people.mpim-bonn.mpg.de\/zagier\/files\/exp-math-2\/fulltext.pdf"],"collections":"Easily explained,The groups group","url":"http:\/\/people.mpim-bonn.mpg.de\/zagier\/files\/exp-math-2\/fulltext.pdf","year":"1986","urldate":"2012-02-27"},{"key":"VanLeeuwaarden2011","type":"article","title":"Random walks reaching against all odds the other side of the quarter plane","author":"van Leeuwaarden, Johan S. H. and Raschel, Kilian","abstract":"For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state $(i_0,j_0)$, we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive an exact expression for the probability of this event, and derive an asymptotic expression for the case when $i_0$ becomes large, a situation in which the event becomes highly unlikely. The exact expression follows from the solution of a boundary value problem and is in terms of an integral that involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model and the asymmetric exclusion process.","comment":"","date_added":"2012-03-01","date_published":"2011-04-01","urls":["http:\/\/arxiv.org\/abs\/1104.3034","http:\/\/arxiv.org\/pdf\/1104.3034v2"],"collections":"Probability and statistics","archivePrefix":"arXiv","arxivId":"1104.3034","eprint":"1104.3034","journal":"Plays","month":"apr","pages":"17","url":"http:\/\/arxiv.org\/abs\/1104.3034 http:\/\/arxiv.org\/pdf\/1104.3034v2","year":"2011","primaryClass":"math.PR","urldate":"2012-03-01"},{"key":"Ferguson","type":"misc","title":"A New Approximation to $\\pi$ (Conclusion)","author":"Ferguson, D. F. and Wrench, John W","abstract":"","comment":"","date_added":"2012-03-14","date_published":"1948-08-06","urls":["http:\/\/www.jstor.org\/discover\/10.2307\/2002657?uid=3738032&uid=2&uid=4&sid=47698756219137"],"collections":"Easily explained","url":"http:\/\/www.jstor.org\/discover\/10.2307\/2002657?uid=3738032&uid=2&uid=4&sid=47698756219137","urldate":"2012-03-14","year":"1948"},{"key":"Giblin1990","type":"article","title":"The bitangent sphere problem","author":"Giblin, PJ","abstract":"","comment":"","date_added":"2012-03-24","date_published":"1990-08-06","urls":["http:\/\/www.jstor.org\/stable\/10.2307\/2323998","http:\/\/dl.acm.org\/citation.cfm?id=87129"],"collections":"Geometry","journal":"American Mathematical Monthly","url":"http:\/\/www.jstor.org\/stable\/10.2307\/2323998 http:\/\/dl.acm.org\/citation.cfm?id=87129","year":"1990","urldate":"2012-03-24"},{"key":"Petersen2011","type":"article","title":"Statistical Laws Governing Fluctuations in Word Use from Word Birth to Word Death","author":"Petersen, Alexander M and Tenenbaum, Joel and Havlin, Shlomo and Stanley, H Eugene","abstract":"We analyze the dynamic properties of 10^7 words recorded in English, Spanish and Hebrew over the period 1800--2008 in order to gain insight into the coevolution of language and culture. We report language independent patterns useful as benchmarks for theoretical models of language evolution. A significantly decreasing (increasing) trend in the birth (death) rate of words indicates a recent shift in the selection laws governing word use. For new words, we observe a peak in the growth-rate fluctuations around 40 years after introduction, consistent with the typical entry time into standard dictionaries and the human generational timescale. Pronounced changes in the dynamics of language during periods of war shows that word correlations, occurring across time and between words, are largely influenced by coevolutionary social, technological, and political factors. We quantify cultural memory by analyzing the long-term correlations in the use of individual words using detrended fluctuation analysis.","comment":"","date_added":"2012-03-27","date_published":"2011-07-01","urls":["http:\/\/arxiv.org\/abs\/1107.3707","http:\/\/arxiv.org\/pdf\/1107.3707v2"],"collections":"Probability and statistics","archivePrefix":"arXiv","arxivId":"1107.3707","eprint":"1107.3707","journal":"Methods","month":"jul","pages":"31","url":"http:\/\/arxiv.org\/abs\/1107.3707 http:\/\/arxiv.org\/pdf\/1107.3707v2","year":"2011","primaryClass":"physics.soc-ph","urldate":"2012-03-27"},{"key":"Ianovski2012","type":"article","title":"Cake Cutting Mechanisms","author":"Ianovski, Egor","abstract":"We examine the history of cake cutting mechanisms and discuss the efficiency of their allocations. In the case of piecewise uniform preferences, we define a game that in the presence of strategic agents has equilibria that are not dominated by the allocations of any mechanism. We identify that the equilibria of this game coincide with the allocations of an existing cake cutting mechanism.","comment":"","date_added":"2012-04-07","date_published":"2012-03-01","urls":["http:\/\/arxiv.org\/abs\/1203.0100","http:\/\/arxiv.org\/pdf\/1203.0100v2"],"collections":"Puzzles,Easily explained,Protocols and strategies,Food","archivePrefix":"arXiv","arxivId":"1203.0100","eprint":"1203.0100","journal":"October","month":"mar","number":"March","url":"http:\/\/arxiv.org\/abs\/1203.0100 http:\/\/arxiv.org\/pdf\/1203.0100v2","year":"2012","primaryClass":"cs.GT","urldate":"2012-04-07"},{"key":"Chelghoum2004","type":"article","title":"Cellular automata in the hyperbolic plane: proposal for a new environment","author":"Chelghoum, Kamel and Margenstern, Maurice and Martin, Beno\\^it","abstract":"","comment":"","date_added":"2012-04-08","date_published":"2004-08-06","urls":["https:\/\/link.springer.com\/chapter\/10.1007\/978-3-540-30479-1_70"],"collections":"Basically computer science","journal":"Cellular Automata","pages":"1--11","url":"https:\/\/link.springer.com\/chapter\/10.1007\/978-3-540-30479-1_70","year":"2004","urldate":"2012-04-08"},{"key":"Lemoine2010","type":"article","title":"Computer analysis of Sprouts with nimbers","author":"Lemoine, Julien and Viennot, Simon","abstract":"Sprouts is a two-player topological game, invented in 1967 in the University of Cambridge by John Conway and Michael Paterson. The game starts with p spots, and ends in at most 3p-1 moves. The first player who cannot play loses. The complexity of the p-spot game is very high, so that the best hand-checked proof only shows who the winner is for the 7-spot game, and the best previous computer analysis reached p=11. We have written a computer program, using mainly two new ideas. The nimber (also known as Sprague-Grundy number) allows us to compute separately independent subgames; and when the exploration of a part of the game tree seems to be too difficult, we can manually force the program to search elsewhere. Thanks to these improvements, we reached up to p=32. The outcome of the 33-spot game is still unknown, but the biggest computed value is the 47-spot game ! All the computed values support the Sprouts conjecture: the first player has a winning strategy if and only if p is 3, 4 or 5 modulo 6. We have also used a check algorithm to reduce the number of positions needed to prove which player is the winner. It is now possible to hand-check all the games until p=11 in a reasonable amount of time.","comment":"","date_added":"2012-04-21","date_published":"2010-08-01","urls":["http:\/\/arxiv.org\/abs\/1008.2320","http:\/\/arxiv.org\/pdf\/1008.2320v1"],"collections":"Unusual arithmetic,Computational complexity of games","archivePrefix":"arXiv","arxivId":"1008.2320","eprint":"1008.2320","journal":"Analysis","month":"aug","pages":"17","url":"http:\/\/arxiv.org\/abs\/1008.2320 http:\/\/arxiv.org\/pdf\/1008.2320v1","year":"2010","primaryClass":"math.CO","urldate":"2012-04-21"},{"key":"Thompson1914","type":"book","title":"Calculus Made Easy","author":"Thompson, Silvanus P","abstract":"Being a very simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the DIFFERENTIAL CALCULUS and the INTEGRAL CALCULUS","comment":"","date_added":"2012-04-26","date_published":"1914-08-06","urls":["http:\/\/www.gutenberg.org\/ebooks\/33283"],"collections":"Education","url":"http:\/\/www.gutenberg.org\/ebooks\/33283","year":"1914","urldate":"2012-04-26"},{"key":"Vecer2009","type":"article","title":"Estimating the Effect of the Red Card in Soccer","author":"Vecer, Jan and Kopriva, Frantisek","abstract":"We study the effect of the red card in a soccer game. A red card is given by a referee to signify that a player has been sent off following a serious misconduct. The player who has been sent off must leave the game immediately and cannot be replaced during the game. His team must continue the game with one player fewer. We estimate the effect of the red card from betting data on the FIFA World Cup 2006 and Euro 2008, showing that the scoring intensity of the penalized team drops significantly, while the scoring intensity of the opposing team increases slightly. We show that a red card typically leads to a smaller number of goals scored during the game when a stronger team is penalized, but it can lead to an increased number of goals when a weaker team is punished. We also show when it is better to commit a red card offense in exchange for the prevention of a goal opportunity.","comment":"","date_added":"2012-05-09","date_published":"2009-08-06","urls":["http:\/\/www.stat.columbia.edu\/~vecer\/redcard.pdf"],"collections":"Easily explained","journal":"Journal of Quantitative Analysis in","number":"1992","pages":"1--13","url":"http:\/\/www.stat.columbia.edu\/~vecer\/redcard.pdf","year":"2009","urldate":"2012-05-09"},{"key":"Baez2012","type":"article","title":"G2 and the Rolling Ball","author":"Baez, John C and Huerta, John","abstract":"Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2: it acts as the symmetries of a 'spinorial ball rolling on a projective plane', again when the ratio of radii is 1:3. We explain this ratio in simple terms using the incidence geometry of G2, and show how a form of geometric quantization applied to this system gives the imaginary split octonions.","comment":"","date_added":"2012-05-14","date_published":"2012-05-01","urls":["http:\/\/arxiv.org\/abs\/1205.2447","http:\/\/arxiv.org\/pdf\/1205.2447v4"],"collections":"Basically physics","archivePrefix":"arXiv","arxivId":"1205.2447","eprint":"1205.2447","month":"may","pages":"28","url":"http:\/\/arxiv.org\/abs\/1205.2447 http:\/\/arxiv.org\/pdf\/1205.2447v4","year":"2012","primaryClass":"math.DG","urldate":"2012-05-14"},{"key":"Geiges2011","type":"article","title":"How to recognise a 4-ball when you see one","author":"Geiges, Hansj\u00f6rg and Zehmisch, Kai","abstract":"We apply the method of filling with holomorphic discs to a 4-dimensional symplectic cobordism with the standard contact 3-sphere as a convex boundary component. We establish the following dichotomy: either the cobordism is diffeomorphic to a ball, or there is a periodic Reeb orbit of quantifiably short period in the concave boundary of the cobordism. This allows us to give a unified treatment of various results concerning Reeb dynamics on contact 3-manifolds, symplectic fillability, the topology of symplectic cobordisms, symplectic non-squeezing, and the non-existence of exact Lagrangian surfaces in standard symplectic 4-space.","comment":"","date_added":"2012-05-19","date_published":"2011-04-01","urls":["http:\/\/arxiv.org\/abs\/1104.1543","http:\/\/arxiv.org\/pdf\/1104.1543v3"],"collections":"Attention-grabbing titles,Geometry","month":"apr","pages":"26","url":"http:\/\/arxiv.org\/abs\/1104.1543 http:\/\/arxiv.org\/pdf\/1104.1543v3","year":"2011","archivePrefix":"arXiv","eprint":"1104.1543","primaryClass":"math.SG","urldate":"2012-05-19"},{"key":"Ravsky2012","type":"article","title":"On an error in the star puzzle by Henry E. Dudeney","author":"Ravsky, Alex","abstract":"We found a solution of the star puzzle (a path on a chessboard from c5 to d4 in 14 straight strokes) in 14 queen moves, which has been claimed by the author as impossible.","comment":"","date_added":"2012-05-22","date_published":"2012-05-01","urls":["http:\/\/arxiv.org\/abs\/1205.0747","http:\/\/arxiv.org\/pdf\/1205.0747v2"],"collections":"Puzzles","archivePrefix":"arXiv","arxivId":"1205.0747","eprint":"1205.0747","journal":"Star","month":"may","pages":"2","url":"http:\/\/arxiv.org\/abs\/1205.0747 http:\/\/arxiv.org\/pdf\/1205.0747v2","year":"2012","primaryClass":"math.HO","urldate":"2012-05-22"},{"key":"Feist2012","type":"article","title":"Topology Explains Why Automobile Sunshades Fold Oddly","author":"Feist, Curtis and Naimi, Ramin","abstract":"We use braids and linking number to explain why automobile shades fold into an odd number of loops.","comment":"","date_added":"2012-05-23","date_published":"2012-05-01","urls":["http:\/\/arxiv.org\/abs\/1205.4797","http:\/\/arxiv.org\/pdf\/1205.4797v1"],"collections":"Easily explained,Basically physics,Fun maths facts","archivePrefix":"arXiv","arxivId":"1205.4797","eprint":"1205.4797","month":"may","pages":"8","url":"http:\/\/arxiv.org\/abs\/1205.4797 http:\/\/arxiv.org\/pdf\/1205.4797v1","year":"2012","primaryClass":"math.GT","urldate":"2012-05-23"},{"key":"Kindler2004","type":"article","title":"On distributions computable by random walks on graphs","author":"Kindler, G","abstract":"","comment":"","date_added":"2012-06-07","date_published":"2004-08-06","urls":["http:\/\/dl.acm.org\/citation.cfm?id=982809"],"collections":"","journal":"of the fifteenth annual ACM-SIAM symposium on","url":"http:\/\/dl.acm.org\/citation.cfm?id=982809","year":"2004","urldate":"2012-06-07"},{"key":"Isaksen2002","type":"article","title":"A cohomological viewpoint on elementary school arithmetic","author":"Isaksen, DC","abstract":"","comment":"","date_added":"2012-06-14","date_published":"2002-08-06","urls":["http:\/\/www.jstor.org\/stable\/10.2307\/3072368"],"collections":"Unusual arithmetic","journal":"The American mathematical monthly","number":"9","pages":"796--805","url":"http:\/\/www.jstor.org\/stable\/10.2307\/3072368","volume":"109","year":"2002","urldate":"2012-06-14"},{"key":"Kraft2001","type":"article","title":"The wobbly garden table","author":"Kraft, Hanspeter","abstract":"","comment":"","date_added":"2012-06-17","date_published":"2001-08-06","urls":["http:\/\/www.amsi.ge\/jbpc\/020101\/full\/9_18KR01F.pdf"],"collections":"Easily explained,Basically physics,Fun maths facts","journal":"J. Biol. Phys. Chem","number":"December","pages":"95--96","url":"http:\/\/www.amsi.ge\/jbpc\/020101\/full\/9_18KR01F.pdf","volume":"1","year":"2001","urldate":"2012-06-17"},{"key":"Baek2010","type":"article","title":"Equilibrium solution to the lowest unique positive integer game","author":"Baek, Seung Ki and Bernhardsson, Sebastian","abstract":"We address the equilibrium concept of a reverse auction game so that no one can enhance the individual payoff by a unilateral change when all the others follow a certain strategy. In this approach the combinatorial possibilities to consider become very much involved even for a small number of players, which has hindered a precise analysis in previous works. We here present a systematic way to reach the solution for a general number of players, and show that this game is an example of conflict between the group and the individual interests.","comment":"","date_added":"2012-06-22","date_published":"2010-01-01","urls":["http:\/\/arxiv.org\/abs\/1001.1065","http:\/\/arxiv.org\/pdf\/1001.1065v1"],"collections":"Protocols and strategies,Fun maths facts","archivePrefix":"arXiv","arxivId":"1001.1065","doi":"10.1142\/S0219477510000071","eprint":"1001.1065","keywords":"game,lowest unique positive integer,nash equilibrium,projection operator","month":"jan","number":"0","pages":"8","url":"http:\/\/arxiv.org\/abs\/1001.1065 http:\/\/arxiv.org\/pdf\/1001.1065v1","volume":"0","year":"2010","primaryClass":"math.CO","urldate":"2012-06-22"},{"key":"Chalcraft","type":"misc","title":"Train Sets","author":"Chalcraft, Adam and Greene, Michael","abstract":"","comment":"","date_added":"2012-06-25","date_published":"1994-08-06","urls":["http:\/\/www.monochrom.at\/turingtrainterminal\/Chalcraft.pdf"],"collections":"Easily explained,Things to make and do","url":"http:\/\/www.monochrom.at\/turingtrainterminal\/Chalcraft.pdf","urldate":"2012-06-25","year":"1994"},{"key":"Dubner2001","type":"article","title":"The top ten prime numbers","author":"Dubner, H","abstract":"","comment":"","date_added":"2012-06-28","date_published":"2001-08-06","urls":["http:\/\/primes.utm.edu\/lists\/top_ten\/topten.pdf"],"collections":"Easily explained,Lists and catalogues,Integerology","url":"http:\/\/primes.utm.edu\/lists\/top_ten\/topten.pdf","year":"2001","urldate":"2012-06-28"},{"key":"Sloane2006","type":"article","title":"Seven Staggering Sequences","author":"Sloane, N J A","abstract":"When the Handbook of Integer Sequences came out in 1973, Philip Morrison gave it an enthusiastic review in the Scientific American and Martin Gardner was kind enough to say in his Mathematical Games column for July 1974 that \"every recreational mathematician should buy a copy forthwith.\" That book contained 2372 sequences. Today the On-Line Encyclopedia of Integer Sequences (or OEIS) contains 117000 sequences. The following are seven that I find especially interesting. Many of them quite literally stagger. The sequences will be labeled with their numbers (such as A064413) in the OEIS. Much more information about them can be found there and in the references cited.","comment":"","date_added":"2012-07-14","date_published":"2006-08-06","urls":["http:\/\/neilsloane.com\/doc\/g4g7.pdf"],"collections":"Easily explained,Integerology","pages":"1--12","url":"http:\/\/neilsloane.com\/doc\/g4g7.pdf","year":"2006","urldate":"2012-07-14"},{"key":"Mamakani2012","type":"article","title":"A New Rose : The First Simple Symmetric 11-Venn Diagram","author":"Mamakani, Khalegh and Ruskey, Frank","abstract":"A symmetric Venn diagram is one that is invariant under rotation, up to a relabeling of curves. A simple Venn diagram is one in which at most two curves intersect at any point. In this paper we introduce a new property of Venn diagrams called crosscut symmetry, which is related to dihedral symmetry. Utilizing a computer search restricted to crosscut symmetry we found many simple symmetric Venn diagrams with 11 curves. This answers an existence question that has been open since the 1960's. The first such diagram that was discovered is shown here.","comment":"","date_added":"2012-08-09","date_published":"2012-07-01","urls":["http:\/\/arxiv.org\/abs\/1207.6452","http:\/\/arxiv.org\/pdf\/1207.6452v1"],"collections":"Art,Easily explained","archivePrefix":"arXiv","arxivId":"1207.6452","eprint":"1207.6452","keywords":"crosscut symmetry,hypercube,symmetric graphs,venn diagram","month":"jul","url":"http:\/\/arxiv.org\/abs\/1207.6452 http:\/\/arxiv.org\/pdf\/1207.6452v1","year":"2012","primaryClass":"cs.CG","urldate":"2012-08-09"},{"key":"Cui2012","type":"article","title":"The Canonical Basis of $\\dot{\\mathbf{U}}$ for Type $A_{2}$","author":"Cui, Weideng","abstract":"The modified quantized enveloping algebra has a remarkable basis, called the canonical basis, which was introduced by Lusztig. In this paper, all these monomial elements of the canonical basis for type $A_{2}$ are determined and we also give a conjecture about all polynomial elements of the canonical basis.","comment":"","date_added":"2012-08-29","date_published":"2012-08-01","urls":["http:\/\/arxiv.org\/abs\/1208.5531","http:\/\/arxiv.org\/pdf\/1208.5531v3"],"collections":"","archivePrefix":"arXiv","arxivId":"1208.5531","eprint":"1208.5531","keywords":"algebra,canonical basis,the modified quantized enveloping,the quasi-","month":"aug","url":"http:\/\/arxiv.org\/abs\/1208.5531 http:\/\/arxiv.org\/pdf\/1208.5531v3","year":"2012","primaryClass":"math.RT","urldate":"2012-08-29"},{"key":"Stuckman2005","type":"article","title":"Mastermind is NP-Complete","author":"Stuckman, Jeff and Zhang, Guo-Qiang","abstract":"In this paper we show that the Mastermind Satisfiability Problem (MSP) is NP-complete. The Mastermind is a popular game which can be turned into a logical puzzle called Mastermind Satisfiability Problem in a similar spirit to the Minesweeper puzzle. By proving that MSP is NP-complete, we reveal its intrinsic computational property that makes it challenging and interesting. This serves as an addition to our knowledge about a host of other puzzles, such as Minesweeper, Mah-Jongg, and the 15-puzzle.","comment":"","date_added":"2012-09-02","date_published":"2005-12-01","urls":["http:\/\/arxiv.org\/abs\/cs\/0512049","http:\/\/arxiv.org\/pdf\/cs\/0512049v1"],"collections":"Puzzles,Basically computer science,Games to play with friends,Computational complexity of games","archivePrefix":"arXiv","arxivId":"cs\/0512049","eprint":"cs\/0512049","journal":"Arxiv preprint cs\/0512049","keywords":"computational complexity,mastermind,theory of computation","month":"dec","pages":"1--7","primaryClass":"cs.CC","url":"http:\/\/arxiv.org\/abs\/cs\/0512049 http:\/\/arxiv.org\/pdf\/cs\/0512049v1","year":"2005","urldate":"2012-09-02"},{"key":"Kahan1998","type":"article","title":"How Java's floating-point hurts everyone everywhere","author":"Kahan, W and Darcy, JD","abstract":"","comment":"","date_added":"2012-09-05","date_published":"1998-08-06","urls":["http:\/\/port70.net\/~nsz\/articles\/float\/kahan_java_hurts_1998.pdf","http:\/\/www.cs.berkeley.edu\/~wkahan\/JAVAhurt.pdf"],"collections":"Basically computer science","journal":"\\ldots 1998 Workshop on Java \\ldots","pages":"1--81","url":"http:\/\/port70.net\/~nsz\/articles\/float\/kahan_java_hurts_1998.pdf http:\/\/www.cs.berkeley.edu\/~wkahan\/JAVAhurt.pdf","year":"1998","urldate":"2012-09-05"},{"key":"Sunic2011","type":"article","title":"Twin Towers of Hanoi","author":"Sunic, Zoran","abstract":"In the Twin Towers of Hanoi version of the well known Towers of Hanoi Problem there are two coupled sets of pegs. In each move, one chooses a pair of pegs in one of the sets and performs the only possible legal transfer of a disk between the chosen pegs (the smallest disk from one of the pegs is moved to the other peg), but also, simultaneously, between the corresponding pair of pegs in the coupled set (thus the same sequence of moves is always used in both sets). We provide upper and lower bounds on the length of the optimal solutions to problems of the following type. Given an initial and a final position of N disks in each of the coupled sets, what is the smallest number of moves needed to simultaneously obtain the final position from the initial one in each set? Our analysis is based on the use of a group, called Hanoi Towers group, of rooted ternary tree automorphisms, which models the original problem in such a way that the configurations on N disks are the vertices at level N of the tree and the action of the generators of the group represents the three possible moves between the three pegs. The twin version of the problem is analyzed by considering the action of Hanoi Towers group on pairs of vertices.","comment":"","date_added":"2012-09-28","date_published":"2011-08-01","urls":["http:\/\/arxiv.org\/abs\/1108.4494","http:\/\/arxiv.org\/pdf\/1108.4494v1"],"collections":"Puzzles,Easily explained","archivePrefix":"arXiv","arxivId":"1108.4494","eprint":"1108.4494","month":"aug","pages":"1--19","url":"http:\/\/arxiv.org\/abs\/1108.4494 http:\/\/arxiv.org\/pdf\/1108.4494v1","year":"2011","primaryClass":"math.CO","urldate":"2012-09-28"},{"key":"Coulbois2012","type":"article","title":"The topology of the minimal regular cover of the Archimedean tessellations","author":"Coulbois, Thierry and Pellicer, Daniel and Raggi, Miguel and Ram\u00edrez, Camilo and Valdez, Ferr\u00e1n","abstract":"In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.","comment":"","date_added":"2012-10-05","date_published":"2012-10-01","urls":["http:\/\/arxiv.org\/abs\/1210.1518","http:\/\/arxiv.org\/pdf\/1210.1518v1"],"collections":"Geometry","archivePrefix":"arXiv","arxivId":"1210.1518","eprint":"1210.1518","month":"oct","pages":"21","url":"http:\/\/arxiv.org\/abs\/1210.1518 http:\/\/arxiv.org\/pdf\/1210.1518v1","year":"2012","primaryClass":"math.GT","urldate":"2012-10-05"},{"key":"Bruijn1981","type":"article","title":"Algebraic theory of Penrose's non-periodic tilings of the plane","author":"Bruijn, NG De","abstract":"","comment":"","date_added":"2012-10-13","date_published":"1981-08-06","urls":["http:\/\/alexandria.tue.nl\/repository\/freearticles\/597566.pdf"],"collections":"Geometry","journal":"Kon. Nederl. Akad. Wetensch. Proc. Ser. A","url":"http:\/\/alexandria.tue.nl\/repository\/freearticles\/597566.pdf","year":"1981","urldate":"2012-10-13"},{"key":"Gerlach2010","type":"article","title":"On sphere-filling ropes","author":"Gerlach, Henryk and von der Mosel, Heiko","abstract":"What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others exhibit a striking resemblance to Turing patterns in chemistry, or to ordered phases of long elastic rods stuffed into spherical shells.","comment":"","date_added":"2012-10-29","date_published":"2010-05-01","urls":["http:\/\/arxiv.org\/abs\/1005.4609","http:\/\/arxiv.org\/pdf\/1005.4609v1"],"collections":"Easily explained,Geometry,Fun maths facts","archivePrefix":"arXiv","arxivId":"1005.4609","eprint":"1005.4609","journal":"Nature","month":"may","pages":"15","url":"http:\/\/arxiv.org\/abs\/1005.4609 http:\/\/arxiv.org\/pdf\/1005.4609v1","year":"2010","primaryClass":"math.GT","urldate":"2012-10-29"},{"key":"Pasles2001","type":"article","title":"The lost squares of Dr. Franklin: Ben Franklin's missing squares and the secret of the magic circle","author":"Pasles, PC","abstract":"","comment":"","date_added":"2012-10-30","date_published":"2001-08-06","urls":["http:\/\/www.jstor.org\/stable\/10.2307\/2695704"],"collections":"Puzzles,History","journal":"The American Mathematical Monthly","number":"6","pages":"489--511","url":"http:\/\/www.jstor.org\/stable\/10.2307\/2695704","volume":"108","year":"2001","urldate":"2012-10-30"},{"key":"Papy1970","type":"article","title":"Papy's Minicomputer","author":"Papy, F","abstract":"","comment":"","date_added":"2012-10-31","date_published":"1970-08-06","urls":["http:\/\/www.rkennes.be\/Papy-Minicomputer\/minicomp-anglais.pdf"],"collections":"Easily explained,Things to make and do,Education","journal":"Math Teaching","url":"http:\/\/www.rkennes.be\/Papy-Minicomputer\/minicomp-anglais.pdf","year":"1970","urldate":"2012-10-31"},{"key":"Khovanova2012","type":"article","title":"Conway's Wizards","author":"Khovanova, Tanya","abstract":"I present and discuss a puzzle about wizards invented by John H. Conway.","comment":"","date_added":"2012-11-03","date_published":"2012-10-01","urls":["http:\/\/arxiv.org\/abs\/1210.5460","http:\/\/arxiv.org\/pdf\/1210.5460v1"],"collections":"Puzzles,Easily explained","archivePrefix":"arXiv","arxivId":"1210.5460","eprint":"1210.5460","month":"oct","pages":"5","url":"http:\/\/arxiv.org\/abs\/1210.5460 http:\/\/arxiv.org\/pdf\/1210.5460v1","year":"2012","primaryClass":"math.HO","urldate":"2012-11-03"},{"key":"Alexander2012","type":"article","title":"Biologically Unavoidable Sequences","author":"Alexander, Samuel","abstract":"A biologically unavoidable sequence is an infinite gender sequence which occurs in every gendered, infinite genealogical network satisfying certain tame conditions. We show that every eventually periodic sequence is biologically unavoidable (this generalizes Koenig's Lemma), and we exhibit some biologically avoidable sequences. Finally we give an application of unavoidable sequences to cellular automata.","comment":"","date_added":"2012-12-16","date_published":"2012-12-01","urls":["http:\/\/arxiv.org\/abs\/1212.0186","http:\/\/arxiv.org\/pdf\/1212.0186v2"],"collections":"Fun maths facts","archivePrefix":"arXiv","arxivId":"1212.0186","eprint":"1212.0186","month":"dec","pages":"9","url":"http:\/\/arxiv.org\/abs\/1212.0186 http:\/\/arxiv.org\/pdf\/1212.0186v2","volume":"63","year":"2012","primaryClass":"math.CO","urldate":"2012-12-16"},{"key":"Wilcox2006","type":"article","title":"Invited commentary: the perils of birth weight--a lesson from directed acyclic graphs.","author":"Wilcox, Allen J","abstract":"The strong association of birth weight with infant mortality is complicated by a paradoxical finding: Small babies in high-risk populations usually have lower risk than small babies in low-risk populations. In this issue of the Journal, Hern\u00e1ndez-D\u00edaz et al. (Am J Epidemiol 2006;164:1115-20) address this \"birth weight paradox\" using directed acyclic graphs (DAGs). They conclude that the paradox is the result of bias created by adjustment for a factor (birth weight) that is affected by the exposure of interest and at the same time shares causes with the outcome (mortality). While this bias has been discussed before, the DAGs presented by Hern\u00e1ndez-D\u00edaz et al. provide more firmly grounded criticism. The DAGs demonstrate (as do many other examples) that seemingly reasonable adjustments can distort epidemiologic results. In this commentary, the birth weight paradox is shown to be an illustration of Simpson's Paradox. It is possible for a factor to be protective within every stratum of a variable and yet be damaging overall. Questions remain as to the causal role of birth weight.","comment":"","date_added":"2013-01-02","date_published":"2006-12-01","urls":["http:\/\/www.ncbi.nlm.nih.gov\/pubmed\/16931545","http:\/\/aje.oxfordjournals.org\/content\/164\/11\/1121.long"],"collections":"Easily explained,Probability and statistics","doi":"10.1093\/aje\/kwj276","issn":"0002-9262","journal":"American journal of epidemiology","keywords":"Causality,Confounding Factors (Epidemiology),Female,Humans,Infant,Infant Mortality,Low Birth Weight,Newborn,Prevalence,Risk Assessment,Risk Factors,Smoking,Smoking: adverse effects,Smoking: epidemiology,United States,United States: epidemiology","month":"dec","number":"11","pages":"1121--3; discussion 1124--5","pmid":"16931545","url":"http:\/\/www.ncbi.nlm.nih.gov\/pubmed\/16931545 http:\/\/aje.oxfordjournals.org\/content\/164\/11\/1121.long","volume":"164","year":"2006","urldate":"2013-01-02"},{"key":"Rothemund2004","type":"article","title":"Algorithmic self-assembly of DNA Sierpinski triangles.","author":"Rothemund, Paul W K and Papadakis, Nick and Winfree, Erik","abstract":"Algorithms and information, fundamental to technological and biological organization, are also an essential aspect of many elementary physical phenomena, such as molecular self-assembly. Here we report the molecular realization, using two-dimensional self-assembly of DNA tiles, of a cellular automaton whose update rule computes the binary function XOR and thus fabricates a fractal pattern--a Sierpinski triangle--as it grows. To achieve this, abstract tiles were translated into DNA tiles based on double-crossover motifs. Serving as input for the computation, long single-stranded DNA molecules were used to nucleate growth of tiles into algorithmic crystals. For both of two independent molecular realizations, atomic force microscopy revealed recognizable Sierpinski triangles containing 100-200 correct tiles. Error rates during assembly appear to range from 1% to 10%. Although imperfect, the growth of Sierpinski triangles demonstrates all the necessary mechanisms for the molecular implementation of arbitrary cellular automata. This shows that engineered DNA self-assembly can be treated as a Turing-universal biomolecular system, capable of implementing any desired algorithm for computation or construction tasks.","comment":"","date_added":"2013-01-18","date_published":"2004-12-01","urls":["http:\/\/dx.plos.org\/10.1371\/journal.pbio.0020424"],"collections":"","issn":"1545-7885","journal":"PLoS biology","keywords":"Algorithms,Base Sequence,Biophysics,Biophysics: methods,Computational Biology,Computational Biology: methods,Computer Simulation,Computers, Molecular,DNA,DNA: chemistry,Genetic Engineering,Microscopy, Atomic Force,Models, Genetic,Reproducibility of Results,Sequence Analysis, DNA,Ultraviolet Rays","month":"dec","number":"12","pages":"e424","publisher":"Public Library of Science","url":"http:\/\/dx.plos.org\/10.1371\/journal.pbio.0020424","volume":"2","year":"2004","urldate":"2013-01-18"},{"key":"Bielczyk2012","type":"article","title":"Delay can stabilize: Love affairs dynamics","author":"Bielczyk, Natalia and Bodnar, Marek and Fory\u015b, Urszula","abstract":"We discuss two models of interpersonal interactions with delay. The first model is linear, and allows the presentation of a rigorous mathematical analysis of stability, while the second is nonlinear and a typical local stability analysis is thus performed. The linear model is a direct extension of the classic Strogatz model. On the other hand, as interpersonal relations are nonlinear dynamical processes, the nonlinear model should better reflect real interactions. Both models involve immediate reaction on partner's state and a correction of the reaction after some time. The models we discuss belong to the class of two-variable systems with one delay for which appropriate delay stabilizes an unstable steady state. We formulate a theorem and prove that stabilization takes place in our case. We conclude that considerable (meaning large enough, but not too large) values of time delay involved in the model can stabilize love affairs dynamics.","comment":"","date_added":"2013-01-19","date_published":"2012-12-01","urls":["http:\/\/dx.doi.org\/10.1016\/j.amc.2012.10.028"],"collections":"Easily explained,Modelling","issn":"00963003","journal":"Applied Mathematics and Computation","keywords":"dyadic interactions,stability switches,stabilization of steady state,the hopf bifurcation,time delay","month":"dec","number":"8","pages":"3923--3937","url":"http:\/\/dx.doi.org\/10.1016\/j.amc.2012.10.028","volume":"219","year":"2012","urldate":"2013-01-19"},{"key":"Fiore2013","type":"article","title":"Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality","author":"Fiore, Thomas M. and Noll, Thomas and Satyendra, Ramon","abstract":"A familiar problem in neo-Riemannian theory is that the P, L, and R operations defined as contextual inversions on pitch-class segments do not produce parsimonious voice leading. We incorporate permutations into T\/I-PLR-duality to resolve this issue and simultaneously broaden the applicability of this duality. More precisely, we construct the dual group to the permutation group acting on n-tuples with distinct entries, and prove that the dual group to permutations adjoined with a group G of invertible affine maps Z12 -\\textgreater Z12 is the internal direct product of the dual to permutations and the dual to G. Musical examples include Liszt, R. W. Venezia, S. 201 and Schoenberg, String Quartet Number 1, Opus 7. We also prove that the Fiore--Noll construction of the dual group in the finite case works, and clarify the relationship of permutations with the RICH transformation.","comment":"","date_added":"2013-01-21","date_published":"2013-01-01","urls":["http:\/\/arxiv.org\/abs\/1301.4136","http:\/\/arxiv.org\/pdf\/1301.4136v1"],"collections":"Music,The groups group","month":"jan","pages":"15","url":"http:\/\/arxiv.org\/abs\/1301.4136 http:\/\/arxiv.org\/pdf\/1301.4136v1","year":"2013","archivePrefix":"arXiv","eprint":"1301.4136","primaryClass":"math.GR","urldate":"2013-01-21"},{"key":"Davis2012","type":"article","title":"Conway's Rational Tangles","author":"Davis, Tom","abstract":"","comment":"","date_added":"2013-02-06","date_published":"2012-08-06","urls":["http:\/\/www.geometer.org\/mathcircles\/tangle.pdf"],"collections":"Easily explained,Things to make and do,Fun maths facts","pages":"1--21","url":"http:\/\/www.geometer.org\/mathcircles\/tangle.pdf","year":"2012","urldate":"2013-02-06"},{"key":"Galvin1993","type":"article","title":"Embedding countable groups in 2-generator groups","author":"Galvin, Fred","abstract":"","comment":"","date_added":"2013-02-19","date_published":"1993-08-06","urls":["http:\/\/www.jstor.org\/stable\/10.2307\/2324618"],"collections":"Fun maths facts,The groups group","journal":"American Mathematical Monthly","number":"6","pages":"578--580","url":"http:\/\/www.jstor.org\/stable\/10.2307\/2324618","volume":"100","year":"1993","urldate":"2013-02-19"},{"key":"Hughes2007","type":"article","title":"The Muddy Children : A logic for public announcement","author":"Hughes, Jesse","abstract":"","comment":"","date_added":"2013-02-19","date_published":"2007-08-06","urls":["http:\/\/phiwumbda.org\/~jesse\/slides\/MuddyChildren.handouts.pdf"],"collections":"","url":"http:\/\/phiwumbda.org\/~jesse\/slides\/MuddyChildren.handouts.pdf","year":"2007","urldate":"2013-02-19"},{"key":"Ghang2013","type":"article","title":"Zeroless Arithmetic: Representing Integers ONLY using ONE","author":"Ghang, EK and Zeilberger, Doron","abstract":"We use recurrence equations (alias difference equations) to enumerate the number of formula-representations of positive integers using only addition and multiplication, and using addition, multiplication, and exponentiation, where all the inputs are ones. We also describe efficient algorithms for the random generation of such representations, and use Dynamical Programming to find a shortest possible formula representing any given positive integer.","comment":"","date_added":"2013-03-07","date_published":"2013-03-01","urls":["http:\/\/arxiv.org\/abs\/1303.0885","http:\/\/arxiv.org\/pdf\/1303.0885v2"],"collections":"Unusual arithmetic,Easily explained,Integerology","archivePrefix":"arXiv","arxivId":"1303.0885","eprint":"1303.0885","isbn":"1111111111","journal":"arXiv preprint arXiv:1303.0885","month":"mar","pages":"1--7","url":"http:\/\/arxiv.org\/abs\/1303.0885 http:\/\/arxiv.org\/pdf\/1303.0885v2","year":"2013","primaryClass":"math.CO","urldate":"2013-03-07"},{"key":"Cherowitzo2006","type":"article","title":"Constructing the Tits ovoid from an elliptic quadric","author":"Cherowitzo, WE","abstract":"","comment":"","date_added":"2013-03-19","date_published":"2006-08-06","urls":["http:\/\/www-math.ucdenver.edu\/~wcherowi\/research\/htmltalks\/ovoids\/ovoids.pdf"],"collections":"Geometry","journal":"To appear","url":"http:\/\/www-math.ucdenver.edu\/~wcherowi\/research\/htmltalks\/ovoids\/ovoids.pdf","year":"2006","urldate":"2013-03-19"},{"key":"Kranakis2010","type":"article","title":"The urinal problem","author":"Kranakis, Evangelos and Krizanc, Danny","abstract":"","comment":"","date_added":"2013-03-28","date_published":"2010-08-06","urls":["http:\/\/link.springer.com\/chapter\/10.1007\/978-3-642-13122-6_28","http:\/\/people.scs.carleton.ca\/~kranakis\/Papers\/urinal.pdf"],"collections":"Puzzles,Easily explained","journal":"Fun with Algorithms","number":"1","url":"http:\/\/link.springer.com\/chapter\/10.1007\/978-3-642-13122-6_28 http:\/\/people.scs.carleton.ca\/~kranakis\/Papers\/urinal.pdf","year":"2010","urldate":"2013-03-28"},{"key":"Goucher","type":"misc","title":"Circuitry in 3D chess","author":"Goucher, Adam","abstract":"","comment":"","date_added":"2013-04-03","date_published":"2013-08-06","urls":["http:\/\/cp4space.wordpress.com\/2013\/04\/03\/circuitry-in-3d-chess\/"],"collections":"","url":"http:\/\/cp4space.wordpress.com\/2013\/04\/03\/circuitry-in-3d-chess\/","urldate":"2013-04-03","year":"2013"},{"key":"Sauerland2013","type":"article","title":"Familial sinistrals avoid exact numbers.","author":"Sauerland, Uli and Gotzner, Nicole","abstract":"We report data from an internet questionnaire of sixty number trivia. Participants were asked for the number of cups in their house, the number of cities they know and 58 other quantities. We compare the answers of familial sinistrals - individuals who are left-handed themselves or have a left-handed close blood-relative - with those of pure familial dextrals - right-handed individuals who reported only having right-handed close blood-relatives. We show that familial sinistrals use rounder numbers than pure familial dextrals in the survey responses. Round numbers in the decimal system are those that are multiples of powers of 10 or of half or a quarter of a power of 10. Roundness is a gradient concept, e.g. 100 is rounder than 50 or 200. We show that very round number like 100 and 1000 are used with 25% greater likelihood by familial sinistrals than by pure familial dextrals, while pure familial dextrals are more likely to use less round numbers such as 25, 60, and 200. We then use Sigurd's (1988, Language in Society) index of the roundness of a number and report that familial sinistrals' responses are significantly rounder on average than those of pure familial dextrals. To explain the difference, we propose that the cognitive effort of using exact numbers is greater for the familial sinistral group because their language and number systems tend to be more distributed over both hemispheres of the brain. Our data support the view that exact and approximate quantities are processed by two separate cognitive systems. Specifically, our behavioral data corroborates the view that the evolutionarily older, approximate number system is present in both hemispheres of the brain, while the exact number system tends to be localized in only one hemisphere.","comment":"","date_added":"2013-04-15","date_published":"2013-01-01","urls":["http:\/\/dx.plos.org\/10.1371\/journal.pone.0059103"],"collections":"Easily explained,Probability and statistics","editor":"Stamatakis, Emmanuel Andreas","issn":"1932-6203","journal":"PloS one","keywords":"Anthropology,Behavior,Biology,Cognition,Cognitive neurology,Cognitive neuroscience,Cognitive psychology,Human families,Linguistic anthropology,Linguistics,Mathematics,Medicine,Mental Health,Neuroethology,Neurolinguistics,Neurological Disorders,Neurology,Neuropsychology,Neuroscience,Number concepts,Number theory,Psycholinguistics,Psychology,Research Article,Social and behavioral sciences,Sociology,Verbal behavior","month":"jan","number":"3","pages":"e59103","publisher":"Public Library of Science","url":"http:\/\/dx.plos.org\/10.1371\/journal.pone.0059103","volume":"8","year":"2013","urldate":"2013-04-15"},{"key":"Vierling-Claassen2013","type":"article","title":"Division of labor in child care: A game-theoretic approach","author":"Vierling-Claassen, a.","abstract":"","comment":"","date_added":"2013-05-07","date_published":"2013-05-01","urls":["http:\/\/rss.sagepub.com\/cgi\/doi\/10.1177\/1043463112473795"],"collections":"Easily explained,Protocols and strategies","doi":"10.1177\/1043463112473795","issn":"1043-4631","journal":"Rationality and Society","month":"may","number":"2","pages":"198--228","url":"http:\/\/rss.sagepub.com\/cgi\/doi\/10.1177\/1043463112473795","volume":"25","year":"2013","urldate":"2013-05-07"},{"key":"Coecke2005","type":"article","title":"Kindergarten Quantum Mechanics","author":"Coecke, Bob","abstract":"These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns `doing quantum mechanics using only pictures of lines, squares, triangles and diamonds'. This picture calculus can be seen as a very substantial extension of Dirac's notation, and has a purely algebraic counterpart in terms of so-called Strongly Compact Closed Categories (introduced by Abramsky and I in quant-ph\/0402130 and [4]) which subsumes my Logic of Entanglement quant-ph\/0402014. For a survey on the `what', the `why' and the `hows' I refer to a previous set of lecture notes quant-ph\/0506132. In a last section we provide some pointers to the body of technical literature on the subject.","comment":"","date_added":"2013-05-28","date_published":"2005-10-01","urls":["http:\/\/arxiv.org\/abs\/quant-ph\/0510032"],"collections":"","archivePrefix":"arXiv","arxivId":"quant-ph\/0510032","eprint":"0510032","keywords":"-a quantum information,-w quantum mechanics,03,65,67,category theory,dirac notation,graphical calculus,logic,pacs,quantum formalism","month":"oct","pages":"18","primaryClass":"quant-ph","url":"http:\/\/arxiv.org\/abs\/quant-ph\/0510032","year":"2005","urldate":"2013-05-28"},{"key":"Popoff2013","type":"article","title":"Using Monoidal Categories in the Transformational Study of Musical Time-Spans and Rhythms","author":"Popoff, Alexandre","abstract":"Transformational musical theory has so far mainly focused on the study of groups acting on musical chords, one of the most famous example being the action of the dihedral group D24 on the set of major and minor chords. Comparatively less work has been devoted to the study of transformations of time-spans and rhythms. D. Lewin was the first to study group actions on time-spans by using a subgroup of the affine group in one dimension. In our previous work, the work of Lewin has been included in the more general framework of group extensions, and generalizations to time-spans on multiple timelines have been proposed. The goal of this paper is to show that such generalizations have a categorical background in free monodical categories generated by a group-as-category. In particular, symmetric monodical categories allow to deal with the possible interexchanges between timelines. We also show that more general time-spans can be considered, in which single time-spans are encapsulated in a \"bracket\" of time-spans, which allow for the description of complex rhythms.","comment":"","date_added":"2013-06-02","date_published":"2013-05-01","urls":["http:\/\/arxiv.org\/abs\/1305.7192","http:\/\/arxiv.org\/pdf\/1305.7192v3"],"collections":"","archivePrefix":"arXiv","arxivId":"1305.7192","eprint":"1305.7192","month":"may","pages":"17","url":"http:\/\/arxiv.org\/abs\/1305.7192 http:\/\/arxiv.org\/pdf\/1305.7192v3","year":"2013","primaryClass":"math.GR","urldate":"2013-06-02"},{"key":"Khovanova2013","type":"article","title":"A Line of Sages","author":"Khovanova, Tanya","abstract":"","comment":"","date_added":"2013-06-20","date_published":"2013-08-06","urls":["http:\/\/tanyakhovanova.com\/publications\/ALineOfWizards.pdf"],"collections":"Puzzles","pages":"1--4","url":"http:\/\/tanyakhovanova.com\/publications\/ALineOfWizards.pdf","year":"2013","urldate":"2013-06-20"},{"key":"Conrey2012","type":"article","title":"Smooth neighbors","author":"Conrey, Brian and Holmstrom, Mark and McLaughlin, Tara","abstract":"We give a new algorithm that quickly finds smooth neighbors.","comment":"","date_added":"2013-07-11","date_published":"2012-12-01","urls":["http:\/\/arxiv.org\/abs\/1212.5161","http:\/\/arxiv.org\/pdf\/1212.5161v1"],"collections":"Integerology","archivePrefix":"arXiv","arxivId":"1212.5161","eprint":"1212.5161","month":"dec","pages":"1--14","url":"http:\/\/arxiv.org\/abs\/1212.5161 http:\/\/arxiv.org\/pdf\/1212.5161v1","year":"2012","primaryClass":"math.NT","urldate":"2013-07-11"},{"key":"Lehmer1964","type":"article","title":"On a problem of St\u00f6rmer","author":"Lehmer, DH","abstract":"","comment":"","date_added":"2013-07-11","date_published":"1964-08-06","urls":["http:\/\/projecteuclid.org\/euclid.ijm\/1256067456"],"collections":"Easily explained,Fun maths facts","journal":"Illinois Journal of Mathematics","url":"http:\/\/projecteuclid.org\/euclid.ijm\/1256067456","year":"1964","urldate":"2013-07-11"},{"key":"Haralambous1999","type":"article","title":"From Unicode to Typography, a Case Study the Greek Script","author":"Haralambous, Yannis","abstract":"","comment":"","date_added":"2013-07-22","date_published":"1999-08-06","urls":["http:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.30.4076","http:\/\/web.archive.org\/web\/20120229131933\/http:\/\/omega.enstb.org\/yannis\/pdf\/boston99.pdf"],"collections":"Basically computer science,Notation and conventions,History","journal":"\\ldots 14th International Unicode Conference, available from \\ldots","pages":"1--36","url":"http:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.30.4076 http:\/\/web.archive.org\/web\/20120229131933\/http:\/\/omega.enstb.org\/yannis\/pdf\/boston99.pdf","year":"1999","urldate":"2013-07-22"},{"key":"Fouhey2013","type":"article","title":"On n-Dimensional Polytope Schemes","author":"Fouhey, David F and Maturana, Daniel","abstract":"","comment":"","date_added":"2013-08-02","date_published":"2013-08-06","urls":["http:\/\/www.oneweirdkerneltrick.com\/polytope.pdf"],"collections":"Geometry","url":"http:\/\/www.oneweirdkerneltrick.com\/polytope.pdf","year":"2013","urldate":"2013-08-02"},{"key":"Braswell2013a","type":"article","title":"Cookie Monster Devours Naccis","author":"Braswell, Leigh Marie and Khovanova, Tanya","abstract":"In 2002, Cookie Monster appeared in The Inquisitive Problem Solver. The hungry monster wants to empty a set of jars filled with various numbers of cookies. On each of his moves, he may choose any subset of jars and take the same number of cookies from each of those jars. The Cookie Monster number is the minimum number of moves Cookie Monster must use to empty all of the jars. This number depends on the initial distribution of cookies in the jars. We discuss bounds of the Cookie Monster number and explicitly find the Cookie Monster number for Fibonacci, Tribonacci and other nacci sequences.","comment":"","date_added":"2013-10-09","date_published":"2013-05-01","urls":["http:\/\/arxiv.org\/abs\/1305.4305","http:\/\/arxiv.org\/pdf\/1305.4305v1"],"collections":"Attention-grabbing titles,Puzzles,Animals,Food,Fibonaccinalia","month":"may","pages":"8","url":"http:\/\/arxiv.org\/abs\/1305.4305 http:\/\/arxiv.org\/pdf\/1305.4305v1","year":"2013","archivePrefix":"arXiv","eprint":"1305.4305","primaryClass":"math.HO","urldate":"2013-10-09"},{"key":"Speyer2004a","type":"article","title":"Perfect Matchings and the Octahedron Recurrence","author":"Speyer, David E","abstract":"We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky.","comment":"","date_added":"2013-10-09","date_published":"2004-02-01","urls":["http:\/\/arxiv.org\/abs\/math\/0402452","http:\/\/arxiv.org\/pdf\/math\/0402452v2"],"collections":"","month":"feb","url":"http:\/\/arxiv.org\/abs\/math\/0402452 http:\/\/arxiv.org\/pdf\/math\/0402452v2","year":"2004","archivePrefix":"arXiv","eprint":"math\/0402452","primaryClass":"math.CO","urldate":"2013-10-09"},{"key":"Buchholz2013","type":"article","title":"An Infinite Set of Heron Triangles with Two Rational Medians","author":"Buchholz, Ralph H and Rathbun, Randall L","abstract":"","comment":"","date_added":"2013-10-09","date_published":"2013-08-06","urls":["http:\/\/www.jstor.org\/stable\/2974977"],"collections":"Geometry,Fun maths facts","number":"2","pages":"107--115","url":"http:\/\/www.jstor.org\/stable\/2974977","volume":"104","year":"2013","urldate":"2013-10-09"},{"key":"Fomin2001","type":"article","title":"The Laurent phenomenon","author":"Fomin, Sergey and Zelevinsky, Andrei","abstract":"A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of applications. In particular, we settle in the affirmative a conjecture of D.Gale and R.Robinson on integrality of generalized Somos sequences, and prove the Laurent property for several multidimensional recurrences, confirming conjectures by J.Propp, N.Elkies, and M.Kleber.","comment":"","date_added":"2013-10-09","date_published":"2001-04-01","urls":["http:\/\/arxiv.org\/abs\/math.CO\/0104241"],"collections":"","month":"apr","pages":"21","url":"http:\/\/arxiv.org\/abs\/math.CO\/0104241","year":"2001","urldate":"2013-10-09"},{"key":"Feinstein2010","type":"article","title":"Swiss cheeses, rational approximation and universal plane curves","author":"Feinstein, JF and Heath, MJ","abstract":"","comment":"","date_added":"2013-11-14","date_published":"2010-08-06","urls":["http:\/\/www.maths.nottingham.ac.uk\/personal\/jff\/Papers\/pdf\/Swisscheeses2.pdf"],"collections":"Attention-grabbing titles,Food,Geometry","journal":"Studia Math","keywords":"and phrases,rational approximation,swiss cheeses,uniform algebras","url":"http:\/\/www.maths.nottingham.ac.uk\/personal\/jff\/Papers\/pdf\/Swisscheeses2.pdf","year":"2010","urldate":"2013-11-14"},{"key":"Bulteau2011","type":"article","title":"Pancake Flipping is Hard","author":"Bulteau, Laurent and Fertin, Guillaume and Rusu, Irena","abstract":"Pancake Flipping is the problem of sorting a stack of pancakes of different\nsizes (that is, a permutation), when the only allowed operation is to insert a\nspatula anywhere in the stack and to flip the pancakes above it (that is, to\nperform a prefix reversal). In the burnt variant, one side of each pancake is\nmarked as burnt, and it is required to finish with all pancakes having the\nburnt side down. Computing the optimal scenario for any stack of pancakes and\ndetermining the worst-case stack for any stack size have been challenges over\nmore than three decades. Beyond being an intriguing combinatorial problem in\nitself, it also yields applications, e.g. in parallel computing and\ncomputational biology. In this paper, we show that the Pancake Flipping\nproblem, in its original (unburnt) variant, is NP-hard, thus answering the\nlong-standing question of its computational complexity.","comment":"","date_added":"2013-11-15","date_published":"2011-11-01","urls":["http:\/\/arxiv.org\/abs\/1111.0434","http:\/\/arxiv.org\/pdf\/1111.0434v2"],"collections":"Attention-grabbing titles,Puzzles,Basically computer science,Easily explained,Food","month":"nov","url":"http:\/\/arxiv.org\/abs\/1111.0434 http:\/\/arxiv.org\/pdf\/1111.0434v2","year":"2011","archivePrefix":"arXiv","eprint":"1111.0434","primaryClass":"cs.CC","urldate":"2013-11-15"},{"key":"Grau2011","type":"article","title":"Giuga Numbers and the arithmetic derivative","author":"Grau, Jos\u00e9 Mar\u00eda and Oller-Marc\u00e9n, Antonio M.","abstract":"We characterize Giuga Numbers as solutions to the equation $n'=an+1$, with $a\n\\in \\mathbb{N}$ and $n'$ being the arithmetic derivative. Although this fact\ndoes not refute Lava's conjecture, it brings doubts about its veracity.","comment":"","date_added":"2013-11-15","date_published":"2011-03-01","urls":["http:\/\/arxiv.org\/abs\/1103.2298","http:\/\/arxiv.org\/pdf\/1103.2298v1"],"collections":"Unusual arithmetic,Integerology","month":"mar","url":"http:\/\/arxiv.org\/abs\/1103.2298 http:\/\/arxiv.org\/pdf\/1103.2298v1","year":"2011","archivePrefix":"arXiv","eprint":"1103.2298","primaryClass":"math.NT","urldate":"2013-11-15"},{"key":"Sloane2013","type":"article","title":"2178 And All That","author":"Sloane, NJA","abstract":"For integers g >= 3, k >= 2, call a number N a (g,k)-reverse multiple if the\nreversal of N in base g is equal to k times N. The numbers 1089 and 2178 are\nthe two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089\nand 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to\nstudy the problem of finding all (g,k)-reverse multiples. By using modified\nversions of her trees, which we call Young graphs, we determine the possible\nvalues of k for bases g = 2 through 100, and then show how to apply the\ntransfer-matrix method to enumerate the (g,k)-reverse multiples with a given\nnumber of base-g digits. These Young graphs are interesting finite directed\ngraphs, whose structure is not at all well understood.","comment":"","date_added":"2013-11-20","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1307.0453","http:\/\/arxiv.org\/pdf\/1307.0453v4"],"collections":"Easily explained,Integerology","journal":"arXiv preprint arXiv:1307.0453","url":"http:\/\/arxiv.org\/abs\/1307.0453 http:\/\/arxiv.org\/pdf\/1307.0453v4","year":"2013","archivePrefix":"arXiv","eprint":"1307.0453","primaryClass":"math.NT","urldate":"2013-11-20"},{"key":"Mestrovic2011","type":"article","title":"Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)","author":"Mestrovic, Romeo","abstract":"In 1862 Wolstenholme proved that for any prime $p\\ge 5$ the numerator of the\nfraction $$ 1+\\frac 12 +\\frac 13+...+\\frac{1}{p-1}\n $$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of\nthe fraction\n $$ 1+\\frac{1}{2^2} +\\frac{1}{3^2}+...+\\frac{1}{(p-1)^2}\n $$ written in reduced form is divisible by $p$. The first of the above\ncongruences, the so called {\\it Wolstenholme's theorem}, is a fundamental\ncongruence in combinatorial number theory. In this article, consisting of 11\nsections, we provide a historical survey of Wolstenholme's type congruences and\nrelated problems. Namely, we present and compare several generalizations and\nextensions of Wolstenholme's theorem obtained in the last hundred and fifty\nyears. In particular, we present more than 70 variations and generalizations of\nthis theorem including congruences for Wolstenholme primes. These congruences\nare discussed here by 33 remarks.\n The Bibliography of this article contains 106 references consisting of 13\ntextbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of\nInteger Sequences. In this article, some results of these references are cited\nas generalizations of certain Wolstenholme's type congruences, but without the\nexpositions of related congruences. The total number of citations given here is\n189.","comment":"","date_added":"2013-11-21","date_published":"2011-11-01","urls":["http:\/\/arxiv.org\/abs\/1111.3057","http:\/\/arxiv.org\/pdf\/1111.3057v2"],"collections":"History,Integerology","month":"nov","url":"http:\/\/arxiv.org\/abs\/1111.3057 http:\/\/arxiv.org\/pdf\/1111.3057v2","year":"2011","archivePrefix":"arXiv","eprint":"1111.3057","primaryClass":"math.NT","urldate":"2013-11-21"},{"key":"Bell2008","type":"article","title":"The mathematics of Septoku","author":"Bell, George I.","abstract":"Septoku is a Sudoku variant invented by Bruce Oberg, played on a hexagonal grid of 37 cells. We show that up to rotations, reflections, and symbol permutations, there are only six valid Septoku boards. In order to have a unique solution, we show that the minimum number of given values is six. We generalize the puzzle to other board shapes, and devise a puzzle on a star-shaped board with 73 cells with six givens which has a unique solution. We show how this puzzle relates to the unsolved Hadwiger-Nelson problem in combinatorial geometry.","comment":"","date_added":"2013-12-22","date_published":"2008-01-01","urls":["http:\/\/arxiv.org\/abs\/0801.3697","http:\/\/arxiv.org\/pdf\/0801.3697v4"],"collections":"Puzzles,Easily explained","month":"jan","pages":"11","url":"http:\/\/arxiv.org\/abs\/0801.3697 http:\/\/arxiv.org\/pdf\/0801.3697v4","year":"2008","archivePrefix":"arXiv","eprint":"0801.3697","primaryClass":"math.CO","urldate":"2013-12-22"},{"key":"Middleton2013","type":"article","title":"Circular orbits on a warped spandex fabric","author":"Middleton, Chad A. and Langston, Michael","abstract":"We present a theoretical and experimental analysis of circular-like orbits made by a marble rolling on a warped spandex fabric. We show that the mass of the fabric interior to the orbital path influences the motion of the marble in a nontrivial way, and can even dominate the orbital characteristics. We also compare a Kepler-like expression for such orbits to similar expressions for orbits about a spherically-symmetric massive object in the presence of a constant vacuum energy, as described by general relativity.","comment":"","date_added":"2014-01-07","date_published":"2013-12-01","urls":["http:\/\/arxiv.org\/abs\/1312.3893","http:\/\/arxiv.org\/pdf\/1312.3893v1"],"collections":"Basically physics,Things to make and do,Geometry","month":"dec","pages":"13","url":"http:\/\/arxiv.org\/abs\/1312.3893 http:\/\/arxiv.org\/pdf\/1312.3893v1","year":"2013","archivePrefix":"arXiv","eprint":"1312.3893","primaryClass":"physics.class-ph","urldate":"2014-01-07"},{"key":"Benjamin2003","type":"article","title":"Linear recurrences through tilings and Markov chains","author":"Benjamin, AT and Hanusa, CRH and Su, FE","abstract":"","comment":"","date_added":"2014-01-21","date_published":"2003-08-06","urls":["http:\/\/math.hmc.edu\/~benjamin\/papers\/hanusa.pdf"],"collections":"","journal":"Utilitas Mathematica","pages":"3--17","url":"http:\/\/math.hmc.edu\/~benjamin\/papers\/hanusa.pdf","volume":"64","year":"2003","urldate":"2014-01-21"},{"key":"Hirsch2014","type":"article","title":"More ties than we thought","author":"Hirsch, Dan and Patterson, Meredith L and Sandberg, Anders and Vejdemo-Johansson, Mikael","abstract":"We extend the existing enumeration of neck tie knots to include tie knots with a textured front, tied with the narrow end of a tie. These tie knots have gained popularity in recent years, based on reconstructions of a costume detail from The Matrix Reloaded, and are explicitly ruled out in the enumeration by Fink and Mao (2000). We show that the relaxed tie knot description language that comprehensively describes these extended tie knot classes is either context sensitive or context free. It has a sub-language that covers all the knots that inspired the work, and that is regular. From this regular sub-language we enumerate 177 147 distinct tie knots that seem tieable with a normal necktie. These are found through an enumeration of 2 046 winding patterns that can be varied by tucking the tie under itself at various points along the winding.","comment":"","date_added":"2014-02-06","date_published":"2014-01-01","urls":["http:\/\/arxiv.org\/abs\/1401.8242","http:\/\/arxiv.org\/pdf\/1401.8242v2"],"collections":"Attention-grabbing titles,Easily explained","month":"jan","url":"http:\/\/arxiv.org\/abs\/1401.8242 http:\/\/arxiv.org\/pdf\/1401.8242v2","year":"2014","archivePrefix":"arXiv","eprint":"1401.8242","primaryClass":"cs.FL","urldate":"2014-02-06"},{"key":"Reifegerste2003","type":"article","title":"On the diagram of 132-avoiding permutations","author":"Reifegerste, Astrid","abstract":"","comment":"","date_added":"2014-03-31","date_published":"2003-08-01","urls":["http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0195669803000659"],"collections":"Combinatorics,The groups group","doi":"10.1016\/S0195-6698(03)00065-9","issn":"01956698","journal":"European Journal of Combinatorics","month":"aug","number":"6","pages":"759--776","url":"http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0195669803000659","volume":"24","year":"2003","urldate":"2014-03-31"},{"key":"Demontigny2013","type":"article","title":"Generalizing Zeckendorf's Theorem to f-decompositions","author":"Demontigny, Philippe and Do, Thao and Kulkarni, Archit and Miller, Steven J. and Moon, David and Varma, Umang","abstract":"A beautiful theorem of Zeckendorf states that every positive integer can be\nuniquely decomposed as a sum of non-consecutive Fibonacci numbers $\\{F_n\\}$,\nwhere $F_1 = 1$, $F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$. For general\nrecurrences $\\{G_n\\}$ with non-negative coefficients, there is a notion of a\nlegal decomposition which again leads to a unique representation, and the\nnumber of summands in the representations of uniformly randomly chosen $m \\in\n[G_n, G_{n+1})$ converges to a normal distribution as $n \\to \\infty$.\n We consider the converse question: given a notion of legal decomposition, is\nit possible to construct a sequence $\\{a_n\\}$ such that every positive integer\ncan be decomposed as a sum of terms from the sequence? We encode a notion of\nlegal decomposition as a function $f:\\N_0\\to\\N_0$ and say that if $a_n$ is in\nan \"$f$-decomposition\", then the decomposition cannot contain the $f(n)$ terms\nimmediately before $a_n$ in the sequence; special choices of $f$ yield many\nwell known decompositions (including base-$b$, Zeckendorf and factorial). We\nprove that for any $f:\\N_0\\to\\N_0$, there exists a sequence\n$\\{a_n\\}_{n=0}^\\infty$ such that every positive integer has a unique\n$f$-decomposition using $\\{a_n\\}$. Further, if $f$ is periodic, then the unique\nincreasing sequence $\\{a_n\\}$ that corresponds to $f$ satisfies a linear\nrecurrence relation. Previous research only handled recurrence relations with\nno negative coefficients. We find a function $f$ that yields a sequence that\ncannot be described by such a recurrence relation. Finally, for a class of\nfunctions $f$, we prove that the number of summands in the $f$-decomposition of\nintegers between two consecutive terms of the sequence converges to a normal\ndistribution.","comment":"","date_added":"2014-04-28","date_published":"2013-09-01","urls":["http:\/\/arxiv.org\/abs\/1309.5599","http:\/\/arxiv.org\/pdf\/1309.5599v1"],"collections":"Unusual arithmetic,Fibonaccinalia,Integerology","month":"sep","url":"http:\/\/arxiv.org\/abs\/1309.5599 http:\/\/arxiv.org\/pdf\/1309.5599v1","year":"2013","archivePrefix":"arXiv","eprint":"1309.5599","primaryClass":"math.NT","urldate":"2014-04-28"},{"key":"Khovanova2014","type":"article","title":"Nim Fractals","author":"Khovanova, Tanya and Xiong, Joshua","abstract":"We enumerate P-positions in the game of Nim in two different ways. In one series of sequences we enumerate them by the maximum number of counters in a pile. In another series of sequences we enumerate them by the total number of counters. We show that the game of Nim can be viewed as a cellular automaton, where the total number of counters divided by 2 can be considered as a generation in which P-positions are born. We prove that the three-pile Nim sequence enumerated by the total number of counters is a famous toothpick sequence based on the Ulam-Warburton cellular automaton. We introduce 10 new sequences.","comment":"","date_added":"2014-06-05","date_published":"2014-05-01","urls":["http:\/\/arxiv.org\/abs\/1405.5942","http:\/\/arxiv.org\/pdf\/1405.5942v1"],"collections":"Games to play with friends,Combinatorics","month":"may","pages":"19","url":"http:\/\/arxiv.org\/abs\/1405.5942 http:\/\/arxiv.org\/pdf\/1405.5942v1","year":"2014","archivePrefix":"arXiv","eprint":"1405.5942","primaryClass":"math.CO","urldate":"2014-06-05"},{"key":"Clifford2008","type":"article","title":"History-dependent random processes","author":"Clifford, P. and Stirzaker, D.","abstract":"Ulam has defined a history-dependent random sequence by the recursion Xn+1=Xn+XU(n), where (U(n); n[≥]1) is a sequence of independent random variables with U(n) uniformly distributed on {1, ..., n} and X1=1. We introduce a new class of continuous-time history-dependent random processes regulated by Poisson processes. The simplest of these, a univariate process regulated by a homogeneous Poisson process, replicates in continuous time the essential properties of Ulam's sequence, and greatly facilitates its analysis. We consider several generalizations and extensions of this, including bivariate and multivariate coupled history-dependent processes, and cases when the dependence on the past is not uniform. The analysis of the discrete-time formulations of these models would be at the very least an extremely formidable project, but we determine the asymptotic growth rates of their means and higher moments with relative ease.","comment":"","date_added":"2014-06-30","date_published":"2008-05-01","urls":["http:\/\/rspa.royalsocietypublishing.org\/content\/464\/2093\/1105"],"collections":"","issn":"1364-5021","journal":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","month":"may","number":"2093","pages":"1105--1124","url":"http:\/\/rspa.royalsocietypublishing.org\/content\/464\/2093\/1105","volume":"464","year":"2008","urldate":"2014-06-30"},{"key":"Petersen2004","type":"article","title":"An arctic circle theorem for groves","author":"Petersen, T. K. and Speyer, D.","abstract":"In earlier work, Jockusch, Propp, and Shor proved a theorem describing the limiting shape of the boundary between the uniformly tiled corners of a random tiling of an Aztec diamond and the more unpredictable `temperate zone' in the interior of the region. The so-called arctic circle theorem made precise a phenomenon observed in random tilings of large Aztec diamonds. Here we examine a related combinatorial model called groves. Created by Carroll and Speyer as combinatorial interpretations for Laurent polynomials given by the cube recurrence, groves have observable frozen regions which we describe precisely via asymptotic analysis of a generating function. Our approach also provides another way to prove the arctic circle theorem for Aztec diamonds.","comment":"","date_added":"2014-07-01","date_published":"2004-07-01","urls":["http:\/\/arxiv.org\/abs\/math\/0407171","http:\/\/arxiv.org\/pdf\/math\/0407171v1"],"collections":"","month":"jul","pages":"25","url":"http:\/\/arxiv.org\/abs\/math\/0407171 http:\/\/arxiv.org\/pdf\/math\/0407171v1","year":"2004","archivePrefix":"arXiv","eprint":"math\/0407171","primaryClass":"math.CO","urldate":"2014-07-01"},{"key":"Papadopoulos2014","type":"article","title":"Mathematics and group theory in music","author":"Papadopoulos, Athanase","abstract":"The purpose of this paper is to show through particular examples how group theory is used in music. The examples are chosen from the theoretical work and from the compositions of Olivier Messiaen (1908-1992), one of the most influential twentieth century composers and pedagogues. Messiaen consciously used mathematical concepts derived from symmetry and groups, in his teaching and in his compositions. Before dwelling on this, I will give a quick overview of the relation between mathematics and music. This will put the discussion on symmetry and group theory in music in a broader context and it will provide the reader of this handbook some background and some motivation for the subject. The relation between mathematics and music, during more than two millennia, was lively, widespread, and extremely enriching for both domains. This paper will appear in the Handbook of Group actions, vol. II (ed. L. Ji, A. Papadopoulos and S.-T. Yau), Higher Eucation Press and International Press.","comment":"","date_added":"2014-07-24","date_published":"2014-07-01","urls":["http:\/\/arxiv.org\/abs\/1407.5757","http:\/\/arxiv.org\/pdf\/1407.5757v1"],"collections":"Music,The groups group","month":"jul","url":"http:\/\/arxiv.org\/abs\/1407.5757 http:\/\/arxiv.org\/pdf\/1407.5757v1","year":"2014","archivePrefix":"arXiv","eprint":"1407.5757","primaryClass":"math.HO","urldate":"2014-07-24"},{"key":"Legendre2013","type":"article","title":"Foldings and Meanders","author":"Legendre, St\u00e9phane","abstract":"We review the stamp folding problem, the number of ways to fold a strip of $n$ stamps, and the related problem of enumerating meander configurations. The study of equivalence classes of foldings and meanders under symmetries allows to characterize and enumerate folding and meander shapes. Symmetric foldings and meanders are described, and relations between folding and meandric sequences are given. Extended tables for these sequences are provided.","comment":"","date_added":"2014-08-19","date_published":"2013-02-01","urls":["http:\/\/arxiv.org\/abs\/1302.2025","http:\/\/arxiv.org\/pdf\/1302.2025v1"],"collections":"Easily explained,Geometry","month":"feb","pages":"18","url":"http:\/\/arxiv.org\/abs\/1302.2025 http:\/\/arxiv.org\/pdf\/1302.2025v1","year":"2013","archivePrefix":"arXiv","eprint":"1302.2025","primaryClass":"math.CO","urldate":"2014-08-19"},{"key":"Mahadevan1993","type":"article","title":"The shape of a Mobius band","author":"Mahadevan, L and Keller, JB","abstract":"","comment":"","date_added":"2014-08-20","date_published":"1993-08-06","urls":["http:\/\/rspa.royalsocietypublishing.org\/content\/440\/1908\/149.short","http:\/\/www.seas.harvard.edu\/softmat\/downloads\/pre2000-20.pdf"],"collections":"Geometry","journal":"Proceedings: Mathematical and Physical Sciences","number":"1908","pages":"149--162","url":"http:\/\/rspa.royalsocietypublishing.org\/content\/440\/1908\/149.short http:\/\/www.seas.harvard.edu\/softmat\/downloads\/pre2000-20.pdf","volume":"440","year":"1993","urldate":"2014-08-20"},{"key":"Martin2013","type":"article","title":"How often should you clean your room?","author":"Martin, Kimball and Shankar, Krishnan","abstract":"We introduce and study a combinatorial optimization problem motivated by the question in the title. In the simple case where you use all objects in your room equally often, we investigate asymptotics of the optimal time to clean up in terms of the number of objects in your room. In particular, we prove a logarithmic upper bound, solve an approximate version of this problem, and conjecture a precise logarithmic asymptotic.","comment":"","date_added":"2014-09-02","date_published":"2013-05-01","urls":["http:\/\/arxiv.org\/abs\/1305.1984","http:\/\/arxiv.org\/pdf\/1305.1984v2"],"collections":"Easily explained","month":"may","pages":"28","url":"http:\/\/arxiv.org\/abs\/1305.1984 http:\/\/arxiv.org\/pdf\/1305.1984v2","year":"2013","archivePrefix":"arXiv","eprint":"1305.1984","primaryClass":"math.CO","urldate":"2014-09-02"},{"key":"Bailey2008","type":"article","title":"Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes","author":"Bailey, RA","abstract":"","comment":"","date_added":"2014-10-02","date_published":"2008-08-06","urls":["http:\/\/www.ingentaconnect.com\/content\/maa\/amm\/2008\/00000115\/00000005\/art00001","http:\/\/www.maths.qmul.ac.uk\/~pjc\/preprints\/sudoku.pdf"],"collections":"Puzzles","journal":"American Mathematical \\ldots","pages":"1--31","url":"http:\/\/www.ingentaconnect.com\/content\/maa\/amm\/2008\/00000115\/00000005\/art00001 http:\/\/www.maths.qmul.ac.uk\/~pjc\/preprints\/sudoku.pdf","year":"2008","urldate":"2014-10-02"},{"key":"Hughes2006","type":"article","title":"Proofs without syntax","author":"Hughes, DJD","abstract":"","comment":"","date_added":"2014-10-09","date_published":"2006-08-06","urls":["http:\/\/www.jstor.org\/stable\/20160016"],"collections":"Notation and conventions,About proof","journal":"Annals of Mathematics","number":"3","pages":"1065--1076","url":"http:\/\/www.jstor.org\/stable\/20160016","volume":"164","year":"2006","urldate":"2014-10-09"},{"key":"Richardson2014","type":"article","title":"The Super Patalan Numbers","author":"Richardson, Thomas M.","abstract":"We introduce the super Patalan numbers, a generalization of the super Catalan\nnumbers in the sense of Gessel, and prove a number of properties analagous to\nthose of the super Catalan numbers. The super Patalan numbers generalize the\nsuper Catalan numbers similarly to how the Patalan numbers generalize the\nCatalan numbers.","comment":"","date_added":"2014-10-23","date_published":"2014-10-01","urls":["http:\/\/arxiv.org\/abs\/1410.5880","http:\/\/arxiv.org\/pdf\/1410.5880v1"],"collections":"Attention-grabbing titles,Easily explained,Integerology","month":"oct","url":"http:\/\/arxiv.org\/abs\/1410.5880 http:\/\/arxiv.org\/pdf\/1410.5880v1","year":"2014","archivePrefix":"arXiv","eprint":"1410.5880","primaryClass":"math.CO","urldate":"2014-10-23"},{"key":"Bell2007","type":"article","title":"Solving Triangular Peg Solitaire","author":"Bell, George I.","abstract":"We consider the one-person game of peg solitaire on a triangular board of\narbitrary size. The basic game begins from a full board with one peg missing\nand finishes with one peg at a specified board location. We develop necessary\nand sufficient conditions for this game to be solvable. For all solvable\nproblems, we give an explicit solution algorithm. On the 15-hole board, we\ncompare three simple solution strategies. We then consider the problem of\nfinding solutions that minimize the number of moves (where a move is one or\nmore consecutive jumps by the same peg), and find the shortest solution to the\nbasic game on all triangular boards with up to 55 holes (10 holes on a side).","comment":"","date_added":"2014-11-10","date_published":"2007-03-01","urls":["http:\/\/arxiv.org\/abs\/math\/0703865v6","http:\/\/arxiv.org\/pdf\/math\/0703865v6"],"collections":"Puzzles,Fun maths facts","month":"mar","url":"http:\/\/arxiv.org\/abs\/math\/0703865v6 http:\/\/arxiv.org\/pdf\/math\/0703865v6","year":"2007","archivePrefix":"arXiv","eprint":"math\/0703865","primaryClass":"math.CO","urldate":"2014-11-10"},{"key":"Miller2009a","type":"article","title":"Irrationality From The Book","author":"Miller, Steven J. and Montague, David","abstract":"We generalize Tennenbaum's geometric proof of the irrationality of sqrt(2) to\nsqrt(n) for n = 3, 5, 6 and 10.","comment":"","date_added":"2014-12-01","date_published":"2009-09-01","urls":["http:\/\/arxiv.org\/abs\/0909.4913","http:\/\/arxiv.org\/pdf\/0909.4913v2"],"collections":"About proof,Fun maths facts","month":"sep","url":"http:\/\/arxiv.org\/abs\/0909.4913 http:\/\/arxiv.org\/pdf\/0909.4913v2","year":"2009","archivePrefix":"arXiv","eprint":"0909.4913","primaryClass":"math.HO","urldate":"2014-12-01"},{"key":"McGuire2012","type":"article","title":"Bells, Motels and Permutation Groups","author":"McGuire, Gary","abstract":"This article is about the mathematics of ringing the changes. We describe the mathematics which arises from a real-world activity, that of ringing the changes on bells. We present Rankin's solution of one of the famous old problems in the subject. This article was written in 2003.","comment":"","date_added":"2014-12-17","date_published":"2012-03-01","urls":["http:\/\/arxiv.org\/abs\/1203.1835","http:\/\/arxiv.org\/pdf\/1203.1835v1"],"collections":"Easily explained,Music,Fun maths facts,Combinatorics,The groups group","month":"mar","url":"http:\/\/arxiv.org\/abs\/1203.1835 http:\/\/arxiv.org\/pdf\/1203.1835v1","year":"2012","archivePrefix":"arXiv","eprint":"1203.1835","primaryClass":"math.GR","urldate":"2014-12-17"},{"key":"Arthan2004","type":"article","title":"The Eudoxus Real Numbers","author":"Arthan, R. D.","abstract":"This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real numbers using Dedekind cuts, the idea is inspired by the ancient Greek theory of proportion, due to Eudoxus. However, unlike the Dedekind construction, the construction proceeds directly from the integers to the real numbers bypassing the intermediate construction of the rational numbers. The construction of the additive group of the reals depends on rather simple algebraic properties of the integers. This part of the construction can be generalised in several natural ways, e.g., with an arbitrary abelian group playing the role of the integers. This raises the question: what does the construction construct? In an appendix we sketch some generalisations and answer this question in some simple cases. The treatment of the main construction is intended to be self-contained and assumes familiarity only with elementary algebra in the ring of integers and with the definitions of the abstract algebraic notions of group, ring and field.","comment":"","date_added":"2015-01-07","date_published":"2004-05-01","urls":["http:\/\/arxiv.org\/abs\/math\/0405454","http:\/\/arxiv.org\/pdf\/math\/0405454v1"],"collections":"Unusual arithmetic","month":"may","pages":"15","url":"http:\/\/arxiv.org\/abs\/math\/0405454 http:\/\/arxiv.org\/pdf\/math\/0405454v1","year":"2004","archivePrefix":"arXiv","eprint":"math\/0405454","primaryClass":"math.HO","urldate":"2015-01-07"},{"key":"Loregian2015","type":"article","title":"This is the (co)end, my only (co)friend","author":"Loregian, Fosco","abstract":"The present note is a recollection of the most striking and useful applications of co\/end calculus. We put a considerable effort in making arguments and constructions rather explicit: after having given a series of preliminary definitions, we characterize co\/ends as particular co\/limits; then we derive a number of results directly from this characterization. The last sections discuss the most interesting examples where co\/end calculus serves as a powerful abstract way to do explicit computations in diverse fields like Algebra, Algebraic Topology and Category Theory. The appendices serve to sketch a number of results in theories heavily relying on co\/end calculus; the reader who dares to arrive at this point, being completely introduced to the mysteries of co\/end fu, can regard basically every statement as a guided exercise.","comment":"","date_added":"2015-01-13","date_published":"2015-01-01","urls":["http:\/\/arxiv.org\/abs\/1501.02503","http:\/\/arxiv.org\/pdf\/1501.02503v2"],"collections":"Attention-grabbing titles","month":"jan","pages":"39","url":"http:\/\/arxiv.org\/abs\/1501.02503 http:\/\/arxiv.org\/pdf\/1501.02503v2","year":"2015","archivePrefix":"arXiv","eprint":"1501.02503","primaryClass":"math.CT","urldate":"2015-01-13"},{"key":"Feldman1993","type":"article","title":"On dice and coins: Models of computation for random generation","author":"Feldman, D and Impagliazzo, R and Naor, M","abstract":"","comment":"","date_added":"2015-01-26","date_published":"1993-08-06","urls":["http:\/\/www.sciencedirect.com\/science\/article\/pii\/S089054018371028X"],"collections":"Easily explained,Things to make and do,Modelling","journal":"Information and \\ldots","url":"http:\/\/www.sciencedirect.com\/science\/article\/pii\/S089054018371028X","year":"1993","urldate":"2015-01-26"},{"key":"Gouyou-Beauchamps2005","type":"article","title":"Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice","author":"Gouyou-Beauchamps, Dominique and Leroux, Pierre","abstract":"We enumerate the symmetry classes of convex polyominoes on the hexagonal (honeycomb) lattice. Here convexity is to be understood as convexity along the three main column directions. We deduce the generating series of free (i.e. up to reflection and rotation) and of asymmetric convex hexagonal polyominoes, according to area and half-perimeter. We give explicit formulas or implicit functional equations for the generating series, which are convenient for computer algebra. Thus, computations can be carried out up to area 70.","comment":"","date_added":"2015-02-02","date_published":"2005-11-01","urls":["http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0304397505004950"],"collections":"Geometry,Combinatorics","issn":"03043975","journal":"Theoretical Computer Science","month":"nov","number":"2-3","pages":"307--334","url":"http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0304397505004950","volume":"346","year":"2005","urldate":"2015-02-02"},{"key":"Calude2006","type":"misc","title":"Exact Approximations of Omega Numbers","author":"Calude, C.S and Dinneen, Michael","abstract":"","comment":"","date_added":"2015-02-03","date_published":"2006-12-01","urls":["https:\/\/researchspace.auckland.ac.nz\/handle\/2292\/3800"],"collections":"","issn":"1178-3540","keywords":"Computing and Communication Sciences,Fields of Research::280000 Information","month":"dec","publisher":"Department of Computer Science, The University of Auckland, New Zealand","url":"https:\/\/researchspace.auckland.ac.nz\/handle\/2292\/3800","year":"2006","urldate":"2015-02-03"},{"key":"Ahlbach2012","type":"article","title":"Efficient Algorithms for Zeckendorf Arithmetic","author":"Ahlbach, Connor and Usatine, Jeremy and Pippenger, Nicholas","abstract":"We study the problem of addition and subtraction using the Zeckendorf representation of integers. We show that both operations can be performed in linear time; in fact they can be performed by combinational logic networks with linear size and logarithmic depth. The implications of these results for multiplication, division and square-root extraction are also discussed.","comment":"","date_added":"2015-02-16","date_published":"2012-07-01","urls":["http:\/\/arxiv.org\/abs\/1207.4497","http:\/\/arxiv.org\/pdf\/1207.4497v1"],"collections":"Easily explained,Fun maths facts,Integerology","month":"jul","url":"http:\/\/arxiv.org\/abs\/1207.4497 http:\/\/arxiv.org\/pdf\/1207.4497v1","year":"2012","archivePrefix":"arXiv","eprint":"1207.4497","primaryClass":"cs.DS","urldate":"2015-02-16"},{"key":"Oller2007","type":"article","title":"The dying rabbit problem revisited","author":"Oller, Antonio M.","abstract":"In this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular we give a general recurrence relation for these sequences (improving the work in the paper Hoggatt, V. E., Jr.; Lind, D. A. \"The dying rabbit problem\". Fibonacci Quart. 7 1969 no. 5, 482--487) and we calculate explicitly their general term (extending the work in the paper Miles, E. P., Jr. Generalized Fibonacci numbers and associated matrices. Amer. Math. Monthly 67 1960 745--752). In passing, and as a technical requirement, we also study the behavior of the positive real roots of the characteristic polynomial of the considered sequences.","comment":"","date_added":"2015-02-18","date_published":"2007-10-01","urls":["http:\/\/arxiv.org\/abs\/0710.2216","http:\/\/arxiv.org\/pdf\/0710.2216v1"],"collections":"Easily explained,Animals,Fibonaccinalia","month":"oct","pages":"8","url":"http:\/\/arxiv.org\/abs\/0710.2216 http:\/\/arxiv.org\/pdf\/0710.2216v1","year":"2007","archivePrefix":"arXiv","eprint":"0710.2216","primaryClass":"math.NT","urldate":"2015-02-18"},{"key":"Chvatal1975","type":"article","title":"A combinatorial theorem in plane geometry","author":"Chv\u00e1tal, V","abstract":"","comment":"","date_added":"2015-02-23","date_published":"1975-02-01","urls":["http:\/\/www.sciencedirect.com\/science\/article\/pii\/0095895675900611"],"collections":"Easily explained,Geometry,Fun maths facts,Combinatorics","issn":"00958956","journal":"Journal of Combinatorial Theory, Series B","month":"feb","number":"1","pages":"39--41","url":"http:\/\/www.sciencedirect.com\/science\/article\/pii\/0095895675900611","volume":"18","year":"1975","urldate":"2015-02-23"},{"key":"Golumbic1998","type":"article","title":"Complexity and Algorithms for Graph and Hypergraph Sandwich Problems","author":"Golumbic, Martin Charles and Wassermann, Amir","abstract":"","comment":"","date_added":"2015-02-24","date_published":"1998-08-01","urls":["http:\/\/link.springer.com\/10.1007\/s003730050028"],"collections":"","issn":"0911-0119","journal":"Graphs and Combinatorics","month":"aug","number":"3","pages":"223--239","url":"http:\/\/link.springer.com\/10.1007\/s003730050028","volume":"14","year":"1998","urldate":"2015-02-24"},{"key":"Cotogno2009","type":"article","title":"A Brief Critique of Pure Hypercomputation","author":"Cotogno, Paolo","abstract":"","comment":"","date_added":"2015-03-05","date_published":"2009-10-01","urls":["http:\/\/link.springer.com\/10.1007\/s11023-009-9161-7"],"collections":"Basically computer science","issn":"0924-6495","journal":"Minds and Machines","month":"oct","number":"3","pages":"391--405","url":"http:\/\/link.springer.com\/10.1007\/s11023-009-9161-7","volume":"19","year":"2009","urldate":"2015-03-05"},{"key":"Anderson2015","type":"article","title":"Finding long chains in kidney exchange using the traveling salesman problem","author":"Anderson, Ross and Ashlagi, Itai and Gamarnik, David and Roth, Alvin E.","abstract":"SignificanceThere are currently more than 100,000 patients on the waiting list in the United States for a kidney transplant from a deceased donor. To address this shortage, kidney exchange programs allow patients with living incompatible donors to exchange donors through cycles and chains initiated by altruistic nondirected donors. To determine which exchanges will take place, kidney exchange programs use algorithms for maximizing the number of transplants under constraints about the size of feasible exchanges. This problem is NP-hard, and algorithms previously used were unable to optimize when chains could be long. We developed two algorithms that use integer programming to solve this problem, one of which is inspired by the traveling salesman, that together can find optimal solutions in practice. As of May 2014 there were more than 100,000 patients on the waiting list for a kidney transplant from a deceased donor. Although the preferred treatment is a kidney transplant, every year there are fewer donors than new patients, so the wait for a transplant continues to grow. To address this shortage, kidney paired donation (KPD) programs allow patients with living but biologically incompatible donors to exchange donors through cycles or chains initiated by altruistic (nondirected) donors, thereby increasing the supply of kidneys in the system. In many KPD programs a centralized algorithm determines which exchanges will take place to maximize the total number of transplants performed. This optimization problem has proven challenging both in theory, because it is NP-hard, and in practice, because the algorithms previously used were unable to optimally search over all long chains. We give two new algorithms that use integer programming to optimally solve this problem, one of which is inspired by the techniques used to solve the traveling salesman problem. These algorithms provide the tools needed to find optimal solutions in practice.","comment":"","date_added":"2015-03-07","date_published":"2015-01-01","urls":["http:\/\/www.pnas.org\/content\/112\/3\/663"],"collections":"Basically computer science,Protocols and strategies","issn":"0027-8424","journal":"Proceedings of the National Academy of Sciences","month":"jan","number":"3","pages":"201421853","url":"http:\/\/www.pnas.org\/content\/112\/3\/663","volume":"112","year":"2015","urldate":"2015-03-07"},{"key":"Mugford2001","type":"article","title":"The accuracy of Buffon's needle: a rule of thumb used by ants to estimate area","author":"Mugford, S. T.","abstract":"Colonies of the ant Leptothorax albipennis naturally inhabit flat rock crevices. Scouts can determine, before initiating an emigration, if a nest has sufficient area to house their colony. They do so with a rule of thumb: the Buffon's needle algorithm. Based on a derivation from the classical statistical geometry of Comte George de Buffon in the 18th century, it can be shown that it is possible to estimate the area of a plane from the frequency of intersections between two sets of randomly scattered lines of known lengths. Our earlier work has shown that individual ants use this Buffon's needle algorithm by laying individual-specific trail pheromones on a first visit to a potential nest site and by assessing the frequency at which they intersect that path on a second visit. Nest area would be inversely proportional to the intersection frequency. The simplest procedure would be for individual ants to keep their first-visit path-length constant regardless of the size of the nest they are visiting. Here we show, for the first time, that this is the case. We also determine the potential quality of information that individual ants might have at their disposal from their own path-laying and path-crossing activities. Hence, we can determine the potential accuracy of nest area estimation by individual ants. Our findings suggest that ants using the Buffon's needle rule of thumb might obtain remarkably accurate assessments of nest area.","comment":"","date_added":"2015-03-09","date_published":"2001-11-01","urls":["http:\/\/beheco.oxfordjournals.org\/content\/12\/6\/655.full"],"collections":"Animals,Probability and statistics","issn":"14657279","journal":"Behavioral Ecology","month":"nov","number":"6","pages":"655--658","url":"http:\/\/beheco.oxfordjournals.org\/content\/12\/6\/655.full","volume":"12","year":"2001","urldate":"2015-03-09"},{"key":"Ferrie2015","type":"article","title":"Have you been using the wrong estimator? These guys bound average fidelity using this one weird trick von Neumann didn't want you to know","author":"Ferrie, Christopher and Kueng, Richard","abstract":"We give bounds on the average fidelity achievable by any quantum state estimator, which is arguably the most prominently used figure of merit in quantum state tomography. Moreover, these bounds can be computed online---that is, while the experiment is running. We show numerically that these bounds are quite tight for relevant distributions of density matrices. We also show that the Bayesian mean estimator is ideal in the sense of performing close to the bound without requiring optimization. Our results hold for all finite dimensional quantum systems.","comment":"","date_added":"2015-03-26","date_published":"2015-03-01","urls":["http:\/\/arxiv.org\/abs\/1503.00677","http:\/\/arxiv.org\/pdf\/1503.00677v2"],"collections":"Attention-grabbing titles,Probability and statistics","month":"mar","url":"http:\/\/arxiv.org\/abs\/1503.00677 http:\/\/arxiv.org\/pdf\/1503.00677v2","year":"2015","archivePrefix":"arXiv","eprint":"1503.00677","primaryClass":"quant-ph","urldate":"2015-03-26"},{"key":"Schumacher2004","type":"article","title":"Reversible quantum cellular automata","author":"Schumacher, B. and Werner, R. F.","abstract":"We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules generating such automata. The same local rules also generate the global time step for automata with periodic boundary conditions. Our main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme. This implies that, in contrast to the classical case, the inverse of a nearest neighbor quantum cellular automaton is again a nearest neighbor automaton. We present several construction methods for quantum cellular automata, based on unitaries commuting with their translates, on the quantization of (arbitrary) reversible classical cellular automata, on quantum circuits, and on Clifford transformations with respect to a description of the single cells by finite Weyl systems. Moreover, we indicate how quantum random walks can be considered as special cases of cellular automata, namely by restricting a quantum lattice gas automaton with local particle number conservation to the single particle sector.","comment":"","date_added":"2015-06-28","date_published":"2004-05-01","urls":["http:\/\/arxiv.org\/abs\/quant-ph\/0405174"],"collections":"Basically computer science","month":"may","pages":"20","url":"http:\/\/arxiv.org\/abs\/quant-ph\/0405174","year":"2004","urldate":"2015-06-28"},{"key":"Gerogiorgakis2015","type":"article","title":"Mind the Croc! Rationality Gaps vis-\u00e0-vis the Crocodile Paradox","author":"Gerogiorgakis, Stamatios","abstract":"This article discusses rationality gaps triggered by self-referential\/cyclic choice, the latter being understood as choosing according to a norm that refers to the choosing itself. The Crocodile Paradox is reformulated and analyzed as a game\u2014named CP\u2014whose Nash equilibrium is shown to trigger a cyclic choice and to invite a rationality gap. It is shown that choosing the Nash equilibrium of CP conforms to the principles Wolfgang Spohn and Haim Gaifman introduced to, allegedly, guarantee acyclicity but, in fact, does not prevent self-referential\/cyclic choice and rationality gaps. It is shown that CP is a counter-example to Gaifman's solution of the rationality gaps problem.","comment":"","date_added":"2015-09-03","date_published":"2015-06-01","urls":["http:\/\/www.tandfonline.com\/doi\/full\/10.1080\/01445340.2015.1046211#.VegVv_lViHg"],"collections":"Attention-grabbing titles,Animals","issn":"0144-5340","journal":"History and Philosophy of Logic","language":"en","month":"jun","pages":"1--13","publisher":"Taylor & Francis","url":"http:\/\/www.tandfonline.com\/doi\/full\/10.1080\/01445340.2015.1046211#.VegVv_lViHg","year":"2015","urldate":"2015-09-03"},{"key":"Diaconis1989a","type":"article","title":"Fair Dice","author":"Diaconis, Persi and Keller, Joseph B","abstract":"","comment":"","date_added":"2015-09-15","date_published":"1989-08-06","urls":["http:\/\/links.jstor.org\/sici?sici=0002-9890%28198904%2996%3A4%3C337%3AFD%3E2.0.CO%3B2-Z"],"collections":"Games to play with friends,Probability and statistics","journal":"The American Mathematical Monthly","number":"4","pages":"337--339","url":"http:\/\/links.jstor.org\/sici?sici=0002-9890%28198904%2996%3A4%3C337%3AFD%3E2.0.CO%3B2-Z","volume":"96","year":"1989","urldate":"2015-09-15"},{"key":"Muller2013","type":"article","title":"Another Proof of Segre's Theorem about Ovals","author":"M\u00fcller, Peter","abstract":"In 1955 B. Segre showed that any oval in a projective plane over a finite field of odd order is a conic. His proof constructs a conic which matches the oval in some points, and then shows that it actually coincides with the oval. Here we give another proof. We describe the oval by a possibly high degree polynomial, and then show that the degree is actually 2.","comment":"","date_added":"2015-09-29","date_published":"2013-11-01","urls":["http:\/\/arxiv.org\/abs\/1311.3082","http:\/\/arxiv.org\/pdf\/1311.3082v1"],"collections":"About proof","month":"nov","pages":"2","url":"http:\/\/arxiv.org\/abs\/1311.3082 http:\/\/arxiv.org\/pdf\/1311.3082v1","year":"2013","archivePrefix":"arXiv","eprint":"1311.3082","primaryClass":"math.NT","urldate":"2015-09-29"},{"key":"Rechnitzer2003","type":"article","title":"Haruspicy and anisotropic generating functions","author":"Rechnitzer, Andrew","abstract":"Guttmann and Enting [Phys. Rev. Lett. 76 (1996) 344\u2013347] proposed the examination of anisotropic generating functions as a test of the solvability of models of bond animals. In this article we describe a technique for examining some properties of anisotropic generating functions. For a\u00a0wide range of solved and unsolved families of bond animals, we show that the coefficients of\u00a0yn is rational, the degree of its numerator is at most that of its denominator, and the denominator is a\u00a0product of cyclotomic polynomials. Further, we are able to find a multiplicative upper bound for these denominators which, by comparison with numerical studies [Jensen, personal communication; Jensen and Guttmann, personal communication], appears to be very tight. These facts can be used to greatly reduce the amount of computation required in generating series expansions. They also have strong and negative implications for the solvability of these problems.","comment":"","date_added":"2015-11-08","date_published":"2003-02-01","urls":["http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0196885802005341"],"collections":"","issn":"01968858","journal":"Advances in Applied Mathematics","month":"feb","number":"1-2","pages":"228--257","url":"http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0196885802005341","volume":"30","year":"2003","urldate":"2015-11-08"},{"key":"Blote1984","type":"article","title":"Spiralling self-avoiding walks: an exact solution","author":"Blote, H W J and Hilhorst, H J","abstract":"","comment":"","date_added":"2015-11-08","date_published":"1984-02-01","urls":["http:\/\/iopscience.iop.org\/article\/10.1088\/0305-4470\/17\/3\/004"],"collections":"","issn":"0305-4470","journal":"Journal of Physics A: Mathematical and General","language":"en","month":"feb","number":"3","pages":"L111--L115","publisher":"IOP Publishing","url":"http:\/\/iopscience.iop.org\/article\/10.1088\/0305-4470\/17\/3\/004","volume":"17","year":"1984","urldate":"2015-11-08"},{"key":"Katz2010","type":"article","title":"When is .999... less than 1?","author":"Katz, Karin Usadi and Katz, Mikhail G.","abstract":"We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is \"an infinite number of 9s\" merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone's \"semicolon\" notation? Is it possible to choose a canonical alternative interpretation? Should unital evaluation of the symbol .999 . . . be inculcated in a pre-limit teaching environment? The problem of the unital evaluation is hereby examined from the pre-R, pre-lim viewpoint of the student.","comment":"","date_added":"2015-11-19","date_published":"2010-07-01","urls":["http:\/\/arxiv.org\/abs\/1007.3018","http:\/\/arxiv.org\/pdf\/1007.3018v1"],"collections":"Unusual arithmetic","month":"jul","pages":"28","url":"http:\/\/arxiv.org\/abs\/1007.3018 http:\/\/arxiv.org\/pdf\/1007.3018v1","year":"2010","archivePrefix":"arXiv","eprint":"1007.3018","primaryClass":"math.HO","urldate":"2015-11-19"},{"key":"Sanders2015","type":"article","title":"The effective content of Reverse Nonstandard Mathematics and the nonstandard content of effective Reverse Mathematics","author":"Sanders, Sam","abstract":"The aim of this paper is to highlight a hitherto unknown computational aspect of Nonstandard Analysis pertaining to Reverse Mathematics (RM). In particular, we shall establish RM-equivalences between theorems from Nonstandard Analysis in a fragment of Nelson's internal set theory. We then extract primitive recursive terms from Goedel's system T (not involving Nonstandard Analysis) from the proofs of the aforementioned nonstandard equivalences. The resulting terms turn out to be witnesses for effective1 equivalences in Kohlenbach's higher-order RM. In other words, from an RM-equivalence in Nonstandard Analysis, we can extract the associated effective higher-order RM-equivalence which does not involve Nonstandard Analysis anymore. Finally, we show that certain effective equivalences in turn give rise to the original nonstandard theorems from which they were derived.","comment":"","date_added":"2015-11-19","date_published":"2015-11-01","urls":["http:\/\/arxiv.org\/abs\/1511.04679","http:\/\/arxiv.org\/pdf\/1511.04679v1"],"collections":"Attention-grabbing titles,About proof","month":"nov","url":"http:\/\/arxiv.org\/abs\/1511.04679 http:\/\/arxiv.org\/pdf\/1511.04679v1","year":"2015","archivePrefix":"arXiv","eprint":"1511.04679","primaryClass":"math.LO","urldate":"2015-11-19"},{"key":"Gordinowicz2015","type":"article","title":"Planar graph is on fire","author":"Gordinowicz, Przemys\u0142aw","abstract":"Let $G$ be any connected graph on $n$ vertices, $n \\ge 2.$ Let $k$ be any positive integer. Suppose that a fire breaks out on some vertex of $G.$ Then in each turn $k$ firefighters can protect vertices of $G$ --- each can protect one vertex not yet on fire; Next a fire spreads to all unprotected neighbours. The $\\$emph{$k$-surviving} rate of G, denoted by $\\rho_k(G),$ is the expected fraction of vertices that can be saved from the fire by $k$ firefighters, provided that the starting vertex is chosen uniformly at random. In this paper, it is shown that for any planar graph $G$ we have $\\rho_3(G) \\ge \\frac{2}{21}.$ Moreover, 3 firefighters are needed for the first step only; after that it is enough to have 2 firefighters per each round. This result significantly improves known solutions to a problem of Cai and Wang (there was no positive bound known for surviving rate of general planar graph with only 3 firefighters). The proof is done using the separator theorem for planar graphs.","comment":"","date_added":"2015-12-02","date_published":"2015-08-01","urls":["http:\/\/arxiv.org\/abs\/1311.1158","http:\/\/arxiv.org\/pdf\/1311.1158v2"],"collections":"Attention-grabbing titles,Puzzles,Fun maths facts","issn":"03043975","journal":"Theoretical Computer Science","month":"aug","pages":"160--164","url":"http:\/\/arxiv.org\/abs\/1311.1158 http:\/\/arxiv.org\/pdf\/1311.1158v2","volume":"593","year":"2015","archivePrefix":"arXiv","eprint":"1311.1158","primaryClass":"math.CO","urldate":"2015-12-02"},{"key":"Diaz2003","type":"article","title":"Comparative kinetics of the snowball respect to other dynamical objects","author":"Diaz, Rodolfo A. and Gonzalez, Diego L. and Marin, Francisco and Martinez, R.","abstract":"We examine the kinetics of a snowball that is gaining mass while is rolling downhill. This dynamical system combines rotational effects with effects involving the variation of mass. In order to understand the consequences of both effects we compare its behavior with the one of some objects in which such effects are absent. Environmental conditions are also included. We conclude that the comparative velocity of the snowball is very sensitive to the hill profile and the retardation factors. We emphasize that the increase of mass (inertia), could surprisingly diminish the retardation effect due to the drag force. Additionally, when an exponential trajectory is assumed, the maximum velocity of the snowball can be reached at an intermediate step of the trip.","comment":"","date_added":"2016-01-18","date_published":"2003-10-01","urls":["http:\/\/arxiv.org\/abs\/physics\/0310010","http:\/\/arxiv.org\/pdf\/physics\/0310010v2"],"collections":"Basically physics","month":"oct","pages":"16","url":"http:\/\/arxiv.org\/abs\/physics\/0310010 http:\/\/arxiv.org\/pdf\/physics\/0310010v2","year":"2003","archivePrefix":"arXiv","eprint":"physics\/0310010","primaryClass":"physics.class-ph","urldate":"2016-01-18"},{"key":"BlancoAbellan2015","type":"misc","title":"On gardeners, dukes and mathematical instruments","author":"Blanco Abell\u00e1n, M\u00f3nica","abstract":"Postprint (author's final draft)","comment":"","date_added":"2016-01-18","date_published":"2015-08-06","urls":["http:\/\/upcommons.upc.edu\/handle\/2117\/78617"],"collections":"History","booktitle":"Bulletin of the Scientific Instrument Society","issn":"0956-8271","keywords":"Mathematical instruments,\u00c0rees tem\u00e0tiques de la UPC::Matem\u00e0tiques i estad\u00eds","language":"eng","pages":"22--28","url":"http:\/\/upcommons.upc.edu\/handle\/2117\/78617","volume":"125","year":"2015","urldate":"2016-01-18"},{"key":"Hung","type":"article","title":"De Bruijn's Combinatorics","author":"Hung, J.W.Nienhuys (Ling-Ju and Eds.), Ton Kloks","abstract":"This is a translation of the handwritten classroom notes taken by Nienhuys of a course in combinatorics given by N.G. de Bruijn at Eindhoven University of Technology, during the 1970s and 1980s.","comment":"","date_added":"2016-02-05","date_published":"2012-08-06","urls":["http:\/\/vixra.org\/abs\/1208.0223"],"collections":"Combinatorics","pages":"192","url":"http:\/\/vixra.org\/abs\/1208.0223","urldate":"2016-02-05","year":"2012"},{"key":"Uspensky1994","type":"article","title":"G\u00f6del's incompleteness theorem","author":"Uspensky, V","abstract":"","comment":"","date_added":"2011-06-01","date_published":"1994-08-01","urls":["http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/0304397594902224"],"collections":"Fun maths facts","url":"http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/0304397594902224","doi":"10.1016\/0304-3975(94)90222-4","issn":"03043975","journal":"Theoretical Computer Science","month":"aug","number":"2","pages":"239--319","volume":"130","year":"1994","urldate":"2011-06-01"},{"key":"FIBONACCIJIGSAWPUZZLE","type":"article","title":"Fibonacci Jigsaw Puzzle","author":"Akio Hizume","abstract":"","comment":"","date_added":"2016-05-19","date_published":"2016-08-06","urls":["http:\/\/starcage.org\/fibonacci_puzzle\/fibonacci_puzzle.html"],"collections":"Puzzles,Easily explained,Fibonaccinalia","url":"http:\/\/starcage.org\/fibonacci_puzzle\/fibonacci_puzzle.html","urldate":"2016-05-19","year":"2016"},{"key":"Bernhardsson2011","type":"article","title":"A Paradoxical Property of the Monkey Book","author":"Bernhardsson, Sebastian and Baek, Seung Ki and Minnhagen, Petter","abstract":"A \"monkey book\" is a book consisting of a random distribution of letters and blanks, where a group of letters surrounded by two blanks is defined as a word. We compare the statistics of the word distribution for a monkey book with the corresponding distribution for the general class of random books, where the latter are books for which the words are randomly distributed. It is shown that the word distribution statistics for the monkey book is different and quite distinct from a typical sampled book or real book. In particular the monkey book obeys Heaps' power law to an extraordinary good approximation, in contrast to the word distributions for sampled and real books, which deviate from Heaps' law in a characteristics way. The somewhat counter-intuitive conclusion is that a \"monkey book\" obeys Heaps' power law precisely because its word-frequency distribution is not a smooth power law, contrary to the expectation based on simple mathematical arguments that if one is a power law, so is the other.","comment":"","date_added":"2011-04-03","date_published":"2011-03-01","urls":["http:\/\/arxiv.org\/abs\/1103.2681","http:\/\/arxiv.org\/pdf\/1103.2681v1.pdf"],"collections":"Animals","url":"http:\/\/arxiv.org\/abs\/1103.2681 http:\/\/arxiv.org\/pdf\/1103.2681v1.pdf","archivePrefix":"arXiv","arxivId":"1103.2681","doi":"10.1088\/1742-5468\/2011\/07\/P07013","eprint":"1103.2681","journal":"Contemporary Physics","month":"mar","pages":"5","year":"2011","urldate":"2011-04-03"},{"key":"Culbertson2012","type":"article","title":"A categorical foundation for Bayesian probability","author":"Culbertson, Jared and Sturtz, Kirk","abstract":"Given two measurable spaces $H$ and $D$ with countably generated $\\sigma$-algebras, a prior probability measure $P_H$ on $H$ and a sampling distribution $\\mcS:H \\rightarrow D$, there is a corresponding inference map $\\mcI:D \\rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $\\mu:1 \\rightarrow D$, a posterior probability $\\hat{P_H}=\\mcI \\circ \\mu$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $\\mcI$ and the process repeats with each additional measurement. The main result shows that the assumption of Polish spaces to obtain regular conditional probabilities is not necessary---countably generated spaces suffice. This less stringent condition then allows for non-trivial decision rules (Eilenberg--Moore algebras) on finite (as well as non finite) spaces, and also provides for a common framework for decision theory and Bayesian probability.","comment":"","date_added":"2012-05-08","date_published":"2012-05-01","urls":["http:\/\/arxiv.org\/abs\/1205.1488","http:\/\/arxiv.org\/pdf\/1205.1488v3"],"collections":"Probability and statistics","url":"http:\/\/arxiv.org\/abs\/1205.1488 http:\/\/arxiv.org\/pdf\/1205.1488v3","archivePrefix":"arXiv","arxivId":"1205.1488","eprint":"1205.1488","keywords":"bayesian probability,categorical foundation for probability,giry,monad,msc 2000 subject,primary 60a05,probabilistic logic,regular conditional probability,secondary 62c10","month":"may","pages":"18","year":"2012","urldate":"2012-05-08"},{"key":"DismalArithmetic","type":"article","title":"Dismal Arithmetic","author":"David Applegate and Marc LeBrun and N. J. A. Sloane","abstract":"Dismal arithmetic is just like the arithmetic you learned in school, only\r\nsimpler: there are no carries, when you add digits you just take the largest,\r\nand when you multiply digits you take the smallest. This paper studies basic\r\nnumber theory in this world, including analogues of the primes, number of\r\ndivisors, sum of divisors, and the partition function.","comment":"","date_added":"2016-05-19","date_published":"2011-08-06","urls":["http:\/\/arxiv.org\/abs\/1107.1130v2","http:\/\/arxiv.org\/pdf\/1107.1130v2"],"collections":"Unusual arithmetic,Easily explained,Integerology","url":"http:\/\/arxiv.org\/abs\/1107.1130v2 http:\/\/arxiv.org\/pdf\/1107.1130v2","urldate":"2016-05-19","archivePrefix":"arXiv","eprint":"1107.1130","primaryClass":"math.NT","year":"2011"},{"key":"TwoNotesOnNotation","type":"article","title":"Two notes on notation","author":"Donald E. Knuth","abstract":"The author advocates two specific mathematical notations from his popular\r\ncourse and joint textbook, \"Concrete Mathematics\". The first of these,\r\nextending an idea of Iverson, is the notation \"[P]\" for the function which is 1\r\nwhen the Boolean condition P is true and 0 otherwise. This notation can\r\nencourage and clarify the use of characteristic functions and Kronecker deltas\r\nin sums and integrals.\r\n The second notation puts Stirling numbers on the same footing as binomial\r\ncoefficients. Since binomial coefficients are written on two lines in\r\nparentheses and read \"n choose k\", Stirling numbers of the first kind should be\r\nwritten on two lines in brackets and read \"n cycle k\", while Stirling numbers\r\nof the second kind should be written in braces and read \"n subset k\". (I might\r\nsay \"n partition k\".) The written form was first suggested by Imanuel Marx. The\r\nvirtues of this notation are that Stirling partition numbers frequently appear\r\nin combinatorics, and that it more clearly presents functional relations\r\nsimilar to those satisfied by binomial coefficients.","comment":"","date_added":"2016-05-19","date_published":"1992-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/9205211v1","http:\/\/arxiv.org\/pdf\/math\/9205211v1"],"collections":"Notation and conventions","url":"http:\/\/arxiv.org\/abs\/math\/9205211v1 http:\/\/arxiv.org\/pdf\/math\/9205211v1","urldate":"2016-05-19","archivePrefix":"arXiv","eprint":"math\/9205211","primaryClass":"math.HO","year":"1992"},{"key":"OntheCookieMonsterProblem","type":"article","title":"On the Cookie Monster Problem","author":"Leigh Marie Braswell and Tanya Khovanova","abstract":"The Cookie Monster Problem supposes that the Cookie Monster wants to empty a\r\nset of jars filled with various numbers of cookies. On each of his moves, he\r\nmay choose any subset of jars and take the same number of cookies from each of\r\nthose jars. The Cookie Monster number of a set is the minimum number of moves\r\nthe Cookie Monster must use to empty all of the jars. This number depends on\r\nthe initial distribution of cookies in the jars. We discuss bounds of the\r\nCookie Monster number and explicitly find the Cookie Monster number for jars\r\ncontaining cookies in the Fibonacci, Tribonacci, n-nacci, and Super-n-nacci\r\nsequences. We also construct sequences of k jars such that their Cookie Monster\r\nnumbers are asymptotically rk, where r is any real number between 0 and 1\r\ninclusive.","comment":"","date_added":"2016-05-19","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1309.5985v1","http:\/\/arxiv.org\/pdf\/1309.5985v1"],"collections":"Puzzles,Animals,Food,Fibonaccinalia,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1309.5985v1 http:\/\/arxiv.org\/pdf\/1309.5985v1","urldate":"2016-05-19","archivePrefix":"arXiv","eprint":"1309.5985","primaryClass":"math.HO","year":"2013"},{"key":"Brumleve2012","type":"article","title":"The mate-in-n problem of infinite chess is decidable","author":"Brumleve, Dan and Hamkins, Joel David and Schlicht, Philipp","abstract":"Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-n problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most n moves. A naive formulation of this problem leads to assertions of high arithmetic complexity with 2n alternating quantifiers---there is a move for white, such that for every black reply, there is a counter-move for white, and so on. In such a formulation, the problem does not appear to be decidable; and one cannot expect to search an infinitely branching game tree even to finite depth. Nevertheless, the main theorem of this article, confirming a conjecture of the first author and C. D. A. Evans, establishes that the mate-in-n problem of infinite chess is computably decidable, uniformly in the position and in n. Furthermore, there is a computable strategy for optimal play from such mate-in-n positions. The proof proceeds by showing that the mate-in-n problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable. Indeed, it is definable in Presburger arithmetic. Unfortunately, this resolution of the mate-in-n problem does not appear to settle the decidability of the more general winning-position problem, the problem of determining whether a designated player has a winning strategy from a given position, since a position may admit a winning strategy without any bound on the number of moves required. This issue is connected with transfinite game values in infinite chess, and the exact value of the omega one of chess is not known.","comment":"","date_added":"2012-03-28","date_published":"2012-01-01","urls":["http:\/\/arxiv.org\/abs\/1201.5597","http:\/\/arxiv.org\/pdf\/1201.5597v4"],"collections":"Puzzles,Games to play with friends,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1201.5597 http:\/\/arxiv.org\/pdf\/1201.5597v4","archivePrefix":"arXiv","arxivId":"1201.5597","eprint":"1201.5597","journal":"New York","month":"jan","pages":"10","year":"2012","urldate":"2012-03-28"},{"key":"Xu2012","type":"article","title":"Survey on fusible numbers","author":"Xu, Junyan","abstract":"We point out that the recursive formula that appears in Erickson's presentation \"Fusible Numbers\" is incorrect, and pose an alternate conjecture about the structure of fusible numbers. Although we are unable to solve the conjecture, we succeed in establishing some basic properties of fusible numbers. We suggest some possible approaches to the conjecture, and list further problems in the final chapter.","comment":"","date_added":"2012-05-02","date_published":"2012-02-01","urls":["http:\/\/arxiv.org\/abs\/1202.5614","http:\/\/arxiv.org\/pdf\/1202.5614v1"],"collections":"Easily explained","url":"http:\/\/arxiv.org\/abs\/1202.5614 http:\/\/arxiv.org\/pdf\/1202.5614v1","archivePrefix":"arXiv","arxivId":"1202.5614","eprint":"1202.5614","month":"feb","pages":"10","year":"2012","urldate":"2012-05-02"},{"key":"Balbiani2008","type":"article","title":"\u2018Knowable' As \u2018Known After an Announcement'","author":"Balbiani, Philippe and Baltag, Alexandru and Ditmarsch, Hans Van and Herzig, Andreas and Hoshi, Tomohiro and De Lima, Tiago","abstract":"","comment":"","date_added":"2010-06-26","date_published":"2008-12-01","urls":["http:\/\/www.journals.cambridge.org\/abstract_S1755020308080210"],"collections":"","url":"http:\/\/www.journals.cambridge.org\/abstract_S1755020308080210","journal":"The Review of Symbolic Logic","month":"dec","number":"03","pages":"305","volume":"1","year":"2008","urldate":"2010-06-26"},{"key":"Galperin2003","type":"article","title":"Playing pool with $\\pi$ (the number $\\pi$ from a billiard point of view)","author":"Galperin, G","abstract":"","comment":"","date_added":"2013-11-14","date_published":"2003-08-06","urls":["http:\/\/www.turpion.org\/php\/reference.phtml?journal_id=rd&paper_id=252","http:\/\/ics.org.ru\/doc?pdf=440"],"collections":"Easily explained,Fun maths facts","url":"http:\/\/www.turpion.org\/php\/reference.phtml?journal_id=rd&paper_id=252 http:\/\/ics.org.ru\/doc?pdf=440","doi":"10.1070\/RD2003v008n04ABEH000252","journal":"Regular and Chaotic Dynamics","pages":"375--394","year":"2003","urldate":"2013-11-14"},{"key":"Hata1993","type":"article","title":"Rational approximations to $\\pi$ and some other numbers","author":"Hata, Masayoshi and Mignotte, M and Chudnovsky, G V and Beukers, F","abstract":"","comment":"","date_added":"2015-02-16","date_published":"1993-08-06","urls":["http:\/\/matwbn.icm.edu.pl\/ksiazki\/aa\/aa63\/aa6344.pdf"],"collections":"Easily explained","url":"http:\/\/matwbn.icm.edu.pl\/ksiazki\/aa\/aa63\/aa6344.pdf","journal":"Acta Arithmetica","number":"4","volume":"63","year":"1993","urldate":"2015-02-16"},{"key":"Chen2010","type":"article","title":"A Closed-Form Algorithm for Converting Hilbert Space-Filling Curve Indices","author":"Chen, Chih-sheng and Lin, Shen-yi and Fan, Min-hsuan and Huang, Chua-huang","abstract":"","comment":"","date_added":"2010-09-01","date_published":"2010-08-06","urls":["http:\/\/www.iaeng.org\/IJCS\/issues_v37\/issue_1\/IJCS_37_1_02.pdf"],"collections":"Basically computer science","url":"http:\/\/www.iaeng.org\/IJCS\/issues_v37\/issue_1\/IJCS_37_1_02.pdf","journal":"IAENG International Journal of Computer Science","keywords":"closed-form,hilbert space-filling curve,program generation,tensor product","number":"February","year":"2010","urldate":"2010-09-01"},{"key":"Renteln","type":"article","title":"Foolproof : A Sampling of Mathematical Folk Humor","author":"Renteln, Paul and Dundes, Alan","abstract":"","comment":"","date_added":"2010-06-18","date_published":"2005-08-06","urls":["http:\/\/www.ams.org\/notices\/200501\/fea-dundes.pdf"],"collections":"Lists and catalogues","url":"http:\/\/www.ams.org\/notices\/200501\/fea-dundes.pdf","journal":"Physics","urldate":"2010-06-18","year":"2005"},{"key":"Scarle2008","type":"misc","title":"Implications of the Turing Completeness of Reaction-Diffusion Models, informed by GPGPU simulations on an XBox 360: Cardiac Arrythmias, Re-entry and the Halting Problem","author":"Scarle, S","abstract":"","comment":"","date_added":"2010-09-30","date_published":"2008-08-06","urls":["http:\/\/www.ncbi.nlm.nih.gov\/pubmed\/19577519","http:\/\/research.microsoft.com\/pubs\/79271\/turing.pdf"],"collections":"Basically computer science,Basically physics,Unusual computers","url":"http:\/\/www.ncbi.nlm.nih.gov\/pubmed\/19577519 http:\/\/research.microsoft.com\/pubs\/79271\/turing.pdf","year":"2008","urldate":"2010-09-30"},{"key":"Flajolet2011","type":"article","title":"On Buffon Machines and Numbers","author":"Flajolet, Philippe","abstract":"The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically \"computes\" the number 2\/pi ~ 0.63661, which is the experiment's probability of success. Generalizing the experiment and simplifying the computational framework, we consider probability distributions, which can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that generate geometric, Poisson, and logarithmic-series distributions. We provide human-accessible Buffon machines, which require a dozen coin flips or less, on average, and produce experiments whose probabilities of success are expressible in terms of numbers such as, exp(-1), log 2, sqrt(3), cos(1\/4), aeta(5). Generally, we develop a collection of constructions based on simple probabilistic mechanisms that enable one to design Buffon experiments involving compositions of exponentials and logarithms, polylogarithms, direct and inverse trigonometric functions, algebraic and hypergeometric functions, as well as functions defined by integrals, such as the Gaussian error function.","comment":"","date_added":"2011-01-12","date_published":"2011-08-06","urls":["http:\/\/arxiv.org\/abs\/0906.5560","http:\/\/arxiv.org\/pdf\/0906.5560v2"],"collections":"Easily explained,Probability and statistics","url":"http:\/\/arxiv.org\/abs\/0906.5560 http:\/\/arxiv.org\/pdf\/0906.5560v2","journal":"Bernoulli","pages":"1--12","year":"2011","urldate":"2011-01-12","archivePrefix":"arXiv","eprint":"0906.5560"},{"key":"West2002","type":"article","title":"Better approximations to cumulative normal functions","author":"West, Graeme","abstract":"","comment":"","date_added":"2011-01-22","date_published":"2002-08-06","urls":["http:\/\/www.codeplanet.eu\/files\/download\/accuratecumnorm.pdf"],"collections":"Probability and statistics","url":"http:\/\/www.codeplanet.eu\/files\/download\/accuratecumnorm.pdf","journal":"Wilmott","pages":"70--76","year":"2002","urldate":"2011-01-22"},{"key":"Tondering2005","type":"article","title":"Surreal Numbers \u2013 An Introduction","author":"T\u00f8ndering, Claus","abstract":"","comment":"","date_added":"2011-02-16","date_published":"2005-08-06","urls":["http:\/\/www.tondering.dk\/download\/sur16.pdf"],"collections":"Unusual arithmetic","url":"http:\/\/www.tondering.dk\/download\/sur16.pdf","number":"January","year":"2005","urldate":"2011-02-16"},{"key":"Ellermeyer2008","type":"article","title":"James Garfield's Proof of the Pythagorean Theorem","author":"Ellermeyer, S F","abstract":"","comment":"","date_added":"2011-02-18","date_published":"2008-08-06","urls":["http:\/\/math.kennesaw.edu\/~sellerme\/sfehtml\/classes\/math1112\/garfieldpro.pdf"],"collections":"History,About proof","url":"http:\/\/math.kennesaw.edu\/~sellerme\/sfehtml\/classes\/math1112\/garfieldpro.pdf","pages":"1--4","year":"2008","urldate":"2011-02-18"},{"key":"Gerh2010","type":"article","title":"Automatic calculation of plane loci using Grobner bases and integration into a Dynamic Geometry System","author":"Gerh, Michael","abstract":"","comment":"","date_added":"2011-03-24","date_published":"2010-08-06","urls":["http:\/\/link.springer.com\/chapter\/10.1007%2F978-3-642-25070-5_4","http:\/\/jsxgraph.uni-bayreuth.de\/talks\/adg10\/presentation.pdf"],"collections":"Basically computer science","url":"http:\/\/link.springer.com\/chapter\/10.1007%2F978-3-642-25070-5_4 http:\/\/jsxgraph.uni-bayreuth.de\/talks\/adg10\/presentation.pdf","booktitle":"Opera","year":"2010","urldate":"2011-03-24"},{"key":"Figueira2002","type":"article","title":"An example of a computable absolutely normal number","author":"Figueira, Santiago","abstract":"","comment":"","date_added":"2011-03-28","date_published":"2002-08-06","urls":["http:\/\/www.glyc.dc.uba.ar\/santiago\/papers\/absnor.pdf"],"collections":"Fun maths facts","url":"http:\/\/www.glyc.dc.uba.ar\/santiago\/papers\/absnor.pdf","journal":"Theoretical Computer Science","pages":"947--958","volume":"270","year":"2002","urldate":"2011-03-28"},{"key":"Chandler","type":"article","title":"Testing Petri Nets for Mobile Robots Using Gr\u00f6bner Bases","author":"Chandler, Angie and Heyworth, Anne and Blair, Lynne and Seward, Derek","abstract":"","comment":"","date_added":"2011-05-09","date_published":"2000-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/0002119v1","http:\/\/citeseerx.ist.psu.edu\/viewdoc\/download;jsessionid=5C9463550D7C2241928C47ED1498EACB?doi=10.1.1.109.7667&rep=rep1&type=pdf","http:\/\/arxiv.org\/pdf\/math\/0002119.pdf"],"collections":"Basically computer science","url":"http:\/\/arxiv.org\/abs\/math\/0002119v1 http:\/\/citeseerx.ist.psu.edu\/viewdoc\/download;jsessionid=5C9463550D7C2241928C47ED1498EACB?doi=10.1.1.109.7667&rep=rep1&type=pdf http:\/\/arxiv.org\/pdf\/math\/0002119.pdf","journal":"Transition","urldate":"2011-05-09","year":"2000"},{"key":"Owens","type":"article","title":"Investigations of Game of Life cellular automata rules on Penrose Tilings : lifetime and ash statistics","author":"Owens, Nick and Stepney, Susan","abstract":"","comment":"","date_added":"2011-08-14","date_published":"2010-08-06","urls":["https:\/\/www-users.cs.york.ac.uk\/susan\/bib\/ss\/nonstd\/auto08b.pdf"],"collections":"Probability and statistics","url":"https:\/\/www-users.cs.york.ac.uk\/susan\/bib\/ss\/nonstd\/auto08b.pdf","pages":"1--34","urldate":"2011-08-14","year":"2010"},{"key":"Bacchus1995","type":"article","title":"Against Conditionalization","author":"Bacchus, Fahiem","abstract":"","comment":"","date_added":"2011-08-28","date_published":"1995-08-06","urls":["http:\/\/www.cs.toronto.edu\/~fbacchus\/Papers\/BKTSYN90.pdf"],"collections":"Probability and statistics","url":"http:\/\/www.cs.toronto.edu\/~fbacchus\/Papers\/BKTSYN90.pdf","year":"1995","urldate":"2011-08-28"},{"key":"Miller2009","type":"article","title":"Irrationality from the book","author":"Miller, Steven J and Montague, David","abstract":"","comment":"","date_added":"2011-08-29","date_published":"2009-08-06","urls":["http:\/\/arxiv.org\/abs\/0909.4913","http:\/\/arxiv.org\/pdf\/0909.4913v2"],"collections":"Easily explained","url":"http:\/\/arxiv.org\/abs\/0909.4913 http:\/\/arxiv.org\/pdf\/0909.4913v2","archivePrefix":"arXiv","arxivId":"arXiv:0909.4913v2","eprint":"arXiv:0909.4913v2","pages":"2--8","year":"2009","urldate":"2011-08-29"},{"key":"Shamos2011","type":"article","title":"Shamos's Catalog of the Real Numbers","author":"Shamos, Michael Ian","abstract":"","comment":"","date_added":"2011-09-08","date_published":"2011-08-06","urls":["http:\/\/euro.ecom.cmu.edu\/people\/faculty\/mshamos\/cat.pdf"],"collections":"Lists and catalogues","url":"http:\/\/euro.ecom.cmu.edu\/people\/faculty\/mshamos\/cat.pdf","booktitle":"Science","isbn":"0671750615","year":"2011","urldate":"2011-09-08"},{"key":"Mccarthy2006","type":"article","title":"Mad Abel : A card game for 2 + players","author":"Mccarthy, Sm\u00e1ri","abstract":"","comment":"","date_added":"2011-09-13","date_published":"2006-08-06","urls":["https:\/\/www.pagat.com\/docs\/madabel.pdf"],"collections":"Unusual arithmetic,Easily explained,Games to play with friends,The groups group","url":"https:\/\/www.pagat.com\/docs\/madabel.pdf","pages":"3--7","year":"2006","urldate":"2011-09-13"},{"key":"Erickson","type":"article","title":"Fusible Numbers","author":"Erickson, Jeff","abstract":"","comment":"","date_added":"2011-09-21","date_published":"2010-08-06","urls":["http:\/\/www.mathpuzzle.com\/fusible.pdf"],"collections":"Unusual arithmetic,Easily explained","url":"http:\/\/www.mathpuzzle.com\/fusible.pdf","urldate":"2011-09-21","year":"2010"},{"key":"Khovanova2010","type":"article","title":"Baron Munchhausen Redeems Himself : Bounds for a Coin-Weighing Puzzle Background","author":"Khovanova, Tanya and Lewis, Joel Brewster","abstract":"We investigate a coin-weighing puzzle that appeared in the Moscow Math Olympiad in 1991. We generalize the puzzle by varying the number of participating coins, and deduce an upper bound on the number of weighings needed to solve the puzzle that is noticeably better than the trivial upper bound. In particular, we show that logarithmically-many weighings on a balance suffice.","comment":"","date_added":"2011-10-14","date_published":"2010-08-06","urls":["http:\/\/arxiv.org\/abs\/1006.4135v1","http:\/\/arxiv.org\/pdf\/1006.4135v1"],"collections":"Attention-grabbing titles,Puzzles","url":"http:\/\/arxiv.org\/abs\/1006.4135v1 http:\/\/arxiv.org\/pdf\/1006.4135v1","archivePrefix":"arXiv","arxivId":"arXiv:1006.4135v1","eprint":"arXiv:1006.4135v1","pages":"1--19","year":"2010","urldate":"2011-10-14"},{"key":"Brenton2008","type":"article","title":"Remainder Wheels and Group Theory","author":"Brenton, Lawrence","abstract":"","comment":"","date_added":"2011-11-02","date_published":"2008-08-06","urls":["http:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Polya\/Brenton.pdf","http:\/\/www.maa.org\/programs\/maa-awards\/writing-awards\/remainder-wheels-and-group-theory"],"collections":"Unusual arithmetic,Easily explained,The groups group","url":"http:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Polya\/Brenton.pdf http:\/\/www.maa.org\/programs\/maa-awards\/writing-awards\/remainder-wheels-and-group-theory","number":"2","pages":"129--135","volume":"39","year":"2008","urldate":"2011-11-02"},{"key":"Demaine2008","type":"article","title":"Tetris is Hard, Even to Approximate","author":"Demaine, Erik D and Hohenberger, Susan and Liben-Nowell, David","abstract":"In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. We prove that in the offline version of Tetris, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p^(1-epsilon), when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of (2 - epsilon), for any epsilon>0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piece sets.","comment":"","date_added":"2011-11-23","date_published":"2008-08-06","urls":["http:\/\/arxiv.org\/abs\/cs\/0210020","http:\/\/arxiv.org\/pdf\/cs\/0210020v1"],"collections":"Attention-grabbing titles,Computational complexity of games","url":"http:\/\/arxiv.org\/abs\/cs\/0210020 http:\/\/arxiv.org\/pdf\/cs\/0210020v1","archivePrefix":"arXiv","arxivId":"arXiv:cs\/0210020v1","eprint":"0210020v1","journal":"Technology","pages":"1--56","primaryClass":"arXiv:cs","year":"2008","urldate":"2011-11-23"},{"key":"Lane","type":"article","title":"What Are the Odds?","author":"Lane, F.C.","abstract":"Gambling Has No Place in Baseball But Every Move on the Diamond Is Governed by the Laws of Chance--- The Successful Manager Is Successful Just So Far As He Knows and Accepts the Odds","comment":"","date_added":"2011-12-09","date_published":"1919-08-06","urls":["http:\/\/library.la84.org\/SportsLibrary\/BBM\/1919\/bbm226k.pdf"],"collections":"Games to play with friends,Probability and statistics","url":"http:\/\/library.la84.org\/SportsLibrary\/BBM\/1919\/bbm226k.pdf","keywords":"1919,Iss. 6,Vol. 22,pgs.337-339","pages":"337--339","urldate":"2011-12-09","year":"1919"},{"key":"Appel2002","type":"article","title":"Deobfuscation is in NP","author":"Appel, Andrew W","abstract":"","comment":"","date_added":"2011-12-12","date_published":"2002-08-06","urls":["https:\/\/www.cs.princeton.edu\/~appel\/papers\/deobfus.pdf"],"collections":"Basically computer science","url":"https:\/\/www.cs.princeton.edu\/~appel\/papers\/deobfus.pdf","journal":"Symposium A Quarterly Journal In Modern Foreign Literatures","pages":"1--2","year":"2002","urldate":"2011-12-12"},{"key":"Gerdes2000","type":"article","title":"Ethnomathematics as a new research field , illustrated by studies of mathematical ideas in African history","author":"Gerdes, Paulus","abstract":"","comment":"","date_added":"2012-01-19","date_published":"2000-08-06","urls":["http:\/\/citeseerx.ist.psu.edu\/viewdoc\/download?doi=10.1.1.551.2369&rep=rep1&type=pdf"],"collections":"History,Art","url":"http:\/\/citeseerx.ist.psu.edu\/viewdoc\/download?doi=10.1.1.551.2369&rep=rep1&type=pdf","journal":"African Philosophy","year":"2000","urldate":"2012-01-19"},{"key":"Deuber2004","type":"article","title":"A note on paradoxical metric spaces","author":"Deuber, W A and Simonovits, M and Os, V T S","abstract":"","comment":"","date_added":"2013-01-26","date_published":"2004-08-06","urls":["http:\/\/renyi.hu\/~miki\/walter07.pdf"],"collections":"","url":"http:\/\/renyi.hu\/~miki\/walter07.pdf","number":"1995","pages":"17--23","volume":"30","year":"2004","urldate":"2013-01-26"},{"key":"Dijkstra1981","type":"article","title":"Fibonacci numbers and Leonardo numbers","author":"Dijkstra, E.W.","abstract":"","comment":"","date_added":"2013-11-19","date_published":"1981-08-06","urls":["https:\/\/www.cs.utexas.edu\/users\/EWD\/transcriptions\/EWD07xx\/EWD797.html"],"collections":"Easily explained,Fibonaccinalia,Integerology","url":"https:\/\/www.cs.utexas.edu\/users\/EWD\/transcriptions\/EWD07xx\/EWD797.html","pages":"1--8","year":"1981","urldate":"2013-11-19"},{"key":"Hampton2009","type":"article","title":"A Mathematical Coloring Book","author":"Hampton, Marshall","abstract":"","comment":"","date_added":"2014-04-24","date_published":"2009-08-06","urls":["http:\/\/www.d.umn.edu\/~mhampton\/mathcolor17b.pdf"],"collections":"Art,Easily explained,Things to make and do","url":"http:\/\/www.d.umn.edu\/~mhampton\/mathcolor17b.pdf","year":"2009","urldate":"2014-04-24"},{"key":"Bialostocki1998","type":"article","title":"An Application of Elementary Group Theory to Central Solitaire","author":"Bialostocki, Arie","abstract":"","comment":"","date_added":"2014-11-10","date_published":"1998-08-06","urls":["http:\/\/www.oswego.edu\/Documents\/wac\/deans_awards_2015\/MAT%20%20Hannahan%20%203-31-15.pdf"],"collections":"","url":"http:\/\/www.oswego.edu\/Documents\/wac\/deans_awards_2015\/MAT%20%20Hannahan%20%203-31-15.pdf","journal":"The College Mathematics Journal","number":"3","pages":"208--212","volume":"29","year":"1998","urldate":"2014-11-10"},{"key":"Casazza","type":"article","title":"A mathematician's survival guide","author":"Casazza, Peter G","abstract":"","comment":"","date_added":"2012-04-29","date_published":"2012-08-06","urls":["https:\/\/faculty.missouri.edu\/~casazzap\/pdf\/140-MAA.pdf"],"collections":"The act of doing maths","url":"https:\/\/faculty.missouri.edu\/~casazzap\/pdf\/140-MAA.pdf","pages":"1--18","urldate":"2012-04-29","year":"2012"},{"key":"Demaine2010","type":"article","title":"Circle Packing for Origami Design Is Hard","author":"Demaine, E.D. and Fekete, S.P. and Lang, R.J.","abstract":"We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems, settling the complexity of these natural packing problems. On the positive side, we show that any set of circles of total area 1 can be packed into a square of size 8\/pi=2.546... These results are motivated by problems arising in the context of origami design.","comment":"","date_added":"2010-08-13","date_published":"2010-08-06","urls":["http:\/\/arxiv.org\/abs\/1008.1224","http:\/\/arxiv.org\/pdf\/1008.1224v2"],"collections":"Geometry","url":"http:\/\/arxiv.org\/abs\/1008.1224 http:\/\/arxiv.org\/pdf\/1008.1224v2","journal":"Arxiv preprint arXiv:1008.1224","pages":"1--17","publisher":"arxiv.org","year":"2010","urldate":"2010-08-13","archivePrefix":"arXiv","eprint":"1008.1224"},{"key":"CountingGroups","type":"article","title":"Counting groups: gnus, moas and other exotica","author":"John H. Conway and Heiko Dietrich and E.A. O\u2019Brien","abstract":"The number of groups of a given order is a fascinating function. We report on\r\nits known values, discuss some of its properties, and study some related functions.","comment":"","date_added":"2016-05-20","date_published":"2008-08-06","urls":["https:\/\/www.math.auckland.ac.nz\/~obrien\/research\/gnu.pdf"],"collections":"Attention-grabbing titles,Animals,The groups group","url":"https:\/\/www.math.auckland.ac.nz\/~obrien\/research\/gnu.pdf","urldate":"2016-05-20","year":"2008"},{"key":"LostMathematics","type":"article","title":"Lectures on lost mathematics","author":"Branko Gr\u00fcnbaum","abstract":"","comment":"","date_added":"2012-05-10","date_published":"2010-08-06","urls":["http:\/\/scholar.google.com\/scholar?hl=en&btnG=Search&q=intitle:Lectures+on+lost+mathematics#0","https:\/\/digital.lib.washington.edu\/researchworks\/bitstream\/handle\/1773\/15700\/Lost%2520Mathematics.pdf?sequence=1"],"collections":"History,The act of doing maths","url":"http:\/\/scholar.google.com\/scholar?hl=en&btnG=Search&q=intitle:Lectures+on+lost+mathematics#0 https:\/\/digital.lib.washington.edu\/researchworks\/bitstream\/handle\/1773\/15700\/Lost%2520Mathematics.pdf?sequence=1","number":"April","year":"2010","urldate":"2012-05-10"},{"key":"Poloblanco2007","type":"article","title":"Theory and History of Geometric Models","author":"Polo-blanco, Irene","abstract":"","comment":"","date_added":"2012-04-15","date_published":"2007-08-06","urls":["http:\/\/dissertations.ub.rug.nl\/FILES\/faculties\/science\/2007\/i.polo.blanco\/thesis.pdf"],"collections":"History,Geometry","url":"http:\/\/dissertations.ub.rug.nl\/FILES\/faculties\/science\/2007\/i.polo.blanco\/thesis.pdf","isbn":"9789078927013","number":"april 1977","year":"2007","urldate":"2012-04-15"},{"key":"OnPellegrinos20CapsinS43","type":"article","title":"On Pellegrino's 20-Caps in $S_{4,3}$","author":"R. Hill","abstract":"Although Pellegrino demonstrated that every 20-cap in $S_{4,3}$ is one of two geometric types, but it is by no means clear how many inequivalent 20-caps are there in each type. This chapter demonstrates that there are in all exactly nine inequivalent 20-caps in $S_{4,3}$. It also shows that just two of these occur as the intersection of a 56-cap in $S_{5,3}$ with a hyperplane. Because any 10-cap in $S_{3,3}$ is an elliptic quadric and is unique up to equivalence, it follows that any choice of E and V is equivalent to any other. However, for a given choice of E and V, there are 310 different r-caps. The seemingly difficult task of finding how many of these are inequivalent is made relatively simple by using the triple transitivity of the group Aut E on the points of E, together with the uniqueness of the ternary Golay code. The chapter identifies those 20-caps that occur as the intersection of a 56-cap in $S_{5,3}$ with a hyperplane and shows that caps of both these types do occur as sections of a 56-cap in $S_{5,3}$.","comment":"","date_added":"2016-06-01","date_published":"1983-08-06","urls":["http:\/\/www.sciencedirect.com\/science\/article\/pii\/S030402080873322X","http:\/\/www.sciencedirect.com\/science\/article\/pii\/S030402080873322X\/pdf?md5=d1b75feaabe33b62c6beb656d86a2a7d&pid=1-s2.0-S030402080873322X-main.pdf"],"collections":"","url":"http:\/\/www.sciencedirect.com\/science\/article\/pii\/S030402080873322X http:\/\/www.sciencedirect.com\/science\/article\/pii\/S030402080873322X\/pdf?md5=d1b75feaabe33b62c6beb656d86a2a7d&pid=1-s2.0-S030402080873322X-main.pdf","urldate":"2016-06-01","year":"1983"},{"key":"ChallengingProblems","type":"article","title":"Challenging mathematical problems with elementary solutions","author":"A.M. Yaglom and I.M. Yaglom","abstract":"","comment":"","date_added":"2016-06-08","date_published":"1964-08-06","urls":["https:\/\/drive.google.com\/file\/d\/0B45juCGJ7U7PNmJhY2M0ZTMtNzU1MC00NGU3LTg3NGItMzE3OTViOThiYmU3\/view"],"collections":"Puzzles","url":"https:\/\/drive.google.com\/file\/d\/0B45juCGJ7U7PNmJhY2M0ZTMtNzU1MC00NGU3LTg3NGItMzE3OTViOThiYmU3\/view","urldate":"2016-06-08","year":"1964"},{"key":"DrMitchillandtheMathematicalTetrodonThePublicDomainReview","type":"article","title":"Dr Mitchill and the Mathematical Tetrodon","author":"Kevin Dann","abstract":"","comment":"","date_added":"2016-06-13","date_published":"2015-08-06","urls":["http:\/\/publicdomainreview.org\/2015\/09\/16\/dr-mitchill-and-the-mathematical-tetrodon\/"],"collections":"Attention-grabbing titles,Art,Animals","url":"http:\/\/publicdomainreview.org\/2015\/09\/16\/dr-mitchill-and-the-mathematical-tetrodon\/","urldate":"2016-06-13","year":"2015"},{"key":"Dudley","type":"article","title":"What to do when the trisector comes","author":"Dudley, Underwood","abstract":"","comment":"","date_added":"2015-11-30","date_published":"1983-08-06","urls":["http:\/\/web.mst.edu\/~lmhall\/WhatToDoWhenTrisectorComes.pdf"],"collections":"Attention-grabbing titles,The act of doing maths","url":"http:\/\/web.mst.edu\/~lmhall\/WhatToDoWhenTrisectorComes.pdf","urldate":"2015-11-30","year":"1983"},{"key":"Thesnaillemma","type":"article","title":"The snail lemma","author":"Enrico M. Vitale","abstract":"The classical snake lemma produces a six terms exact sequence starting from\r\na commutative square with one of the edge being a regular epimorphism. We establish\r\na new diagram lemma, that we call snail lemma, removing such a condition. We also\r\nshow that the snail lemma subsumes the snake lemma and we give an interpretation of\r\nthe snail lemma in terms of strong homotopy kernels. Our results hold in any pointed\r\nregular protomodular category.","comment":"","date_added":"2016-06-13","date_published":"2016-08-06","urls":["http:\/\/www.tac.mta.ca\/tac\/volumes\/31\/19\/31-19abs.html","http:\/\/www.tac.mta.ca\/tac\/volumes\/31\/19\/31-19.pdf"],"collections":"Attention-grabbing titles,Animals","url":"http:\/\/www.tac.mta.ca\/tac\/volumes\/31\/19\/31-19abs.html http:\/\/www.tac.mta.ca\/tac\/volumes\/31\/19\/31-19.pdf","urldate":"2016-06-13","year":"2016"},{"key":"Yato2003","type":"article","title":"Complexity and Completeness of Finding Another solution and its Application to Puzzles","author":"Takayushi Yato","abstract":"The Another Solution Problem (ASP) of a problem $\\Pi$ is the following problem: for a given instance $x$ of $\\Pi$ and a solution $s$ to it, find a solution to $x$ other than $s$. The notion of ASP as a new class of problems was first introduced by Ueda and Nagao. They also pointed out that polynomial-time parsimonious reductions which allow polynomial-time transformation of solutions can derive the NP-completeness of ASP of a certain problem from that of ASP of another. They used this property to show the NP-completeness of ASP of Nonogram, a sort of puzzle. Following it, Seta considered the problem to find another solution when $n$\r\nsolutions are given. (We call the problem $n$-ASP.) He proved the NP-completeness of $n$-ASP of some problems, including Cross Sum, for any $n$.\r\n\r\nIn this thesis we establish a rigid formalization of $n$-ASPs to investigate their characteristics more clearly. In particular we introduce ASP-completeness, the completeness with respect to the reductions satisfying the properties mentioned above, and show that ASP-completeness of a problem implies NP-completeness of $n$-ASP of the problem for all $n$. Moreover we research the relation between ASPs and other versions of problems, such as counting problems and enumeration problems, and show the equivalence of the class of problems which allow enumerations of solutions in polynomial time and the class of problems of which $n$-ASP is\r\nsolvable in polynomial time.\r\n\r\nAs Ueda and Nagao pointed out, the complexity of ASPs has a relation with the difficulty of designing puzzles. We prove the ASP-completeness of three popular puzzles: Slither Link, Number Place and Fillomino. The ASP-completeness of Slither Link is shown via a reduction from the Hamiltonian circuit problem for restricted graphs, that of Number Place is from the problem of Latin square completion, and that of Fillomino is from planar 3SAT. Since ASP=completeness implies NP-completeness as is mentioned above, these results can be regarded as new results of NP-completeness proof of puzzles.","comment":"","date_added":"2016-06-18","date_published":"2003-08-06","urls":["http:\/\/www-imai.is.s.u-tokyo.ac.jp\/~yato\/data2\/MasterThesis.pdf"],"collections":"Puzzles,Basically computer science,About proof","url":"http:\/\/www-imai.is.s.u-tokyo.ac.jp\/~yato\/data2\/MasterThesis.pdf","urldate":"2016-06-18","year":"2003"},{"key":"AnIrrationalityMeasureforRegularPaperfoldingNumbers","type":"article","title":"An Irrationality Measure for Regular Paperfolding Numbers","author":"Michael Coons and Paul Vrbik","abstract":"Let $F(z) = \\sum_{n \\geq 1} f_n z^n$ be the generating series of the regular paperfolding sequence. For a real number $\\alpha$ the irrationality exponent $\\mu(\\alpha)$, of $\\alpha$, is defined as the supremum of the set of real numbers $\\mu$ such that the inequality $\\lvert \\alpha - p\/q \\rvert \\lt q-\\mu$ has infinitely many solutions $(p,q) \\in Z \\times N$. In this paper, using a method introduced by Bugeaud, we prove that\r\n\r\n\\[ \\mu(F(1\/b)) \\leq 275331112987\/137522851840 = 2.002075359 \\ldots \\]\r\n\r\nfor all integers $b \\geq 2$. This improves upon the previous bound of $\\mu(F(1\/b)) \\leq 5$ given by Adamczewski and Rivoal.","comment":"","date_added":"2016-07-11","date_published":"2012-08-06","urls":["https:\/\/cs.uwaterloo.ca\/journals\/JIS\/VOL15\/Coons\/coons3.html","https:\/\/cs.uwaterloo.ca\/journals\/JIS\/VOL15\/Coons\/coons3.pdf"],"collections":"Things to make and do,Integerology","url":"https:\/\/cs.uwaterloo.ca\/journals\/JIS\/VOL15\/Coons\/coons3.html https:\/\/cs.uwaterloo.ca\/journals\/JIS\/VOL15\/Coons\/coons3.pdf","urldate":"2016-07-11","year":"2012"},{"key":"RationalPolynomialsThatTakeIntegerValuesattheFibonacciNumbers","type":"article","title":"Rational Polynomials That Take Integer Values at the Fibonacci Numbers","author":"Keith Johnson and Kira Scheibelhut","abstract":"An integer-valued polynomial on a subset $S$ of $\\mathbb{Z}$ is a polynomial $f(x) \\in \\mathbb{Q}[x]$ with the property $f(S) \\subseteq \\mathbb{Z}$. This article describes the ring of such polynomials in the special case that $S$ is the Fibonacci numbers. An algorithm is described for finding a regular basis, i.e., an ordered sequence of polynomials, the $n$th one of degree $n$, with which any such polynomial can be expressed as a unique integer linear combination.","comment":"","date_added":"2016-08-02","date_published":"2016-08-06","urls":["http:\/\/www.jstor.org\/stable\/10.4169\/amer.math.monthly.123.4.338"],"collections":"Fibonaccinalia,Fun maths facts,Integerology","url":"http:\/\/www.jstor.org\/stable\/10.4169\/amer.math.monthly.123.4.338","urldate":"2016-08-02","year":"2016"},{"key":"TenLessonsRota","type":"article","title":"Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations","author":"Giancarlo Rota","abstract":"","comment":"An entertainingly indignant rant about the way DEs are taught.","date_added":"2016-08-03","date_published":"1997-08-06","urls":["https:\/\/web.williams.edu\/Mathematics\/lg5\/Rota.pdf"],"collections":"Attention-grabbing titles,The act of doing maths","url":"https:\/\/web.williams.edu\/Mathematics\/lg5\/Rota.pdf","urldate":"2016-08-03","year":"1997"},{"key":"MattersComputational","type":"book","title":"Matters Computational - Ideas, Algorithms, Source Code","author":"J\u00f6rg Arndt","abstract":"This is the book \"Matters Computational\" (formerly titled \"Algorithms for Programmers\"), published with Springer.","comment":"Lots and lots of algorithms!","date_added":"2016-08-03","date_published":"2010-08-06","urls":["http:\/\/www.jjj.de\/fxt\/fxtbook.pdf","http:\/\/www.jjj.de\/fxt\/#fxtbook"],"collections":"Basically computer science","url":"http:\/\/www.jjj.de\/fxt\/fxtbook.pdf http:\/\/www.jjj.de\/fxt\/#fxtbook","urldate":"2016-08-03","year":"2010"},{"key":"Thedenominatorsofconvergentsforcontinuedfractions","type":"article","title":"The denominators of convergents for continued fractions","author":"Lulu Fang and Min Wu and Bing Li","abstract":"For any real number $x \\in [0,1)$, we denote by $q_n(x)$ the denominator of\r\nthe $n$-th convergent of the continued fraction expansion of $x$ $(n \\in\r\n\\mathbb{N})$. It is well-known that the Lebesgue measure of the set of points\r\n$x \\in [0,1)$ for which $\\log q_n(x)\/n$ deviates away from $\\pi^2\/(12\\log2)$\r\ndecays to zero as $n$ tends to infinity. In this paper, we study the rate of\r\nthis decay by giving an upper bound and a lower bound. What is interesting is\r\nthat the upper bound is closely related to the Hausdorff dimensions of the\r\nlevel sets for $\\log q_n(x)\/n$. As a consequence, we obtain a large deviation\r\ntype result for $\\log q_n(x)\/n$, which indicates that the rate of this decay is\r\nexponential.","comment":"","date_added":"2016-08-06","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1608.01246v1","http:\/\/arxiv.org\/pdf\/1608.01246v1"],"collections":"","url":"http:\/\/arxiv.org\/abs\/1608.01246v1 http:\/\/arxiv.org\/pdf\/1608.01246v1","urldate":"2016-08-06","archivePrefix":"arXiv","eprint":"1608.01246","primaryClass":"math.NT","year":"2016"},{"key":"BeckettGrayCodes","type":"article","title":"Beckett-Gray Codes","author":"Mark Cooke and Chris North and Megan Dewar and Brett Stevens","abstract":"In this paper we discuss a natural mathematical structure that is derived\r\nfrom Samuel Beckett's play \"Quad\". This structure is called a binary\r\nBeckett-Gray code. Our goal is to formalize the definition of a binary\r\nBeckett-Gray code and to present the work done to date. In addition, we\r\ndescribe the methodology used to obtain enumeration results for binary\r\nBeckett-Gray codes of order $n = 6$ and existence results for binary\r\nBeckett-Gray codes of orders $n = 7,8$. We include an estimate, using Knuth's\r\nmethod, for the size of the exhaustive search tree for $n=7$. Beckett-Gray\r\ncodes can be realized as successive states of a queue data structure. We show\r\nthat the binary reflected Gray code can be realized as successive states of two\r\nstack data structures.","comment":"","date_added":"2016-08-24","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1608.06001v1","http:\/\/arxiv.org\/pdf\/1608.06001v1"],"collections":"","url":"http:\/\/arxiv.org\/abs\/1608.06001v1 http:\/\/arxiv.org\/pdf\/1608.06001v1","urldate":"2016-08-24","archivePrefix":"arXiv","eprint":"1608.06001","primaryClass":"math.CO","year":"2016"},{"key":"TopologicallyDistinctSetsofNonintersectingCirclesinthePlane","type":"article","title":"Topologically Distinct Sets of Non-intersecting Circles in the Plane","author":"Richard J. Mathar","abstract":"Nested parentheses are forms in an algebra which define orders of\r\nevaluations. A class of well-formed sets of associated opening and closing\r\nparentheses is well studied in conjunction with Dyck paths and Catalan numbers.\r\nNested parentheses also represent cuts through circles on a line. These become\r\ntopologies of non-intersecting circles in the plane if the underlying algebra\r\nis commutative.\r\n This paper generalizes the concept and answers quantitatively - as\r\nrecurrences and generating functions of matching rooted forests - the\r\nquestions: how many different topologies of nested circles exist in the plane\r\nif (i) pairs of circles may intersect, or (ii) even triples of circles may\r\nintersect. That analysis is driven by examining the symmetry properties of the\r\ninner regions of the fundamental type(s) of the intersecting pairs and triples.","comment":"","date_added":"2016-08-25","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1603.00077v1","http:\/\/arxiv.org\/pdf\/1603.00077v1"],"collections":"Easily explained,Geometry","url":"http:\/\/arxiv.org\/abs\/1603.00077v1 http:\/\/arxiv.org\/pdf\/1603.00077v1","urldate":"2016-08-25","archivePrefix":"arXiv","eprint":"1603.00077","primaryClass":"math.CO","year":"2016"},{"key":"WinklerSevenPuzzles","type":"article","title":"Seven Puzzles You Think You Must Not Have Heard Correctly","author":"Peter Winkler","abstract":"A typical mathematical puzzle sounds tricky but solvable \u2014 if not by you, then perhaps by the\r\ngenius down the hall. But sometimes the task at hand is so obviously impossible that you are moved\r\nto ask whether you understood the problem correctly, and other times, the task seems so trivial\r\nthat you are sure you must have missed something.\r\nHere, I have compiled seven puzzles which have often been greeted by words similar to \u201cWait\r\na minute \u2014 I must not have heard that correctly.\u201d Some seem too hard, some too easy; after you've\r\nworked on them for a while, you may find that the hard ones now seem easy and vice versa.","comment":"","date_added":"2016-08-30","date_published":"2006-08-06","urls":["https:\/\/math.dartmouth.edu\/~pw\/solutions.pdf"],"collections":"Attention-grabbing titles,Puzzles","url":"https:\/\/math.dartmouth.edu\/~pw\/solutions.pdf","urldate":"2016-08-30","year":"2006"},{"key":"PonytailMotion","type":"article","title":"Ponytail Motion","author":"Joseph B. Keller","abstract":"A jogger's ponytail sways from side to side as the jogger runs, although her head does not move from side to side. The jogger's head just moves up and down, forcing the ponytail to do so also. We show in two ways that this vertical motion is unstable to lateral perturbations. First we treat the ponytail as a rigid pendulum, and then we treat it as a flexible string; in each case, it is hanging from a support which is moving up and down periodically, and we solve the linear equation for small lateral oscillation. The angular displacement of the pendulum and the amplitude of each mode of the string satisfy Hill's equation. This equation has solutions which grow exponentially in time when the natural frequency of the pendulum, or that of a mode of the string, is close to an integer multiple of half the frequency of oscillation of the support. Then the vertical motion is unstable, and the ponytail sways.","comment":"","date_added":"2016-09-19","date_published":"2010-08-06","urls":["http:\/\/epubs.siam.org\/doi\/abs\/10.1137\/090760477","http:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090760477"],"collections":"Easily explained,Basically physics","url":"http:\/\/epubs.siam.org\/doi\/abs\/10.1137\/090760477 http:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090760477","urldate":"2016-09-19","year":"2010"},{"key":"HistoricalMethodsForMultiplication","type":"article","title":"Historical methods for multiplication","author":"Bj\u00f8rn Smestad and Konstantinos Nikolantonakis","abstract":"This paper summarizes the contents of our workshop. In this workshop, we presented and discussed the \"Greek\" multiplication, given by Eutokios of Ascalon in his commentary on The Measurement of a Circle. We discussed part of the text from the treatise of Eutokios. Our basic thesis is that we think that this historical method for multiplication is part of the algorithms friendly to the user (based on the ideas that the children use in their informal mental strategies). The important idea is that the place value of numbers is maintained and the students act with quantities and not with isolated symbols as it happens with the classic algorithm. This helps students to control their thought at every stage of calculation. We also discussed the Russian method and the method by the cross (basically the same as \"Casting out nines\") to control the execution of the operations.","comment":"","date_added":"2016-09-22","date_published":"2010-08-06","urls":["https:\/\/www.researchgate.net\/publication\/263733700_Historical_methods_for_multiplication"],"collections":"Notation and conventions,History,Easily explained,Integerology","url":"https:\/\/www.researchgate.net\/publication\/263733700_Historical_methods_for_multiplication","urldate":"2016-09-22","year":"2010"},{"key":"Fractalgeometryofacomplexplumagetraitrevealsbirdsquality","type":"article","title":"Fractal geometry of a complex plumage trait reveals bird's quality","author":"Lorenzo P\u00e9rez-Rodr\u00edguez and Roger Jovani and Fran\\ccois Mougeot","abstract":"Animal coloration is key in natural and sexual selection, playing significant roles in intra- and interspecific communication because of its linkage to individual behaviour, genetics and physiology. Simple animal traits such as the area or the colour intensity of homogeneous patches have been profusely studied. More complex patterns are widespread in nature, but they escape our understanding because their variation is difficult to capture effectively by standard, simple measures. Here, we used fractal geometry to quantify inter-individual variation in the expression of a complex plumage trait, the heterogeneous black bib of the red-legged partridge (Alectoris rufa). We show that a higher bib fractal dimension (FD) predicted better individual body condition, as well as immune responsiveness, which is condition-dependent in our study species. Moreover, when food intake was experimentally reduced during moult as a means to reduce body condition, the bib's FD significantly decreased. Fractal geometry therefore provides new opportunities for the study of complex animal colour patterns and their roles in animal communication.","comment":"","date_added":"2016-09-27","date_published":"2013-08-06","urls":["http:\/\/rspb.royalsocietypublishing.org\/content\/280\/1755\/20122783"],"collections":"Animals,Geometry","url":"http:\/\/rspb.royalsocietypublishing.org\/content\/280\/1755\/20122783","urldate":"2016-09-27","year":"2013"},{"key":"AvoidingSquaresandOverlapsOvertheNaturalNumbers","type":"article","title":"Avoiding Squares and Overlaps Over the Natural Numbers","author":"Mathieu Guay-Paquet and Jeffrey Shallit","abstract":"We consider avoiding squares and overlaps over the natural numbers, using a\r\ngreedy algorithm that chooses the least possible integer at each step; the word\r\ngenerated is lexicographically least among all such infinite words. In the case\r\nof avoiding squares, the word is 01020103..., the familiar ruler function, and\r\nis generated by iterating a uniform morphism. The case of overlaps is more\r\nchallenging. We give an explicitly-defined morphism phi : N* -> N* that\r\ngenerates the lexicographically least infinite overlap-free word by iteration.\r\nFurthermore, we show that for all h,k in N with h <= k, the word phi^{k-h}(h)\r\nis the lexicographically least overlap-free word starting with the letter h and\r\nending with the letter k, and give some of its symmetry properties.","comment":"","date_added":"2016-10-03","date_published":"2009-08-06","urls":["http:\/\/arxiv.org\/abs\/0901.1397v1","http:\/\/arxiv.org\/pdf\/0901.1397v1"],"collections":"Integerology","url":"http:\/\/arxiv.org\/abs\/0901.1397v1 http:\/\/arxiv.org\/pdf\/0901.1397v1","urldate":"2016-10-03","archivePrefix":"arXiv","eprint":"0901.1397","primaryClass":"math.CO","year":"2009"},{"key":"ADiscreteandBoundedEnvyFreeCakeCuttingProtocolforAnyNumberofAgents","type":"article","title":"A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents","author":"Haris Aziz and Simon Mackenzie","abstract":"We consider the well-studied cake cutting problem in which the goal is to\r\nfind an envy-free allocation based on queries from $n$ agents. The problem has\r\nreceived attention in computer science, mathematics, and economics. It has been\r\na major open problem whether there exists a discrete and bounded envy-free\r\nprotocol. We resolve the problem by proposing a discrete and bounded envy-free\r\nprotocol for any number of agents. The maximum number of queries required by\r\nthe protocol is $n^{n^{n^{n^{n^n}}}}$. We additionally show that even if we do\r\nnot run our protocol to completion, it can find in at most $n^{n+1}$ queries a\r\npartial allocation of the cake that achieves proportionality (each agent gets\r\nat least $1\/n$ of the value of the whole cake) and envy-freeness. Finally we\r\nshow that an envy-free partial allocation can be computed in $n^{n+1}$ queries\r\nsuch that each agent gets a connected piece that gives the agent at least\r\n$1\/(3n)$ of the value of the whole cake.","comment":"","date_added":"2016-10-13","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1604.03655v10","http:\/\/arxiv.org\/pdf\/1604.03655v10"],"collections":"Attention-grabbing titles,Easily explained,Protocols and strategies,Food,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1604.03655v10 http:\/\/arxiv.org\/pdf\/1604.03655v10","urldate":"2016-10-13","archivePrefix":"arXiv","eprint":"1604.03655","primaryClass":"cs.DS","year":"2016"},{"key":"GeometricMechanicsofCurvedCreaseOrigami","type":"article","title":"Geometric Mechanics of Curved Crease Origami","author":"Marcelo A. Dias and Levi H. Dudte and L. Mahadevan and Christian D. Santangelo","abstract":"Folding a sheet of paper along a curve can lead to structures seen in\r\ndecorative art and utilitarian packing boxes. Here we present a theory for the\r\nsimplest such structure: an annular circular strip that is folded along a\r\ncentral circular curve to form a three-dimensional buckled structure driven by\r\ngeometrical frustration. We quantify this shape in terms of the radius of the\r\ncircle, the dihedral angle of the fold and the mechanical properties of the\r\nsheet of paper and the fold itself. When the sheet is isometrically deformed\r\neverywhere except along the fold itself, stiff folds result in creases with\r\nconstant curvature and oscillatory torsion. However, relatively softer folds\r\ninherit the broken symmetry of the buckled shape with oscillatory curvature and\r\ntorsion. Our asymptotic analysis of the isometrically deformed state is\r\ncorroborated by numerical simulations which allow us to generalize our analysis\r\nto study multiply folded structures.","comment":"","date_added":"2016-10-14","date_published":"2012-08-06","urls":["http:\/\/arxiv.org\/abs\/1206.0461v2","http:\/\/arxiv.org\/pdf\/1206.0461v2"],"collections":"Art,Basically physics,Things to make and do,Geometry","url":"http:\/\/arxiv.org\/abs\/1206.0461v2 http:\/\/arxiv.org\/pdf\/1206.0461v2","urldate":"2016-10-14","archivePrefix":"arXiv","eprint":"1206.0461","primaryClass":"cond-mat.soft","year":"2012"},{"key":"ASpaceEfficientAlgorithmfortheCalculationoftheDigitDistributionintheKolakoskiSequence","type":"article","title":"A Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence","author":"Johan Nilsson","abstract":"With standard algorithms for generating the classical Kolakoski sequence, the\r\nnumerical calculation of the digit distribution requires a linear amount of\r\nspace. Here, we present an algorithm for calculating the distribution of the\r\ndigits in the classical Kolakoski sequence, that only requires a logarithmic\r\namount of space and still runs in linear time. The algorithm is easily\r\nadaptable to generalised Kolakoski sequences.","comment":"","date_added":"2016-10-14","date_published":"2011-08-06","urls":["http:\/\/arxiv.org\/abs\/1110.4228v2","http:\/\/arxiv.org\/pdf\/1110.4228v2"],"collections":"","url":"http:\/\/arxiv.org\/abs\/1110.4228v2 http:\/\/arxiv.org\/pdf\/1110.4228v2","urldate":"2016-10-14","archivePrefix":"arXiv","eprint":"1110.4228","primaryClass":"math.CO","year":"2011"},{"key":"Tworemarksonevenandoddtownproblems","type":"article","title":"Two remarks on even and oddtown problems","author":"Benny Sudakov and Pedro Vieira","abstract":"A family $\\mathcal A$ of subsets of an $n$-element set is called an eventown\r\n(resp. oddtown) if all its sets have even (resp. odd) size and all pairwise\r\nintersections have even size. Using tools from linear algebra, it was shown by\r\nBerlekamp and Graver that the maximum size of an eventown is $2^{\\left\\lfloor\r\nn\/2\\right\\rfloor}$. On the other hand (somewhat surprisingly), it was proven by\r\nBerlekamp, that oddtowns have size at most $n$. Over the last four decades,\r\nmany extensions of this even\/oddtown problem have been studied. In this paper\r\nwe present new results on two such extensions. First, extending a result of Vu,\r\nwe show that a $k$-wise eventown (i.e., intersections of $k$ sets are even) has\r\nfor $k \\geq 3$ a unique extremal configuration and obtain a stability result\r\nfor this problem. Next we improve some known bounds for the defect version of\r\nan $\\ell$-oddtown problem. In this problem we consider sets of size $\\not\\equiv\r\n0 \\pmod \\ell$ where $\\ell$ is a prime number $\\ell$ (not necessarily $2$) and\r\nallow a few pairwise intersections to also have size $\\not\\equiv 0 \\pmod \\ell$.","comment":"","date_added":"2016-10-27","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1610.07907v1","http:\/\/arxiv.org\/pdf\/1610.07907v1"],"collections":"Combinatorics","url":"http:\/\/arxiv.org\/abs\/1610.07907v1 http:\/\/arxiv.org\/pdf\/1610.07907v1","urldate":"2016-10-27","archivePrefix":"arXiv","eprint":"1610.07907","primaryClass":"math.CO","year":"2016"},{"key":"QuasipracticalNumbers","type":"article","title":"Quasipractical Numbers","author":"Harvey J. Hindin","abstract":"","comment":"","date_added":"2016-11-28","date_published":"1980-08-06","urls":["http:\/\/ieeexplore.ieee.org\/document\/1090205\/"],"collections":"Unusual arithmetic,Integerology","url":"http:\/\/ieeexplore.ieee.org\/document\/1090205\/","urldate":"2016-11-28","year":"1980"},{"key":"DevelopingaMathematicalModelforBobbinLace","type":"article","title":"Developing a Mathematical Model for Bobbin Lace","author":"Veronika Irvine and Frank Ruskey","abstract":"Bobbin lace is a fibre art form in which intricate and delicate patterns are\r\ncreated by braiding together many threads. An overview of how bobbin lace is\r\nmade is presented and illustrated with a simple, traditional bookmark design.\r\nResearch on the topology of textiles and braid theory form a base for the\r\ncurrent work and is briefly summarized. We define a new mathematical model that\r\nsupports the enumeration and generation of bobbin lace patterns using an\r\nintelligent combinatorial search. Results of this new approach are presented\r\nand, by comparison to existing bobbin lace patterns, it is demonstrated that\r\nthis model reveals new patterns that have never been seen before. Finally, we\r\napply our new patterns to an original bookmark design and propose future areas\r\nfor exploration.","comment":"","date_added":"2016-11-28","date_published":"2014-08-06","urls":["http:\/\/arxiv.org\/abs\/1406.1532v3","http:\/\/arxiv.org\/pdf\/1406.1532v3"],"collections":"Art,Geometry,Modelling","url":"http:\/\/arxiv.org\/abs\/1406.1532v3 http:\/\/arxiv.org\/pdf\/1406.1532v3","urldate":"2016-11-28","archivePrefix":"arXiv","eprint":"1406.1532","primaryClass":"math.CO","year":"2014"},{"key":"SequencesOfConsecutiveNNivenNumbers","type":"article","title":"Sequences of consecutive \\(n\\)-Niven numbers","author":"H.G. Grundman","abstract":"A Niven number is a positive integer that is divisible by the sum of its digits. In 1982, Kennedy showed that there do not exist sequences of more than 21 consecutive Niven numbers. In 1992, Cooper & Kennedy improved this result by proving that there does not exist a sequence of more than 20 consecutive Niven numbers. They also proved that this bound is the best possible by producing an infinite family of sequences of 20 consecutive Niven numbers. For any positive integer \\(n \\gt 2\\), define an \\(n\\)-Niven number to be a positive integer that is divisible by the sum of the digits in its base \\(n\\) expansion. This paper examines the maximal possible\r\nlengths of sequences of consecutive \\(n\\)-Niven numbers. The main result is given in the following theorem. ","comment":"","date_added":"2016-12-05","date_published":"1992-08-06","urls":["http:\/\/www.fq.math.ca\/Scanned\/32-2\/grundman.pdf"],"collections":"Integerology","url":"http:\/\/www.fq.math.ca\/Scanned\/32-2\/grundman.pdf","urldate":"2016-12-05","year":"1992"},{"key":"Everynaturalnumberisthesumoffortyninepalindromes","type":"article","title":"Every natural number is the sum of forty-nine palindromes","author":"William D. Banks","abstract":"It is shown that the set of decimal palindromes is an additive basis for the\r\nnatural numbers. Specifically, we prove that every natural number can be\r\nexpressed as the sum of forty-nine (possibly zero) decimal palindromes.","comment":"","date_added":"2016-12-17","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1508.04721v1","http:\/\/arxiv.org\/pdf\/1508.04721v1"],"collections":"Fun maths facts,Integerology","url":"http:\/\/arxiv.org\/abs\/1508.04721v1 http:\/\/arxiv.org\/pdf\/1508.04721v1","urldate":"2016-12-17","archivePrefix":"arXiv","eprint":"1508.04721","primaryClass":"math.NT","year":"2015"},{"key":"TwoshortproofsofthePerfectForestTheorem","type":"article","title":"Two short proofs of the Perfect Forest Theorem","author":"Yair Caro and Josef Lauri and Christina Zarb","abstract":"A perfect forest is a spanning forest of a connected graph $G$, all of whose\r\ncomponents are induced subgraphs of $G$ and such that all vertices have odd\r\ndegree in the forest. A perfect forest generalised a perfect matching since, in\r\na matching, all components are trees on one edge. Scott first proved the\r\nPerfect Forest Theorem, namely, that every connected graph of even order has a\r\nperfect forest. Gutin then gave another proof using linear algebra.\r\n We give here two very short proofs of the Perfect Forest Theorem which use\r\nonly elementary notions from graph theory. Both our proofs yield\r\npolynomial-time algorithms for finding a perfect forest in a connected graph of\r\neven order.","comment":"","date_added":"2016-12-18","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1612.05004v1","http:\/\/arxiv.org\/pdf\/1612.05004v1"],"collections":"About proof,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1612.05004v1 http:\/\/arxiv.org\/pdf\/1612.05004v1","urldate":"2016-12-18","archivePrefix":"arXiv","eprint":"1612.05004","primaryClass":"math.CO","year":"2016"},{"key":"BalloonPolyhedra","type":"article","title":"Balloon Polyhedra","author":"Erik D. Demaine and Martin L. Demaine and Vi Hart","abstract":"","comment":"","date_added":"2016-12-21","date_published":"2013-08-06","urls":["http:\/\/erikdemaine.org\/papers\/Balloons_ShapingSpace2\/paper.pdf"],"collections":"Art,Easily explained,Things to make and do,Geometry","url":"http:\/\/erikdemaine.org\/papers\/Balloons_ShapingSpace2\/paper.pdf","urldate":"2016-12-21","year":"2013"},{"key":"ASingularMathematicalPromenade","type":"article","title":"A Singular Mathematical Promenade","author":"Etienne Ghys","abstract":"This is neither an elementary introduction to singularity theory nor a\r\nspecialized treatise containing many new theorems. The purpose of this little\r\nbook is to invite the reader on a mathematical promenade. We will pay a visit\r\nto Hipparchus, Newton and Gauss, but also to many contemporary mathematicians.\r\nWe will play with a bit of algebra, topology, geometry, complex analysis and\r\ncomputer science. Hopefully, some motivated undergraduates and some more\r\nadvanced mathematicians will enjoy some of these panoramas.","comment":"A delight!","date_added":"2016-12-22","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1612.06373v1","http:\/\/arxiv.org\/pdf\/1612.06373v1"],"collections":"History","url":"http:\/\/arxiv.org\/abs\/1612.06373v1 http:\/\/arxiv.org\/pdf\/1612.06373v1","urldate":"2016-12-22","archivePrefix":"arXiv","eprint":"1612.06373","primaryClass":"math.AG","year":"2016"},{"key":"HumanInferencesaboutSequencesAMinimalTransitionProbabilityModel","type":"article","title":"Human Inferences about Sequences: A Minimal Transition Probability Model","author":"Florent Meyniel and Maxime Maheu and Stanislas Dehaene","abstract":"The brain constantly infers the causes of the inputs it receives and uses these inferences to generate statistical expectations about future observations. Experimental evidence for these expectations and their violations include explicit reports, sequential effects on reaction times, and mismatch or surprise signals recorded in electrophysiology and functional MRI. Here, we explore the hypothesis that the brain acts as a near-optimal inference device that constantly attempts to infer the time-varying matrix of transition probabilities between the stimuli it receives, even when those stimuli are in fact fully unpredictable. This parsimonious Bayesian model, with a single free parameter, accounts for a broad range of findings on surprise signals, sequential effects and the perception of randomness. Notably, it explains the pervasive asymmetry between repetitions and alternations encountered in those studies. Our analysis suggests that a neural machinery for inferring transition probabilities lies at the core of human sequence knowledge.","comment":"","date_added":"2017-01-09","date_published":"2016-08-06","urls":["http:\/\/journals.plos.org\/ploscompbiol\/article?id=10.1371\/journal.pcbi.1005260"],"collections":"The act of doing maths,Probability and statistics,Modelling","url":"http:\/\/journals.plos.org\/ploscompbiol\/article?id=10.1371\/journal.pcbi.1005260","urldate":"2017-01-09","year":"2016"},{"key":"RulesforFoldingPolyminoesfromOneLeveltoTwoLevels","type":"article","title":"Rules for Folding Polyminoes from One Level to Two Levels","author":"Julia Martin and Elizabeth Wilcox","abstract":"Polyominoes have been the focus of many recreational and research\r\ninvestigations. In this article, the authors investigate whether a paper cutout\r\nof a polyomino can be folded to produce a second polyomino in the same shape as\r\nthe original, but now with two layers of paper. For the folding, only \"corner\r\nfolds\" and \"half edge cuts\" are allowed, unless the polyomino forms a closed\r\nloop, in which case one is allowed to completely cut two squares in the\r\npolyomino apart. With this set of allowable moves, the authors present\r\nalgorithms for folding different types of polyominoes and prove that certain\r\npolyominoes can successfully be folded to two layers. The authors also\r\nestablish that other polyominoes cannot be folded to two layers if only these\r\nmoves are allowed.","comment":"","date_added":"2017-01-16","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1701.03461v1","http:\/\/arxiv.org\/pdf\/1701.03461v1"],"collections":"Easily explained,Things to make and do,Geometry,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1701.03461v1 http:\/\/arxiv.org\/pdf\/1701.03461v1","urldate":"2017-01-16","archivePrefix":"arXiv","eprint":"1701.03461","primaryClass":"math.CO","year":"2017"},{"key":"OntheExistenceofOrdinaryTriangles","type":"article","title":"On the Existence of Ordinary Triangles","author":"Radoslav Fulek and Hossein Nassajian Mojarrad and M\u00e1rton Nasz\u00f3di and J\u00f3zsef Solymosi and Sebastian U. Stich and May Szedl\u00e1k","abstract":"Let $P$ be a finite point set in the plane. A $c$-ordinary triangle in $P$ is\r\na subset of $P$ consisting of three non-collinear points such that each of the\r\nthree lines determined by the three points contains at most $c$ points of $P$.\r\nWe prove that there exists a constant $c>0$ such that $P$ contains a\r\n$c$-ordinary triangle, provided that $P$ is not contained in the union of two\r\nlines. Furthermore, the number of $c$-ordinary triangles in $P$ is\r\n$\\Omega(|P|)$.","comment":"","date_added":"2017-02-06","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1701.08183v1","http:\/\/arxiv.org\/pdf\/1701.08183v1"],"collections":"Geometry,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1701.08183v1 http:\/\/arxiv.org\/pdf\/1701.08183v1","urldate":"2017-02-06","archivePrefix":"arXiv","eprint":"1701.08183","primaryClass":"math.CO","year":"2017"},{"key":"TransfiniteVersionofWeltersGame","type":"article","title":"Transfinite Version of Welter's Game","author":"Tomoaki Abuku","abstract":"We study the transfinite version of Welter's Game, a combinatorial game,\r\nwhich is played on the belt divided into squares with general ordinal numbers\r\nextended from natural numbers.\r\n In particular, we obtain a straight-forward solution for the transfinite\r\nversion based on those of the transfinite version of Nim and the original\r\nversion of Welter's Game.","comment":"","date_added":"2017-02-06","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1701.08928v1","http:\/\/arxiv.org\/pdf\/1701.08928v1"],"collections":"Protocols and strategies,Games to play with friends,Combinatorics","url":"http:\/\/arxiv.org\/abs\/1701.08928v1 http:\/\/arxiv.org\/pdf\/1701.08928v1","urldate":"2017-02-06","archivePrefix":"arXiv","eprint":"1701.08928","primaryClass":"math.CO","year":"2017"},{"key":"PlanepartitionsintheworkofRichardStanleyandhisschool","type":"article","title":"Plane partitions in the work of Richard Stanley and his school","author":"C. Krattenthaler","abstract":"These notes provide a survey of the theory of plane partitions, seen through\r\nthe glasses of the work of Richard Stanley and his school.","comment":"","date_added":"2017-02-06","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1503.05934v2","http:\/\/arxiv.org\/pdf\/1503.05934v2"],"collections":"Art,Combinatorics","url":"http:\/\/arxiv.org\/abs\/1503.05934v2 http:\/\/arxiv.org\/pdf\/1503.05934v2","urldate":"2017-02-06","archivePrefix":"arXiv","eprint":"1503.05934","primaryClass":"math.CO","year":"2015"},{"key":"RandomTrianglesandPolygonsinthePlane","type":"article","title":"Random Triangles and Polygons in the Plane","author":"Jason Cantarella and Tom Needham and Clayton Shonkwiler and Gavin Stewart","abstract":"We consider the problem of finding the probability that a random triangle is\r\nobtuse, which was first raised by Lewis Caroll. Our investigation leads us to a\r\nnatural correspondence between plane polygons and the Grassmann manifold of\r\n2-planes in real $n$-space proposed by Allen Knutson and Jean-Claude Hausmann.\r\nThis correspondence defines a natural probability measure on plane polygons. In\r\nthese terms, we answer Caroll's question. We then explore the Grassmannian\r\ngeometry of planar quadrilaterals, providing an answer to Sylvester's\r\nfour-point problem, and describing explicitly the moduli space of unordered\r\nquadrilaterals. All of this provides a concrete introduction to a family of\r\nmetrics used in shape classification and computer vision.","comment":"","date_added":"2017-02-06","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1702.01027v1","http:\/\/arxiv.org\/pdf\/1702.01027v1"],"collections":"Probability and statistics,Geometry","url":"http:\/\/arxiv.org\/abs\/1702.01027v1 http:\/\/arxiv.org\/pdf\/1702.01027v1","urldate":"2017-02-06","archivePrefix":"arXiv","eprint":"1702.01027","primaryClass":"math.MG","year":"2017"},{"key":"ThreeThoughtsonPrimeSimplicity","type":"article","title":"Three Thoughts on \u201cPrime Simplicity\u201d","author":"Michael Hardy","abstract":"In 2009, Catherine Woodgold and I published \u2018\u2018Prime Simplicity\u2019\u2019, examining the belief that Euclid\u2019s famous proof of the infinitude of prime numbers was by contradiction. We demonstrated that that belief is widespread among mathematicians and is false: Euclid\u2019s proof is simpler and better than the frequently seen proof by contradiction. The extra complication of the indirect proof serves no purpose and has pitfalls that can mislead the reader.","comment":"","date_added":"2017-02-27","date_published":"2012-08-06","urls":["https:\/\/link.springer.com\/article\/10.1007\/s00283-012-9322-z"],"collections":"About proof,Integerology","url":"https:\/\/link.springer.com\/article\/10.1007\/s00283-012-9322-z","urldate":"2017-02-27","year":"2012"},{"key":"PrimeSimplicity","type":"article","title":"Prime Simplicity","author":"Michael Hardy and Catherine Woodgold","abstract":"","comment":"Euclid's proof of the infinitude of the primes wasn't a proof by contradiction!","date_added":"2017-02-27","date_published":"2009-08-06","urls":["https:\/\/link.springer.com\/article\/10.1007\/s00283-009-9064-8"],"collections":"About proof,Integerology","url":"https:\/\/link.springer.com\/article\/10.1007\/s00283-009-9064-8","urldate":"2017-02-27","year":"2009"},{"key":"StatisticsDoneWrong","type":"article","title":"Statistics Done Wrong","author":"Alex Reinhart","abstract":"If you\u2019re a practicing scientist, you probably use statistics to analyze your data. From basic t tests and standard error calculations to Cox proportional hazards models and propensity score matching, we rely on statistics to give answers to scientific problems.\r\n\r\nThis is unfortunate, because statistical errors are rife.\r\n\r\nStatistics Done Wrong is a guide to the most popular statistical errors and slip-ups committed by scientists every day, in the lab and in peer-reviewed journals. Many of the errors are prevalent in vast swaths of the published literature, casting doubt on the findings of thousands of papers. Statistics Done Wrong assumes no prior knowledge of statistics, so you can read it before your first statistics course or after thirty years of scientific practice.","comment":"","date_added":"2017-03-01","date_published":"2015-08-06","urls":["https:\/\/www.statisticsdonewrong.com\/"],"collections":"Attention-grabbing titles,Lists and catalogues,Probability and statistics,Education","url":"https:\/\/www.statisticsdonewrong.com\/","urldate":"2017-03-01","year":"2015"},{"key":"Themathematicsoflecturehallpartitions","type":"article","title":"The mathematics of lecture hall partitions","author":"Carla D. Savage","abstract":"Over the past twenty years, lecture hall partitions have emerged as\r\nfundamental combinatorial structures, leading to new generalizations and\r\ninterpretations of classical theorems and new results. In recent years,\r\ngeometric approaches to lecture hall partitions have used polyhedral geometry\r\nto discover further properties of these rich combinatorial objects.\r\n In this paper we give an overview of some of the surprising connections that\r\nhave surfaced in the process of trying to understand the lecture hall\r\npartitions.","comment":"","date_added":"2017-03-08","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1607.01765v1","http:\/\/arxiv.org\/pdf\/1607.01765v1"],"collections":"Easily explained,Combinatorics","url":"http:\/\/arxiv.org\/abs\/1607.01765v1 http:\/\/arxiv.org\/pdf\/1607.01765v1","urldate":"2017-03-08","archivePrefix":"arXiv","eprint":"1607.01765","primaryClass":"math.CO","year":"2016"},{"key":"CrazySequentialRepresentationNumbersfrom0to11111intermsofIncreasingandDecreasingOrdersof1to9","type":"article","title":"Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9","author":"Inder J. Taneja","abstract":"Natural numbers from 0 to 11111 are written in terms of 1 to 9 in two\r\ndifferent ways. The first one in increasing order of 1 to 9, and the second one\r\nin decreasing order. This is done by using the operations of addition,\r\nmultiplication, subtraction, potentiation, and division. In both the situations\r\nthere are no missing numbers, except one, i.e., 10958 in the increasing case.","comment":"","date_added":"2017-03-12","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1302.1479v5","http:\/\/arxiv.org\/pdf\/1302.1479v5"],"collections":"Easily explained,Integerology","url":"http:\/\/arxiv.org\/abs\/1302.1479v5 http:\/\/arxiv.org\/pdf\/1302.1479v5","urldate":"2017-03-12","archivePrefix":"arXiv","eprint":"1302.1479","primaryClass":"math.HO","year":"2013"},{"key":"BeyondFloatingPoint","type":"article","title":"Beyond Floating Point: Next-Generation Computer Arithmetic ","author":"John L. Gustafson","abstract":"","comment":"About posits, which beat IEEE floats on pretty much everything. Lots of nice pictures!","date_added":"2017-03-21","date_published":"2017-08-06","urls":["http:\/\/web.stanford.edu\/class\/ee380\/Abstracts\/170201-slides.pdf"],"collections":"Basically computer science,Unusual arithmetic","url":"http:\/\/web.stanford.edu\/class\/ee380\/Abstracts\/170201-slides.pdf","urldate":"2017-03-21","year":"2017"},{"key":"SENEWSPAPAC00","type":"article","title":"PAPAC-00, a Do-It-Yourself Paper Computer","author":"Rollin P. Mayer","abstract":"","comment":"","date_added":"2017-03-21","date_published":"1958-08-06","urls":["http:\/\/ieeexplore.ieee.org\/stamp\/stamp.jsp?arnumber=5222588"],"collections":"Basically computer science,Easily explained,Things to make and do,Unusual computers","url":"http:\/\/ieeexplore.ieee.org\/stamp\/stamp.jsp?arnumber=5222588","urldate":"2017-03-21","year":"1958"},{"key":"Whitt2013","type":"article","title":"The Maximum Throughput Rate for Each Hole on a Golf Course","author":"Whitt, Ward","abstract":"","comment":"","date_added":"2013-11-25","date_published":"2015-08-06","urls":["http:\/\/www.columbia.edu\/~ww2040\/golf_throughput_POMS_2015.pdf"],"collections":"Games to play with friends,Basically physics,Probability and statistics","url":"http:\/\/www.columbia.edu\/~ww2040\/golf_throughput_POMS_2015.pdf","urldate":"2013-11-25","year":"2015"},{"key":"item61","type":"misc","title":"Prime numbers in certain arithmetic progressions","author":"Ram Murty and Nithum Thain","abstract":"We discuss to what extent Euclid's elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet's theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod $k$) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.","comment":"","date_added":"2016-05-07","date_published":"2001-08-06","urls":["http:\/\/projecteuclid.org\/download\/pdf_1\/euclid.facm\/1229442627"],"collections":"Fun maths facts,Integerology","url":"http:\/\/projecteuclid.org\/download\/pdf_1\/euclid.facm\/1229442627","urldate":"2016-05-07","year":"2001"},{"key":"Plagge1978","type":"article","title":"Fractions without Quotients: Arithmetic of Repeating Decimals","author":"Plagge, Richard","abstract":"","comment":"","date_added":"2012-02-10","date_published":"1978-08-06","urls":["https:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Polya\/00494925.di020680.02p0417i.pdf","http:\/\/www.jstor.org\/stable\/10.2307\/3026549"],"collections":"Notation and conventions,Unusual arithmetic,Easily explained","url":"https:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Polya\/00494925.di020680.02p0417i.pdf http:\/\/www.jstor.org\/stable\/10.2307\/3026549","urldate":"2012-02-10","year":"1978"},{"key":"Fink1999","type":"article","title":"Designing tie knots by random walks","author":"Thomas M. Fink and Yong Mao","abstract":"The simplest of conventional tie knots, the four-in-hand, has its origins in late-nineteenth-century England. The Duke of Windsor, as King Edward VIII became after abdicating in 1936, is credited with introducing what is now known as the Windsor knot, from which its smaller derivative, the half-Windsor, evolved. In 1989, the Pratt knot, the first new knot to appear in fifty years, was revealed on the front page of The New York Times.","comment":"","date_added":"2011-12-09","date_published":"1999-08-06","urls":["http:\/\/www.nature.com\/nature\/journal\/v398\/n6722\/abs\/398031a0.html"],"collections":"Probability and statistics","url":"http:\/\/www.nature.com\/nature\/journal\/v398\/n6722\/abs\/398031a0.html","urldate":"2011-12-09","year":"1999"},{"key":"NaguraOntheintervalcontainingatleastoneprimenumber","type":"article","title":"On the interval containing at least one prime number","author":"Jitsuro Nagura","abstract":"","comment":"","date_added":"2016-11-14","date_published":"1952-08-06","urls":["https:\/\/projecteuclid.org\/euclid.pja\/1195570997","https:\/\/projecteuclid.org\/download\/pdf_1\/euclid.pja\/1195570997"],"collections":"Fun maths facts,Integerology","url":"https:\/\/projecteuclid.org\/euclid.pja\/1195570997 https:\/\/projecteuclid.org\/download\/pdf_1\/euclid.pja\/1195570997","urldate":"2016-11-14","year":"1952"},{"key":"item58","type":"misc","title":"Area and Hausdorff Dimension of Julia Sets of Entire Functions","author":"Curt McMullen","abstract":"We show the Julia set of $\\lambda \\sin(z)$ has positive area and the action of $\\lambda \\sin(z)$ on its Julia set is not ergodic; the Julia set of $\\lambda \\exp(z)$ has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero.","comment":"","date_added":"2015-12-17","date_published":"1987-08-06","urls":["http:\/\/www.math.harvard.edu\/~ctm\/papers\/home\/text\/papers\/entire\/entire.pdf"],"collections":"","url":"http:\/\/www.math.harvard.edu\/~ctm\/papers\/home\/text\/papers\/entire\/entire.pdf","urldate":"2015-12-17","year":"1987"},{"key":"Kolliopoulos","type":"article","title":"The 1-Hyperbolic Projection for User Interfaces","author":"Kolliopoulos, Alexander","abstract":"The problem of dealing with representations of information that does not fit conveniently within allotted screen space is pervasive in graphical interfaces. While there are techniques for dealing with this problem in various ways, some properties of such existing techniques are not satisfying. For example, global structure of information may be lost in favor of local focus, or information may not be mapped into a rectangular area. The 1-hyperbolic interface is proposed to deal with some of these deficiencies, and the mathematics involved in display and interaction are derived. The calculations necessary for this interface are easy to implement, and can run reasonably even on slow devices. A fully functional prototype for displaying tree structures has been developed to compare the effects of this new interface\r\nto those of a standard interface. The results of usability experiments conducted with this prototype are also presented and analyzed.","comment":"","date_added":"2011-04-12","date_published":"2003-08-06","urls":["http:\/\/www.dgp.toronto.edu\/~alexk\/hyperproj.pdf"],"collections":"Basically computer science","url":"http:\/\/www.dgp.toronto.edu\/~alexk\/hyperproj.pdf","urldate":"2011-04-12","year":"2003"},{"key":"item36","type":"article","title":"Solving Differential Equations by Symmetry Groups","author":"John Starret","abstract":"","comment":"","date_added":"2013-12-06","date_published":"2007-08-06","urls":["https:\/\/www.jstor.org\/stable\/27642331"],"collections":"","url":"https:\/\/www.jstor.org\/stable\/27642331","urldate":"2013-12-06","year":"2007"},{"key":"Mercer2009","type":"article","title":"On Furstenberg's Proof of the Infinitude of Primes","author":"Mercer, Idris D","abstract":"","comment":"","date_added":"2010-09-09","date_published":"2009-08-06","urls":["http:\/\/www.idmercer.com\/monthly355-356-mercer.pdf"],"collections":"About proof,Integerology","url":"http:\/\/www.idmercer.com\/monthly355-356-mercer.pdf","urldate":"2010-09-09","year":"2009"},{"key":"item17","type":"online","title":"Light reflecting off Christmas-tree balls","author":"Joseph O'Rourke","abstract":"'Twas the night before Christmas and under the tree Was a heap of new balls, stacked tight as can be. The balls so gleaming, they reflect all light rays, Which bounce in the stack every which way. When, what to my wondering mind does occur: A question of interest; I hope you concur! From each point outside, I wondered if light Could reach deep inside through gaps so tight?","comment":"","date_added":"2012-04-08","date_published":"2010-08-06","urls":["http:\/\/mathoverflow.net\/questions\/50150\/light-reflecting-off-christmas-tree-balls"],"collections":"Easily explained","url":"http:\/\/mathoverflow.net\/questions\/50150\/light-reflecting-off-christmas-tree-balls","urldate":"2012-04-08","year":"2010"},{"key":"CountingCasesInMarchingCubes","type":"article","title":"Counting Cases in Marching Cubes: Towards a Generic Algorithm for Producing Substitopes","author":"David C. Banks and Stephen Linton","abstract":"We describe how to count the cases that arise in a family of visualization techniques, including marching cubes, sweeping simplices, contour meshing, interval volumes, and separating surfaces. Counting the cases is the first step toward developing a generic visualization algorithm to produce substitopes (geometric substitution of polytopes). We demonstrate the method using a software system (\"GAP\") for computational group theory. The case-counts are organized into a table that provides taxonomy of members of the family; numbers in the table are derived from actual lists of cases, which are computed by our methods. The calculation confirms previously reported case-counts for large dimensions that are too large to check by hand, and predicts the number of cases that will arise in algorithms that have not yet been invented.","comment":"","date_added":"2016-09-28","date_published":"2003-08-06","urls":["https:\/\/www.evl.uic.edu\/cavern\/rg\/20040525_renambot\/Visualization-papers\/papers\/03\/01250354.pdf","http:\/\/ieeexplore.ieee.org\/document\/1250354"],"collections":"Basically computer science","url":"https:\/\/www.evl.uic.edu\/cavern\/rg\/20040525_renambot\/Visualization-papers\/papers\/03\/01250354.pdf http:\/\/ieeexplore.ieee.org\/document\/1250354","urldate":"2016-09-28","year":"2003"},{"key":"Diaconis2006","type":"book","title":"Methods for studying coincidences","author":"Diaconis, P and Mosteller, Frederick","abstract":"This article illustrates basic statistical techniques for studying coincidences. These include data-gathering methods (informal anecdotes, case studies, observational studies, and experiments) and methods of analysis (exploratory and confirmatory data analysis, special analytic techniques, and probabilistic modeling, both general and special purpose). We develop a version of the birthday problem general enough to include dependence, inhomogeneity, and almost multiple matches. We review Fisher\u2019s techniques for giving partial credit for close matches. We develop a model for studying coincidences involving newly learned words. Once we set aside coincidences having apparent causes, four principles account for large numbers of remaining coincidences: hidden cause; psychology, including memory and perception; multiplicity of endpoints, including the counting of \u201cclose\u201d or nearly alike events as if they were identical; and the law of truly large numbers which says that when enormous numbers of events and people and their interactions cumulate over time, almost any outrageous event is bound to occur. These sources account for much of the force of synchronicity.","comment":"","date_added":"2014-10-07","date_published":"2006-08-06","urls":["http:\/\/link.springer.com\/chapter\/10.1007\/978-0-387-44956-2_39"],"collections":"Easily explained,Probability and statistics,Modelling","url":"http:\/\/link.springer.com\/chapter\/10.1007\/978-0-387-44956-2_39","urldate":"2014-10-07","year":"2006"},{"key":"Shuai","type":"article","title":"Does Quantum Interference exist in Twitter?","author":"Shuai, Xin and Ding, Ying and Busemeyer, Jerome and Sun, Yuyin and Chen, Shanshan and Tang, Jie","abstract":"It becomes more difficult to explain the social information transfer phenomena using the classic models based merely on Shannon Information Theory (SIT) and Classic Probability Theory (CPT), because the transfer process in the social world is rich of semantic and highly contextualized. This paper aims to use twitter data to explore whether the traditional models can interpret information transfer in social networks, and whether quantum-like phenomena can be spotted in social networks. Our main contributions are: (1) SIT and CPT fail to interpret the information transfer occurring in Twitter; and (2) Quantum interference exists in Twitter, and (3) a mathematical model is proposed to elucidate the spotted quantum phenomena.","comment":"","date_added":"2011-09-08","date_published":"2011-08-06","urls":["http:\/\/arxiv.org\/abs\/1107.0681","http:\/\/arxiv.org\/pdf\/1107.0681v1"],"collections":"Basically computer science,Basically physics","url":"http:\/\/arxiv.org\/abs\/1107.0681 http:\/\/arxiv.org\/pdf\/1107.0681v1","urldate":"2011-09-08","year":"2011"},{"key":"Saloffcoste","type":"article","title":"Random Walks on Finite Groups","author":"Saloff-coste, Laurent","abstract":"Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the \u201cmixing time\u201d of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as the cut-off phenomenon asserts that this often happens abruptly so that it really makes sense to talk about \u201cthe mixing time\u201d. Random walks on finite groups generalize card shuffling models by replacing the symmetric group by other finite groups. One then would like to understand how the structure of a particular class of groups relates to the mixing time of natural random walks on those groups. It turns out that this is an extremely rich problem which is very far to be understood. Techniques from a great variety of different fields \u2013 Probability, Algebra, Representation Theory, Functional Analysis, Geometry, Combinatorics \u2013 have been used to attack special instances of this problem. This article gives a general overview of this area of research.","comment":"","date_added":"2012-01-05","date_published":"2004-08-06","urls":["http:\/\/statweb.stanford.edu\/~cgates\/PERSI\/papers\/rwfg.pdf"],"collections":"Probability and statistics,The groups group","url":"http:\/\/statweb.stanford.edu\/~cgates\/PERSI\/papers\/rwfg.pdf","urldate":"2012-01-05","year":"2004"},{"key":"NonSexistMenage","type":"article","title":"Non-sexist solution of the m\u00e9nage problem","author":"Kenneth P. Bogart","abstract":"The m\u00e9nage problem asks for the number of ways of seating \\(n\\) couples at a circular table, with men and women alternating, so that no one sits next to his or her partner. We present a straight-forward solution to this problem. What distinguishes our approach is that we do not seat the ladies first.","comment":"","date_added":"2013-06-16","date_published":"1985-08-06","urls":["http:\/\/www.math.dartmouth.edu\/~doyle\/docs\/menage\/menage\/menage.html"],"collections":"Puzzles,Easily explained","url":"http:\/\/www.math.dartmouth.edu\/~doyle\/docs\/menage\/menage\/menage.html","urldate":"2013-06-16","year":"1985"},{"key":"Martin1998","type":"article","title":"Denser Egyptian Fractions","author":"Martin, Greg","abstract":"An Egyptian fraction is a sum of distinct unit fractions (reciprocals of positive integers). We show that every rational number has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest denominator, improving a result from an earlier paper to best-possible form. We also settle, in best-possible form, the related problem of how small M_t(r) can be such that there is an Egyptian fraction representation of r with exactly t terms, the denominators of which are all at most M_t(r). We also consider the following problems posed by Erd\u0151s and Graham: the set of integers that cannot be the largest denominator of an Egyptian fraction representation of 1 is infinite - what is its order of growth? How about those integers that cannot be the second-largest (third-largest, etc.) denominator of such a representation? In the latter case, we show that only finitely many integers cannot be the second-largest (third-largest, etc.) denominator of such a representation; while in the former case, we show that the set of integers that cannot be the largest denominator of such a representation has density zero, and establish its order of growth. Both results extend to representations of any positive rational number.","comment":"","date_added":"2015-07-23","date_published":"1998-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/9811112","http:\/\/arxiv.org\/pdf\/math\/9811112v1"],"collections":"Easily explained,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/math\/9811112 http:\/\/arxiv.org\/pdf\/math\/9811112v1","urldate":"2015-07-23","year":"1998"},{"key":"item5","type":"online","title":"Four questions about fuzzy rankings","author":"Brian Hayes","abstract":"","comment":"","date_added":"2011-01-12","date_published":"2010-08-06","urls":["http:\/\/bit-player.org\/2010\/four-questions-about-fuzzy-rankings"],"collections":"","url":"http:\/\/bit-player.org\/2010\/four-questions-about-fuzzy-rankings","urldate":"2011-01-12","year":"2010"},{"key":"Onsubsetswithintersectionsofevencardinality","type":"article","title":"On subsets with intersections of even cardinality","author":"E.R. Berlekamp","abstract":"This paper solves a question by Paul Erd\u0151s","comment":"","date_added":"2016-11-01","date_published":"1969-08-06","urls":["https:\/\/cms.math.ca\/10.4153\/CMB-1969-059-3","https:\/\/cms.math.ca\/openaccess\/cmb\/v12\/cmb1969v12.0471-0474.pdf"],"collections":"Fun maths facts","url":"https:\/\/cms.math.ca\/10.4153\/CMB-1969-059-3 https:\/\/cms.math.ca\/openaccess\/cmb\/v12\/cmb1969v12.0471-0474.pdf","urldate":"2016-11-01","year":"1969"},{"key":"item54","type":"article","title":"How to Beat Your Wythoff Games' Opponent on Three Fronts","author":"Aviezri S. Fraenkel","abstract":"","comment":"","date_added":"2015-10-23","date_published":"1982-08-06","urls":["http:\/\/www.jstor.org\/stable\/2321643"],"collections":"Protocols and strategies,Games to play with friends","url":"http:\/\/www.jstor.org\/stable\/2321643","urldate":"2015-10-23","year":"1982"},{"key":"BestLaidPlansofLionsandMen","type":"article","title":"Best Laid Plans of Lions and Men","author":"Mikkel Abrahamsen and Jacob Holm and Eva Rotenberg and Christian Wulff-Nilsen","abstract":"We answer the following question dating back to J.E. Littlewood (1885 -\r\n1977): Can two lions catch a man in a bounded area with rectifiable lakes? The\r\nlions and the man are all assumed to be points moving with at most unit speed.\r\nThat the lakes are rectifiable means that their boundaries are finitely long.\r\nThis requirement is to avoid pathological examples where the man survives\r\nforever because any path to the lions is infinitely long. We show that the\r\nanswer to the question is not always \"yes\" by giving an example of a region $R$\r\nin the plane where the man has a strategy to survive forever. $R$ is a\r\npolygonal region with holes and the exterior and interior boundaries are\r\npairwise disjoint, simple polygons. Our construction is the first truly\r\ntwo-dimensional example where the man can survive.\r\n Next, we consider the following game played on the entire plane instead of a\r\nbounded area: There is any finite number of unit speed lions and one fast man\r\nwho can run with speed $1+\\varepsilon$ for some value $\\varepsilon>0$. Can the\r\nman always survive? We answer the question in the affirmative for any constant\r\n$\\varepsilon>0$.","comment":"","date_added":"2017-03-22","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1703.03687v1","http:\/\/arxiv.org\/pdf\/1703.03687v1"],"collections":"Attention-grabbing titles,Protocols and strategies,Animals,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1703.03687v1 http:\/\/arxiv.org\/pdf\/1703.03687v1","urldate":"2017-03-22","archivePrefix":"arXiv","eprint":"1703.03687","primaryClass":"cs.CG","year":"2017"},{"key":"Gale1991","type":"article","title":"The Strange and Surprising Saga of the Somos Sequences","author":"Gale, David","abstract":"","comment":"","date_added":"2013-10-09","date_published":"1991-03-01","urls":["http:\/\/link.springer.com\/10.1007\/BF03024070"],"collections":"Fun maths facts,Integerology","url":"http:\/\/link.springer.com\/10.1007\/BF03024070","urldate":"2013-10-09","doi":"10.1007\/BF03024070","issn":"0343-6993","journal":"The Mathematical Intelligencer","month":"mar","number":"1","pages":"40--43","volume":"13","year":"1991"},{"key":"Flajolet1998","type":"article","title":"Continued fraction algorithms, functional operators, and structure constants","author":"Flajolet, P. and Vall\u00e9e, B.","abstract":"","comment":"","date_added":"2011-01-20","date_published":"1998-08-06","urls":["http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0304397597001230"],"collections":"","url":"http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0304397597001230","urldate":"2011-01-20","issn":"0304-3975","journal":"Theoretical Computer Science","number":"1-2","pages":"1--34","publisher":"Elsevier","volume":"194","year":"1998"},{"key":"Bauer","type":"online","title":"Constructive gem: juggling exponentials","author":"Bauer, Andrej","abstract":"","comment":"","date_added":"2010-07-15","date_published":"2009-08-06","urls":["http:\/\/math.andrej.com\/2009\/09\/09\/constructive-gem-juggling-exponentials\/"],"collections":"Basically computer science","url":"http:\/\/math.andrej.com\/2009\/09\/09\/constructive-gem-juggling-exponentials\/","urldate":"2010-07-15","year":"2009"},{"key":"AnalysisofCarriesinSignedDigitExpansions","type":"article","title":"Analysis of Carries in Signed Digit Expansions","author":"Clemens Heuberger and Sara Kropf and Helmut Prodinger","abstract":"The number of positive and negative carries in the addition of two\r\nindependent random signed digit expansions of given length is analyzed\r\nasymptotically for the $(q, d)$-system and the symmetric signed digit\r\nexpansion. The results include expectation, variance, covariance between the\r\npositive and negative carries and a central limit theorem.\r\n Dependencies between the digits require determining suitable transition\r\nprobabilities to obtain equidistribution on all expansions of given length. A\r\ngeneral procedure is described to obtain such transition probabilities for\r\narbitrary regular languages.\r\n The number of iterations in von Neumann's parallel addition method for the\r\nsymmetric signed digit expansion is also analyzed, again including expectation,\r\nvariance and convergence to a double exponential limiting distribution. This\r\nanalysis is carried out in a general framework for sequences of generating\r\nfunctions.","comment":"","date_added":"2017-02-06","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1503.08816v3","http:\/\/arxiv.org\/pdf\/1503.08816v3"],"collections":"Basically computer science","url":"http:\/\/arxiv.org\/abs\/1503.08816v3 http:\/\/arxiv.org\/pdf\/1503.08816v3","urldate":"2017-02-06","archivePrefix":"arXiv","eprint":"1503.08816","primaryClass":"math.CO","year":"2015"},{"key":"Miquel2007","type":"article","title":"The experimental effectiveness of mathematical proof","author":"Miquel, Alexandre","abstract":"","comment":"","date_added":"2011-10-30","date_published":"2007-08-06","urls":["https:\/\/www.fing.edu.uy\/~amiquel\/publis\/effectiveness.pdf"],"collections":"History,The act of doing maths,About proof","url":"https:\/\/www.fing.edu.uy\/~amiquel\/publis\/effectiveness.pdf","urldate":"2011-10-30","pages":"1--42","year":"2007"},{"key":"CajoriNotations","type":"book","title":"A history of mathematical notations","author":"Florian Cajori","abstract":"","comment":"","date_added":"2011-01-12","date_published":"1993-08-06","urls":["http:\/\/www.maths.ed.ac.uk\/~aar\/papers\/cajorinot.pdf"],"collections":"History,Notation and conventions,Lists and catalogues","url":"http:\/\/www.maths.ed.ac.uk\/~aar\/papers\/cajorinot.pdf","urldate":"2011-01-12","year":"1993"},{"key":"item51","type":"misc","title":"Representations of Palindromic, Prime and Number Patterns","author":"Inder J. Taneja","abstract":"","comment":"","date_added":"2015-06-12","date_published":"2015-08-06","urls":["http:\/\/rgmia.org\/papers\/v18\/v18a77.pdf"],"collections":"Unusual arithmetic,Integerology","url":"http:\/\/rgmia.org\/papers\/v18\/v18a77.pdf","urldate":"2015-06-12","year":"2015"},{"key":"Kahan","type":"misc","title":"Beastly Numbers","author":"Kahan, W","abstract":"It seems unlikely that two computers, designed by different people 1800 miles apart, would be upset in the same way by the same two floating-point numbers 65535... and 4294967295... , but it has happened.","comment":"","date_added":"2012-09-05","date_published":"1996-08-06","urls":["http:\/\/www.cs.berkeley.edu\/~wkahan\/tests\/numbeast.ps"],"collections":"Basically computer science","url":"http:\/\/www.cs.berkeley.edu\/~wkahan\/tests\/numbeast.ps","urldate":"2012-09-05","year":"1996"},{"key":"Srinivasan1948","type":"article","title":"Practical numbers","author":"Srinivasan, A.K.","abstract":"","comment":"","date_added":"2013-10-31","date_published":"1948-06-01","urls":["http:\/\/www.currentscience.ac.in\/Downloads\/article_id_017_06_0179_0180_0.pdf","http:\/\/www.currentscience.ac.in\/php\/toc.php?vol=017&issue=06"],"collections":"Easily explained,Integerology","url":"http:\/\/www.currentscience.ac.in\/Downloads\/article_id_017_06_0179_0180_0.pdf http:\/\/www.currentscience.ac.in\/php\/toc.php?vol=017&issue=06","urldate":"2013-10-31","journal":"Current science","year":"1948","month":"jun","volume":"17","issue":"6"},{"key":"Akleman2011","type":"article","title":"Cyclic twill-woven objects","author":"Akleman, Ergun and Chen, Jianer and Chen, YenLin and Xing, Qing and Gross, Jonathan L.","abstract":"","comment":"","date_added":"2013-05-16","date_published":"2011-06-01","urls":["http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0097849311000422"],"collections":"Basically computer science","url":"http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0097849311000422","urldate":"2013-05-16","doi":"10.1016\/j.cag.2011.03.003","issn":"00978493","journal":"Computers & Graphics","month":"jun","number":"3","pages":"623--631","volume":"35","year":"2011"},{"key":"TheGeneralCounterfeitCoinProblem","type":"article","title":"The general counterfeit coin problem","author":"Lorenz Halbeisen and Norbert Hungerb\u00fchler","abstract":"Given $c$ nickels among which there may be a counterfeit coin, which can only be told\r\napart by its weight being different from the others, and moreover $b$ balances. What is the minimal number of weighings to decide whether there is a counterfeit nickel, if so which one it is and whether it is heavier or lighter than a genuine nickel. We give an answer to this question for sequential and nonsequential strategies and we will consider the problem of more than one counterfeit coin.","comment":"","date_added":"2016-08-24","date_published":"1995-08-06","urls":["http:\/\/user.math.uzh.ch\/halbeisen\/publications\/pdf\/coin.pdf"],"collections":"Puzzles,Easily explained","url":"http:\/\/user.math.uzh.ch\/halbeisen\/publications\/pdf\/coin.pdf","urldate":"2016-08-24","year":"1995"},{"key":"MidpointOfAnInterval","type":"article","title":"How do you compute the midpoint of an interval?","author":"Fr\u00e9d\u00e9ric Goualard","abstract":"","comment":"","date_added":"2016-07-04","date_published":"2014-08-06","urls":["https:\/\/hal.archives-ouvertes.fr\/file\/index\/docid\/576641\/filename\/computing-midpoint.pdf"],"collections":"Basically computer science","url":"https:\/\/hal.archives-ouvertes.fr\/file\/index\/docid\/576641\/filename\/computing-midpoint.pdf","urldate":"2016-07-04","year":"2014"},{"key":"Dividingbyzerohowbadisitreally","type":"article","title":"Dividing by zero - how bad is it, really?","author":"Takayuki Kihara and Arno Pauly","abstract":"In computable analysis testing a real number for being zero is a fundamental\r\nexample of a non-computable task. This causes problems for division: We cannot\r\nensure that the number we want to divide by is not zero. In many cases, any\r\nreal number would be an acceptable outcome if the divisor is zero - but even\r\nthis cannot be done in a computable way.\r\n In this note we investigate the strength of the computational problem \"Robust\r\ndivision\": Given a pair of real numbers, the first not greater than the other,\r\noutput their quotient if well-defined and any real number else. The formal\r\nframework is provided by Weihrauch reducibility. One particular result is that\r\nhaving later calls to the problem depending on the outcomes of earlier ones is\r\nstrictly more powerful than performing all calls concurrently. However, having\r\na nesting depths of two already provides the full power. This solves an open\r\nproblem raised at a recent Dagstuhl meeting on Weihrauch reducibility.\r\n As application for \"Robust division\", we show that it suffices to execute\r\nGaussian elimination.","comment":"","date_added":"2016-06-17","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1606.04126v1","http:\/\/arxiv.org\/pdf\/1606.04126v1"],"collections":"Basically computer science","url":"http:\/\/arxiv.org\/abs\/1606.04126v1 http:\/\/arxiv.org\/pdf\/1606.04126v1","urldate":"2016-06-17","archivePrefix":"arXiv","eprint":"1606.04126","primaryClass":"cs.LO","year":"2016"},{"key":"item53","type":"online","title":"Dice - Numericana","author":"G\u00e9rard P. Michon","abstract":"","comment":"","date_added":"2015-10-17","date_published":"2000-08-06","urls":["http:\/\/www.numericana.com\/answer\/dice.htm"],"collections":"Easily explained,Games to play with friends,Lists and catalogues","url":"http:\/\/www.numericana.com\/answer\/dice.htm","urldate":"2015-10-17","keywords":"cast,casting,combinatorics,dice,die,enumeration,roll,rolling,toss,tossing","year":"2000"},{"key":"item47","type":"misc","title":"On Legendre's Prime Number Formula","author":"Janos Pintz","abstract":"","comment":"","date_added":"2015-03-06","date_published":"1980-08-06","urls":["http:\/\/www.jstor.org\/stable\/2321863?seq=1#page_scan_tab_contents"],"collections":"History,Integerology","url":"http:\/\/www.jstor.org\/stable\/2321863?seq=1#page_scan_tab_contents","urldate":"2015-03-06","year":"1980"},{"key":"Fink2013","type":"article","title":"LIM is not slim","author":"Fink, Alex and Fraenkel, Aviezri S. and Santos, Carlos","abstract":"In this paper LIM, a recently proposed impartial combinatorial ruleset, is analyzed. A formula to describe the $G$-values of LIM positions is given, by way of analyzing an equivalent combinatorial ruleset LIM\u2019, closely related to the classical nim. Also, an enumeration of $P$-positions of LIM with $n$ stones, and its relation to the Ulam-Warburton cellular automaton, is presented.","comment":"","date_added":"2014-06-11","date_published":"2013-05-01","urls":["http:\/\/link.springer.com\/10.1007\/s00182-013-0380-z"],"collections":"Attention-grabbing titles,Combinatorics","url":"http:\/\/link.springer.com\/10.1007\/s00182-013-0380-z","urldate":"2014-06-11","issn":"0020-7276","journal":"International Journal of Game Theory","month":"may","number":"2","pages":"269--281","volume":"43","year":"2013"},{"key":"item40","type":"misc","title":"A number system with an irrational base","author":"George Bergman","abstract":"","comment":"","date_added":"2014-03-12","date_published":"1957-08-06","urls":["http:\/\/www.jstor.org\/discover\/10.2307\/3029218?uid=3738032&uid=2&uid=4&sid=21103722830183"],"collections":"Unusual arithmetic,Fun maths facts,Integerology","url":"http:\/\/www.jstor.org\/discover\/10.2307\/3029218?uid=3738032&uid=2&uid=4&sid=21103722830183","urldate":"2014-03-12","year":"1957"},{"key":"Silva2007","type":"article","title":"Mathematical Games","author":"Silva, Jorge Nuno","abstract":"","comment":"Some slides about mathematical games throughout history.","date_added":"2014-01-24","date_published":"2007-08-06","urls":["http:\/\/www.cim.pt\/files\/cim_games.pdf"],"collections":"History,Games to play with friends","url":"http:\/\/www.cim.pt\/files\/cim_games.pdf","urldate":"2014-01-24","year":"2007"},{"key":"item34","type":"article","title":"Is POPL Mathematics or Science?","author":"Andrew W. Appel","abstract":"","comment":"","date_added":"2013-11-06","date_published":"1992-08-06","urls":["http:\/\/www.cs.princeton.edu\/~appel\/papers\/science.pdf"],"collections":"The act of doing maths,Probability and statistics","url":"http:\/\/www.cs.princeton.edu\/~appel\/papers\/science.pdf","urldate":"2013-11-06","year":"1992"},{"key":"Althofer2013","type":"article","title":"Random Structures from Lego Bricks and Analog Monte Carlo Procedures","author":"Alth\u00f6fer, I","abstract":"","comment":"","date_added":"2013-11-13","date_published":"2013-08-06","urls":["http:\/\/www.althofer.de\/random-lego-structures.pdf"],"collections":"Attention-grabbing titles,Basically computer science,Easily explained,Basically physics","url":"http:\/\/www.althofer.de\/random-lego-structures.pdf","urldate":"2013-11-13","journal":"Update","keywords":"analog monte carlo,artificial life,interactive art,procedure,random lego structures","year":"2013"},{"key":"item33","type":"misc","title":"The Ubiquitous Thue-Morse Sequence","author":"Jeffrey Shallit","abstract":"","comment":"","date_added":"2013-10-21","date_published":"2006-08-06","urls":["https:\/\/cs.uwaterloo.ca\/~shallit\/Talks\/green3.pdf"],"collections":"Easily explained,Fun maths facts","url":"https:\/\/cs.uwaterloo.ca\/~shallit\/Talks\/green3.pdf","urldate":"2013-10-21","year":"2006"},{"key":"Castellanos2013","type":"article","title":"The Ubiquitous Pi","author":"Castellanos, Dario","abstract":"","comment":"","date_added":"2013-06-11","date_published":"2013-08-06","urls":["http:\/\/www.jstor.org\/stable\/2690037"],"collections":"History,Easily explained,Fun maths facts","url":"http:\/\/www.jstor.org\/stable\/2690037","urldate":"2013-06-11","number":"2","pages":"67--98","volume":"61","year":"2013"},{"key":"Kalman","type":"misc","title":"Six Ways to Sum a Series","author":"Kalman, Dan","abstract":"A discussion of the sum of squares of the reciprocals of the positive integers with a review of several proofs.","comment":"","date_added":"2013-06-10","date_published":"1994-08-06","urls":["http:\/\/mathdl.maa.org\/mathDL\/22\/?pa=content&sa=viewDocument&nodeId=2687","http:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Polya\/07468342.di020759.02p00026.pdf"],"collections":"Easily explained,About proof,Fun maths facts","url":"http:\/\/mathdl.maa.org\/mathDL\/22\/?pa=content&sa=viewDocument&nodeId=2687 http:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Polya\/07468342.di020759.02p00026.pdf","urldate":"2013-06-10","year":"1994"},{"key":"DatingRecommenderSystems","type":"online","title":"Online Dating Recommender Systems: The Split-complex Number Approach","author":"J\u00e9r\u00f4me Kunegis","abstract":"","comment":"","date_added":"2012-09-13","date_published":"2012-08-06","urls":["http:\/\/www.slideshare.net\/kunegis\/presentation-2012rsweb"],"collections":"Basically computer science,Unusual arithmetic,Easily explained,Probability and statistics","url":"http:\/\/www.slideshare.net\/kunegis\/presentation-2012rsweb","urldate":"2012-09-13","year":"2012"},{"key":"Arkin2006","type":"article","title":"The Snowblower Problem","author":"Arkin, Esther M. and Bender, Michael A. and Mitchell, Joseph S. B. and Polishchuk, Valentin","abstract":"We introduce the snowblower problem (SBP), a new optimization problem that is closely related to milling problems and to some material-handling problems. The objective in the SBP is to compute a short tour for the snowblower to follow to remove all the snow from a domain (driveway, sidewalk, etc.). When a snowblower passes over each region along the tour, it displaces snow into a nearby region. The constraint is that if the snow is piled too high, then the snowblower cannot clear the pile. We give an algorithmic study of the SBP. We show that in general, the problem is NP-complete, and we present polynomial-time approximation algorithms for removing snow under various assumptions about the operation of the snowblower. Most commercially-available snowblowers allow the user to control the direction in which the snow is thrown. We differentiate between the cases in which the snow can be thrown in any direction, in any direction except backwards, and only to the right. For all cases, we give constant-factor approximation algorithms; the constants increase as the throw direction becomes more restricted. Our results are also applicable to robotic vacuuming (or lawnmowing) with bounded capacity dust bin and to some versions of material-handling problems, in which the goal is to rearrange cartons on the floor of a warehouse.","comment":"","date_added":"2012-01-22","date_published":"2006-03-01","urls":["http:\/\/arxiv.org\/abs\/cs\/0603026","http:\/\/arxiv.org\/pdf\/cs\/0603026v1","http:\/\/www.ams.sunysb.edu\/~jsbm\/papers\/snowblowing.pdf"],"collections":"Basically computer science,Puzzles,Easily explained","url":"http:\/\/arxiv.org\/abs\/cs\/0603026 http:\/\/arxiv.org\/pdf\/cs\/0603026v1 http:\/\/www.ams.sunysb.edu\/~jsbm\/papers\/snowblowing.pdf","urldate":"2012-01-22","month":"mar","pages":"19","year":"2006","archivePrefix":"arXiv","eprint":"cs\/0603026","primaryClass":"cs.DS"},{"key":"Faure2008","type":"article","title":"Hierarchical Position Based Dynamics","author":"Faure, F. and Teschner, M.","abstract":"","comment":"","date_added":"2011-03-11","date_published":"2008-08-06","urls":["http:\/\/www.matthiasmueller.info\/publications\/hpbd.pdf"],"collections":"Basically computer science","url":"http:\/\/www.matthiasmueller.info\/publications\/hpbd.pdf","urldate":"2011-03-11","journal":"matthiasmueller.info","year":"2008"},{"key":"Freeman1967","type":"article","title":"Detection of transposition errors in decimal numbers","author":"Freeman, H","abstract":"","comment":"","date_added":"2010-08-20","date_published":"1967-08-06","urls":["http:\/\/ieeexplore.ieee.org\/xpls\/abs_all.jsp?arnumber=1447797"],"collections":"Basically computer science","url":"http:\/\/ieeexplore.ieee.org\/xpls\/abs_all.jsp?arnumber=1447797","urldate":"2010-08-20","journal":"Proceedings of the IEEE","number":"8","pages":"1500--1501","volume":"55","year":"1967"},{"key":"Berkel2009","type":"article","title":"On a curious property of 3435","author":"Berkel, Daan Van","abstract":"Folklore tells us that there are no uninteresting natural numbers. But some natural numbers are more interesting then others. In this article we will explain why 3435 is one of the more interesting natural numbers around. We will show that 3435 is a Munchausen number in base 10, and we will explain what we mean by that. We will further show that for every base there are finitely many Munchausen numbers in that base.","comment":"","date_added":"2011-02-02","date_published":"2009-08-06","urls":["http:\/\/arxiv.org\/abs\/0911.3038","http:\/\/arxiv.org\/pdf\/0911.3038v2"],"collections":"Easily explained,Fun maths facts,Integerology","url":"http:\/\/arxiv.org\/abs\/0911.3038 http:\/\/arxiv.org\/pdf\/0911.3038v2","urldate":"2011-02-02","archivePrefix":"arXiv","arxivId":"0911.3038v2","eprint":"0911.3038v2","journal":"Folklore","pages":"1--5","year":"2009"},{"key":"HuntingRabbitsontheHypercube","type":"article","title":"Hunting Rabbits on the Hypercube","author":"Jessalyn Bolkema and Corbin Groothuis","abstract":"We explore the Hunters and Rabbits game on the hypercube. In the process, we\r\nfind the solution for all classes of graphs with an isoperimetric nesting\r\nproperty and find the exact hunter number of $Q^n$ to be\r\n$1+\\sum\\limits_{i=0}^{n-2} \\binom{i}{\\lfloor i\/2 \\rfloor}$. In addition, we\r\nextend results to the situation where we allow the rabbit to not move between\r\nshots.","comment":"","date_added":"2017-02-06","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1701.08726v1","http:\/\/arxiv.org\/pdf\/1701.08726v1"],"collections":"Attention-grabbing titles,Easily explained,Protocols and strategies,Animals,Food","url":"http:\/\/arxiv.org\/abs\/1701.08726v1 http:\/\/arxiv.org\/pdf\/1701.08726v1","urldate":"2017-02-06","archivePrefix":"arXiv","eprint":"1701.08726","primaryClass":"math.CO","year":"2017"},{"key":"item32","type":"article","title":"Missing Data: Instrument-Level Heffalumps and Item-Level Woozles","author":"Philip L. Roth and Fred S. Switzer III","abstract":"The purpose of this paper is to provide a brief overview of each of two missing data situations, and try to show the importance of considering which elusive creature a researcher might be hunting. We find that much of the previous literature does not consider the distinction between missing data at the item level or instrument level. Failure to make this distinction can partially muddle one\u2019s treatment of missing data in important situations.","comment":"","date_added":"2013-06-26","date_published":"1999-08-06","urls":["https:\/\/archive.is\/8Jq0O","http:\/\/division.aomonline.org\/rm\/1999_RMD_Forum_Missing_Data.htm"],"collections":"Attention-grabbing titles","url":"https:\/\/archive.is\/8Jq0O http:\/\/division.aomonline.org\/rm\/1999_RMD_Forum_Missing_Data.htm","urldate":"2013-06-26","year":"1999"},{"key":"Oliva2010","type":"article","title":"What Sequential Games , the Tychonoff Theorem and the Double-Negation Shift have in Common","author":"Oliva, Paulo and Escardo, Martin","abstract":"This is a tutorial for mathematically inclined functional programmers, based on previously published, peered reviewed theoretical work. We discuss a higher-type functional, written here in the functional programming language Haskell, which (1) optimally plays sequential games, (2) implements a computational version of the Tychonoff Theorem from topology, and (3) realizes the Double Negation Shift from logic and proof theory. The functional makes sense for finite and infinite (lazy) lists, and in the binary case it amounts to an operation that is available in any (strong) monad. In fact, once we define this monad in Haskell, it turns out that this amazingly versatile functional is already available in Haskell, in the standard prelude, called sequence, which iterates this binary operation.\r\nTherefore Haskell proves that this functional is even more versatile than anticipated, as the function sequence was introduced for other purposes by the language designers, in particular the iteration of a list of monadic effects (but effects are not what we discuss here)","comment":"","date_added":"2010-08-04","date_published":"2010-08-06","urls":["http:\/\/www.cs.bham.ac.uk\/~mhe\/papers\/msfp2010\/Escardo-Oliva-MSFP2010.pdf"],"collections":"Basically computer science,Protocols and strategies,Games to play with friends","url":"http:\/\/www.cs.bham.ac.uk\/~mhe\/papers\/msfp2010\/Escardo-Oliva-MSFP2010.pdf","urldate":"2010-08-04","journal":"Topology","keywords":"agda,axiom of choice,dependent type,exhaustible set,foundations,functional programming,game theory,haskell,infinite data,logic,monad,optimal strategy,search,topology","year":"2010"},{"key":"Janson2008","type":"article","title":"Plane recursive trees, Stirling permutations and an urn model","author":"Janson, Svante","abstract":"","comment":"","date_added":"2010-02-25","date_published":"2008-08-06","urls":["https:\/\/arxiv.org\/abs\/0803.1129v1","http:\/\/arxiv.org\/pdf\/0803.1129"],"collections":"Probability and statistics,Combinatorics","url":"https:\/\/arxiv.org\/abs\/0803.1129v1 http:\/\/arxiv.org\/pdf\/0803.1129","urldate":"2010-02-25","journal":"Preprint","pages":"1--8","year":"2008"},{"key":"Ito2015","type":"article","title":"On the Existence of Generalized Parking Spaces for Complex Reflection Groups","author":"Ito, Yosuke and Okada, Soichi","abstract":"Let $W$ be an irreducible finite complex reflection group acting on a complex vector space $V$. For a positive integer $k$, we consider a class function $\\varphi_k$ given by $\\varphi_k(w) = k^{\\dim V^w}$ for $w \\in W$, where $V^w$ is the fixed-point subspace of $w$. If $W$ is the symmetric group of $n$ letters and $k=n+1$, then $\\varphi_{n+1}$ is the permutation character on (classical) parking functions. In this paper, we give a complete answer to the question when $\\varphi_k$ (resp. its $q$-analogue) is the character of a representation (resp. the graded character of a graded representation) of $W$. As a key to the proof in the symmetric group case, we find the greatest common divisors of specialized Schur functions. And we propose a unimodality conjecture of the coefficients of certain quotients of principally specialized Schur functions.","comment":"","date_added":"2015-09-06","date_published":"2015-08-01","urls":["http:\/\/arxiv.org\/abs\/1508.06846","http:\/\/arxiv.org\/pdf\/1508.06846v1"],"collections":"Attention-grabbing titles,The groups group","url":"http:\/\/arxiv.org\/abs\/1508.06846 http:\/\/arxiv.org\/pdf\/1508.06846v1","urldate":"2015-09-06","month":"aug","year":"2015","archivePrefix":"arXiv","eprint":"1508.06846","primaryClass":"math.CO"},{"key":"Knauer2011","type":"article","title":"How to eat 4\/9 of a pizza","author":"Knauer, Kolja and Micek, Piotr and Ueckerdt, Torsten","abstract":"Given two players alternately picking pieces of a pizza sliced by radial cuts, in such a way that after the first piece is taken every subsequent chosen piece is adjacent to some previously taken piece, we provide a strategy for the starting player to get 4\/9 of the pizza. This is best possible and settles a conjecture of Peter Winkler.","comment":"","date_added":"2012-12-12","date_published":"2011-12-01","urls":["http:\/\/arxiv.org\/abs\/0812.2870","http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0012365X11001154","http:\/\/arxiv.org\/pdf\/0812.2870v4"],"collections":"Easily explained,Protocols and strategies,Puzzles","url":"http:\/\/arxiv.org\/abs\/0812.2870 http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0012365X11001154 http:\/\/arxiv.org\/pdf\/0812.2870v4","urldate":"2012-12-12","archivePrefix":"arXiv","arxivId":"0812.2870","eprint":"0812.2870","journal":"Discrete Mathematics","month":"dec","pages":"1--15","year":"2011","primaryClass":"cs.DM"},{"key":"Hadley1992","type":"article","title":"Problems to sharpen the young","author":"Hadley, John and Singmaster, David","abstract":"An annotated translation of Propositiones ad acuendos juvenes, the oldest mathematical problem collection in Latin, attributed to Alcuin of York.","comment":"","date_added":"2013-03-18","date_published":"1992-08-06","urls":["http:\/\/www.jstor.org\/stable\/10.2307\/3620384"],"collections":"Easily explained,Puzzles","url":"http:\/\/www.jstor.org\/stable\/10.2307\/3620384","urldate":"2013-03-18","journal":"The Mathematical Gazette","number":"475","pages":"102--126","volume":"76","year":"1992"},{"key":"Flocchini2010","type":"article","title":"Mapping an unfriendly subway system","author":"Flocchini, Paola and Kellett, Matthew and Mason, P.","abstract":"We consider a class of highly dynamic networks modelled on an urban subway system. We examine the problem of creating a map of such a subway in less than ideal conditions, where the local residents are not enthusiastic about the process and there is a limited ability to communicate amongst the mappers. More precisely, we study the problem of a team of asynchronous computational entities (the mapping agents) determining the location of black holes in a highly dynamic graph, whose edges are defined by the asynchronous movements ofmobile entities (the subway carriers). We present and analyze a solution protocol. The algorithm solves the problem with the minimum number of agents possible. We also establish lower bounds on the number of carrier moves in the worst case, showing that our protocol is also move-optimal.","comment":"","date_added":"2012-02-07","date_published":"2010-08-06","urls":["https:\/\/link.springer.com\/chapter\/10.1007\/978-3-642-13122-6_20","http:\/\/people.scs.carleton.ca\/~santoro\/Reports\/Subway.pdf"],"collections":"Protocols and strategies","url":"https:\/\/link.springer.com\/chapter\/10.1007\/978-3-642-13122-6_20 http:\/\/people.scs.carleton.ca\/~santoro\/Reports\/Subway.pdf","urldate":"2012-02-07","journal":"Fun with Algorithms","pages":"190--201","publisher":"Springer","year":"2010"},{"key":"Bazargan2007","type":"article","title":"A linear programming approach for aircraft boarding strategy","author":"Bazargan, M","abstract":"","comment":"","date_added":"2011-10-04","date_published":"2007-11-01","urls":["http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0377221706010137"],"collections":"Basically computer science,Protocols and strategies","url":"http:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0377221706010137","urldate":"2011-10-04","doi":"10.1016\/j.ejor.2006.09.071","issn":"03772217","journal":"European Journal of Operational Research","keywords":"boarding strategy,integer programming,or in airlines,transportation","month":"nov","number":"1","pages":"394--411","volume":"183","year":"2007"},{"key":"item13","type":"article","title":"London Calling Philosophy and Engineering: WPE 2008","author":"Glen Miller","abstract":"","comment":"","date_added":"2012-01-24","date_published":"2009-08-06","urls":["http:\/\/www.ncbi.nlm.nih.gov\/pubmed\/19543813"],"collections":"","url":"http:\/\/www.ncbi.nlm.nih.gov\/pubmed\/19543813","urldate":"2012-01-24","booktitle":"The Annals of occupational hygiene","isbn":"1903496780","issn":"0003-4878","number":"2","pmid":"3752842","volume":"30","year":"2009"},{"key":"Ge2007","type":"article","title":"Interpolating Solid Orientations with a $C^2$ -Continuous B-Spline Quaternion Curve","author":"Ge, Wenbing and Huang, Zhangjin and Wang, Guoping","abstract":"","comment":"","date_added":"2011-01-12","date_published":"2007-08-06","urls":["http:\/\/link.springer.com\/chapter\/10.1007%2F978-3-540-73011-8_58"],"collections":"Basically computer science","url":"http:\/\/link.springer.com\/chapter\/10.1007%2F978-3-540-73011-8_58","urldate":"2011-01-12","keywords":"b-spline curve,c 2 -continuous,computer,interpolation,quaternion","pages":"606--615","year":"2007"},{"key":"Stallings1966","type":"article","title":"How not to prove the Poincar\u00e9 conjecture","author":"Stallings, JR","abstract":"I have committed the sin of falsely proving Poincar\u00e9's Conjecture. But that was in another country; and besides, until now no one has known about it. Now, in hope of deterring others from making similar mistakes, I shall describe my mistaken proof. Who knows but that somehow a small change, a new interpretation, and this line of proof may be rectified!","comment":"","date_added":"2014-11-17","date_published":"1966-08-06","urls":["http:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.32.3404","http:\/\/math.berkeley.edu\/~stall\/notPC.ps"],"collections":"Attention-grabbing titles,About proof","url":"http:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.32.3404 http:\/\/math.berkeley.edu\/~stall\/notPC.ps","urldate":"2014-11-17","year":"1966"},{"key":"item41","type":"misc","title":"Useful inequalities cheat sheet","author":"L\u00e1szl\u00f3 Kozma","abstract":"This is a collection of some of the most important mathematical inequalities. I tried to include non-trivial inequalities that can be useful in solving problems or proving theorems. I omitted many details, in some cases even necessary conditions (hopefully only when they were obvious). If you are not sure whether an inequality can be applied in some context, try to find a more detailed source for the exact definition. For lack of space I omitted proofs and discussions on when equality holds.","comment":"","date_added":"2014-04-28","date_published":"2011-08-06","urls":["http:\/\/lkozma.net\/inequalities_cheat_sheet\/"],"collections":"Lists and catalogues,About proof","url":"http:\/\/lkozma.net\/inequalities_cheat_sheet\/","urldate":"2014-04-28","year":"2011"},{"key":"Duran2014","type":"article","title":"The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?","author":"Dur\u00e1n, Antonio J and P\u00e9rez, Mario and Varona, Juan L","abstract":"","comment":"","date_added":"2014-10-23","date_published":"2014-08-06","urls":["http:\/\/www.ams.org\/notices\/201410\/rnoti-p1249.pdf"],"collections":"Drama!","url":"http:\/\/www.ams.org\/notices\/201410\/rnoti-p1249.pdf","urldate":"2014-10-23","isbn":"1584704349","number":"November","pages":"1249--1252","year":"2014"},{"key":"Frieze2013","type":"article","title":"The topology of competitively constructed graphs","author":"Frieze, Alan and Pegden, Wesley","abstract":"We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for this game: for the case $k=3$ a player can ensure the resulting graph is planar, while for the case $k=4$, a player can force the appearance of arbitrarily large clique minors.","comment":"","date_added":"2013-12-30","date_published":"2013-12-01","urls":["http:\/\/arxiv.org\/abs\/1312.0964","http:\/\/arxiv.org\/pdf\/1312.0964v2"],"collections":"Games to play with friends,Protocols and strategies","url":"http:\/\/arxiv.org\/abs\/1312.0964 http:\/\/arxiv.org\/pdf\/1312.0964v2","urldate":"2013-12-30","month":"dec","pages":"9","year":"2013","archivePrefix":"arXiv","eprint":"1312.0964","primaryClass":"math.CO"},{"key":"Kikuchi1994","type":"article","title":"A Note on Boolos' Proof of the Incompleteness Theorem","author":"Kikuchi, Makoto","abstract":"We give a proof of G\u00f6del's first incompleteness theorem based on Berry's paradox, and from it we also derive the second incompleteness theorem model-theoretically.","comment":"","date_added":"2010-09-03","date_published":"1994-08-06","urls":["http:\/\/doi.wiley.com\/10.1002\/malq.19940400409"],"collections":"About proof","url":"http:\/\/doi.wiley.com\/10.1002\/malq.19940400409","urldate":"2010-09-03","journal":"Mathematical Logic Quarterly","number":"4","pages":"528--532","volume":"40","year":"1994"},{"key":"Rechnitzer2006","type":"article","title":"Haruspicy 3: The anisotropic generating function of directed bond-animals is not D-finite","author":"Rechnitzer, Andrew","abstract":"While directed site-animals have been solved on several lattices, directed bond-animals remain unsolved on any nontrivial lattice. In this paper we demonstrate that the anisotropic generating function of directed bond-animals on the square lattice is fundamentally different from that of directed site-animals in that it is not differentiably finite. We also extend this result to directed bond-animals on hypercubic lattices. This indicates that directed bond-animals are unlikely to be solved by similar methods to those used in the solution of directed site-animals. It also implies that a solution cannot be conjectured using computer packages such as Gfun [A Maple package developed by B. Salvy, P. Zimmermann, E. Murray at INRIA, France, available from http:\/\/algo.inria.fr\/libraries\/ at time of submission; B. Salvy, P. Zimmermann, Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (2) (1994) 163\u2013177] or differential approximants [A.J. Guttmann, Asymptotic analysis of coefficients, in: C. Domb, J. Lebowitz (Eds.), Phase Transit. Crit. Phenom., vol. 13, Academic Press, London, 1989, pp. 1\u2013234, programs available from http:\/\/www.ms.unimelb.edu.au\/~tonyg].","comment":"","date_added":"2015-11-08","date_published":"2006-08-01","urls":["http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0097316505001883"],"collections":"Animals,Attention-grabbing titles","url":"http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0097316505001883","urldate":"2015-11-08","issn":"00973165","journal":"Journal of Combinatorial Theory, Series A","keywords":"Bond animals,Differentiably finite power series,Enumeration,Solvability","month":"aug","number":"6","pages":"1031--1049","volume":"113","year":"2006"},{"key":"Williams2012","type":"article","title":"The paramagnetic and glass transitions in sudoku","author":"Williams, Alex and Ackland, Graeme . J.","abstract":"We study the statistical mechanics of a model glassy system based on a familiar and popular mathematical puzzle. Sudoku puzzles provide a very rare example of a class of frustrated systems with a unique groundstate without symmetry. Here, the puzzle is recast as thermodynamic system where the number of violated rules defines the energy. We use Monte Carlo simulation to show that the \"Sudoku Hamiltonian\" exhibits two transitions as a function of temperature, a paramagnetic and a glass transition. Of these, the intermediate condensed phase is the only one which visits the ground state (i.e. it solves the puzzle, though this is not the purpose of the study). Both transitions are associated with an entropy change, paramagnetism measured from the dynamics of the Monte Carlo run, showing a peak in specific heat, while the residual glass entropy is determined by finding multiple instances of the glass by repeated annealing. There are relatively few such simple models for frustrated or glassy systems which exhibit both ordering and glass transitions, sudoku puzzles are unique for the ease with which they can be obtained with the proof of the existence of a unique ground state via the satisfiability of all constraints. Simulations suggest that in the glass phase there is an increase in information entropy with lowering temperature. In fact, we have shown that sudoku have the type of rugged energy landscape with multiple minima which typifies glasses in many physical systems, and this puzzling result is a manifestation of the paradox of the residual glass entropy. These readily-available puzzles can now be used as solvable model Hamiltonian systems for studying the glass transition.","comment":"","date_added":"2012-12-29","date_published":"2012-12-01","urls":["http:\/\/arxiv.org\/abs\/1212.1649","http:\/\/arxiv.org\/pdf\/1212.1649v1"],"collections":"Basically physics","url":"http:\/\/arxiv.org\/abs\/1212.1649 http:\/\/arxiv.org\/pdf\/1212.1649v1","urldate":"2012-12-29","archivePrefix":"arXiv","arxivId":"1212.1649","doi":"10.1103\/PhysRevE.86.031109","eprint":"1212.1649","month":"dec","year":"2012","primaryClass":"math-ph"},{"key":"Kac","type":"article","title":"Can One Hear the Shape of a Drum?","author":"Kac, Mark","abstract":"","comment":"","date_added":"2011-01-12","date_published":"1966-08-06","urls":["http:\/\/www.jstor.org\/stable\/2313748"],"collections":"Basically physics,Easily explained,Music,Geometry,Fun maths facts","url":"http:\/\/www.jstor.org\/stable\/2313748","urldate":"2011-01-12","year":"1966"},{"key":"MeaninginClassicalMathematicsIsitatOddswithIntuitionism","type":"article","title":"Meaning in Classical Mathematics: Is it at Odds with Intuitionism?","author":"Karin Usadi Katz and Mikhail G. Katz","abstract":"We examine the classical\/intuitionist divide, and how it reflects on modern\r\ntheories of infinitesimals. When leading intuitionist Heyting announced that\r\n\"the creation of non-standard analysis is a standard model of important\r\nmathematical research\", he was fully aware that he was breaking ranks with\r\nBrouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a\r\ncomparative textual analysis of three of Bishop's texts, we analyze the\r\nideological and\/or pedagogical nature of his objections to infinitesimals a la\r\nRobinson. Bishop's famous \"debasement\" comment at the 1974 Boston workshop,\r\npublished as part of his Crisis lecture, in reality was never uttered in front\r\nof an audience. We compare the realist and the anti-realist intuitionist\r\nnarratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and\r\nTennant. Variational principles are important physical applications, currently\r\nlacking a constructive framework. We examine the case of the Hawking-Penrose\r\nsingularity theorem, already analyzed by Hellman in the context of the\r\nQuine-Putnam indispensability thesis.","comment":"","date_added":"2017-02-27","date_published":"2011-08-06","urls":["http:\/\/arxiv.org\/abs\/1110.5456v1","http:\/\/arxiv.org\/pdf\/1110.5456v1"],"collections":"History,The act of doing maths","url":"http:\/\/arxiv.org\/abs\/1110.5456v1 http:\/\/arxiv.org\/pdf\/1110.5456v1","urldate":"2017-02-27","archivePrefix":"arXiv","eprint":"1110.5456","primaryClass":"math.LO","year":"2011"},{"key":"Coman","type":"article","title":"The Math Encyclopedia of Smarandache Type Notions","author":"Coman, Marius","abstract":"About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name) and literature, it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache's mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. We structured this book using numbered Definitions, Theorems, Conjectures, Notes and Comments, in order to facilitate an easier reading but also to facilitate references to a specific paragraph. We divided the Bibliography in two parts, Writings by Florentin Smarandache (indexed by the name of books and articles) and Writings on Smarandache notions (indexed by the name of authors). We treated, in this book, about 130 Smarandache type sequences, about 50 Smarandache type functions and many solved or open problems of number theory. We also have, at the end of this book, a proposal for a new Smarandache type notion, id est the concept of \u201ca set of Smarandache-Coman divisors of order k of a composite positive integer n with m prime factors\u201d, notion that seems to have promising applications, at a first glance at least in the study of absolute and relative Fermat pseudoprimes, Carmichael numbers and Poulet numbers. This encyclopedia is both for researchers that will have on hand a tool that will help them \u201cnavigate\u201d in the universe of Smarandache type notions and for young math enthusiasts: many of them will be attached by this wonderful branch of mathematics, number theory, reading the works of Florentin Smarandache.","comment":"","date_added":"2013-11-21","date_published":"2013-08-06","urls":["http:\/\/vixra.org\/abs\/1311.0097"],"collections":"Lists and catalogues","url":"http:\/\/vixra.org\/abs\/1311.0097","urldate":"2013-11-21","pages":"135","year":"2013"},{"key":"Gauvrit2011","type":"article","title":"Sloane's Gap: Do Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS?","author":"Gauvrit, Nicolas and Delahaye, Jean-Paul and Zenil, Hector","abstract":"The Online Encyclopedia of Integer Sequences (OEIS) is made up of thousands of numerical sequences considered particularly interesting by some mathematicians. The graphic representation of the frequency with which a number n as a function of n appears in that database shows that the underlying function decreases fast, and that the points are distributed in a cloud, seemingly split into two by a clear zone that will be referred to here as \"Sloane's Gap\". The decrease and general form are explained by mathematics, but an explanation of the gap requires further considerations.","comment":"","date_added":"2013-10-17","date_published":"2011-01-01","urls":["http:\/\/arxiv.org\/abs\/1101.4470","http:\/\/arxiv.org\/pdf\/1101.4470v2"],"collections":"Easily explained,Probability and statistics,The act of doing maths,Integerology","url":"http:\/\/arxiv.org\/abs\/1101.4470 http:\/\/arxiv.org\/pdf\/1101.4470v2","urldate":"2013-10-17","month":"jan","year":"2011","archivePrefix":"arXiv","eprint":"1101.4470","primaryClass":"math.PR"},{"key":"Fathauer","type":"article","title":"Statistical Modeling of Gang Violence in Los Angeles","author":"Fathauer, Chris","abstract":"","comment":"","date_added":"2012-04-13","date_published":"2010-08-06","urls":["http:\/\/paleo.sscnet.ucla.edu\/SIURO_revised.pdf"],"collections":"Basically computer science,Probability and statistics,Modelling","url":"http:\/\/paleo.sscnet.ucla.edu\/SIURO_revised.pdf","urldate":"2012-04-13","pages":"1--25","year":"2010"},{"key":"Poonen2012","type":"article","title":"Undecidable problems: a sampler","author":"Poonen, Bjorn","abstract":"After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.","comment":"","date_added":"2012-04-04","date_published":"2012-04-01","urls":["http:\/\/arxiv.org\/abs\/1204.0299","http:\/\/arxiv.org\/pdf\/1204.0299v2"],"collections":"Probability and statistics","url":"http:\/\/arxiv.org\/abs\/1204.0299 http:\/\/arxiv.org\/pdf\/1204.0299v2","urldate":"2012-04-04","archivePrefix":"arXiv","arxivId":"1204.0299","eprint":"1204.0299","keywords":"and phrases,decision problem,undecidability","month":"apr","pages":"28","year":"2012","primaryClass":"math.LO"},{"key":"JewishProblems","type":"article","title":"Jewish Problems","author":"Tanya Khovanova and Alexey Radul","abstract":"This is a special collection of problems that were given to select applicants\r\nduring oral entrance exams to the math department of Moscow State University.\r\nThese problems were designed to prevent Jews and other undesirables from\r\ngetting a passing grade. Among problems that were used by the department to\r\nblackball unwanted candidate students, these problems are distinguished by\r\nhaving a simple solution that is difficult to find. Using problems with a\r\nsimple solution protected the administration from extra complaints and appeals.\r\nThis collection therefore has mathematical as well as historical value.","comment":"","date_added":"2017-03-23","date_published":"2011-08-06","urls":["http:\/\/arxiv.org\/abs\/1110.1556v2","http:\/\/arxiv.org\/pdf\/1110.1556v2"],"collections":"Attention-grabbing titles,History,Puzzles","url":"http:\/\/arxiv.org\/abs\/1110.1556v2 http:\/\/arxiv.org\/pdf\/1110.1556v2","urldate":"2017-03-23","archivePrefix":"arXiv","eprint":"1110.1556","primaryClass":"math.HO","year":"2011"},{"key":"item3","type":"misc","title":"openttd logic gates","author":"Heikki Kallasjoki","abstract":"Here's a rather old (and probably outdated) look at how one could simulate digital logic circuits with OpenTTD. Includes the fastest four-bit ripple-carry adder ever: takes about two months (of in-game time) for the carry information to propagate.","comment":"","date_added":"2010-09-30","date_published":"2005-08-06","urls":["http:\/\/zem.fi\/ttd_logic\/"],"collections":"Basically computer science,Unusual computers","url":"http:\/\/zem.fi\/ttd_logic\/","urldate":"2010-09-30","year":"2005"},{"key":"item4","type":"misc","title":"Push-pull LEGO logic gates","author":"Randomwraith","abstract":"","comment":"","date_added":"2010-09-30","date_published":"1999-08-06","urls":["https:\/\/www.randomwraith.com\/logic.html"],"collections":"Basically computer science,Unusual computers","url":"https:\/\/www.randomwraith.com\/logic.html","urldate":"2010-09-30","year":"1999"},{"key":"item6","type":"article","title":"Zaphod Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle","author":"Andrew Granville","abstract":"","comment":"","date_added":"2010-10-31","date_published":"1992-08-06","urls":["http:\/\/www.dms.umontreal.ca\/~andrew\/PDF\/beeb.pdf"],"collections":"Attention-grabbing titles","url":"http:\/\/www.dms.umontreal.ca\/~andrew\/PDF\/beeb.pdf","urldate":"2010-10-31","year":"1992"},{"key":"item9","type":"online","title":"Laying train tracks","author":"Danny Calegari","abstract":"This morning I was playing trains with my son Felix. At the moment he is much more interested in laying the tracks than putting the trains on and moving them around, but he doesn\u2019t tend to get concerned about whether the track closes up to make a loop.","comment":"Group theory applied to BRIO train tracks","date_added":"2011-12-03","date_published":"2011-08-06","urls":["http:\/\/lamington.wordpress.com\/2011\/12\/02\/laying-train-tracks\/"],"collections":"Easily explained,Things to make and do,Fun maths facts,The groups group","url":"http:\/\/lamington.wordpress.com\/2011\/12\/02\/laying-train-tracks\/","urldate":"2011-12-03","year":"2011"},{"key":"item10","type":"online","title":"Understanding Monads With JavaScript","author":"Ionu\u021b G. Stan","abstract":"For the past weeks I've been working hard studying monads. I'm still learning Haskell, and to be honest I thought I knew what monads are all about, but when I wanted to write a little Haskell library, just to sharpen up my skills, I realized that while I understood the way monadic bind (>>=) and return work, I had no understanding of where that state comes from. So, most likely I had no understanding at all. As a result of this I thought I rediscover monads myself using JavaScript. The plan was basically the same as that used when I derived the Y Combinator: start from the initial problem (dealing with explicit immutable state in this case), and work my way up to the solution by applying simple code transformations.","comment":"","date_added":"2012-01-18","date_published":"2011-08-06","urls":["http:\/\/igstan.ro\/posts\/2011-05-02-understanding-monads-with-javascript.html"],"collections":"Basically computer science","url":"http:\/\/igstan.ro\/posts\/2011-05-02-understanding-monads-with-javascript.html","urldate":"2012-01-18","year":"2011"},{"key":"item11","type":"book","title":"Drawings from Angola: living mathematics","author":"Paulus Gerdes","abstract":"For children from age 8 to 14.\"Drawings from Angola\" present an introduction to an African story telling tradition. The tales are illustrated with marvelous drawings made in the sand. The book conveys the stories of the stork and the leopard, the hunter and the dog, the rooster and the fox, and others. It explains how to execute the drawings. The reader is invited to draw tortoises, antelopes, lions, and other animals. The activities proposed throughout the book invite the reader to experiment and to explore the 'rhythm' and symmetry of the illustrations. Surprising results will be playfully obtained, such as in arithmetic, a way to calculate quickly the sum of a sequence of odd numbers. Children will live the beautiful mathematics of the Angolan sanddrawings.Answers to the activities are provided.The book can be used both in classrooms and at home.","comment":"","date_added":"2012-01-19","date_published":"2007-08-06","urls":["http:\/\/books.google.com\/books?id=rRDbdeWoZEMC&pgis=1"],"collections":"Art,Easily explained,Unusual arithmetic","url":"http:\/\/books.google.com\/books?id=rRDbdeWoZEMC&pgis=1","urldate":"2012-01-19","pages":"71","publisher":"Lulu.com","year":"2007"},{"key":"item14","type":"online","title":"Poe, E.: Near A Raven","author":"Mike Keith","abstract":"At the time of its writing in 1995, this composition in Standard Pilish, a retelling of Edgar Allan Poe's \"The Raven\", was one of the longest texts ever written using the \u03c0 constraint, in which the number of letters in each successive word \"spells out\" the digits of \u03c0 (740 digits in this example). For length this poem was subsequently outdone by the nearly-4000-digit Cadaeic Cadenza, whose first section is just Near A Raven with the first three words altered, but since this version is fairly well-known by itself (for example, it was reprinted in Berggren, Borwein and Borwein's \"Pi: A Source Book\"), we have decided to give it its own web page.","comment":"","date_added":"2012-02-25","date_published":"1995-08-06","urls":["http:\/\/www.cadaeic.net\/naraven.htm"],"collections":"Animals,Art","url":"http:\/\/www.cadaeic.net\/naraven.htm","urldate":"2012-02-25","year":"1995"},{"key":"item15","type":"online","title":"Gaussian prime spirals","author":"Joseph O'Rourke","abstract":"Imagine a particle in the complex plane, starting at $c_0$, a Gaussian integer, moving initially $\\pm$ in the horizontal or vertical directions. When it hits a Gaussian prime, it turns left $90^{\\circ}$.","comment":"","date_added":"2012-03-24","date_published":"2012-08-06","urls":["http:\/\/mathoverflow.net\/questions\/91423\/gaussian-prime-spirals"],"collections":"Easily explained","url":"http:\/\/mathoverflow.net\/questions\/91423\/gaussian-prime-spirals","urldate":"2012-03-24","year":"2012"},{"key":"item18","type":"article","title":"Nim multiplication","author":"H. W. Lenstra, Jr.","abstract":"","comment":"","date_added":"2012-04-21","date_published":"1978-08-06","urls":["https:\/\/openaccess.leidenuniv.nl\/bitstream\/handle\/1887\/2125\/346_027.pdf?sequence=1"],"collections":"Games to play with friends,Unusual arithmetic","url":"https:\/\/openaccess.leidenuniv.nl\/bitstream\/handle\/1887\/2125\/346_027.pdf?sequence=1","urldate":"2012-04-21","year":"1978"},{"key":"item24","type":"online","title":"Magic: the Gathering is Turing Complete","author":"Alex Churchill","abstract":"We always knew Magic: the Gathering was a complex game. But now it's proven: you could assemble a computer out of Magic cards.","comment":"","date_added":"2012-09-11","date_published":"2012-08-06","urls":["http:\/\/www.toothycat.net\/~hologram\/Turing\/"],"collections":"Basically computer science,Games to play with friends,Unusual computers","url":"http:\/\/www.toothycat.net\/~hologram\/Turing\/","urldate":"2012-09-11","year":"2012"},{"key":"item22","type":"online","title":"A Hamiltonian circuit for Rubik's Cube","author":"cuBerBruce","abstract":"At last, the Hamiltonian circuit problem for Rubik's Cube has a solution! To be a little more mathematically precise, a Hamiltonian circuit of the quarter-turn metric Cayley graph for the Rubik's Cube group has been found.","comment":"","date_added":"2012-09-04","date_published":"2012-08-06","urls":["http:\/\/bruce.cubing.net\/ham333\/rubikhamiltonexplanation.html"],"collections":"Easily explained,Puzzles,Fun maths facts","url":"http:\/\/bruce.cubing.net\/ham333\/rubikhamiltonexplanation.html","urldate":"2012-09-04","year":"2012"},{"key":"item29","type":"review","title":"Review of \"Groups\" by Georges Papy in New Scientist","author":"T. H. O'Beirne","abstract":"","comment":"","date_added":"2013-03-17","date_published":"1964-09-01","urls":["http:\/\/books.google.co.uk\/books?id=zhboHCUC6LsC&pg=PA582&lpg=PA582&dq=groups+georges+papy&source=bl&ots=SA1veajk7J&sig=JCidC0RAq5u_E4AzahyYlBoLuZU&hl=en&sa=X&ei=F6lFUbyqJoaW0QXI3oDACg&ved=0CDoQ6AEwAQ#v=onepage&q=groups","georges","papy&f=false"],"collections":"The groups group","url":"http:\/\/books.google.co.uk\/books?id=zhboHCUC6LsC&pg=PA582&lpg=PA582&dq=groups+georges+papy&source=bl&ots=SA1veajk7J&sig=JCidC0RAq5u_E4AzahyYlBoLuZU&hl=en&sa=X&ei=F6lFUbyqJoaW0QXI3oDACg&ved=0CDoQ6AEwAQ#v=onepage&q=groups georges papy&f=false","urldate":"2013-03-17","page":"582","year":"1964","month":"sep","day":"3"},{"key":"item37","type":"online","title":"Fair but irregular polyhedral dice","author":"Joseph O'Rourke","abstract":"","comment":"","date_added":"2013-12-16","date_published":"2010-08-06","urls":["http:\/\/mathoverflow.net\/questions\/46684\/fair-but-irregular-polyhedral-dice"],"collections":"Probability and statistics,Geometry","url":"http:\/\/mathoverflow.net\/questions\/46684\/fair-but-irregular-polyhedral-dice","urldate":"2013-12-16","year":"2010"},{"key":"item39","type":"online","title":"Rithmomachia","author":"Daniel U. Thibault and Michel Boutin","abstract":"This complex chess-like game appeared in the western world around the year 1000. The game knew a great burst of popularity in the 15th century, because of some rules changes. When chess also saw its rules change (particularly when the Queen started to move in its modern fashion instead of its previous King-like motion), Rithmomachia started fading rapidly, at the close of the 16th century. The rules given here are those established in 1556 by Claude de Boissi\u00e8re, a Frenchman.","comment":"","date_added":"2014-01-24","date_published":"1984-08-06","urls":["http:\/\/www.gamecabinet.com\/rules\/Rithmomachia.html"],"collections":"Games to play with friends,History","url":"http:\/\/www.gamecabinet.com\/rules\/Rithmomachia.html","urldate":"2014-01-24","keywords":"Board,GURPS,Games,Medieval,Rithmomachia","year":"1984"},{"key":"item42","type":"article","title":"The Number-Pad Game","author":"Alex Fink and Richard Guy","abstract":"","comment":"","date_added":"2014-06-11","date_published":"2007-08-06","urls":["http:\/\/www.maths.qmul.ac.uk\/~fink\/numbpad2f.pdf"],"collections":"Easily explained,Games to play with friends,Puzzles","url":"http:\/\/www.maths.qmul.ac.uk\/~fink\/numbpad2f.pdf","urldate":"2014-06-11","year":"2007"},{"key":"item44","type":"article","title":"A Fresh Look at Peg Solitaire","author":"George I. Bell","abstract":"","comment":"","date_added":"2014-08-26","date_published":"2007-08-06","urls":["http:\/\/recmath.org\/pegsolitaire\/papers\/Bell_AFreshLookatPegSolitaire_MathMag2007.pdf"],"collections":"Easily explained,Puzzles","url":"http:\/\/recmath.org\/pegsolitaire\/papers\/Bell_AFreshLookatPegSolitaire_MathMag2007.pdf","urldate":"2014-08-26","year":"2007"},{"key":"item45","type":"online","title":"Pondering an Artist's Perplexing Tribute to the Pythagorean Theorem","author":"Ivars Peterson","abstract":"","comment":"","date_added":"2014-08-28","date_published":"2009-08-06","urls":["http:\/\/mathtourist.blogspot.co.uk\/2009\/02\/pondering-artists-perplexing-tribute-to.html"],"collections":"Art,Easily explained,Drama!","url":"http:\/\/mathtourist.blogspot.co.uk\/2009\/02\/pondering-artists-perplexing-tribute-to.html","urldate":"2014-08-28","year":"2009"},{"key":"item49","type":"misc","title":"Maximum Matching and a Polyhedron With 0,1-Vertices","author":"Jack Edmonds","abstract":"A matching in a graph $G$ is a subset of edges in $G$ such that no two meet the same node in $G$. The convex polyhedron $C$ is characterised, where the extreme points of $C$ correspond to the matchings in $G$. Where each edge of $G$ carries a real numerical weight, an efficient algorithm is described for finding a matching in $G$ with maximum weight-sum.","comment":"","date_added":"2015-03-07","date_published":"1964-08-06","urls":["http:\/\/nvlpubs.nist.gov\/nistpubs\/jres\/69B\/jresv69Bn1-2p125_A1b.pdf"],"collections":"","url":"http:\/\/nvlpubs.nist.gov\/nistpubs\/jres\/69B\/jresv69Bn1-2p125_A1b.pdf","urldate":"2015-03-07","year":"1964"},{"key":"item50","type":"article","title":"The Lost Calculus (1637-1670): Tangency and Optimization without Limits","author":"Jeff Suzuki","abstract":"An examination of the evolution of the lost calculus from its beginnings in the work of Descartes and its subsequent development by Hudde, and the possibility that nearly every problem of calculus could have been solved using algorithms entirely free from the limit concept.","comment":"","date_added":"2015-03-12","date_published":"2005-08-06","urls":["http:\/\/www.maa.org\/programs\/maa-awards\/writing-awards\/the-lost-calculus-1637-1670-tangency-and-optimization-without-limits"],"collections":"History","url":"http:\/\/www.maa.org\/programs\/maa-awards\/writing-awards\/the-lost-calculus-1637-1670-tangency-and-optimization-without-limits","urldate":"2015-03-12","year":"2005"},{"key":"item52","type":"article","title":"Proposal to Encode the Ganda Currency Mark for Bengali in ISO\/IEC 10646","author":"Anshuman Pandey","abstract":"","comment":"","date_added":"2015-09-30","date_published":"2007-08-06","urls":["http:\/\/std.dkuug.dk\/JTC1\/SC2\/wg2\/docs\/n3311.pdf"],"collections":"Easily explained,Notation and conventions","url":"http:\/\/std.dkuug.dk\/JTC1\/SC2\/wg2\/docs\/n3311.pdf","urldate":"2015-09-30","year":"2007"},{"key":"item55","type":"article","title":"On the Number of Times an Integer Occurs as a Binomial Coefficient","author":"H. L. Abbott and P. Erd\u0151s and D. Hanson","abstract":"","comment":"","date_added":"2015-11-30","date_published":"1974-08-06","urls":["http:\/\/www.jstor.org\/stable\/2319526?seq=4#page_scan_tab_contents"],"collections":"Integerology","url":"http:\/\/www.jstor.org\/stable\/2319526?seq=4#page_scan_tab_contents","urldate":"2015-11-30","year":"1974"},{"key":"item57","type":"article","title":"Approaches to the Enumerative Theory of Meanders","author":"Michael La Croix","abstract":"","comment":"","date_added":"2015-12-14","date_published":"2003-08-06","urls":["http:\/\/www.math.uwaterloo.ca\/~malacroi\/Latex\/Meanders.pdf"],"collections":"","url":"http:\/\/www.math.uwaterloo.ca\/~malacroi\/Latex\/Meanders.pdf","urldate":"2015-12-14","year":"2003"},{"key":"item60","type":"article","title":"Transposable integers in arbitrary bases","author":"Anne L. Ludington","abstract":"","comment":"","date_added":"2016-04-19","date_published":"1985-08-06","urls":["http:\/\/www.fq.math.ca\/Scanned\/25-3\/ludington.pdf"],"collections":"Integerology","url":"http:\/\/www.fq.math.ca\/Scanned\/25-3\/ludington.pdf","urldate":"2016-04-19","year":"1985"},{"key":"Entanglement","type":"online","title":"A game for budding knot theorists","author":"Dave Richeson","abstract":"","comment":"","date_added":"2010-08-25","date_published":"2010-08-01","urls":["http:\/\/divisbyzero.com\/2010\/08\/24\/a-game-for-budding-knot-theorists\/"],"collections":"Games to play with friends","url":"http:\/\/divisbyzero.com\/2010\/08\/24\/a-game-for-budding-knot-theorists\/","urldate":"2010-08-25","year":"2010","month":"aug","day":"24"},{"key":"item56","type":"misc","title":"The Theory of Heaps and the Cartier-Foata Monoid","author":"C. Krattenthaler","abstract":"We present Viennot\u2019s theory of heaps of pieces, show that heaps are equivalent to elements in the partially commutative monoid of Cartier and Foata, and illustrate the main results of the theory by reproducing its application to the enumeration of parallelogram polyominoes due to Bousquet\u2013M\u00e9lou and Viennot.","comment":"","date_added":"2015-12-03","date_published":"2006-08-06","urls":["http:\/\/www.mat.univie.ac.at\/~kratt\/artikel\/heaps.pdf"],"collections":"Combinatorics","url":"http:\/\/www.mat.univie.ac.at\/~kratt\/artikel\/heaps.pdf","urldate":"2015-12-03","year":"2006"},{"key":"NotesontheFourthDimensionThePublicDomainReview","type":"article","title":"Notes on the Fourth Dimension","author":"Jon Crabb ","abstract":"Hyperspace, ghosts, and colourful cubes \u2014 Jon Crabb on the work of Charles Howard Hinton and the cultural history of higher dimensions.","comment":"","date_added":"2016-06-13","date_published":"2015-08-06","urls":["http:\/\/publicdomainreview.org\/2015\/10\/28\/notes-on-the-fourth-dimension\/"],"collections":"Art,Geometry","url":"http:\/\/publicdomainreview.org\/2015\/10\/28\/notes-on-the-fourth-dimension\/","urldate":"2016-06-13","year":"2015"},{"key":"item20","type":"online","title":"To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction","author":"David C Keenan","abstract":"The lambda calculus, and the closely related theory of combinators, are important in the foundations of mathematics, logic and computer science. This paper provides an informal and entertaining introduction by means of an animated graphical notation.","comment":"","date_added":"2012-06-03","date_published":"1996-08-06","urls":["http:\/\/dkeenan.com\/Lambda\/index.htm"],"collections":"Animals,Basically computer science","url":"http:\/\/dkeenan.com\/Lambda\/index.htm","urldate":"2012-06-03","year":"1996"},{"key":"PhotoelectricNumberSieve","type":"online","title":"Photoelectric Number Sieve Machine (\"Gear Machine\")","author":"D. H. Lehmer and Robert Canepa","abstract":"This gear number sieve was constructed to solve number theory problems such as factoring and determining if a number is prime. The machine also had a photo detector and powerful amplifier which was not included in the gift.","comment":"","date_added":"2016-06-13","date_published":"1932-08-06","urls":["http:\/\/www.computerhistory.org\/collections\/catalog\/X85.82"],"collections":"Easily explained,Things to make and do,Integerology","url":"http:\/\/www.computerhistory.org\/collections\/catalog\/X85.82","urldate":"2016-06-13","year":"1932"},{"key":"Feferman","type":"misc","title":"Penrose's Godelian argument","author":"Feferman, Solomon","abstract":"","comment":"","date_added":"2011-05-16","date_published":"1996-08-06","urls":["http:\/\/math.stanford.edu\/~feferman\/papers\/penrose.pdf"],"collections":"","url":"http:\/\/math.stanford.edu\/~feferman\/papers\/penrose.pdf","urldate":"2011-05-16","year":"1996"},{"key":"Levien2008","type":"article","title":"The Euler spiral: a mathematical history","author":"Levien, Raph","abstract":"The beautiful Euler spiral, de\ufb01ned by the linear relationship between curvature and arclength, was \ufb01rst proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, \ufb01rst by Augustin Fresnel to compute diffraction of light through a slit, and again by Arthur Talbot to produce an ideal shape for a railway transition curve connecting a straight section with a section of given curvature. Though it has gathered many names throughout its history, the curve retains its aesthetic and mathematical beauty as Euler had clearly visualized. Its equation is related to the Gamma function, the Gauss error function (erf), and is a special case of the con\ufb02uent hypergeometric function.","comment":"I love the Euler spiral. I don't know why. Maybe it's because I first learnt of it as the \"clothoid\", which is an excellent name, or maybe it's because it gives me something to think about when I'm driving.This shortish essay by Raph Levien gives a readable potted history of the spiral's multiple discoveries and applications, illustrated with some lovely sparse diagrams of the sort that maths-illiterate Etsy craftspeople love.","date_added":"2012-04-07","date_published":"2008-08-06","urls":["http:\/\/raph.levien.com\/phd\/euler_hist.pdf"],"collections":"History,Geometry","url":"http:\/\/raph.levien.com\/phd\/euler_hist.pdf","urldate":"2012-04-07","journal":"Opera","pages":"1--14","year":"2008"},{"key":"ScoopingLoopSnooper","type":"online","title":"Scooping the Loop Snooper","author":"Geoffrey K. Pullum","abstract":"","comment":"A proof that the Halting Problem is undecidable.","date_added":"2012-01-20","date_published":"2000-08-06","urls":["http:\/\/www.lel.ed.ac.uk\/~gpullum\/loopsnoop.html"],"collections":"Attention-grabbing titles,Easily explained,About proof","url":"http:\/\/www.lel.ed.ac.uk\/~gpullum\/loopsnoop.html","urldate":"2012-01-20","year":"2000"},{"key":"Good1819","type":"article","title":"Pantologia. A new (cabinet) cyclop\u00e6dia, by J.M. Good, O. Gregory, and N. Bosworth assisted by other gentlemen of eminence","author":"Good, John Mason and Gregory, Olinthus Gilbert","abstract":"","comment":"Contains something on the continued surd.","date_added":"2012-02-10","date_published":"1819-08-06","urls":["http:\/\/books.google.com\/books?id=72vgv4yVQOAC&pgis=1"],"collections":"History,Lists and catalogues","url":"http:\/\/books.google.com\/books?id=72vgv4yVQOAC&pgis=1","urldate":"2012-02-10","year":"1819"},{"key":"Williams2004","type":"article","title":"High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams","author":"Williams, Hugh C. and Poorten, A. J. Van Der and Stein, Andreas","abstract":"This volume consists of a selection of papers based on presentations made at the international conference on number theory held in honor of Hugh Williams' sixtieth birthday. The papers address topics in the areas of computational and explicit number theory and its applications. The material is suitable for graduate students and researchers interested in number theory.","comment":"I\u00a0think I just kept this because of the title. Tha's a good reason because it's a good title, but I can't remember how I found it. Maybe it contains something interesting.","date_added":"2012-04-12","date_published":"2004-08-06","urls":["http:\/\/books.google.com\/books?id=udr3tHHwBl0C&pgis=1"],"collections":"Attention-grabbing titles,Integerology","url":"http:\/\/books.google.com\/books?id=udr3tHHwBl0C&pgis=1","urldate":"2012-04-12","pages":"392","publisher":"American Mathematical Soc.","year":"2004"},{"key":"Guilbaud1963","type":"article","title":"Analyse alg\u00e9brique d'un scrutin","author":"Guilbaud, GT and Rosenstiehl, P","abstract":"","comment":"Found while looking at the permutohedron.","date_added":"2013-12-01","date_published":"1963-08-06","urls":["http:\/\/www.numdam.org\/article\/MSH_1963__4__9_0.pdf","http:\/\/www.ehess.fr\/revue-msh\/recherche.php?page=157&lignes=1272&action=recherchelignes=1272&action=recherche","http:\/\/archive.numdam.org\/ARCHIVE\/MSH\/MSH_1963__4_\/MSH_1963__4__9_0\/MSH_1963__4__9_0.pdf"],"collections":"","url":"http:\/\/www.numdam.org\/article\/MSH_1963__4__9_0.pdf http:\/\/www.ehess.fr\/revue-msh\/recherche.php?page=157&lignes=1272&action=recherchelignes=1272&action=recherche http:\/\/archive.numdam.org\/ARCHIVE\/MSH\/MSH_1963__4_\/MSH_1963__4__9_0\/MSH_1963__4__9_0.pdf","urldate":"2013-12-01","journal":"Math. Sci. Hum","pages":"9--33","volume":"4","year":"1963"},{"key":"item48","type":"online","title":"Magic squares of seventh powers","author":"Christian Boyer","abstract":"","comment":"Magic square whose sum is the first 52 digits of $\\pi$.","date_added":"2015-03-06","date_published":"2004-08-06","urls":["http:\/\/www.multimagie.com\/English\/SquaresOfSeventhPowers.htm"],"collections":"Puzzles","url":"http:\/\/www.multimagie.com\/English\/SquaresOfSeventhPowers.htm","urldate":"2015-03-06","year":"2004"},{"key":"PauliPascalPyramidsPauliFibonacciNumbersandPauliJacobsthalNumbers","type":"article","title":"Pauli Pascal Pyramids, Pauli Fibonacci Numbers, and Pauli Jacobsthal Numbers","author":"Martin Erik Horn","abstract":"The three anti-commutative two-dimensional Pauli Pascal triangles can be generalized into multi-dimensional Pauli Pascal hyperpyramids. Fibonacci and Jacobsthal numbers are then generalized into Pauli Fibonacci numbers, Pauli\r\nJacobsthal numbers, and Pauli Fibonacci numbers of higher order. And the question is: are Pauli rabbits killer rabbits?","comment":"","date_added":"2017-03-24","date_published":"2007-08-06","urls":["http:\/\/arxiv.org\/abs\/0711.4030v1","http:\/\/arxiv.org\/pdf\/0711.4030v1"],"collections":"Attention-grabbing titles,Fibonaccinalia","url":"http:\/\/arxiv.org\/abs\/0711.4030v1 http:\/\/arxiv.org\/pdf\/0711.4030v1","urldate":"2017-03-24","archivePrefix":"arXiv","eprint":"0711.4030","primaryClass":"math.GM","year":"2007"},{"key":"ApprovalVotinginProductSocieties","type":"article","title":"Approval Voting in Product Societies","author":"Kristen Mazur and Mutiara Sondjaja and Matthew Wright and Carolyn Yarnall","abstract":"In approval voting, individuals vote for all platforms that they find\r\nacceptable. In this situation it is natural to ask: When is agreement possible?\r\nWhat conditions guarantee that some fraction of the voters agree on even a\r\nsingle platform? Berg et. al. found such conditions when voters are asked to\r\nmake a decision on a single issue that can be represented on a linear spectrum.\r\nIn particular, they showed that if two out of every three voters agree on a\r\nplatform, there is a platform that is acceptable to a majority of the voters.\r\nHardin developed an analogous result when the issue can be represented on a\r\ncircular spectrum. We examine scenarios in which voters must make two decisions\r\nsimultaneously. For example, if voters must decide on the day of the week to\r\nhold a meeting and the length of the meeting, then the space of possible\r\noptions forms a cylindrical spectrum. Previous results do not apply to these\r\nmulti-dimensional voting societies because a voter's preference on one issue\r\noften impacts their preference on another. We present a general lower bound on\r\nagreement in a two-dimensional voting society, and then examine specific\r\nresults for societies whose spectra are cylinders and tori.","comment":"","date_added":"2017-03-30","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1703.09870v1","http:\/\/arxiv.org\/pdf\/1703.09870v1"],"collections":"Protocols and strategies","url":"http:\/\/arxiv.org\/abs\/1703.09870v1 http:\/\/arxiv.org\/pdf\/1703.09870v1","urldate":"2017-03-30","archivePrefix":"arXiv","eprint":"1703.09870","primaryClass":"math.CO","year":"2017"},{"key":"item8","type":"misc","title":"Packing circles and spheres on surfaces","author":"Alexander Schiftner and Mathias H\u00f6binger and Johannes Wallner and Helmut Pottmann","abstract":"Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces\u2019 incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate circle packings, hexagonal meshes which\r\ncarry a torsion-free support structure, hybrid tri-hex meshes, and others. We show how triangle meshes can be optimized so as to have the incircle packing property. We explain their relation to conformal geometry and implications on solvability of optimization. The examples we give confirm that this kind of meshes is a rich\r\nsource of geometric structures relevant to architectural geometry.","comment":"","date_added":"2011-09-04","date_published":"2009-08-06","urls":["http:\/\/dl.acm.org\/citation.cfm?id=1618485","http:\/\/www.geometrie.tugraz.at\/wallner\/packing.pdf"],"collections":"Basically computer science,Geometry","url":"http:\/\/dl.acm.org\/citation.cfm?id=1618485 http:\/\/www.geometrie.tugraz.at\/wallner\/packing.pdf","urldate":"2011-09-04","year":"2009"},{"key":"Tropicaltotallypositivematrices","type":"article","title":"Tropical totally positive matrices","author":"St\u00e9phane Gaubert and Adi Niv","abstract":"We investigate the tropical analogues of totally positive and totally\r\nnonnegative matrices. These arise when considering the images by the\r\nnonarchimedean valuation of the corresponding classes of matrices over a real\r\nnonarchimedean valued field, like the field of real Puiseux series. We show\r\nthat the nonarchimedean valuation sends the totally positive matrices precisely\r\nto the Monge matrices. This leads to explicit polyhedral representations of the\r\ntropical analogues of totally positive and totally nonnegative matrices. We\r\nalso show that tropical totally nonnegative matrices with a finite permanent\r\ncan be factorized in terms of elementary matrices. We finally determine the\r\neigenvalues of tropical totally nonnegative matrices, and relate them with the\r\neigenvalues of totally nonnegative matrices over nonarchimedean fields.","comment":"","date_added":"2017-05-02","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1606.00238v3","http:\/\/arxiv.org\/pdf\/1606.00238v3"],"collections":"Attention-grabbing titles,Unusual arithmetic","url":"http:\/\/arxiv.org\/abs\/1606.00238v3 http:\/\/arxiv.org\/pdf\/1606.00238v3","urldate":"2017-05-02","archivePrefix":"arXiv","eprint":"1606.00238","primaryClass":"math.AC","year":"2016"},{"key":"OnFibonacciQuaternions","type":"article","title":"On Fibonacci Quaternions","author":"Serpil Halici","abstract":"In this paper, we investigate the Fibonacci and Lucas quaternions. We give the generating functions and Binet formulas for these quaternions. Moreover, we derive some sums formulas for them.","comment":"","date_added":"2017-05-02","date_published":"2012-08-06","urls":["https:\/\/link.springer.com\/article\/10.1007\/s00006-011-0317-1"],"collections":"Fibonaccinalia","url":"https:\/\/link.springer.com\/article\/10.1007\/s00006-011-0317-1","urldate":"2017-05-02","doi":"10.1007\/s00006-011-0317-1","year":"2012","issue":"2","volume":"22"},{"key":"PVCPolyhedra","type":"article","title":"PVC Polyhedra","author":"David Glickenstein","abstract":"We describe how to construct a dodecahedron, tetrahedron, cube, and\r\noctahedron out of pvc pipes using standard fittings.","comment":"","date_added":"2017-05-02","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1705.00100v1","http:\/\/arxiv.org\/pdf\/1705.00100v1"],"collections":"Easily explained,Things to make and do,Geometry","url":"http:\/\/arxiv.org\/abs\/1705.00100v1 http:\/\/arxiv.org\/pdf\/1705.00100v1","urldate":"2017-05-02","archivePrefix":"arXiv","eprint":"1705.00100","primaryClass":"math.HO","year":"2017"},{"key":"Homotopytypetheorythelogicofspace","type":"article","title":"Homotopy type theory: the logic of space","author":"Michael Shulman","abstract":"This is an introduction to type theory, synthetic topology, and homotopy type\r\ntheory from a category-theoretic and topological point of view, written as a\r\nchapter for the book \"New Spaces for Mathematics and Physics\" (ed. Gabriel\r\nCatren and Mathieu Anel).","comment":"","date_added":"2017-05-03","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1703.03007v1","http:\/\/arxiv.org\/pdf\/1703.03007v1"],"collections":"","url":"http:\/\/arxiv.org\/abs\/1703.03007v1 http:\/\/arxiv.org\/pdf\/1703.03007v1","urldate":"2017-05-03","archivePrefix":"arXiv","eprint":"1703.03007","primaryClass":"math.CT","year":"2017"},{"key":"Theopaquesquare","type":"article","title":"The opaque square","author":"Adrian Dumitrescu and Minghui Jiang","abstract":"The problem of finding small sets that block every line passing through a\r\nunit square was first considered by Mazurkiewicz in 1916. We call such a set\r\n{\\em opaque} or a {\\em barrier} for the square. The shortest known barrier has\r\nlength $\\sqrt{2}+ \\frac{\\sqrt{6}}{2}= 2.6389\\ldots$. The current best lower\r\nbound for the length of a (not necessarily connected) barrier is $2$, as\r\nestablished by Jones about 50 years ago. No better lower bound is known even if\r\nthe barrier is restricted to lie in the square or in its close vicinity. Under\r\na suitable locality assumption, we replace this lower bound by $2+10^{-12}$,\r\nwhich represents the first, albeit small, step in a long time toward finding\r\nthe length of the shortest barrier. A sharper bound is obtained for interior\r\nbarriers: the length of any interior barrier for the unit square is at least $2\r\n+ 10^{-5}$. Two of the key elements in our proofs are: (i) formulas established\r\nby Sylvester for the measure of all lines that meet two disjoint planar convex\r\nbodies, and (ii) a procedure for detecting lines that are witness to the\r\ninvalidity of a short bogus barrier for the square.","comment":"","date_added":"2017-05-22","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1311.3323v1","http:\/\/arxiv.org\/pdf\/1311.3323v1"],"collections":"Easily explained,Geometry","url":"http:\/\/arxiv.org\/abs\/1311.3323v1 http:\/\/arxiv.org\/pdf\/1311.3323v1","urldate":"2017-05-22","archivePrefix":"arXiv","eprint":"1311.3323","primaryClass":"math.CO","year":"2013"},{"key":"CuckooFilterSimplificationandAnalysis","type":"article","title":"Cuckoo Filter: Simplification and Analysis","author":"David Eppstein","abstract":"The cuckoo filter data structure of Fan, Andersen, Kaminsky, and Mitzenmacher\r\n(CoNEXT 2014) performs the same approximate set operations as a Bloom filter in\r\nless memory, with better locality of reference, and adds the ability to delete\r\nelements as well as to insert them. However, until now it has lacked\r\ntheoretical guarantees on its performance. We describe a simplified version of\r\nthe cuckoo filter using fewer hash function calls per query. With this\r\nsimplification, we provide the first theoretical performance guarantees on\r\ncuckoo filters, showing that they succeed with high probability whenever their\r\nfingerprint length is large enough.","comment":"","date_added":"2017-05-23","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1604.06067v1","http:\/\/arxiv.org\/pdf\/1604.06067v1"],"collections":"Animals,Basically computer science","url":"http:\/\/arxiv.org\/abs\/1604.06067v1 http:\/\/arxiv.org\/pdf\/1604.06067v1","urldate":"2017-05-23","archivePrefix":"arXiv","eprint":"1604.06067","primaryClass":"cs.DS","year":"2016"},{"key":"Gunji2011","type":"article","title":"Robust Soldier Crab Ball Gate","author":"Gunji, YP and Nishiyama, Y","abstract":"Based on the field observation of soldier crabs, we previously proposed a model for a swarm of soldier crabs. Here, we describe the interaction of coherent swarms in the simulation model, which is implemented in a logical gate. Because a swarm is generated by inherent perturbation, a swarm can be generated and maintained under highly perturbed conditions. Thus, the model reveals a robust logical gate rather than stable one. In addition, we show that the logical gate of swarms is also implemented by real soldier crabs (Mictyris guinotae).","comment":"","date_added":"2012-04-15","date_published":"2011-08-06","urls":["https:\/\/arxiv.org\/abs\/1204.1749","https:\/\/arxiv.org\/pdf\/1204.1749"],"collections":"Animals,Attention-grabbing titles,Basically computer science,Easily explained,Unusual computers","url":"https:\/\/arxiv.org\/abs\/1204.1749 https:\/\/arxiv.org\/pdf\/1204.1749","urldate":"2012-04-15","journal":"AIP Conference Proceedings","year":"2011"},{"key":"Arithmeticalstructuresongraphs","type":"article","title":"Arithmetical structures on graphs","author":"Hugo Corrales and Carlos E. Valencia","abstract":"Arithmetical structures on a graph were introduced by Lorenzini as some\r\nintersection matrices that arise in the study of degenerating curves in\r\nalgebraic geometry. In this article we study these arithmetical structures, in\r\nparticular we are interested in the arithmetical structures on complete graphs,\r\npaths, and cycles. We begin by looking at the arithmetical structures on a\r\nmultidigraph from the general perspective of $M$-matrices. As an application,\r\nwe recover the result of Lorenzini about the finiteness of the number of\r\narithmetical structures on a graph. We give a description on the arithmetical\r\nstructures on the graph obtained by merging and splitting a vertex of a graph\r\nin terms of its arithmetical structures. On the other hand, we give a\r\ndescription of the arithmetical structures on the clique--star transform of a\r\ngraph, which generalizes the subdivision of a graph. As an application of this\r\nresult we obtain an explicit description of all the arithmetical structures on\r\nthe paths and cycles and we show that the number of the arithmetical structures\r\non a path is a Catalan number.","comment":"","date_added":"2017-06-14","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1604.02502v3","http:\/\/arxiv.org\/pdf\/1604.02502v3"],"collections":"Unusual arithmetic","url":"http:\/\/arxiv.org\/abs\/1604.02502v3 http:\/\/arxiv.org\/pdf\/1604.02502v3","urldate":"2017-06-14","archivePrefix":"arXiv","eprint":"1604.02502","primaryClass":"math.CO","year":"2016"},{"key":"Randomlyjugglingbackwards","type":"article","title":"Randomly juggling backwards","author":"Allen Knutson","abstract":"We recall the directed graph of _juggling states_, closed walks within which\r\ngive juggling patterns, as studied by Ron Graham in [w\/Chung, w\/Butler].\r\nVarious random walks in this graph have been studied before by several authors,\r\nand their equilibrium distributions computed. We motivate a random walk on the\r\nreverse graph (and an enrichment thereof) from a very classical linear algebra\r\nproblem, leading to a particularly simple equilibrium: a Boltzmann distribution\r\nclosely related to the Poincar\\'e series of the b-Grassmannian in\r\ninfinite-dimensional space.\r\n We determine the most likely asymptotic state in the limit of many balls,\r\nwhere in the limit the probability of a 0-throw is kept fixed.","comment":"","date_added":"2017-06-14","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1601.06391v1","http:\/\/arxiv.org\/pdf\/1601.06391v1"],"collections":"Attention-grabbing titles,Probability and statistics","url":"http:\/\/arxiv.org\/abs\/1601.06391v1 http:\/\/arxiv.org\/pdf\/1601.06391v1","urldate":"2017-06-14","archivePrefix":"arXiv","eprint":"1601.06391","primaryClass":"math.CO","year":"2016"},{"key":"AformulagoestocourtPartisangerrymanderingandtheefficiencygap","type":"article","title":"A formula goes to court: Partisan gerrymandering and the efficiency gap","author":"Mira Bernstein and Moon Duchin","abstract":"Recently, a proposal has been advanced to detect unconstitutional partisan\r\ngerrymandering with a simple formula called the efficiency gap. The efficiency\r\ngap is now working its way towards a possible landmark case in the Supreme\r\nCourt. This note explores some of its mathematical properties in light of the\r\nfact that it reduces to a straight proportional comparison of votes to seats.\r\nThough we offer several critiques, we assess that EG can still be a useful\r\ncomponent of a courtroom analysis. But a famous formula can take on a life of\r\nits own and this one will need to be watched closely.","comment":"","date_added":"2017-06-16","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1705.10812v1","http:\/\/arxiv.org\/pdf\/1705.10812v1"],"collections":"Protocols and strategies,Geometry","url":"http:\/\/arxiv.org\/abs\/1705.10812v1 http:\/\/arxiv.org\/pdf\/1705.10812v1","urldate":"2017-06-16","archivePrefix":"arXiv","eprint":"1705.10812","primaryClass":"physics.soc-ph","year":"2017"},{"key":"Everypositiveintegerisasumofthreepalindromes","type":"article","title":"Every positive integer is a sum of three palindromes","author":"Javier Cilleruelo and Florian Luca and Lewis Baxter","abstract":"For integer $g\\ge 5$, we prove that any positive integer can be written as a\r\nsum of three palindromes in base $g$.","comment":"","date_added":"2017-06-21","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1602.06208v2","http:\/\/arxiv.org\/pdf\/1602.06208v2"],"collections":"Easily explained,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1602.06208v2 http:\/\/arxiv.org\/pdf\/1602.06208v2","urldate":"2017-06-21","archivePrefix":"arXiv","eprint":"1602.06208","primaryClass":"math.NT","year":"2016"},{"key":"OnthedateofCauchyscontributionstothefoundingofthetheoryofgroups","type":"article","title":"On the date of Cauchy's contributions to the founding of the theory of groups","author":"Peter M. Neumann","abstract":"Evidence from published sources is used to show that Cauchy's group-theoretical work was all produced in a few months of intense activity starting in September 1845.","comment":"","date_added":"2017-06-21","date_published":"1989-08-06","urls":["https:\/\/www.cambridge.org\/core\/journals\/bulletin-of-the-australian-mathematical-society\/article\/on-the-date-of-cauchys-contributions-to-the-founding-of-the-theory-of-groups\/6801D1A609AA974ACB44B90C81533507"],"collections":"History,Drama!","url":"https:\/\/www.cambridge.org\/core\/journals\/bulletin-of-the-australian-mathematical-society\/article\/on-the-date-of-cauchys-contributions-to-the-founding-of-the-theory-of-groups\/6801D1A609AA974ACB44B90C81533507","urldate":"2017-06-21","doi":"10.1017\/S000497270000438X","year":"1989"},{"key":"ArrangementsOfStarsOnTheAmericanFlag","type":"article","title":"Arrangements of Stars on the American Flag","author":"Dimitris Koukoulopoulos and Johann Thiel","abstract":"In this article, we examine the existence of nice arrangements of stars on the American flag. We show that despite the existence of such arrangements for any number of stars from 1 to 100, with the exception of 29, 69 and 87, they are rare as the number of stars increases.","comment":"","date_added":"2017-06-12","date_published":"2012-08-06","urls":["http:\/\/www.maa.org\/sites\/default\/files\/pdf\/pubs\/monthly_june12-stars.pdf"],"collections":"Art,Easily explained,Geometry","url":"http:\/\/www.maa.org\/sites\/default\/files\/pdf\/pubs\/monthly_june12-stars.pdf","urldate":"2017-06-12","year":"2012"},{"key":"Polylogarithmicladdershypergeometricseriesandthetenmillionthdigitsof3and5","type":"article","title":"Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $\u03b6(3)$ and $\u03b6(5)$","author":"D. J. Broadhurst","abstract":"We develop ladders that reduce $\\zeta(n):=\\sum_{k>0}k^{-n}$, for\r\n$n=3,5,7,9,11$, and $\\beta(n):=\\sum_{k\\ge0}(-1)^k(2k+1)^{-n}$, for $n=2,4,6$,\r\nto convergent polylogarithms and products of powers of $\\pi$ and $\\log2$. Rapid\r\ncomputability results because the required arguments of ${\\rm\r\nLi}_n(z)=\\sum_{k>0}z^k\/k^n$ satisfy $z^8=1\/16^p$, with $p=1,3,5$. We prove that\r\n$G:=\\beta(2)$, $\\pi^3$, $\\log^32$, $\\zeta(3)$, $\\pi^4$, $\\log^42$, $\\log^52$,\r\n$\\zeta(5)$, and six products of powers of $\\pi$ and $\\log2$ are constants whose\r\n$d$th hexadecimal digit can be computed in time~$=O(d\\log^3d)$ and\r\nspace~$=O(\\log d)$, as was shown for $\\pi$, $\\log2$, $\\pi^2$ and $\\log^22$ by\r\nBailey, Borwein and Plouffe. The proof of the result for $\\zeta(5)$ entails\r\ndetailed analysis of hypergeometric series that yield Euler sums, previously\r\nstudied in quantum field theory. The other 13 results follow more easily from\r\nKummer's functional identities. We compute digits of $\\zeta(3)$ and $\\zeta(5)$,\r\nstarting at the ten millionth hexadecimal place. These constants result from\r\ncalculations of massless Feynman diagrams in quantum chromodynamics. In a\r\nrelated paper, hep-th\/9803091, we show that massive diagrams also entail\r\nconstants whose base of super-fast computation is $b=3$.","comment":"","date_added":"2017-06-29","date_published":"1998-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/9803067v1","http:\/\/arxiv.org\/pdf\/math\/9803067v1"],"collections":"Attention-grabbing titles","url":"http:\/\/arxiv.org\/abs\/math\/9803067v1 http:\/\/arxiv.org\/pdf\/math\/9803067v1","urldate":"2017-06-29","archivePrefix":"arXiv","eprint":"math\/9803067","primaryClass":"math.CA","year":"1998"},{"key":"SteinhausLongimeter","type":"online","title":"Steinhaus Longimeter","author":"Chris Staecker","abstract":"The longimeter, invented by Hugo Steinhaus, is a device for measuring the length of a curve drawn on paper.\r\n\r\nIt's a strange grid on transparency that is laid over the curve. The grid is constructed so that the number of times the curve crosses the grid is the length of the curve in millimeters.","comment":"","date_added":"2017-07-02","date_published":"2017-08-06","urls":["http:\/\/cstaecker.fairfield.edu\/~cstaecker\/machines\/longimeter.html","http:\/\/cstaecker.fairfield.edu\/~cstaecker\/machines\/longimeter%20mm.pdf","https:\/\/www.youtube.com\/watch?v=-ZE2Iv-8tsA"],"collections":"Easily explained,Things to make and do","url":"http:\/\/cstaecker.fairfield.edu\/~cstaecker\/machines\/longimeter.html http:\/\/cstaecker.fairfield.edu\/~cstaecker\/machines\/longimeter%20mm.pdf https:\/\/www.youtube.com\/watch?v=-ZE2Iv-8tsA","urldate":"2017-07-02","year":"2017"},{"key":"UnunfoldablePolyhedrawithConvexFaces","type":"article","title":"Ununfoldable Polyhedra with Convex Faces","author":"Marshall Bern and Erik D. Demaine and David Eppstein and Eric Kuo and Andrea Mantler and Jack Snoeyink","abstract":"Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that \u201copen\u201d polyhedra with triangular faces may not be unfoldable no matter how they are cut.","comment":"","date_added":"2017-06-26","date_published":"2003-08-06","urls":["http:\/\/erikdemaine.org\/papers\/Ununfoldable\/"],"collections":"Things to make and do","url":"http:\/\/erikdemaine.org\/papers\/Ununfoldable\/","urldate":"2017-06-26","year":"2003"},{"key":"TheternarycalculatingmachineofThomasFowler","type":"online","title":"The ternary calculating machine of Thomas Fowler","author":"Mark Glusker","abstract":"A large, wooden calculating machine was built in 1840 by Thomas Fowler in his workshop in Great Torrington, Devon, England. In what may have been one of the first uses of lower bases for computing machinery, Fowler chose balanced ternary to represent the numbers in his machine. Very little evidence of this machine has survived.","comment":"","date_added":"2017-06-22","date_published":"1997-08-06","urls":["http:\/\/www.mortati.com\/glusker\/fowler\/index.htm"],"collections":"Basically computer science,History,Unusual arithmetic,Unusual computers,Integerology","url":"http:\/\/www.mortati.com\/glusker\/fowler\/index.htm","urldate":"2017-06-22","year":"1997"},{"key":"SetunSimulator","type":"software","title":"\u0421\u0435\u0442\u0443\u043d\u044c \u0412\u0421 (Setun Web Simulator)","author":"Trinarygroup","abstract":"","comment":"A simulator of the balanced-ternary Setun computer. ","date_added":"2017-06-22","date_published":"2017-08-06","urls":["http:\/\/trinary.ru\/projects\/setunws\/"],"collections":"Basically computer science,History,Unusual computers","url":"http:\/\/trinary.ru\/projects\/setunws\/","urldate":"2017-06-22","year":"2017"},{"key":"Friedman1998","type":"article","title":"Long finite sequences","author":"Friedman, Harvey M","abstract":"Let k be a positive integer. There is a longest finite sequence x1,...,xn in k letters in which no consecutive block xi,...,x2i is a subsequence of any other consecutive block xj,...,x2j. Let n(k) be this longest length. We prove that n(1) = 3, n(2) = 11, and n(3) is incomprehensibly large. We give a lower bound for n(3) in terms of the familiar Ackerman hierarchy. We also give asymptotic upper and lower bounds for n(k). We view n(3) as a particularly elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for n(3).","comment":"","date_added":"2012-04-24","date_published":"1998-08-06","urls":["http:\/\/u.osu.edu\/friedman.8\/files\/2014\/01\/LongFinSeq98-2f0wmq3.pdf"],"collections":"","url":"http:\/\/u.osu.edu\/friedman.8\/files\/2014\/01\/LongFinSeq98-2f0wmq3.pdf","urldate":"2012-04-24","pages":"1--50","year":"1998"},{"key":"item26","type":"online","title":"Earliest Uses of Symbols of Calculus","author":"Jeff Miller","abstract":"","comment":"","date_added":"2012-10-09","date_published":"2008-08-06","urls":["http:\/\/jeff560.tripod.com\/calculus.html"],"collections":"Easily explained,History,Lists and catalogues,Notation and conventions","url":"http:\/\/jeff560.tripod.com\/calculus.html","urldate":"2012-10-09","year":"2008"},{"key":"OvalsandEggCurves","type":"article","title":"Ovals and Egg Curves","author":"J\u00fcrgen K\u00f6ller ","abstract":"","comment":"","date_added":"2017-04-03","date_published":"2000-08-06","urls":["http:\/\/www.mathematische-basteleien.de\/eggcurves.htm"],"collections":"Art,Things to make and do,Food,Geometry","url":"http:\/\/www.mathematische-basteleien.de\/eggcurves.htm","urldate":"2017-04-03","year":"2000"},{"key":"CONRAD","type":"article","title":"Proofs by Descent","author":"Keith Conrad","abstract":"","comment":"","date_added":"2013-11-04","date_published":"2010-08-06","urls":["http:\/\/www.math.uconn.edu\/~kconrad\/blurbs\/ugradnumthy\/descent.pdf"],"collections":"About proof","url":"http:\/\/www.math.uconn.edu\/~kconrad\/blurbs\/ugradnumthy\/descent.pdf","urldate":"2013-11-04","year":"2010"},{"key":"item23","type":"article","title":"Modi\ufb01ed Pascal Triangle and Pascal Surfaces","author":"Rely Pellicer and David Alvo","abstract":"","comment":"","date_added":"2012-09-05","date_published":"2012-08-06","urls":["http:\/\/uai.academia.edu\/DavidAlvo\/Papers\/995009\/Modi_ed_Pascal_Triangle_and_Pascal_Surfaces"],"collections":"","url":"http:\/\/uai.academia.edu\/DavidAlvo\/Papers\/995009\/Modi_ed_Pascal_Triangle_and_Pascal_Surfaces","urldate":"2012-09-05","year":"2012"},{"key":"item28","type":"online","title":"What are some of the most ridiculous proofs in mathematics?","author":"Anonymous","abstract":"","comment":"","date_added":"2013-02-17","date_published":"2012-08-06","urls":["http:\/\/www.quora.com\/Mathematics\/What-are-some-of-the-most-ridiculous-proofs-in-mathematics"],"collections":"The act of doing maths,About proof","url":"http:\/\/www.quora.com\/Mathematics\/What-are-some-of-the-most-ridiculous-proofs-in-mathematics","urldate":"2013-02-17","year":"2012"},{"key":"item27","type":"online","title":"Figures for \"Impossible fractals\"","author":"Cameron Browne","abstract":"","comment":"","date_added":"2012-12-18","date_published":"2008-08-06","urls":["http:\/\/www.cameronius.com\/graphics\/impossible-fractals-figures\/"],"collections":"Art,Easily explained,Geometry","url":"http:\/\/www.cameronius.com\/graphics\/impossible-fractals-figures\/","urldate":"2012-12-18","year":"2008"},{"key":"CSCheatSheet","type":"article","title":"Theoretical Computer Science Cheat Sheet","author":"Steve Seiden","abstract":"","comment":"","date_added":"2011-03-26","date_published":"2003-08-06","urls":["http:\/\/www.tug.org\/texshowcase\/cheat.pdf"],"collections":"Basically computer science,Lists and catalogues","url":"http:\/\/www.tug.org\/texshowcase\/cheat.pdf","urldate":"2011-03-26","journal":"TeX showcase","year":"2003"},{"key":"Allis1988","type":"article","title":"A knowledge-based approach of connect-four","author":"Allis, Victor","abstract":"A Shannon C-type strategy program, VICTOR, is written for Connect-Four, based on nine strategic rules. Each of these rules is proven to be correct, implying that conclusions made by VICTOR are correct. Using VICTOR, strategic rules where found which can be used by Black to at least draw the game, on each 7 \u00d7 (2n) board, provided that White does not start at the middle column, as well as on any 6 \u00d7 (2n) board. In combination with conspiracy-number search, search tables and depth-first search, VICTOR was able to show that White can win on the standard 7 \u00d7 6 board. Using a database of approximately half a million positions, VICTOR can play real time against opponents on the 7 \u00d7 6 board, always winning with White.","comment":"","date_added":"2013-12-03","date_published":"1988-08-06","urls":["http:\/\/www.informatik.uni-trier.de\/~fernau\/DSL0607\/Masterthesis-Viergewinnt.pdf"],"collections":"Games to play with friends","url":"http:\/\/www.informatik.uni-trier.de\/~fernau\/DSL0607\/Masterthesis-Viergewinnt.pdf","urldate":"2013-12-03","year":"1988"},{"key":"Computationalcomplexityand3manifoldsandzombies","type":"article","title":"Computational complexity and 3-manifolds and zombies","author":"Greg Kuperberg and Eric Samperton","abstract":"We show the problem of counting homomorphisms from the fundamental group of a\r\nhomology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is\r\n#P-complete, in the case that $G$ is fixed and $M$ is the computational input.\r\nSimilarly, deciding if there is a non-trivial homomorphism is NP-complete. In\r\nboth reductions, we can guarantee that every non-trivial homomorphism is a\r\nsurjection. As a corollary, for any fixed integer $m \\ge 5$, it is NP-complete\r\nto decide whether $M$ admits a connected $m$-sheeted covering.\r\n Our construction is inspired by universality results in topological quantum\r\ncomputation. Given a classical reversible circuit $C$, we construct $M$ so that\r\nevaluations of $C$ with certain initialization and finalization conditions\r\ncorrespond to homomorphisms $\\pi_1(M) \\to G$. An intermediate state of $C$\r\nlikewise corresponds to a homomorphism $\\pi_1(\\Sigma_g) \\to G$, where\r\n$\\Sigma_g$ is a pointed Heegaard surface of $M$ of genus $g$. We analyze the\r\naction on these homomorphisms by the pointed mapping class group\r\n$\\text{MCG}_*(\\Sigma_g)$ and its Torelli subgroup $\\text{Tor}_*(\\Sigma_g)$. By\r\nresults of Dunfield-Thurston, the action of $\\text{MCG}_*(\\Sigma_g)$ is as\r\nlarge as possible when $g$ is sufficiently large; we can pass to the Torelli\r\ngroup using the congruence subgroup property of $\\text{Sp}(2g,\\mathbb{Z})$. Our\r\nresults can be interpreted as a sharp classical universality property of an\r\nassociated combinatorial $(2+1)$-dimensional TQFT.","comment":"","date_added":"2017-07-13","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1707.03811v1","http:\/\/arxiv.org\/pdf\/1707.03811v1"],"collections":"Attention-grabbing titles,Basically computer science","url":"http:\/\/arxiv.org\/abs\/1707.03811v1 http:\/\/arxiv.org\/pdf\/1707.03811v1","urldate":"2017-07-13","archivePrefix":"arXiv","eprint":"1707.03811","primaryClass":"math.GT","year":"2017"},{"key":"Overcurvature","type":"article","title":"Overcurvature describes the buckling and folding of rings from curved origami to foldable tents","author":"Pierre-Olivier Mouthuy and Michael Coulombier and Thomas Pardoen and Jean-Pierre Raskin and Alain M. Jonas","abstract":"Daily-life foldable items, such as popup tents, the curved origami sculptures exhibited in the Museum of Modern Art of New York, overstrained bicycle wheels, released bilayered microrings and strained cyclic macromolecules, are made of rings buckled or folded in tridimensional saddle shapes. Surprisingly, despite their popularity and their technological and artistic importance, the design of such rings remains essentially empirical. Here we study experimentally the tridimensional buckling of rings on folded paper rings, lithographically processed foldable microrings, human-size wood sculptures or closed arcs of Slinky springs. The general shape adopted by these rings can be described by a single continuous parameter, the overcurvature. An analytical model based on the minimization of the energy of overcurved rings reproduces quantitatively their shape and buckling behaviour. The model also provides guidelines on how to efficiently fold rings for the design of space-saving objects.","comment":"","date_added":"2017-07-20","date_published":"2012-08-06","urls":["https:\/\/www.nature.com\/articles\/ncomms2311","https:\/\/www.nature.com\/articles\/ncomms2311.pdf"],"collections":"Art,Basically physics,Easily explained,Things to make and do,Geometry,Modelling","url":"https:\/\/www.nature.com\/articles\/ncomms2311 https:\/\/www.nature.com\/articles\/ncomms2311.pdf","urldate":"2017-07-20","year":"2012","journal":"Nature"},{"key":"item43","type":"article","title":"The Nesting and Roosting Habits of The Laddered Parenthesis","author":"R. K. Guy and J. L. Selfridge","abstract":"","comment":"","date_added":"2016-09-23","date_published":"1973-08-06","urls":["http:\/\/www.jstor.org\/stable\/2319392","http:\/\/oeis.org\/A003018\/a003018.pdf"],"collections":"Attention-grabbing titles,Easily explained,Combinatorics","url":"http:\/\/www.jstor.org\/stable\/2319392 http:\/\/oeis.org\/A003018\/a003018.pdf","urldate":"2016-09-23","year":"1973"},{"key":"AvianeggshapeFormfunctionandevolution","type":"article","title":"Avian egg shape: Form, function, and evolution","author":"Mary Caswell Stoddard and Ee Hou Yong and Derya Akkaynak and Catherine Sheard and Joseph A. Tobias and L. Mahadevan","abstract":"Avian egg shape is generally explained as an adaptation to life history, yet we currently lack a global synthesis of how egg-shape differences arise and evolve. Here, we apply morphometric, mechanistic, and macroevolutionary analyses to the egg shapes of 1400 bird species. We characterize egg-shape diversity in terms of two biologically relevant variables, asymmetry and ellipticity, allowing us to quantify the observed morphologies in a two-dimensional morphospace. We then propose a simple mechanical model that explains the observed egg-shape diversity based on geometric and material properties of the egg membrane. Finally, using phylogenetic models, we show that egg shape correlates with flight ability on broad taxonomic scales, suggesting that adaptations for flight may have been critical drivers of egg-shape variation in birds.","comment":"","date_added":"2017-07-24","date_published":"2017-08-06","urls":["http:\/\/science.sciencemag.org\/content\/356\/6344\/1249","http:\/\/science.sciencemag.org\/content\/356\/6344\/1249.full.pdf"],"collections":"Animals,Food,Geometry,Modelling","url":"http:\/\/science.sciencemag.org\/content\/356\/6344\/1249 http:\/\/science.sciencemag.org\/content\/356\/6344\/1249.full.pdf","urldate":"2017-07-24","year":"2017","doi":"10.1126\/science.aaj1945"},{"key":"Linusson1998","type":"article","title":"A Smaller Sleeping Bag For A Baby Snake","author":"Linusson, Svante and ASTLUND, JW","abstract":"","comment":"","date_added":"2013-03-20","date_published":"1998-08-06","urls":["http:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.31.7066","http:\/\/www.math.chalmers.se\/~wastlund\/sleepingBag.pdf"],"collections":"Animals,Attention-grabbing titles,Easily explained,Puzzles,Geometry","url":"http:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.31.7066 http:\/\/www.math.chalmers.se\/~wastlund\/sleepingBag.pdf","urldate":"2013-03-20","keywords":"blanket,sleeping bag,snake,worm","pages":"1--5","year":"1998","doi":"10.1.1.31.7066"},{"key":"AnExplicitIsometricReductionoftheUnitSphereintoanArbitrarilySmallBall","type":"article","title":"An Explicit Isometric Reduction of the Unit Sphere into an Arbitrarily Small Ball","author":"Evangelis Bartzos and Vincent Borrelli and Roland Denis and Francis Lazarus and Damien Rohmer and Boris Thibert","abstract":"Spheres are known to be rigid geometric objects: they cannot be deformed isometrically, i.e., while preserving the length of curves, in a twice differentiable way. An unexpected result by Nash (Ann Math 60:383\u2013396, 1954) and Kuiper (Indag Math 17:545\u2013555, 1955) shows that this is no longer the case if one requires the deformations to be only continuously differentiable. A remarkable consequence of their result makes possible the isometric reduction of a unit sphere inside an arbitrarily small ball. In particular, if one views the Earth as a round sphere, the theory allows to reduce its diameter to that of a terrestrial globe while preserving geodesic distances. Here, we describe the first explicit construction and visualization of such a reduced sphere. The construction amounts to solve a nonlinear PDE with boundary conditions. The resulting surface consists of two unit spherical caps joined by a C1 fractal equatorial belt. An intriguing question then arises about the transition between the smooth and the C1 fractal geometries. We show that this transition is similar to the one observed when connecting a Koch curve to a line segment.","comment":"","date_added":"2017-07-25","date_published":"2017-08-06","urls":["https:\/\/link.springer.com\/article\/10.1007\/s10208-017-9360-1","http:\/\/math.univ-lyon1.fr\/homes-www\/borrelli\/Articles\/Focm2017.pdf"],"collections":"Geometry","url":"https:\/\/link.springer.com\/article\/10.1007\/s10208-017-9360-1 http:\/\/math.univ-lyon1.fr\/homes-www\/borrelli\/Articles\/Focm2017.pdf","urldate":"2017-07-25","year":"2017","doi":"10.1007\/s10208-017-9360-1"},{"key":"Doyle2006","type":"article","title":"Division by three","author":"Doyle, Peter G. and Conway, John Horton","abstract":"We prove without appeal to the Axiom of Choice that for any sets A and B, if there is a one-to-one correspondence between 3 cross A and 3 cross B then there is a one-to-one correspondence between A and B. The first such proof, due to Lindenbaum, was announced by Lindenbaum and Tarski in 1926, and subsequently `lost'; Tarski published an alternative proof in 1949. We argue that the proof presented here follows Lindenbaum's original.","comment":"","date_added":"2014-11-17","date_published":"2006-05-01","urls":["https:\/\/arxiv.org\/abs\/math\/0605779v1","https:\/\/arxiv.org\/pdf\/math\/0605779v1.pdf"],"collections":"Fun maths facts","url":"https:\/\/arxiv.org\/abs\/math\/0605779v1 https:\/\/arxiv.org\/pdf\/math\/0605779v1.pdf","urldate":"2014-11-17","month":"may","year":"2006"},{"key":"ProofofConwaysLostCosmologicalTheorem","type":"article","title":"Proof of Conway's Lost Cosmological Theorem","author":"Shalosh B. Ekhad and Doron Zeilberger","abstract":"John Horton Conway's Cosmological Theorem, about Audioactive sequences, for\r\nwhich no extant proof existed, is given a computer-generated proof, hopefully\r\nfor good.","comment":"","date_added":"2017-07-26","date_published":"1998-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/9808077v1","http:\/\/arxiv.org\/pdf\/math\/9808077v1"],"collections":"Attention-grabbing titles,Basically computer science,Easily explained,The act of doing maths,About proof","url":"http:\/\/arxiv.org\/abs\/math\/9808077v1 http:\/\/arxiv.org\/pdf\/math\/9808077v1","urldate":"2017-07-26","archivePrefix":"arXiv","eprint":"math\/9808077","primaryClass":"math.CO","year":"1998"},{"key":"TheCurlingNumberConjecture","type":"article","title":"The Curling Number Conjecture","author":"Benjamin Chaffin and N. J. A. Sloane","abstract":"Given a finite nonempty sequence of integers S, by grouping adjacent terms it\r\nis always possible to write it, possibly in many ways, as S = X Y^k, where X\r\nand Y are sequences and Y is nonempty. Choose the version which maximizes the\r\nvalue of k: this k is the curling number of S. The Curling Number Conjecture is\r\nthat if one starts with any initial sequence S, and extends it by repeatedly\r\nappending the curling number of the current sequence, the sequence will\r\neventually reach 1. The conjecture remains open, but we will report on some\r\nnumerical results and conjectures in the case when S consists of only 2's and\r\n3's.","comment":"","date_added":"2017-07-31","date_published":"2009-08-06","urls":["http:\/\/arxiv.org\/abs\/0912.2382v5","http:\/\/arxiv.org\/pdf\/0912.2382v5"],"collections":"Easily explained,Integerology","url":"http:\/\/arxiv.org\/abs\/0912.2382v5 http:\/\/arxiv.org\/pdf\/0912.2382v5","urldate":"2017-07-31","archivePrefix":"arXiv","eprint":"0912.2382","primaryClass":"math.CO","year":"2009"},{"key":"MaximumgenusofthegeneralizedJengagame","type":"article","title":"Maximum genus of the generalized Jenga game","author":"Rika Akiyama and Nozomi Abe and Hajime Fujita and Yukie Inaba and Mari Hataoka and Shiori Ito and Satomi Seita","abstract":"We treat the boundary of the union of blocks in the Jenga game as a surface\r\nwith a polyhedral structure and consider its genus. We generalize the game and\r\ndetermine the maximum genus of the generalized game.","comment":"","date_added":"2017-08-07","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1708.01503v1","http:\/\/arxiv.org\/pdf\/1708.01503v1"],"collections":"Attention-grabbing titles,Games to play with friends,Things to make and do","url":"http:\/\/arxiv.org\/abs\/1708.01503v1 http:\/\/arxiv.org\/pdf\/1708.01503v1","urldate":"2017-08-07","archivePrefix":"arXiv","eprint":"1708.01503","primaryClass":"math.HO","year":"2017"},{"key":"TheBulgariansolitaireandthemathematicsaroundit","type":"article","title":"The Bulgarian solitaire and the mathematics around it","author":"Vesselin Drensky","abstract":"The Bulgarian solitaire is a mathematical card game played by one person. A\r\npack of \\(n\\) cards is divided into several decks (or \"piles\"). Each move consists\r\nof the removing of one card from each deck and collecting the removed cards to\r\nform a new deck. The game ends when the same position occurs twice. It has\r\nturned out that when \\(n=k(k+1)\/2\\) is a triangular number, the game reaches the\r\nsame stable configuration with size of the piles \\(1,2,\\ldots,k\\). The purpose of the\r\npaper is to tell the (quite amusing) story of the game and to discuss\r\nmathematical problems related with the Bulgarian solitaire.\r\n\r\nThe paper is dedicated to the memory of Borislav Bojanov (1944-2009), a great\r\nmathematician, person, and friend, and one of the main protagonists in the\r\nstory of the Bulgarian solitaire.","comment":"","date_added":"2017-08-10","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1503.00885v1","http:\/\/arxiv.org\/pdf\/1503.00885v1"],"collections":"Easily explained,Games to play with friends,History,Puzzles","url":"http:\/\/arxiv.org\/abs\/1503.00885v1 http:\/\/arxiv.org\/pdf\/1503.00885v1","urldate":"2017-08-10","archivePrefix":"arXiv","eprint":"1503.00885","primaryClass":"math.CO","year":"2015"},{"key":"Frustrationsolitaire","type":"article","title":"Frustration solitaire","author":"Peter G. Doyle and Charles M. Grinstead and J. Laurie Snell","abstract":"In this expository article, we discuss the rank-derangement problem, which\r\nasks for the number of permutations of a deck of cards such that each card is\r\nreplaced by a card of a different rank. This combinatorial problem arises in\r\ncomputing the probability of winning the game of `frustration solitaire'. The\r\nsolution is a prime example of the method of inclusion and exclusion. We also\r\ndiscuss and announce the solution to Montmort's `Probleme du Treize', a related\r\nproblem dating back to circa 1708.","comment":"","date_added":"2017-08-10","date_published":"2007-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/0703900v2","http:\/\/arxiv.org\/pdf\/math\/0703900v2"],"collections":"Easily explained,Games to play with friends,Probability and statistics,Puzzles","url":"http:\/\/arxiv.org\/abs\/math\/0703900v2 http:\/\/arxiv.org\/pdf\/math\/0703900v2","urldate":"2017-08-10","archivePrefix":"arXiv","eprint":"math\/0703900","primaryClass":"math.PR","year":"2007"},{"key":"OfficiallyHomePlateDoesntExist","type":"online","title":"Officially, Home Plate doesn\u2019t exist.","author":"Bill Gosper","abstract":"The official Major League and Little League rule books require the two \u201cslanty\u201d sides to be 12\u201d long and meet at a right angle at the rear corner toward the catcher. This is where the foul lines meet. The left and right sides of Home Plate must poke into fair territory by half the width of the plate, which is 8\u00bd\u201d (17\u201d divided by 2).\r\n\r\nThere is no such shape!","comment":"","date_added":"2017-08-16","date_published":"2017-08-06","urls":["http:\/\/gosper.org\/homeplate.html"],"collections":"Easily explained,Geometry","url":"http:\/\/gosper.org\/homeplate.html","urldate":"2017-08-16","year":"2017"},{"key":"Mosher2013","type":"article","title":"Lone Axes in Outer Space","author":"Mosher, Lee and Pfaff, Catherine","abstract":"Handel and Mosher define the axis bundle for a fully irreducible outer\r\nautomorphism in \"Axes in Outer Space.\" In this paper we give a necessary and\r\nsufficient condition for the axis bundle to consist of a unique periodic fold\r\nline. As a consequence, we give a setting, and means for identifying in this\r\nsetting, when two elements of an outer automorphism group $Out(F_r)$ have\r\nconjugate powers.","comment":"","date_added":"2013-11-29","date_published":"2013-11-01","urls":["http:\/\/arxiv.org\/abs\/1311.6855","http:\/\/arxiv.org\/pdf\/1311.6855v3"],"collections":"Attention-grabbing titles","url":"http:\/\/arxiv.org\/abs\/1311.6855 http:\/\/arxiv.org\/pdf\/1311.6855v3","urldate":"2013-11-29","month":"nov","year":"2013","archivePrefix":"arXiv","eprint":"1311.6855","primaryClass":"math.GR"},{"key":"FactoringintheChickenMcNuggetmonoid","type":"article","title":"Factoring in the Chicken McNugget monoid","author":"Scott Chapman and Christopher O'Neill","abstract":"Every day, 34 million Chicken McNuggets are sold worldwide. At most McDonalds\r\nlocations in the United States today, Chicken McNuggets are sold in packs of 4,\r\n6, 10, 20, 40, and 50 pieces. However, shortly after their introduction in 1979\r\nthey were sold in packs of 6, 9, and 20. The use of these latter three numbers\r\nspawned the so-called Chicken McNugget problem, which asks: \"what numbers of\r\nChicken McNuggets can be ordered using only packs with 6, 9, or 20 pieces?\" In\r\nthis paper, we present an accessible introduction to this problem, as well as\r\nseveral related questions whose motivation comes from the theory of non-unique\r\nfactorization.","comment":"","date_added":"2017-09-06","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1709.01606v1","http:\/\/arxiv.org\/pdf\/1709.01606v1"],"collections":"Animals,Attention-grabbing titles,Easily explained,Unusual arithmetic,Food,Integerology","url":"http:\/\/arxiv.org\/abs\/1709.01606v1 http:\/\/arxiv.org\/pdf\/1709.01606v1","urldate":"2017-09-06","archivePrefix":"arXiv","eprint":"1709.01606","primaryClass":"math.AC","year":"2017"},{"key":"Hsupermagiclabelingsforfirecrackersbananatreesandflowers","type":"article","title":"$H$-supermagic labelings for firecrackers, banana trees and flowers","author":"Rachel Wulan Nirmalasari Wijaya and Andrea Semani\u010dov\u00e1-Fe\u0148ov\u010d\u00edkov\u00e1 and Joe Ryan and Thomas Kalinowski","abstract":"A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ is\r\ncontained in a subgraph $H'=(V',E')$ of $G$ which is isomorphic to $H$. In this\r\ncase we say that $G$ is $H$-supermagic if there is a bijection $f:V\\cup\r\nE\\to\\{1,\\ldots\\lvert V\\rvert+\\lvert E\\rvert\\}$ such that\r\n$f(V)=\\{1,\\ldots,\\lvert V\\rvert\\}$ and $\\sum_{v\\in V(H')}f(v)+\\sum_{e\\in\r\nE(H')}f(e)$ is constant over all subgraphs $H'$ of $G$ which are isomorphic to\r\n$H$. In this paper, we show that for odd $n$ and arbitrary $k$, the firecracker\r\n$F_{k,n}$ is $F_{2,n}$-supermagic, the banana tree $B_{k,n}$ is\r\n$B_{1,n}$-supermagic and the flower $F_n$ is $C_3$-supermagic.","comment":"","date_added":"2017-09-11","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1607.07911v2","http:\/\/arxiv.org\/pdf\/1607.07911v2"],"collections":"Attention-grabbing titles,Basically computer science,Food","url":"http:\/\/arxiv.org\/abs\/1607.07911v2 http:\/\/arxiv.org\/pdf\/1607.07911v2","urldate":"2017-09-11","archivePrefix":"arXiv","eprint":"1607.07911","primaryClass":"cs.DM","year":"2016"},{"key":"Becker2003","type":"article","title":"Topic-based vector space model","author":"J\u00f6rg Becker and Dominik Kuropka","abstract":"This paper motivates and presents the Topic-based Vector Space Model (TVSM), a new vector-based approach for document comparison. The approach does not assume independence between terms and it is flexible regarding the specification of term-similarities. Stop-word-list, stemming and thesaurus can be fully integrated into the model. This paper shows further how the TVSM can be fully implemented within the context of relational databases. This facilitates the use of this approach by generic applications. At the end short comparisons with other vector-based approaches namely the Vector Space Model (VSM) and the Generalized Vector Space Model (GVSM) are presented.","comment":"","date_added":"2012-03-24","date_published":"2003-08-06","urls":["http:\/\/www.kuropka.net\/files\/TVSM.pdf"],"collections":"Basically computer science","url":"http:\/\/www.kuropka.net\/files\/TVSM.pdf","urldate":"2012-03-24","year":"2003"},{"key":"Thetaildoesnotdeterminethesizeofthegiant","type":"article","title":"The tail does not determine the size of the giant","author":"Maria Deijfen and Sebastian Rosengren and Pieter Trapman","abstract":"The size of the giant component in the configuration model is given by a\r\nwell-known expression involving the generating function of the degree\r\ndistribution. In this note, we argue that the size of the giant is not\r\ndetermined by the tail behavior of the degree distribution but rather by the\r\ndistribution over small degrees. Upper and lower bounds for the component size\r\nare derived for an arbitrary given distribution over small degrees $d\\leq L$\r\nand given expected degree, and numerical implementations show that these bounds\r\nare very close already for small values of $L$. On the other hand, examples\r\nillustrate that, for a fixed degree tail, the component size can vary\r\nsubstantially depending on the distribution over small degrees. Hence the\r\ndegree tail does not play the same crucial role for the size of the giant as it\r\ndoes for many other properties of the graph.","comment":"","date_added":"2017-10-04","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1710.01208v1","http:\/\/arxiv.org\/pdf\/1710.01208v1"],"collections":"Attention-grabbing titles","url":"http:\/\/arxiv.org\/abs\/1710.01208v1 http:\/\/arxiv.org\/pdf\/1710.01208v1","urldate":"2017-10-04","archivePrefix":"arXiv","eprint":"1710.01208","primaryClass":"math.PR","year":"2017"},{"key":"AMidsummerKnotsDream","type":"article","title":"A Midsummer Knot's Dream","author":"Allison Henrich and No\u00ebl MacNaughton and Sneha Narayan and Oliver Pechenik and Robert Silversmith and Jennifer Townsend","abstract":"In this paper, we introduce playing games on shadows of knots. We demonstrate\r\ntwo novel games, namely, To Knot or Not to Knot and Much Ado about Knotting. We\r\nalso discuss winning strategies for these games on certain families of knot\r\nshadows. Finally, we suggest variations of these games for further study.","comment":"","date_added":"2020-09-21","date_published":"2010-08-06","urls":["http:\/\/arxiv.org\/abs\/1003.4494v1","http:\/\/arxiv.org\/pdf\/1003.4494v1"],"collections":"attention-grabbing-titles,easily-explained,games-to-play-with-friends,protocols-and-strategies","url":"http:\/\/arxiv.org\/abs\/1003.4494v1 http:\/\/arxiv.org\/pdf\/1003.4494v1","year":"2010","urldate":"2020-09-21","archivePrefix":"arXiv","eprint":"1003.4494","primaryClass":"math.GT"},{"key":"BadgroupsinthesenseofCherlin","type":"article","title":"Bad groups in the sense of Cherlin","author":"Olivier Fr\u00e9con","abstract":"There exists no bad group (in the sense of Gregory Cherlin), namely any\r\nsimple group of Morley rank 3 is isomorphic to $\\mathrm{PSL_2}(K)$ for an algebraically\r\nclosed field $K$.","comment":"","date_added":"2016-08-02","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1607.02994v1","http:\/\/arxiv.org\/pdf\/1607.02994v1"],"collections":"Attention-grabbing titles,The groups group","url":"http:\/\/arxiv.org\/abs\/1607.02994v1 http:\/\/arxiv.org\/pdf\/1607.02994v1","urldate":"2016-08-02","year":"2016"},{"key":"Curiositiesofarithmeticgases","type":"article","title":"Curiosities of arithmetic gases","author":"Ioannis Bakas and Mark J. Bowick","abstract":"Statistical mechanical systems with an exponential density of states are considered. The arithmetic analog of parafermions of arbitrary order is constructed and a formula for boson\u2010parafermion equivalence is obtained using properties of the Riemann zeta function. Interactions (nontrivial mixing) among arithmetic gases using the concept of twisted convolutions are also introduced. Examples of exactly solvable models are discussed in detail.","comment":"","date_added":"2017-10-16","date_published":"1991-08-06","urls":["http:\/\/aip.scitation.org\/doi\/10.1063\/1.529511","http:\/\/aip.scitation.org\/doi\/pdf\/10.1063\/1.529511"],"collections":"Unusual arithmetic","url":"http:\/\/aip.scitation.org\/doi\/10.1063\/1.529511 http:\/\/aip.scitation.org\/doi\/pdf\/10.1063\/1.529511","urldate":"2017-10-16","year":"1991"},{"key":"Mathemagics","type":"article","title":"Mathemagics","author":"Pierre Cartier","abstract":"My thesis is:there is another way of doing mathematics, equally successful, and the two methods should supplement each other and not fight.","comment":"","date_added":"2017-10-24","date_published":"2000-08-06","urls":["http:\/\/ftp.gwdg.de\/pub\/misc\/EMIS\/journals\/SLC\/wpapers\/s44cartier1.pdf"],"collections":"Notation and conventions,The act of doing maths","url":"http:\/\/ftp.gwdg.de\/pub\/misc\/EMIS\/journals\/SLC\/wpapers\/s44cartier1.pdf","urldate":"2017-10-24","year":"2000"},{"key":"TwodimensionalphotonicaperiodiccrystalsbasedonThueMorsesequence","type":"article","title":"Two-dimensional photonic aperiodic crystals based on Thue-Morse sequence","author":"Luigi Moretti and Vito Mocella ","abstract":"We investigate from a theoretical point of view the photonic properties of a two dimensional photonic aperiodic crystal. These structures are obtained by removing the lattice points from a square arrangement, following the inflation rules emerging from the Thue-Morse sequence. The photonic bandgap analysis is performed by means of the density of states calculation. The mechanism of bandgap formation is investigated adopting the single scattering model, and the Mie scattering. The electromagnetic field distribution can be represented as quasi-localized states. Finally, a generalized method to obtain aperiodic photonic structures has been proposed.","comment":"","date_added":"2017-11-13","date_published":"2007-08-06","urls":["https:\/\/www.osapublishing.org\/oe\/abstract.cfm?uri=oe-15-23-15314"],"collections":"Basically physics,Geometry","url":"https:\/\/www.osapublishing.org\/oe\/abstract.cfm?uri=oe-15-23-15314","urldate":"2017-11-13","year":"2007"},{"key":"Thegameofplatesandolives","type":"article","title":"The game of plates and olives","author":"Teena Carroll and David Galvin","abstract":"The game of plates and olives, introduced by Nicolaescu, begins with an empty\r\ntable. At each step either an empty plate is put down, an olive is put down on\r\na plate, an olive is removed, an empty plate is removed, or the olives on one\r\nplate are moved to another plate and the resulting empty plate is removed.\r\nPlates are indistinguishable from one another, as are olives, and there is an\r\ninexhaustible supply of each.\r\n The game derives from the consideration of Morse functions on the $2$-sphere.\r\nSpecifically, the number of topological equivalence classes of excellent Morse\r\nfunctions on the $2$-sphere that have order $n$ (that is, that have $2n+2$\r\ncritical points) is the same as the number of ways of returning to an empty\r\ntable for the first time after exactly $2n+2$ steps. We call this number $M_n$.\r\n Nicolaescu gave the lower bound $M_n \\geq (2n-1)!! = (2\/e)^{n+o(n)}n^n$ and\r\nspeculated that $\\log M_n \\sim n\\log n$. In this note we confirm this\r\nspeculation, showing that $M_n \\leq (4\/e)^{n+o(n)}n^n$.","comment":"","date_added":"2017-11-30","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1711.10670v1","http:\/\/arxiv.org\/pdf\/1711.10670v1"],"collections":"Attention-grabbing titles,Easily explained,Food","url":"http:\/\/arxiv.org\/abs\/1711.10670v1 http:\/\/arxiv.org\/pdf\/1711.10670v1","urldate":"2017-11-30","archivePrefix":"arXiv","eprint":"1711.10670","primaryClass":"math.CO","year":"2017"},{"key":"CryptographicProtocolsWithEverydayObjects","type":"article","title":"Cryptographic Protocols with Everyday Objects","author":"James Heather and Steve Schneider and Vanessa Teague","abstract":"Most security protocols appearing in the literature make use of cryptographic primitives that assume that the participants have access\r\nto some sort of computational device. \r\n\r\nHowever, there are times when there is need for a security mechanism\r\nto evaluate some result without leaking sensitive information, but computational devices are unavailable. We discuss here various protocols for\r\nsolving cryptographic problems using everyday objects: coins, dice, cards, and envelopes.","comment":"","date_added":"2016-11-17","date_published":"2011-08-06","urls":["http:\/\/www.computing.surrey.ac.uk\/personal\/st\/S.Schneider\/papers\/2011\/cryptoforma11a.pdf"],"collections":"Easily explained,Protocols and strategies,Things to make and do","url":"http:\/\/www.computing.surrey.ac.uk\/personal\/st\/S.Schneider\/papers\/2011\/cryptoforma11a.pdf","urldate":"2016-11-17","year":"2011"},{"key":"MechanismsbyTchebyshev","type":"online","title":"Mechanisms by Tchebyshev","author":"","abstract":"This project gathers all the mechanisms created by a great Russian mathematician Pafnuty Lvovich Tchebyshev (1821\u20141894).\r\n\r\nSome of them have been stored in museums: twenty are in the Polytechnical museum (Moscow), five are in the Museum of the History of Saint Petersburg State University, some are in The Mus\u00e9e des Arts et M\u00e9tiers in Paris and in Science Museum (London). There are only photos or descriptions left for some of the mechanisms.\r\n\r\nThe aim of this project is to preserve this heritage by constructing high-quality computer models of the mechanisms that remain and reconstruct those that have disappeared according to archive documents. By agreement with Museums the models are based on accurate measurements of all the original parameters. Any mechanism should be provided with existing photos, computer models and a movie explaining how the mechanisms work and showing it in action.","comment":"","date_added":"2017-12-04","date_published":"2009-08-06","urls":["http:\/\/en.tcheb.ru\/"],"collections":"Basically physics,History,Things to make and do,Geometry","url":"http:\/\/en.tcheb.ru\/","urldate":"2017-12-04","year":"2009"},{"key":"AcompilationofLEGOTechnicpartstosupportlearningexperimentsonlinkages","type":"article","title":"A compilation of LEGO Technic parts to support learning experiments on linkages","author":"Zolt\u00e1n Kov\u00e1cs and Benedek Kov\u00e1cs","abstract":"We present a compilation of LEGO Technic parts to provide easy-to-build\r\nconstructions of basic planar linkages. Some technical issues and their\r\npossible solutions are discussed. Fine details -- like deciding whether the\r\nmotion is an exactly straight line or not -- are forwarded to the dynamic\r\nmathematics software tool GeoGebra.","comment":"","date_added":"2017-12-04","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1712.00440v1","http:\/\/arxiv.org\/pdf\/1712.00440v1"],"collections":"Basically physics,Easily explained,Things to make and do,Geometry","url":"http:\/\/arxiv.org\/abs\/1712.00440v1 http:\/\/arxiv.org\/pdf\/1712.00440v1","urldate":"2017-12-04","archivePrefix":"arXiv","eprint":"1712.00440","primaryClass":"math.HO","year":"2017"},{"key":"SpotitSolitaire","type":"article","title":"Spot it(R) Solitaire","author":"Donna A. Dietz","abstract":"The game of Spot it(R) is based on an order 7 finite projective plane. This\r\narticle presents a solitaire challenge: extract an order 7 affine plane and\r\narrange those 49 cards into a square such that the symmetries of the affine and\r\nprojective planes are obvious. The objective is not to simply create such a\r\ndeck already in this solved position. Rather, it is to solve the inverse\r\nproblem of arranging the cards of such a deck which has already been created\r\nshuffled.","comment":"","date_added":"2017-12-18","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1301.7058v1","http:\/\/arxiv.org\/pdf\/1301.7058v1"],"collections":"Games to play with friends,Puzzles","url":"http:\/\/arxiv.org\/abs\/1301.7058v1 http:\/\/arxiv.org\/pdf\/1301.7058v1","urldate":"2017-12-18","archivePrefix":"arXiv","eprint":"1301.7058","primaryClass":"math.HO","year":"2013","keywords":"dobble"},{"key":"AnUnusualCubicRepresentationProblem","type":"article","title":"An unusual cubic representation problem","author":"Andrew Bremner and Allan Macleod","abstract":"For a non-zero integer \\(N\\), we consider the problem of finding \\(3\\) integers\r\n\\( (a, b, c) \\) such that\r\n\r\n\\[ N = \\frac{a}{b+c} + \\frac{b}{c+a} + \\frac{c}{a+b}. \\]\r\n\r\nWe show that the existence of solutions is related to points of infinite order on a family of elliptic curves. We discuss strictly positive solutions and prove the surprising fact that such solutions do not exist for \\(N\\) odd, even though there may exist solutions with one of \\(a, b, c\\) negative. We also show that, where a strictly positive solution does exist, it can be of enormous size (trillions of digits, even in the range we consider).","comment":"","date_added":"2017-08-07","date_published":"2014-08-06","urls":["http:\/\/ami.ektf.hu\/uploads\/papers\/finalpdf\/AMI_43_from29to41.pdf"],"collections":"Puzzles","url":"http:\/\/ami.ektf.hu\/uploads\/papers\/finalpdf\/AMI_43_from29to41.pdf","urldate":"2017-08-07","year":"2014"},{"key":"FoldingPolyominoesintoPolyCubes","type":"article","title":"Folding Polyominoes into (Poly)Cubes","author":"Oswin Aichholzer and Michael Biro and Erik D. Demaine and Martin L. Demaine and David Eppstein and S\u00e1ndor P. Fekete and Adam Hesterberg and Irina Kostitsyna and Christiane Schmidt","abstract":"We study the problem of folding a polyomino $P$ into a polycube $Q$, allowing\r\nfaces of $Q$ to be covered multiple times. First, we define a variety of\r\nfolding models according to whether the folds (a) must be along grid lines of\r\n$P$ or can divide squares in half (diagonally and\/or orthogonally), (b) must be\r\nmountain or can be both mountain and valley, (c) can remain flat (forming an\r\nangle of $180^\\circ$), and (d) must lie on just the polycube surface or can\r\nhave interior faces as well. Second, we give all the inclusion relations among\r\nall models that fold on the grid lines of $P$. Third, we characterize all\r\npolyominoes that can fold into a unit cube, in some models. Fourth, we give a\r\nlinear-time dynamic programming algorithm to fold a tree-shaped polyomino into\r\na constant-size polycube, in some models. Finally, we consider the triangular\r\nversion of the problem, characterizing which polyiamonds fold into a regular\r\ntetrahedron.","comment":"","date_added":"2018-01-03","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1712.09317v1","http:\/\/arxiv.org\/pdf\/1712.09317v1"],"collections":"Puzzles,Things to make and do,Geometry","url":"http:\/\/arxiv.org\/abs\/1712.09317v1 http:\/\/arxiv.org\/pdf\/1712.09317v1","urldate":"2018-01-03","year":"2017","archivePrefix":"arXiv","eprint":"1712.09317","primaryClass":"cs.CG"},{"key":"Moler2003","type":"article","title":"Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later","author":"Moler, Cleve and Van Loan, C.","abstract":"","comment":"","date_added":"2012-02-09","date_published":"2003-08-06","urls":["http:\/\/www.jstor.org\/stable\/10.2307\/25054364","http:\/\/www.cs.cornell.edu\/cv\/researchpdf\/19ways+.pdf"],"collections":"Attention-grabbing titles,Basically computer science","url":"http:\/\/www.jstor.org\/stable\/10.2307\/25054364 http:\/\/www.cs.cornell.edu\/cv\/researchpdf\/19ways+.pdf","urldate":"2012-02-09","year":"2003","journal":"SIAM review","keywords":"1,15a15,65f15,65f30,65l99,ams subject classifications,and eco-,biological,condition,exponential,introduction,mathematical models of many,matrix,physical,pii,roundoff error,s0036144502418010,truncation error","number":"1","pages":"3--49","publisher":"JSTOR","volume":"45"},{"key":"AnyMonotoneBooleanFunctionCanBeRealizedByInterlockedPolygons","type":"article","title":"Any Monotone Boolean Function Can Be Realized by Interlocked Polygons","author":"Erik D. Demaine and Martin L. Demaine and Ryuhei Uehara","abstract":"We show how to construct interlocked collections of simple polygons in the plane that fall apart upon removing certain combinations of pieces. Precisely, interior-disjoint simple planar polygons are interlocked if no subset can be separated arbitrarily far from the rest, moving each polygon as a rigid object as in a sliding-block puzzle. Removing a subset \\(S\\) of these polygons might keep them interlocked or free the polygons, allowing them to separate. Clearly freeing removal sets satisfy monotonicity: if \\(S \\subseteq S\u2032\\) and removing \\(S\\) frees the polygons, then so does \\(S\u2032\\). In this paper, we show that any monotone Boolean function \\(f\\) on \\(n\\) variables can be described by \\(m > n\\) interlocked polygons: \\(n\\) of the \\(m\\) polygons represent the \\(n\\) variables, and removing a subset of these \\(n\\) polygons frees the remaining polygons if and only if \\(f\\) is 1 when the corresponding variables are 1.","comment":"","date_added":"2018-01-08","date_published":"2010-08-06","urls":["http:\/\/erikdemaine.org\/papers\/InterlockedPolygons_CCCG2010\/","http:\/\/erikdemaine.org\/papers\/InterlockedPolygons_CCCG2010\/paper.pdf"],"collections":"Basically computer science,Easily explained,Geometry,Fun maths facts","url":"http:\/\/erikdemaine.org\/papers\/InterlockedPolygons_CCCG2010\/ http:\/\/erikdemaine.org\/papers\/InterlockedPolygons_CCCG2010\/paper.pdf","year":"2010","urldate":"2018-01-08"},{"key":"Randomrailwaysmodeledasrandom3regulargraphs","type":"article","title":"Random railways modeled as random 3-regular graphs","author":"Hans Garmo","abstract":"In a cubic multigraph certain restrictions on the paths are made. Due to these restrictions a special kind of connectivity is defined. The asymptotic probability of this connectivity is calculated in a random cubic multigraph and is shown to be 1\/3.","comment":"","date_added":"2018-01-09","date_published":"1996-08-06","urls":["http:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/(SICI)1098-2418(199608\/09)9:1\/2%3C113::AID-RSA8%3E3.0.CO;2-%23\/abstract"],"collections":"Attention-grabbing titles,Probability and statistics,Modelling","url":"http:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/(SICI)1098-2418(199608\/09)9:1\/2%3C113::AID-RSA8%3E3.0.CO;2-%23\/abstract","year":"1996","urldate":"2018-01-09"},{"key":"WhatdidRyserConjecture","type":"article","title":"What did Ryser Conjecture?","author":"Darcy Best and Ian M. Wanless","abstract":"Two prominent conjectures by Herbert J. Ryser have been falsely attributed to\r\na somewhat obscure conference proceedings that he wrote in German. Here we\r\nprovide a translation of that paper and try to correct the historical record at\r\nleast as far as what was conjectured in it. The two conjectures relate to\r\ntransversals in Latin squares of odd order and to the relationship between the\r\ncovering number and the matching number of multipartite hypergraphs.","comment":"","date_added":"2018-01-10","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1801.02893v1","http:\/\/arxiv.org\/pdf\/1801.02893v1"],"collections":"History,Drama!","url":"http:\/\/arxiv.org\/abs\/1801.02893v1 http:\/\/arxiv.org\/pdf\/1801.02893v1","year":"2018","urldate":"2018-01-10","archivePrefix":"arXiv","eprint":"1801.02893","primaryClass":"math.CO"},{"key":"Richeson2013","type":"article","title":"Circular reasoning: who first proved that $C\/d$ is a constant?","author":"Richeson, David","abstract":"We answer the question: who first proved that $C\/d$ is a constant? We argue that Archimedes proved that the ratio of the circumference of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant ($C\/d=A\/r^{2}$). He stated neither result explicitly, but both are implied by his work. His proof required the addition of two axioms beyond those in Euclid's Elements; this was the first step toward a rigorous theory of arc length. We also discuss how Archimedes's work coexisted with the 2000-year belief -- championed by scholars from Aristotle to Descartes -- that it is impossible to find the ratio of a curved line to a straight line.","comment":"","date_added":"2013-03-06","date_published":"2013-03-01","urls":["http:\/\/arxiv.org\/abs\/1303.0904","http:\/\/arxiv.org\/pdf\/1303.0904v2"],"collections":"Geometry,About proof","url":"http:\/\/arxiv.org\/abs\/1303.0904 http:\/\/arxiv.org\/pdf\/1303.0904v2","urldate":"2013-03-06","year":"2013","archivePrefix":"arXiv","arxivId":"1303.0904","eprint":"1303.0904","keywords":"and phrases,arc,archimedes,aristotle,circle,descartes,history,pi","month":"mar","pages":"17","primaryClass":"math.HO"},{"key":"Markowsky1992","type":"article","title":"Misconceptions about the Golden Ratio","author":"Markowsky, George","abstract":"","comment":"","date_added":"2010-09-29","date_published":"1992-01-01","urls":["http:\/\/www.jstor.org\/stable\/2686193?origin=crossref"],"collections":"Easily explained,History","url":"http:\/\/www.jstor.org\/stable\/2686193?origin=crossref","urldate":"2010-09-29","year":"1992","journal":"The College Mathematics Journal","month":"jan","number":"1","pages":"2","volume":"23"},{"key":"PlayingGameswithAlgorithmsAlgorithmicCombinatorialGameTheory","type":"article","title":"Playing Games with Algorithms: Algorithmic Combinatorial Game Theory","author":"Erik D. Demaine and Robert A. Hearn","abstract":"Combinatorial games lead to several interesting, clean problems in algorithms\r\nand complexity theory, many of which remain open. The purpose of this paper is\r\nto provide an overview of the area to encourage further research. In\r\nparticular, we begin with general background in Combinatorial Game Theory,\r\nwhich analyzes ideal play in perfect-information games, and Constraint Logic,\r\nwhich provides a framework for showing hardness. Then we survey results about\r\nthe complexity of determining ideal play in these games, and the related\r\nproblems of solving puzzles, in terms of both polynomial-time algorithms and\r\ncomputational intractability results. Our review of background and survey of\r\nalgorithmic results are by no means complete, but should serve as a useful\r\nprimer.","comment":"","date_added":"2018-01-23","date_published":"2001-08-06","urls":["http:\/\/arxiv.org\/abs\/cs\/0106019v2","http:\/\/arxiv.org\/pdf\/cs\/0106019v2"],"collections":"Basically computer science,Games to play with friends,Combinatorics,Computational complexity of games","url":"http:\/\/arxiv.org\/abs\/cs\/0106019v2 http:\/\/arxiv.org\/pdf\/cs\/0106019v2","year":"2001","urldate":"2018-01-23","archivePrefix":"arXiv","eprint":"cs\/0106019","primaryClass":"cs.CC"},{"key":"Cormode2004","type":"article","title":"The hardness of the Lemmings game, or Oh no, more NP-completeness proofs","author":"Cormode, Graham","abstract":"","comment":"","date_added":"2012-01-27","date_published":"2004-08-06","urls":["https:\/\/pdfs.semanticscholar.org\/9413\/3f598974f93b089ca7d2875d59d0dab1e1eb.pdf"],"collections":"Animals,Attention-grabbing titles,Basically computer science,Games to play with friends,About proof,Computational complexity of games","url":"https:\/\/pdfs.semanticscholar.org\/9413\/3f598974f93b089ca7d2875d59d0dab1e1eb.pdf","urldate":"2012-01-27","year":"2004","booktitle":"Proceedings of the 3rd International Conference on FUN with Algorithms","pages":"65--76"},{"key":"Thegrasshopperproblem","type":"article","title":"The grasshopper problem","author":"Olga Goulko and Adrian Kent","abstract":"We introduce and physically motivate the following problem in geometric\r\ncombinatorics, originally inspired by analysing Bell inequalities. A\r\ngrasshopper lands at a random point on a planar lawn of area one. It then jumps\r\nonce, a fixed distance $d$, in a random direction. What shape should the lawn\r\nbe to maximise the chance that the grasshopper remains on the lawn after\r\njumping? We show that, perhaps surprisingly, a disc shaped lawn is not optimal\r\nfor any $d>0$. We investigate further by introducing a spin model whose ground\r\nstate corresponds to the solution of a discrete version of the grasshopper\r\nproblem. Simulated annealing and parallel tempering searches are consistent\r\nwith the hypothesis that for $ d < \\pi^{-1\/2}$ the optimal lawn resembles a\r\ncogwheel with $n$ cogs, where the integer $n$ is close to $ \\pi ( \\arcsin (\r\n\\sqrt{\\pi} d \/2 ) )^{-1}$. We find transitions to other shapes for $d \\gtrsim\r\n\\pi^{-1\/2}$.","comment":"","date_added":"2018-01-24","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1705.07621v3","http:\/\/arxiv.org\/pdf\/1705.07621v3"],"collections":"Animals,Probability and statistics,Puzzles,Geometry","url":"http:\/\/arxiv.org\/abs\/1705.07621v3 http:\/\/arxiv.org\/pdf\/1705.07621v3","year":"2017","urldate":"2018-01-24","archivePrefix":"arXiv","eprint":"1705.07621","primaryClass":"cond-mat.stat-mech"},{"key":"Anemptyexercise","type":"article","title":"An empty exercise","author":"Carl de Boor","abstract":"The exercise in question concerns the rules which should govern the treatment of empty matrices in a matrix-oriented computing environment like MATLAB. This provides students of Linear Algebra with an unusual test of their understanding of the standard definitions and rules governing matrices.","comment":"","date_added":"2018-01-24","date_published":"1990-08-06","urls":["https:\/\/dl.acm.org\/citation.cfm?id=122273","http:\/\/ftp.cs.wisc.edu\/Approx\/empty.pdf"],"collections":"Attention-grabbing titles,Basically computer science,Notation and conventions","url":"https:\/\/dl.acm.org\/citation.cfm?id=122273 http:\/\/ftp.cs.wisc.edu\/Approx\/empty.pdf","year":"1990","urldate":"2018-01-24"},{"key":"MechanicalComputingSystemsUsingOnlyLinksandRotaryJoints","type":"article","title":"Mechanical Computing Systems Using Only Links and Rotary Joints","author":"Ralph C. Merkle and Robert A. Freitas Jr. and Tad Hogg and Thomas E. Moore and Matthew S. Moses and James Ryley","abstract":"A new paradigm for mechanical computing is demonstrated that requires only\r\ntwo basic parts, links and rotary joints. These basic parts are combined into\r\ntwo main higher level structures, locks and balances, and suffice to create all\r\nnecessary combinatorial and sequential logic required for a Turing-complete\r\ncomputational system. While working systems have yet to be implemented using\r\nthis new paradigm, the mechanical simplicity of the systems described may lend\r\nthemselves better to, e.g., microfabrication, than previous mechanical\r\ncomputing designs. Additionally, simulations indicate that if molecular-scale\r\nimplementations could be realized, they would be far more energy-efficient than\r\nconventional electronic computers.","comment":"","date_added":"2018-01-30","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1801.03534v1","http:\/\/arxiv.org\/pdf\/1801.03534v1"],"collections":"Basically computer science,Things to make and do,Unusual computers","url":"http:\/\/arxiv.org\/abs\/1801.03534v1 http:\/\/arxiv.org\/pdf\/1801.03534v1","year":"2018","urldate":"2018-01-30","archivePrefix":"arXiv","eprint":"1801.03534","primaryClass":"cs.ET"},{"key":"TheMuffinProblem","type":"article","title":"The Muffin Problem","author":"Guangiqi Cui and John Dickerson and Naveen Durvasula and William Gasarch and Erik Metz and Naveen Raman and Sung Hyun Yoo","abstract":"You have $m$ muffins and $s$ students. You want to divide the muffins into\r\npieces and give the shares to students such that every student has\r\n$\\frac{m}{s}$ muffins. Find a divide-and-distribute protocol that maximizes the\r\nminimum piece. Let $f(m,s)$ be the minimum piece in the optimal protocol. We\r\nprove that $f(m,s)$ exists, is rational, and finding it is computable (though\r\npossibly difficult). We show that $f(m,s)$ can be derived from $f(s,m)$; hence\r\nwe need only consider $m\\ge s$. For $1\\le s\\le 6$ we find nice formulas for\r\n$f(m,s)$. We also find a nice formula for $f(s+1,s)$. We give a function\r\n$FC(m,s)$ such that, for $m\\ge s+2$, $f(m,s)\\le FC(m,s)$. This function\r\npermeates the entire paper since it is often the case that $f(m,s)=FC(m,s)$.\r\nMore formally, for all $s$ there is a nice formula $FORM(m,s)$ such that, for\r\nall but a finite number of $m$, $f(m,s)=FC(m,s)=FORM(m,s)$. For those finite\r\nnumber of exceptions we have another function $INT(m,s)$ such that $f(m,s)\\le\r\nINT(m,s)$. It seems to be the case that when $m\\ge s+2$,\r\n$f(m,s)=\\min\\{f(m,s),INT(m,s)\\}$. For $s=7$ to 60 we have conjectured formulas\r\nfor $f(m,s)$ that include exceptions.","comment":"","date_added":"2018-01-30","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1709.02452v2","http:\/\/arxiv.org\/pdf\/1709.02452v2"],"collections":"Easily explained,Food,Protocols and strategies","url":"http:\/\/arxiv.org\/abs\/1709.02452v2 http:\/\/arxiv.org\/pdf\/1709.02452v2","year":"2017","urldate":"2018-01-30","archivePrefix":"arXiv","eprint":"1709.02452","primaryClass":"math.CO"},{"key":"Straightknots","type":"article","title":"Straight knots","author":"Nicholas Owad","abstract":"We introduce a new invariant, the straight number of a knot. We give some\r\nrelations to crossing number and petal number. Then we discuss the methods we\r\nused to compute the straight numbers for all the knots in the standard knot\r\ntable and present some interesting questions and the full table.","comment":"","date_added":"2018-02-01","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1801.10428v1","http:\/\/arxiv.org\/pdf\/1801.10428v1"],"collections":"Attention-grabbing titles,Easily explained","url":"http:\/\/arxiv.org\/abs\/1801.10428v1 http:\/\/arxiv.org\/pdf\/1801.10428v1","year":"2018","urldate":"2018-02-01","archivePrefix":"arXiv","eprint":"1801.10428","primaryClass":"math.GT"},{"key":"Fingerprintdatabasesfortheorems","type":"article","title":"Fingerprint databases for theorems","author":"Sara C. Billey and Bridget E. Tenner","abstract":"We discuss the advantages of searchable, collaborative, language-independent\r\ndatabases of mathematical results, indexed by \"fingerprints\" of small and\r\ncanonical data. Our motivating example is Neil Sloane's massively influential\r\nOn-Line Encyclopedia of Integer Sequences. We hope to encourage the greater\r\nmathematical community to search for the appropriate fingerprints within each\r\ndiscipline, and to compile fingerprint databases of results wherever possible.\r\nThe benefits of these databases are broad - advancing the state of knowledge,\r\nenhancing experimental mathematics, enabling researchers to discover unexpected\r\nconnections between areas, and even improving the refereeing process for\r\njournal publication.","comment":"","date_added":"2018-02-12","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1304.3866v1","http:\/\/arxiv.org\/pdf\/1304.3866v1"],"collections":"Lists and catalogues,The act of doing maths","url":"http:\/\/arxiv.org\/abs\/1304.3866v1 http:\/\/arxiv.org\/pdf\/1304.3866v1","year":"2013","urldate":"2018-02-12","archivePrefix":"arXiv","eprint":"1304.3866","primaryClass":"math.HO"},{"key":"NotablePropertiesofSpecificNumbers","type":"online","title":"Notable Properties of Specific Numbers","author":"Robert Munafo","abstract":"","comment":"","date_added":"2018-03-05","date_published":null,"urls":["http:\/\/www.mrob.com\/pub\/math\/numbers.html"],"collections":"Lists and catalogues,Integerology","url":"http:\/\/www.mrob.com\/pub\/math\/numbers.html","urldate":"2018-03-05","year":""},{"key":"HowdoyoufixanOvalTrackPuzzle","type":"article","title":"How do you fix an Oval Track Puzzle?","author":"David A. Nash and Sara Randall","abstract":"The oval track group, $OT_{n,k}$, is the subgroup of the symmetric group,\r\n$S_n$, generated by the basic moves available in a generalized oval track\r\npuzzle with $n$ tiles and a turntable of size $k$. In this paper we completely\r\ndescribe the oval track group for all possible $n$ and $k$ and use this\r\ninformation to answer the following question: If the tiles are removed from an\r\noval track puzzle, how must they be returned in order to ensure that the puzzle\r\nis still solvable? As part of this discussion we introduce the parity subgroup\r\nof $S_n$ in the case when $n$ is even.","comment":"","date_added":"2018-03-13","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1612.04476v3","http:\/\/arxiv.org\/pdf\/1612.04476v3"],"collections":"Easily explained,Puzzles","url":"http:\/\/arxiv.org\/abs\/1612.04476v3 http:\/\/arxiv.org\/pdf\/1612.04476v3","year":"2016","urldate":"2018-03-13","archivePrefix":"arXiv","eprint":"1612.04476","primaryClass":"math.GR"},{"key":"NumeralSystemsoftheWorld","type":"online","title":"Numeral Systems of the World","author":"Bernard Comrie and Eugene Chan","abstract":"The principal purpose of this web site is to document the various numeral systems used by the currently spoken 7,099 human languages, focusing especially on little-known, undescribed and endangered languages, to record and preserve the traditional counting systems before they fall out of use.","comment":"","date_added":"2018-03-25","date_published":"2006-08-06","urls":["https:\/\/mpi-lingweb.shh.mpg.de\/numeral\/"],"collections":"Lists and catalogues,Notation and conventions","url":"https:\/\/mpi-lingweb.shh.mpg.de\/numeral\/","year":"2006","urldate":"2018-03-25"},{"key":"APuzzleForPirates","type":"article","title":"A Puzzle for Pirates","author":"Ian Stewart","abstract":"A generalisation of the puzzle where pirates divide up a stash of coins by proposing splits in decreasing order of seniority. If a split is voted down, the proposing pirate is thrown overboard.","comment":"","date_added":"2018-04-03","date_published":"1999-08-06","urls":["https:\/\/omohundro.files.wordpress.com\/2009\/03\/stewart99_a_puzzle_for_pirates.pdf"],"collections":"Puzzles","url":"https:\/\/omohundro.files.wordpress.com\/2009\/03\/stewart99_a_puzzle_for_pirates.pdf","year":"1999","urldate":"2018-04-03"},{"key":"Thematerialityofmathematicspresentingmathematicsattheblackboard","type":"article","title":"The materiality of mathematics: presenting mathematics at the blackboard","author":"Christian Greiffenhagen","abstract":"Sociology has been accused of neglecting the importance of material things in human life and the material aspects of social practices. Efforts to correct this have recently been made, with a growing concern to demonstrate the materiality of social organization, not least through attention to objects and the body.As a result, there have been a plethora of studies reporting the social construction and effects of a variety of material objects as well as studies that have explored the material dimensions of a diversity of practices. In different ways these studies have questioned the Cartesian dualism of a strict separation of \u2018mind\u2019 and \u2018body\u2019. However, it could be argued that the idea of the mind as immaterial has not been entirely banished and lingers when it comes to discussing abstract thinking and reasoning. The aim of this article is to extend the material turn to abstract thought, using mathematics as a paradigmatic example. This paper explores how writing mathematics (on paper, blackboards, or even in the air) is indispensable for doing and thinking mathematics.The paper is based on video recordings of lectures in formal logic and investigates how mathematics is presented at the blackboard. The paper discusses the iconic character of blackboards in mathematics and describes in detail a number of inscription practices of presenting mathematics at the blackboard (such as the use of lines and boxes, the designation of particular regions for specific mathematical purposes, as well as creating an \u2018architecture\u2019 visualizing the overall structure of the proof). The paper argues that doing mathematics really is \u2018thinking with eyes and hands\u2019 (Latour 1986). Thinking in mathematics is inextricably interwoven with writing mathematics.","comment":"","date_added":"2018-04-06","date_published":"2014-08-06","urls":["https:\/\/dspace.lboro.ac.uk\/dspace-jspui\/handle\/2134\/14295","https:\/\/dspace.lboro.ac.uk\/dspace-jspui\/bitstream\/2134\/14295\/3\/greiffenhagen-materiality_20120511.pdf"],"collections":"The act of doing maths","url":"https:\/\/dspace.lboro.ac.uk\/dspace-jspui\/handle\/2134\/14295 https:\/\/dspace.lboro.ac.uk\/dspace-jspui\/bitstream\/2134\/14295\/3\/greiffenhagen-materiality_20120511.pdf","urldate":"2018-04-06","year":"2014"},{"key":"ChocolategamesthatsatisfytheinequalityforandGrundynumbers","type":"article","title":"Chocolate games that satisfy the inequality \\(y \\leq \\left \\lfloor \\frac{z}{k} \\right\\rfloor\\) for \\(k=1,2\\) and Grundy numbers","author":"Shunsuke Nakamura and Ryo Hanafusa and Wataru Ogasa and Takeru Kitagawa and Ryohei Miyadera","abstract":"We study chocolate games that are variants of a game of Nim. We can cut the chocolate games in 3 directions, and we represent the chocolates with coordinates \\( \\{x,y,z\\}\\) , where \\( x,y,z \\) are the maximum times you can cut them in each direction.\r\nThe coordinates \\( \\{x,y,z\\}\\) of the chocolates satisfy the inequalities \\( y\\leq \\lfloor \\frac{z}{k} \\rfloor \\) for \\( k = 1,2\\) .\r\nFor \\( k = 2\\) we prove a theorem for the L-state (loser's state), and the proof of this theorem can be easily generalized to the case of an arbitrary even number \\(k\\).\r\nFor \\(k = 1\\) we prove a theorem for the L-state (loser's state), and we need the theory of Grundy numbers to prove the theorem. The generalization of the case of \\( k = 1\\) to the case of an arbitrary odd number is an open problem. The authors present beautiful graphs made by Grundy numbers of these chocolate games.","comment":"","date_added":"2017-10-18","date_published":"2013-08-06","urls":["http:\/\/www.mi.sanu.ac.rs\/vismath\/miyadera2013\/index.html"],"collections":"Attention-grabbing titles,Food,Games to play with friends","url":"http:\/\/www.mi.sanu.ac.rs\/vismath\/miyadera2013\/index.html","urldate":"2017-10-18","year":"2013"},{"key":"NiceNeighbours","type":"article","title":"Nice Neighbors: A Brief Adventure in Mathematical Gamification","author":"Chris Staecker","abstract":"Last year I came across a strange graph theory problem from digital topology. I turned it into a video game to help wrap my mind around it. It was fun to play, so I made it into a web game that other people could play. I took 3,500 unsolved math problems, made each one into a level of the game, and waited to see if people would solve my problems for me. Within two months, hundreds of people and at least one nonperson played the game, and together they solved every level. I\u2019ll describe the mathematics behind this game and some of the surprises along the way that still have me scratching my head. ","comment":"","date_added":"2018-04-17","date_published":"2016-08-06","urls":["http:\/\/cstaecker.fairfield.edu\/~cstaecker\/files\/research\/nnpublished.pdf"],"collections":"Easily explained,Puzzles","url":"http:\/\/cstaecker.fairfield.edu\/~cstaecker\/files\/research\/nnpublished.pdf","year":"2016","urldate":"2018-04-17"},{"key":"TheMathematicalColoringBook","type":"article","title":"The Mathematical Coloring Book","author":"Alexander Soifer","abstract":"Due to the author's correspondence with Van der Waerden, Erd\u00f6s, Baudet, members of the Schur Circle, and others, and due to voluminous archival materials uncovered by the author over 18 years of his work on the book, this book contains material that has never before been published.","comment":"","date_added":"2018-04-18","date_published":"2009-08-06","urls":["https:\/\/link.springer.com\/book\/10.1007%2F978-0-387-74642-5"],"collections":"Attention-grabbing titles","url":"https:\/\/link.springer.com\/book\/10.1007%2F978-0-387-74642-5","year":"2009","urldate":"2018-04-18"},{"key":"AnInvitationtoInverseGroupTheory","type":"article","title":"An Invitation to Inverse Group Theory","author":"Jo\u00e3o Ara\u00fajo and Peter J. Cameron and Francesco Matucci","abstract":"In group theory there are many constructions which produce a new group from a\r\ngiven one. Often the result is a subgroup: the derived group, centre, socle,\r\nFrattini subgroup, Hall subgroup, Fitting subgroup, and so on. Other\r\nconstructions may produce groups in other ways, for example quotients (solvable\r\nresidual, derived quotient) or cohomology groups (Schur multiplier). Inverse\r\ngroup theory refers to problems in which a construction and the resulting group\r\nis given and we want information about the possible original group or groups;\r\nexamples are the {\\em inverse Schur multiplier problem} (given a finite abelian\r\ngroup is it the Schur multiplier of some finite group?), or the {\\em inverse\r\nderived group} (given a group $G$ is there a group $H$ such that $H'=G$?). In\r\n1956 B. H. Neumann sent a first invitation to inverse group theory, but\r\napparently the topic did not receive the attention it deserves, so that we\r\nattempt here at repeating that invitation. Many of the inverse group problems\r\nassociated with the constructions referred to above are trivial, but some are\r\nnot. Like Neumann we will work mainly on inverse derived groups. We also\r\nexplain how the main questions about inverse Frattini subgroups have been\r\nsettled.\r\n An integral of a group $G$ is a group $H$ such that the derived group of $H$\r\nis $G$. Our first goal is to prove a number of general facts about the\r\nintegrals of finite groups, and to raise some open questions. Our results\r\nconcern orders of non-integrable groups (we give a complete description of the\r\nset of such numbers), the smallest integral of a group (in particular, we show\r\nthat if a finite group is integrable it has a finite integral), and groups\r\nwhich can be integrated infinitely often, a problem already tackled by Neumann.\r\nWe also consider integrals of infinite groups. Regarding inverse Frattini, we\r\nexplain Neumann's and Eick's results.","comment":"","date_added":"2018-04-19","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1803.10179v2","http:\/\/arxiv.org\/pdf\/1803.10179v2"],"collections":"Unusual arithmetic,The groups group","url":"http:\/\/arxiv.org\/abs\/1803.10179v2 http:\/\/arxiv.org\/pdf\/1803.10179v2","year":"2018","urldate":"2018-04-19","archivePrefix":"arXiv","eprint":"1803.10179","primaryClass":"math.GR"},{"key":"Whatisthesmallestprime","type":"article","title":"What is the smallest prime?","author":"Chris K. Caldwell and Yeng Xiong","abstract":"What is the first prime? It seems that the number two should be the obvious\r\nanswer, and today it is, but it was not always so. There were times when and\r\nmathematicians for whom the numbers one and three were acceptable answers. To\r\nfind the first prime, we must also know what the first positive integer is.\r\nSurprisingly, with the definitions used at various times throughout history,\r\none was often not the first positive integer (some started with two, and a few\r\nwith three). In this article, we survey the history of the primality of one,\r\nfrom the ancient Greeks to modern times. We will discuss some of the reasons\r\ndefinitions changed, and provide several examples. We will also discuss the\r\nlast significant mathematicians to list the number one as prime.","comment":"","date_added":"2016-07-05","date_published":"2012-08-06","urls":["http:\/\/arxiv.org\/abs\/1209.2007v2","http:\/\/arxiv.org\/pdf\/1209.2007v2"],"collections":"Easily explained,History,Notation and conventions,Integerology","url":"http:\/\/arxiv.org\/abs\/1209.2007v2 http:\/\/arxiv.org\/pdf\/1209.2007v2","urldate":"2016-07-05","year":"2012","archivePrefix":"arXiv","eprint":"1209.2007","primaryClass":"math.HO"},{"key":"Mostprimitivegroupshavemessyinvariants","type":"article","title":"Most primitive groups have messy invariants","author":"W.C. Huffman and N.J.A. Sloane","abstract":"Suppose \\(G\\) is a finite group of complex \\(n \\times n\\) matrices, and let \\(R^G\\) be the ring of invariants of \\(G\\): i.e., those polynomials fixed by \\(G\\). Many authors, from Klein to the present day, have described \\(R^G\\) by writing it as a direct sum \\(\\sum_{j=1}^\\delta \\eta_j\\mathrm{C}[\\theta_1, \\ldots, \\ltheta_n]\\). For example, if $G$ is a unitary group generated by reflections, \\(\\delta = 1\\). In this note we show that in general this approach is hopeless by proving that, for any \\(\\epsilon > 0\\), the smallest possible \\(delta\\) is greater than \\(|G|^{n-1-\\epsilon}\\) for almost all primitive groups. Since for any group we can choose \\(\\delta \\leq |G|^{n-1}\\), this means that most primitive groups are about as bad as they can be. The upper bound on \\(delta\\) follows from Dade's theorem that the \\(\\theta_i\\) can be chosen to have degrees dividing \\(|G\\).","comment":"","date_added":"2018-05-06","date_published":"1979-08-06","urls":["https:\/\/www.sciencedirect.com\/science\/article\/pii\/0001870879900380","http:\/\/neilsloane.com\/doc\/Me61.pdf"],"collections":"Attention-grabbing titles,The groups group","url":"https:\/\/www.sciencedirect.com\/science\/article\/pii\/0001870879900380 http:\/\/neilsloane.com\/doc\/Me61.pdf","year":"1979","urldate":"2018-05-06"},{"key":"Redefiningtheintegral","type":"article","title":"Redefining the integral","author":"Derek Orr","abstract":"In this paper, we discuss a similar functional to that of a standard\r\nintegral. The main difference is in its definition: instead of taking a sum, we\r\nare taking a product. It turns out this new \"star-integral\" may be written in\r\nterms of the standard integral but it has many different (and similar)\r\ninteresting properties compared to the regular integral. Further, we define a\r\n\"star-derivative\" and discuss its relationship to the \"star-integral\".","comment":"","date_added":"2018-05-08","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1805.01861v1","http:\/\/arxiv.org\/pdf\/1805.01861v1"],"collections":"Unusual arithmetic","url":"http:\/\/arxiv.org\/abs\/1805.01861v1 http:\/\/arxiv.org\/pdf\/1805.01861v1","year":"2018","urldate":"2018-05-08","archivePrefix":"arXiv","eprint":"1805.01861","primaryClass":"math.CA"},{"key":"MathCounterexamples","type":"online","title":"Math Counterexamples","author":"Jean-Pierre Merx","abstract":"I initiated this website because for years I have been passionated about Mathematics as a hobby and also by \u201cstrange objects\u201d. Mathematical counterexamples combine both topics.\r\n\r\nThe first counterexample I was exposed with is the one of an unbounded positive continuous function with a convergent integral. I took time to find such a counterexample\u2026 but that was a positive experience to raise my interest in counterexamples.\r\n\r\nAccording to Wikipedia a counterexample is an exception to a proposed general rule or law. And in mathematics, it is (by a slight abuse) also sometimes used for examples illustrating the necessity of the full hypothesis of a theorem, by considering a case where a part of the hypothesis is not verified, and where one can show that the conclusion does not hold.\r\n\r\nBy extension, I call a counterexample any example whose role is not that of illustrating a true theorem. For instance, a polynomial as an example of a continuous function is not a counterexample, but a polynomial as an example of a function that fails to be bounded or of a function that fails to be periodic is a counterexample.\r\n\r\nWhile I\u2019m particularly interested in Topology and Analysis, I will also try to cover Logic and Algebra counterexamples.","comment":"","date_added":"2018-05-09","date_published":"2014-08-06","urls":["http:\/\/www.mathcounterexamples.net\/"],"collections":"Lists and catalogues","url":"http:\/\/www.mathcounterexamples.net\/","year":"2014","urldate":"2018-05-09"},{"key":"Howlongdoesittaketocatchawildkangaroo","type":"article","title":"How long does it take to catch a wild kangaroo?","author":"Ravi Montenegro and Prasad Tetali","abstract":"We develop probabilistic tools for upper and lower bounding the expected time\r\nuntil two independent random walks on $\\ZZ$ intersect each other. This leads to\r\nthe first sharp analysis of a non-trivial Birthday attack, proving that\r\nPollard's Kangaroo method solves the discrete logarithm problem $g^x=h$ on a\r\ncyclic group in expected time $(2+o(1))\\sqrt{b-a}$ for an average\r\n$x\\in_{uar}[a,b]$. Our methods also resolve a conjecture of Pollard's, by\r\nshowing that the same bound holds when step sizes are generalized from powers\r\nof 2 to powers of any fixed $n$.","comment":"","date_added":"2018-05-12","date_published":"2008-08-06","urls":["http:\/\/arxiv.org\/abs\/0812.0789v2","http:\/\/arxiv.org\/pdf\/0812.0789v2"],"collections":"Animals,Easily explained,Probability and statistics","url":"http:\/\/arxiv.org\/abs\/0812.0789v2 http:\/\/arxiv.org\/pdf\/0812.0789v2","year":"2008","urldate":"2018-05-12","archivePrefix":"arXiv","eprint":"0812.0789","primaryClass":"math.PR"},{"key":"ExactEnumerationOfGardenOfEdenPartitions","type":"article","title":"Exact Enumeration of Garden of Eden Partitions","author":"Brian Hopkins and James A. Sellers","abstract":"We give two proofs for a formula that counts the number of partitions of \\(n\\) that have rank \u22122 or less (which we call Garden of Eden partitions). These partitions arise naturally in analyzing the game Bulgarian solitaire, summarized in Section 1. Section 2 presents a generating function argument for the formula based on Dyson\u2019s original paper where the rank of a partition is defined. Section 3 gives a combinatorial proof of the result, based on a bijection on Bressoud and Zeilberger.","comment":"","date_added":"2018-05-13","date_published":"2006-08-06","urls":["https:\/\/www.emis.de\/journals\/INTEGERS\/papers\/a19int2005\/a19int2005.Abstract.html","https:\/\/www.emis.de\/journals\/INTEGERS\/papers\/a19int2005\/a19int2005.pdf","http:\/\/www.personal.psu.edu\/jxs23\/HS_integers_final.pdf"],"collections":"Attention-grabbing titles,Easily explained,Combinatorics","url":"https:\/\/www.emis.de\/journals\/INTEGERS\/papers\/a19int2005\/a19int2005.Abstract.html https:\/\/www.emis.de\/journals\/INTEGERS\/papers\/a19int2005\/a19int2005.pdf http:\/\/www.personal.psu.edu\/jxs23\/HS_integers_final.pdf","year":"2006","urldate":"2018-05-13"},{"key":"AdescriptionoftheouterautomorphismofS6andtheinvariantsofsixpointsinprojectivespace","type":"article","title":"A description of the outer automorphism of \\(S_6\\), and the invariants of six points in projective space","author":"Ben Howard and John Millson and Andrew Snowden and Ravi Vakil","abstract":"We use a simple description of the outer automorphism of \\(S_6\\) to cleanly describe the invariant theory of six points in \\(\\mathbb{P}^1\\), \\(\\mathbb{P}^2\\), and \\(\\mathbb{P}^3\\).","comment":"The nobbly wobbly toy is based on this.","date_added":"2018-05-14","date_published":"2008-08-06","urls":["https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0097316508000125","http:\/\/math.stanford.edu\/~vakil\/files\/sixjan2308.pdf"],"collections":"Things to make and do,The groups group","url":"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0097316508000125 http:\/\/math.stanford.edu\/~vakil\/files\/sixjan2308.pdf","urldate":"2018-05-14","year":"2008"},{"key":"TheRearrangementNumber","type":"article","title":"The Rearrangement Number","author":"Andreas Blass and J\u00f6rg Brendle and Will Brian and Joel David Hamkins and Michael Hardy and Paul B. Larson","abstract":"How many permutations of the natural numbers are needed so that every\r\nconditionally convergent series of real numbers can be rearranged to no longer\r\nconverge to the same sum? We show that the minimum number of permutations\r\nneeded for this purpose, which we call the rearrangement number, is\r\nuncountable, but whether it equals the cardinal of the continuum is independent\r\nof the usual axioms of set theory. We compare the rearrangement number with\r\nseveral natural variants, for example one obtained by requiring the rearranged\r\nseries to still converge but to a new, finite limit. We also compare the\r\nrearrangement number with several well-studied cardinal characteristics of the\r\ncontinuum. We present some new forcing constructions designed to add\r\npermutations that rearrange series from the ground model in particular ways,\r\nthereby obtaining consistency results going beyond those that follow from\r\ncomparisons with familiar cardinal characteristics. Finally we deal briefly\r\nwith some variants concerning rearrangements by a special sort of permutations\r\nand with rearranging some divergent series to become (conditionally)\r\nconvergent.","comment":"","date_added":"2018-05-16","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1612.07830v1","http:\/\/arxiv.org\/pdf\/1612.07830v1"],"collections":"Combinatorics","url":"http:\/\/arxiv.org\/abs\/1612.07830v1 http:\/\/arxiv.org\/pdf\/1612.07830v1","year":"2016","urldate":"2018-05-16","archivePrefix":"arXiv","eprint":"1612.07830","primaryClass":"math.LO"},{"key":"SovietStreetMathematicsLandausLicensePlateGame","type":"article","title":"Soviet Street Mathematics: Landau\u2019s License Plate Game","author":"Harun \u0160iljak","abstract":"Lev Landau is considered one of the greatest physicists of the 20th century. Books by Landau and his student and collaborator Evgeny Lifshitz are a must-read in physics education around the world, and there are quite a few terms in physics that bear the name of the great Landau. However, this article is about something almost trivial: a mathematical game he enjoyed playing.","comment":"","date_added":"2018-05-17","date_published":"2018-08-06","urls":["https:\/\/link.springer.com\/article\/10.1007%2Fs00283-017-9743-9"],"collections":"Easily explained,Games to play with friends,Puzzles","url":"https:\/\/link.springer.com\/article\/10.1007%2Fs00283-017-9743-9","year":"2018","urldate":"2018-05-17"},{"key":"Neumbering","type":"article","title":"Neumbering","author":"O.G. Cassani and John H. Conway","abstract":"The importance of starting at 0 when counting has not often been discussed, nor has the incompatibility between this way of numbering and the\r\nusual adjectives first, second, third ... In fact, if the first number is zero, then the fifth is four and the ninth is eight, which is perfectly coherent with the traditional way of numbering, but it\u2019s confusing if we start from zero. This is a good reason to introduce John von Neumann\u2019s convention, which we call \u2018\u2018Neumbering.\u2019\u2019 The authors have been using this name privately, and we apologise for being slangy. This part of the paper starts by using it publicly.","comment":"","date_added":"2018-05-17","date_published":"2018-08-06","urls":["https:\/\/link.springer.com\/article\/10.1007\/s00283-017-9720-3"],"collections":"Easily explained,Notation and conventions","url":"https:\/\/link.springer.com\/article\/10.1007\/s00283-017-9720-3","year":"2018","urldate":"2018-05-17"},{"key":"RenyisParkingProblemRevisited","type":"article","title":"Renyi's Parking Problem Revisited","author":"Matthew P. Clay and Nandor J. Simanyi","abstract":"R\\'enyi's parking problem (or $1D$ sequential interval packing problem) dates\r\nback to 1958, when R\\'enyi studied the following random process: Consider an\r\ninterval $I$ of length $x$, and sequentially and randomly pack disjoint unit\r\nintervals in $I$ until the remaining space prevents placing any new segment.\r\nThe expected value of the measure of the covered part of $I$ is $M(x)$, so that\r\nthe ratio $M(x)\/x$ is the expected filling density of the random process.\r\nFollowing recent work by Gargano {\\it et al.} \\cite{GWML(2005)}, we studied the\r\ndiscretized version of the above process by considering the packing of the $1D$\r\ndiscrete lattice interval $\\{1,2,...,n+2k-1\\}$ with disjoint blocks of $(k+1)$\r\nintegers but, as opposed to the mentioned \\cite{GWML(2005)} result, our\r\nexclusion process is symmetric, hence more natural. Furthermore, we were able\r\nto obtain useful recursion formulas for the expected number of $r$-gaps ($0\\le\r\nr\\le k$) between neighboring blocks. We also provided very fast converging\r\nseries and extensive computer simulations for these expected numbers, so that\r\nthe limiting filling density of the long line segment (as $n\\to \\infty$) is\r\nR\\'enyi's famous parking constant, $0.7475979203...$.","comment":"","date_added":"2018-05-21","date_published":"2014-08-06","urls":["http:\/\/arxiv.org\/abs\/1406.1781v2","http:\/\/arxiv.org\/pdf\/1406.1781v2"],"collections":"Easily explained","url":"http:\/\/arxiv.org\/abs\/1406.1781v2 http:\/\/arxiv.org\/pdf\/1406.1781v2","year":"2014","urldate":"2018-05-21","archivePrefix":"arXiv","eprint":"1406.1781","primaryClass":"math.PR"},{"key":"OneParameterIsAlwaysEnough","type":"article","title":"One parameter is always enough","author":"Steven T. Piantadosi","abstract":"We construct an elementary equation with a single real valued parameter that is capable of fitting any \u201cscatter plot\u201d on any number of points to within a fixed precision. Specifically, given given a fixed \\(\\epsilon \\gt 0\\), we may construct \\(f_\\theta\\) so that for any collection of ordered pairs \\( \\{(x_j,y_j)\\}_{j=0}^n \\) with \\(n,x_j \\in \\mathbb{N}\\) and \\(y_j \\in (0,1)\\), there exists a \\(\\theta \\in [0,1]\\) giving \\(|f_\\theta(x_j)-y_j| \\lt \\epsilon\\) for all \\(j\\) simultaneously. To achieve this, we apply prior results about the logistic map, an iterated map in dynamical systems theory that can be solved exactly. The existence of an equation \\(f_\\theta\\) with this property highlights that \u201cparameter counting\u201d fails as a measure of\r\nmodel complexity when the class of models under consideration is only slightly broad.","comment":"","date_added":"2018-06-06","date_published":"2018-08-06","urls":["https:\/\/colala.bcs.rochester.edu\/papers\/piantadosi2018one.pdf"],"collections":"Attention-grabbing titles,Probability and statistics,Fun maths facts,Modelling","url":"https:\/\/colala.bcs.rochester.edu\/papers\/piantadosi2018one.pdf","year":"2018","urldate":"2018-06-06"},{"key":"Programmingquantumcomputersusing3Dpuzzlescoffeecupsanddoughnuts","type":"article","title":"Programming quantum computers using 3-D puzzles, coffee cups, and doughnuts","author":"Simon J. Devitt","abstract":"The task of programming a quantum computer is just as strange as quantum\r\nmechanics itself. But it now looks like a simple 3D puzzle may be the future\r\ntool of quantum software engineers.","comment":"","date_added":"2016-09-23","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1609.06628v1","http:\/\/arxiv.org\/pdf\/1609.06628v1"],"collections":"Unusual computers","url":"http:\/\/arxiv.org\/abs\/1609.06628v1 http:\/\/arxiv.org\/pdf\/1609.06628v1","urldate":"2016-09-23","year":"2016","archivePrefix":"arXiv","eprint":"1609.06628","primaryClass":"quant-ph"},{"key":"WhenareMultiplesofPolygonalNumbersagainPolygonalNumbers","type":"article","title":"When are Multiples of Polygonal Numbers again Polygonal Numbers?","author":"Jasbir S. Chahal and Nathan Priddis","abstract":"Euler showed that there are infinitely many triangular numbers that are three\r\ntimes another triangular number. In general, as we prove, it is an easy\r\nconsequence of the Pell equation that for a given square-free m > 1, the\r\nrelation D = mD' is satisfied by infinitely many pairs of triangular numbers D,\r\nD'. However, due to the erratic behavior of the fundamental solution to the\r\nPell equation, this problem is more difficult for more general polygonal\r\nnumbers. We will show that if one solution exists, then infinitely many exist.\r\nWe give an example, however, showing that there are cases where no solution\r\nexists. Finally, we also show in this paper that, given m > n > 1 with obvious\r\nexceptions, the simultaneous relations P = mP', P = nP\" has only finitely many\r\npossibilities not just for triangular numbers, but for triplets P, P', P\" of\r\npolygonal numbers.","comment":"","date_added":"2018-06-25","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1806.07981v1","http:\/\/arxiv.org\/pdf\/1806.07981v1"],"collections":"Easily explained,Fun maths facts,Integerology","url":"http:\/\/arxiv.org\/abs\/1806.07981v1 http:\/\/arxiv.org\/pdf\/1806.07981v1","urldate":"2018-06-25","year":"2018","archivePrefix":"arXiv","eprint":"1806.07981","primaryClass":"math.NT"},{"key":"AsurprisinglysimpledeBruijnsequenceconstruction","type":"article","title":"A surprisingly simple de Bruijn sequence construction","author":"Joe Sawada and Aaron Williams and DennisWong","abstract":"Pick any length \\(n\\) binary string \\(b_1 b_2 \\dots b_n\\) and remove the first bit \\(b_1\\). If \\(b_2 b_3 \\dots b_n 1\\) is a necklace then append the complement of \\(b_1\\) to the end of the remaining string; otherwise append \\(b_1\\). By repeating this process, eventually all \\(2^n\\) binary strings will be visited cyclically. This shift rule leads to a new de Bruijn sequence construction that can be generated in \\(O(1)\\)-amortized time per bit.","comment":"","date_added":"2018-06-25","date_published":"2016-08-06","urls":["https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0012365X15002873"],"collections":"Basically computer science,Combinatorics","url":"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0012365X15002873","year":"2016","urldate":"2018-06-25"},{"key":"ProofwithoutWordsFairAllocationofaPizza","type":"article","title":"Proof without Words: Fair Allocation of a Pizza","author":"Larry Carter and Stan Wagon","abstract":"","comment":"","date_added":"2018-06-26","date_published":"1994-08-06","urls":["https:\/\/www.jstor.org\/stable\/2690845"],"collections":"Easily explained,Protocols and strategies,Things to make and do,About proof,Fun maths facts","url":"https:\/\/www.jstor.org\/stable\/2690845","urldate":"2018-06-26","year":"1994"},{"key":"Howtoheartheshapeofabilliardtable","type":"article","title":"How to hear the shape of a billiard table","author":"Aaron Calderon and Solly Coles and Diana Davis and Justin Lanier and Andre Oliveira","abstract":"The bounce spectrum of a polygonal billiard table is the collection of all\r\nbi-infinite sequences of edge labels corresponding to billiard trajectories on\r\nthe table. We give methods for reconstructing from the bounce spectrum of a\r\npolygonal billiard table both the cyclic ordering of its edge labels and the\r\nsizes of its angles. We also show that it is impossible to reconstruct the\r\nexact shape of a polygonal billiard table from any finite collection of finite\r\nwords from its bounce spectrum.","comment":"","date_added":"2018-06-27","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1806.09644v1","http:\/\/arxiv.org\/pdf\/1806.09644v1"],"collections":"Basically physics,Easily explained,Geometry,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1806.09644v1 http:\/\/arxiv.org\/pdf\/1806.09644v1","year":"2018","urldate":"2018-06-27","archivePrefix":"arXiv","eprint":"1806.09644","primaryClass":"math.DS"},{"key":"NationalCurveBank","type":"article","title":"National Curve Bank","author":"Shirley B. Gray and Stewart Venit and Russ Abbott","abstract":"The National Curve Bank is a resource for students of mathematics. We strive to provide features - for example, animation and interaction - that a printed page cannot offer. We also include geometrical, algebraic, and historical aspects of curves, the kinds of attributes that make the mathematics special and enrich classroom learning.","comment":"","date_added":"2018-07-02","date_published":"2002-08-06","urls":["http:\/\/web.calstatela.edu\/curvebank\/home\/home.htm"],"collections":"Lists and catalogues,Geometry","url":"http:\/\/web.calstatela.edu\/curvebank\/home\/home.htm","year":"2002","urldate":"2018-07-02"},{"key":"OnSomeRegularToroids","type":"article","title":"On Some Regular Toroids","author":"Lajos Szilassi","abstract":"As it is known, in a regular polyhedron every face has the same number of edges and every vertex has the same number of edges, as well. A polyhedron is called topologically regular if further conditions (e.g. on the angle of the faces or the edges) are not imposed. An ordinary polyhedron is called a toroid if it is topologically torus-like (i.e. it can be converted to a torus by continuous deformation), and its faces are simple polygons. A toroid is said to be regular if it is topologically regular. It is easy to see, that the regular toroids can be classified into three classes, according to the number of edges of a vertex and of a face. There are infinitely many regular toroids in each class, because the number of the faces and vertices can be arbitrarily large. Hence, we study mainly those regular toroids, whose number of faces or vertices is minimal, or that ones, which have any other special properties. Among these polyhedra, we take special attention to the so called \"Cs\u00e1sz\u00e1r-polyhedron\", which has no diagonal, i.e. each pair of vertices are neighbouring, and its dual polyhedron (in topological sense) the so called \"Szilassi-polyhedron\", whose each pair of faces are neighbouring. The first one was found by \u00c1kos Cs\u00e1sz\u00e1r in 1949, and the latter one was found by the author, in 1977.","comment":"","date_added":"2018-07-26","date_published":"2001-08-06","urls":["http:\/\/symmetry-us.com\/Journals\/visbook\/szilassi\/"],"collections":"Easily explained,Things to make and do,Geometry","url":"http:\/\/symmetry-us.com\/Journals\/visbook\/szilassi\/","year":"2001","urldate":"2018-07-26"},{"key":"Euclidstheoremontheinfinitudeofprimesahistoricalsurveyofitsproofs300BC2017andanothernewproof","type":"article","title":"Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2017) and another new proof","author":"Romeo Me\u0161trovi\u0107","abstract":"In this article, we provide a comprehensive historical survey of 183\r\ndifferent proofs of famous Euclid's theorem on the infinitude of prime numbers.\r\nThe author is trying to collect almost all the known proofs on infinitude of\r\nprimes, including some proofs that can be easily obtained as consequences of\r\nsome known problems or divisibility properties. Furthermore, here are listed\r\nnumerous elementary proofs of the infinitude of primes in different arithmetic\r\nprogressions.\r\n All the references concerning the proofs of Euclid's theorem that use similar\r\nmethods and ideas are exposed subsequently. Namely, presented proofs are\r\ndivided into 8 subsections of Section 2 in dependence of the methods that are\r\nused in them. {\\bf Related new 14 proofs (2012-2017) are given in the last\r\nsubsection of Section 2.} In the next section, we survey mainly elementary\r\nproofs of the infinitude of primes in different arithmetic progressions.\r\nPresented proofs are special cases of Dirichlet's theorem. In Section 4, we\r\ngive a new simple \"Euclidean's proof\" of the infinitude of primes.","comment":"","date_added":"2018-08-21","date_published":"2012-08-06","urls":["http:\/\/arxiv.org\/abs\/1202.3670v3","http:\/\/arxiv.org\/pdf\/1202.3670v3"],"collections":"Easily explained,History,Lists and catalogues,The act of doing maths,About proof,Integerology","url":"http:\/\/arxiv.org\/abs\/1202.3670v3 http:\/\/arxiv.org\/pdf\/1202.3670v3","year":"2012","urldate":"2018-08-21","archivePrefix":"arXiv","eprint":"1202.3670","primaryClass":"math.HO"},{"key":"TheNamerClaimergame","type":"article","title":"The Namer-Claimer game","author":"Ben Barber","abstract":"In each round of the Namer-Claimer game, Namer names a distance d, then\r\nClaimer claims a subset of [n] that does not contain two points that differ by\r\nd. Claimer wins once they have claimed sets covering [n]. I show that the\r\nlength of this game is of order log log n with optimal play from each side.","comment":"","date_added":"2018-09-04","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1808.10800v1","http:\/\/arxiv.org\/pdf\/1808.10800v1"],"collections":"Easily explained,Games to play with friends","url":"http:\/\/arxiv.org\/abs\/1808.10800v1 http:\/\/arxiv.org\/pdf\/1808.10800v1","year":"2018","urldate":"2018-09-04","archivePrefix":"arXiv","eprint":"1808.10800","primaryClass":"math.CO"},{"key":"MathematicalWriting","type":"article","title":"Mathematical Writing","author":"Donald E. Knuth and Tracy Larrabee and Paul M. Roberts","abstract":"This report is based on a course of the same name given at Stanford University during autumn quarter, 1987. Here\u2019s the catalog description:\r\n\r\nCS 209. Mathematical Writing \u2014 Issues of technical writing and the effective presentation of mathematics and computer science. Preparation of theses, papers, books, and \u201cliterate\u201d computer programs. A term paper on a topic of your choice; this paper may be used for credit in another course.","comment":"","date_added":"2018-09-19","date_published":"1987-08-06","urls":["http:\/\/jmlr.csail.mit.edu\/reviewing-papers\/knuth_mathematical_writing.pdf"],"collections":"Notation and conventions,The act of doing maths,Education","url":"http:\/\/jmlr.csail.mit.edu\/reviewing-papers\/knuth_mathematical_writing.pdf","year":"1987","urldate":"2018-09-19"},{"key":"TheSplittingAlgorithmforEgyptianFractions","type":"article","title":"The Splitting Algorithm for Egyptian Fractions","author":"L. Beeckmans","abstract":"The purpose of this paper is to answer a question raised by Stewart in 1964; we prove that the so-called splitting algorithm for Egyptian fractions based on the identity 1\/x = 1\/(x + 1) + 1\/x(x + 1) terminates.","comment":"","date_added":"2018-09-20","date_published":"1991-08-06","urls":["https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0022314X83710152?via%3Dihub"],"collections":"Easily explained","url":"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0022314X83710152?via%3Dihub","year":"1991","urldate":"2018-09-20"},{"key":"FindingtheBanditinaGraphSequentialSearchandStop","type":"article","title":"Finding the Bandit in a Graph: Sequential Search-and-Stop","author":"Pierre Perrault and Vianney Perchet and Michal Valko","abstract":"We consider the problem where an agent wants to find a hidden object that is\r\nrandomly located in some vertex of a directed acyclic graph (DAG) according to\r\na fixed but possibly unknown distribution. The agent can only examine vertices\r\nwhose in-neighbors have already been examined. In scheduling theory, this\r\nproblem is denoted by $1|prec|\\sum w_jC_j$. However, in this paper, we address\r\nlearning setting where we allow the agent to stop before having found the\r\nobject and restart searching on a new independent instance of the same problem.\r\nThe goal is to maximize the total number of hidden objects found under a time\r\nconstraint. The agent can thus skip an instance after realizing that it would\r\nspend too much time on it. Our contributions are both to the search theory and\r\nmulti-armed bandits. If the distribution is known, we provide a quasi-optimal\r\ngreedy strategy with the help of known computationally efficient algorithms for\r\nsolving $1|prec|\\sum w_jC_j$ under some assumption on the DAG. If the\r\ndistribution is unknown, we show how to sequentially learn it and, at the same\r\ntime, act near-optimally in order to collect as many hidden objects as\r\npossible. We provide an algorithm, prove theoretical guarantees, and\r\nempirically show that it outperforms the na\\\"ive baseline.","comment":"","date_added":"2018-09-22","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1806.02282v1","http:\/\/arxiv.org\/pdf\/1806.02282v1"],"collections":"Attention-grabbing titles,Protocols and strategies","url":"http:\/\/arxiv.org\/abs\/1806.02282v1 http:\/\/arxiv.org\/pdf\/1806.02282v1","year":"2018","urldate":"2018-09-22","archivePrefix":"arXiv","eprint":"1806.02282","primaryClass":"stat.ML"},{"key":"Thelargestsmallhexagon","type":"article","title":"The largest small hexagon","author":"R. L. Graham","abstract":"The problem of determining the largest area a plane hexagon of unit diameter can have, raised some 20 years ago by H. Lenz, is settled. It is shown that such a hexagon is unique and has an area exceeding that of a regular hexagon of unit diameter by about 4%.","comment":"","date_added":"2018-09-22","date_published":"1975-08-06","urls":["https:\/\/www.sciencedirect.com\/science\/article\/pii\/0097316575900047"],"collections":"Attention-grabbing titles,Easily explained,Geometry,Fun maths facts","url":"https:\/\/www.sciencedirect.com\/science\/article\/pii\/0097316575900047","year":"1975","urldate":"2018-09-22"},{"key":"Powerlawdistributionsinempiricaldata","type":"article","title":"Power-law distributions in empirical data","author":"Aaron Clauset and Cosma Rohilla Shalizi and M. E. J. Newman","abstract":"Power-law distributions occur in many situations of scientific interest and\r\nhave significant consequences for our understanding of natural and man-made\r\nphenomena. Unfortunately, the detection and characterization of power laws is\r\ncomplicated by the large fluctuations that occur in the tail of the\r\ndistribution -- the part of the distribution representing large but rare events\r\n-- and by the difficulty of identifying the range over which power-law behavior\r\nholds. Commonly used methods for analyzing power-law data, such as\r\nleast-squares fitting, can produce substantially inaccurate estimates of\r\nparameters for power-law distributions, and even in cases where such methods\r\nreturn accurate answers they are still unsatisfactory because they give no\r\nindication of whether the data obey a power law at all. Here we present a\r\nprincipled statistical framework for discerning and quantifying power-law\r\nbehavior in empirical data. Our approach combines maximum-likelihood fitting\r\nmethods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic\r\nand likelihood ratios. We evaluate the effectiveness of the approach with tests\r\non synthetic data and give critical comparisons to previous approaches. We also\r\napply the proposed methods to twenty-four real-world data sets from a range of\r\ndifferent disciplines, each of which has been conjectured to follow a power-law\r\ndistribution. In some cases we find these conjectures to be consistent with the\r\ndata while in others the power law is ruled out.","comment":"","date_added":"2018-09-24","date_published":"2007-08-06","urls":["http:\/\/arxiv.org\/abs\/0706.1062v2","http:\/\/arxiv.org\/pdf\/0706.1062v2"],"collections":"Drama!,Probability and statistics","url":"http:\/\/arxiv.org\/abs\/0706.1062v2 http:\/\/arxiv.org\/pdf\/0706.1062v2","year":"2007","urldate":"2018-09-24","archivePrefix":"arXiv","eprint":"0706.1062","primaryClass":"physics.data-an"},{"key":"Searchingforgeneralizedbinarynumbersystems","type":"online","title":"Searching for generalized binary number systems","author":"Attila Kov\u00e1cs","abstract":"The aim of the project is to find all the generalized binary number systems up to dimension 11. Below we give a short description of the number system concept and mention a few possible applications.","comment":"","date_added":"2016-08-22","date_published":"2017-08-06","urls":["http:\/\/szdg.lpds.sztaki.hu\/assets\/desc_numsys_es.php"],"collections":"Unusual arithmetic,Fun maths facts,Integerology","url":"http:\/\/szdg.lpds.sztaki.hu\/assets\/desc_numsys_es.php","urldate":"2016-08-22","year":"2017"},{"key":"SettingLinearAlgebraProblems","type":"article","title":"Setting linear algebra problems","author":"John D. Steele","abstract":"In this report I collect together some of the techniques I have evolved for setting linear algebra problems, with particular attention paid towards ensuring relatively easy arithmetic. Some are given as MAPLE routines.","comment":"","date_added":"2018-09-27","date_published":"1997-08-06","urls":["http:\/\/web.maths.unsw.edu.au\/~jds\/Papers\/linalg.pdf"],"collections":"Education","url":"http:\/\/web.maths.unsw.edu.au\/~jds\/Papers\/linalg.pdf","year":"1997","urldate":"2018-09-27"},{"key":"NearMissPolyhedra","type":"online","title":"Near Miss Polyhedra","author":"Jim McNeill","abstract":"The polyhedra on this page are not quite regular, but as they are close I present them here as 'near misses'. ","comment":"","date_added":"2018-01-10","date_published":"2002-08-06","urls":["http:\/\/www.orchidpalms.com\/polyhedra\/acrohedra\/nearmiss\/nearmiss.htm"],"collections":"The act of doing maths,Geometry","url":"http:\/\/www.orchidpalms.com\/polyhedra\/acrohedra\/nearmiss\/nearmiss.htm","urldate":"2018-01-10","year":"2002"},{"key":"Thenumberdictionary","type":"online","title":"The number dictionary","author":"","abstract":"The purpose is to provide an opportunity to show properties of numbers.","comment":"","date_added":"2017-04-12","date_published":"2008-08-06","urls":["http:\/\/numdic.com\/"],"collections":"Lists and catalogues,Integerology","url":"http:\/\/numdic.com\/","urldate":"2017-04-12","year":"2008"},{"key":"Jerabek2016","type":"article","title":"Division by zero","author":"Emil Je\u0159\u00e1bek","abstract":"As a consequence of the MRDP theorem, the set of Diophantine equations provably unsolvable in any sufficiently strong theory of arithmetic is algorithmically undecidable. In contrast, we show the decidability of Diophantine equations provably unsolvable in Robinson's arithmetic Q. The argument hinges on an analysis of a particular class of equations, hitherto unexplored in Diophantine literature. We also axiomatize the universal fragment of Q in the process.","comment":"","date_added":"2016-04-26","date_published":"2016-04-01","urls":["http:\/\/arxiv.org\/abs\/1604.07309","http:\/\/arxiv.org\/pdf\/1604.07309v1"],"collections":"About proof","url":"http:\/\/arxiv.org\/abs\/1604.07309 http:\/\/arxiv.org\/pdf\/1604.07309v1","urldate":"2016-04-26","year":"2016","month":"apr","pages":"12","archivePrefix":"arXiv","eprint":"1604.07309","primaryClass":"math.LO"},{"key":"VanHarreveld1939","type":"article","title":"Doubly-, triply-, quadruply- and quintuply-innervated crustacean muscles","author":"van Harreveld, A.","abstract":"","comment":"","date_added":"2011-08-09","date_published":"1939-04-01","urls":["http:\/\/doi.wiley.com\/10.1002\/cne.900700208"],"collections":"Animals","url":"http:\/\/doi.wiley.com\/10.1002\/cne.900700208","urldate":"2011-08-09","year":"1939","issn":"0021-9967","journal":"The Journal of Comparative Neurology","month":"apr","number":"2","pages":"285--296","volume":"70"},{"key":"Szekely2005","type":"article","title":"Half of a coin: negative probabilities","author":"Sz\u00e9kely, GJ","abstract":"","comment":"","date_added":"2013-07-17","date_published":"2005-08-06","urls":["http:\/\/citeseerx.ist.psu.edu\/viewdoc\/download;jsessionid=31A106EA94D27A532BC5142A6E7F621C?doi=10.1.1.592.2043&rep=rep1&type=pdf"],"collections":"Probability and statistics,Unusual arithmetic,Fun maths facts","url":"http:\/\/citeseerx.ist.psu.edu\/viewdoc\/download;jsessionid=31A106EA94D27A532BC5142A6E7F621C?doi=10.1.1.592.2043&rep=rep1&type=pdf","urldate":"2013-07-17","year":"2005","journal":"Wilmott Magazine","pages":"66--68"},{"key":"Pierce2009","type":"article","title":"The Circle-Squaring Problem Decomposed","author":"Pierce, Pamela and Ramsay, John","abstract":"","comment":"","date_added":"2013-03-17","date_published":"2009-08-06","urls":["https:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Evans\/Pierce.pdf"],"collections":"Geometry","url":"https:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Evans\/Pierce.pdf","urldate":"2013-03-17","year":"2009","journal":"Math \\ldots","number":"November","pages":"19--22"},{"key":"Hutter2002","type":"article","title":"The Fastest and Shortest Algorithm for All Well-Defined Problems","author":"Hutter, Marcus","abstract":"An algorithm M is described that solves any well-defined problem p as quickly as the fastest algorithm computing a solution to p, save for a factor of 5 and low-order additive terms. M optimally distributes resources between the execution of provably correct p-solving programs and an enumeration of all proofs, including relevant proofs of program correctness and of time bounds on program runtimes. M avoids Blum's speed-up theorem by ignoring programs without correctness proof. M has broader applicability and can be faster than Levin's universal search, the fastest method for inverting functions save for a large multiplicative constant. An extension of Kolmogorov complexity and two novel natural measures of function complexity are used to show that the most efficient program computing some function f is also among the shortest programs provably computing f. \r\n\r\n","comment":"","date_added":"2012-08-14","date_published":"2002-08-06","urls":["http:\/\/www.hutter1.net\/ai\/pfastprg.htm","http:\/\/arxiv.org\/abs\/cs.CC\/0206022","https:\/\/arxiv.org\/pdf\/cs\/0206022.pdf"],"collections":"Attention-grabbing titles,Basically computer science","url":"http:\/\/www.hutter1.net\/ai\/pfastprg.htm http:\/\/arxiv.org\/abs\/cs.CC\/0206022 https:\/\/arxiv.org\/pdf\/cs\/0206022.pdf","urldate":"2012-08-14","year":"2002","journal":"International Journal of Foundations of Computer Science","keywords":"Kolmogorov complexity,acceleration,algorithmic information theory,blum's speed-up theorem,computational complexity,levin search","number":"3","pages":"431--443","volume":"13"},{"key":"Shelah1992","type":"article","title":"Cardinal arithmetic for skeptics","author":"Shelah, Saharon","abstract":"When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with \"consistency\" rather than \"truth\" may be felt to give the subject an air of unreality. Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of $2^{\\aleph_0}$, cannot be settled on the basis of the usual axioms of set theory (ZFC). Although much can be said in favor of such independence results, rather than undertaking to challenge such prejudices, we have a more modest goal; we wish to point out an area of contemporary set theory in which theorems are abundant, although the conventional wisdom views the subject as dominated by independence results, namely, cardinal arithmetic.","comment":"","date_added":"2012-05-02","date_published":"1992-04-01","urls":["http:\/\/www.ams.org\/journals\/bull\/1992-26-02\/S0273-0979-1992-00261-6\/S0273-0979-1992-00261-6.pdf","http:\/\/arxiv.org\/pdf\/math\/9201251v1.pdf"],"collections":"Attention-grabbing titles,The act of doing maths,Unusual arithmetic","url":"http:\/\/www.ams.org\/journals\/bull\/1992-26-02\/S0273-0979-1992-00261-6\/S0273-0979-1992-00261-6.pdf http:\/\/arxiv.org\/pdf\/math\/9201251v1.pdf","urldate":"2012-05-02","year":"1992","doi":"10.1090\/S0273-0979-1992-00261-6","issn":"0273-0979","journal":"Bulletin of the American Mathematical Society","month":"apr","number":"2","pages":"197--211","volume":"26"},{"key":"Shoemake1985","type":"article","title":"Animating rotation with quaternion curves","author":"Shoemake, Ken","abstract":"Solid bodies roll and tumble through space. In computer animation, so do cameras. The rotations of these objects are best described using a four coordinate system, quaternions, as is shown in this paper. Of all quaternions, those on the unit sphere are most suitable for animation, but the question of how to construct curves on spheres has not been much explored. This paper gives one answer by presenting a new kind of spline curve, created on a sphere, suitable for smoothly in-betweening (i.e. interpolating) sequences of arbitrary rotations. Both theory and experiment show that the motion generated is smooth and natural, without quirks found in earlier methods.","comment":"","date_added":"2011-01-12","date_published":"1985-08-06","urls":["http:\/\/portal.acm.org\/citation.cfm?doid=325334.325242"],"collections":"Basically computer science","url":"http:\/\/portal.acm.org\/citation.cfm?doid=325334.325242","urldate":"2011-01-12","year":"1985","journal":"International Conference on Computer Graphics and Interactive Techniques","keywords":"B-spline,B\u00e9zier curve,animation,approximation,in-betweening,interpolation,quaternion,rotation,spherical geometry,spline","number":"3","pages":"245","volume":"19"},{"key":"item38","type":"misc","title":"The Stick Problem","author":"Augustine Bertagnolli","abstract":"Given sticks of possible sizes one through six, what is the smallest number of sticks you can have\r\nto ensure that you are able to form a perfect square? The Pigeonhole Principle tells us that if we have\r\nnineteen sticks we would have at least four of one of the sizes, but can we do better if we take partitions\r\ninto account? This is one case of the stick problem which, though simple in statement, proves to be\r\nnot so simple in solution. In this paper, we define the stick problem clearly, discuss our methods for\r\napproaching and simplifying the problem, provide an algorithm for generating solutions, and present\r\nsome computer generated solutions for specific cases.","comment":"","date_added":"2014-01-13","date_published":"2013-08-06","urls":["http:\/\/ajbertagnolli.com\/wp-content\/uploads\/2013\/10\/sticks2.pdf"],"collections":"Easily explained,Geometry,Puzzles","url":"http:\/\/ajbertagnolli.com\/wp-content\/uploads\/2013\/10\/sticks2.pdf","urldate":"2014-01-13","year":"2013"},{"key":"Romano","type":"article","title":"Deriving Uniform Polyhedra with Wythoff's Construction","author":"Romano, Don","abstract":"","comment":"","date_added":"2011-05-10","date_published":"2010-08-06","urls":["http:\/\/math.ucdenver.edu\/~mferrara\/seminar\/F2010\/Don_Romano_Seminar.pdf"],"collections":"Geometry","url":"http:\/\/math.ucdenver.edu\/~mferrara\/seminar\/F2010\/Don_Romano_Seminar.pdf","urldate":"2011-05-10","year":"2010"},{"key":"CalculatorForensics","type":"online","title":"Calculator Forensics","author":"Mike Sebastian","abstract":"Results from the evaluation of this equation in degrees mode: arcsin (arccos (arctan (tan (cos (sin (9) ) ) ) ) ) ","comment":"","date_added":"2018-10-16","date_published":"2000-08-06","urls":["http:\/\/www.rskey.org\/~mwsebastian\/miscprj\/results.htm"],"collections":"Basically computer science,Easily explained,Lists and catalogues","url":"http:\/\/www.rskey.org\/~mwsebastian\/miscprj\/results.htm","year":"2000","urldate":"2018-10-16"},{"key":"Shima2012","type":"article","title":"How far can Tarzan jump?","author":"Shima, Hiroyuki","abstract":"The tree-based rope swing is a popular recreation facility, often installed in outdoor areas, giving pleasure to thrill-seekers. In the setting, one drops down from a high platform, hanging from a rope, then swings at a great speed like \"Tarzan\", and finally jumps ahead to land on the ground. The question now arises: How far can Tarzan jump by the swing? In this article, I present an introductory analysis of the Tarzan swing mechanics, a big pendulum-like swing with Tarzan himself attached as weight. The analysis enables determination of how farther forward Tarzan can jump using a given swing apparatus. The discussion is based on elementary mechanics and, therefore, expected to provide rich opportunities for investigations using analytic and numerical methods.","comment":"","date_added":"2012-09-02","date_published":"2012-08-01","urls":["http:\/\/arxiv.org\/abs\/1208.4355","http:\/\/arxiv.org\/pdf\/1208.4355v1"],"collections":"Basically physics,Easily explained","url":"http:\/\/arxiv.org\/abs\/1208.4355 http:\/\/arxiv.org\/pdf\/1208.4355v1","urldate":"2012-09-02","year":"2012","archivePrefix":"arXiv","arxivId":"1208.4355","eprint":"1208.4355","month":"aug","pages":"8","primaryClass":"physics.pop-ph"},{"key":"Henderson","type":"misc","title":"The Collatz Fractal","author":"Henderson, Xander","abstract":"","comment":"","date_added":"2012-01-15","date_published":"2012-08-06","urls":["http:\/\/yozh.org\/2012\/01\/12\/the_collatz_fractal\/"],"collections":"Easily explained,Fun maths facts","url":"http:\/\/yozh.org\/2012\/01\/12\/the_collatz_fractal\/","urldate":"2012-01-15","year":"2012"},{"key":"Enumerationofmarycacti","type":"article","title":"Enumeration of m-ary cacti","author":"Miklos Bona and Michel Bousquet and Gilbert Labelle and Pierre Leroux","abstract":"The purpose of this paper is to enumerate various classes of cyclically\r\ncolored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is\r\nmotivated by the topological classification of complex polynomials having at\r\nmost m critical values, studied by Zvonkin and others. We obtain explicit\r\nformulae for both labelled and unlabelled m-ary cacti, according to i) the\r\nnumber of polygons, ii) the vertex-color distribution, iii) the vertex-degree\r\ndistribution of each color. We also enumerate m-ary cacti according to the\r\norder of their automorphism group. Using a generalization of Otter's formula,\r\nwe express the species of m-ary cacti in terms of rooted and of pointed cacti.\r\nA variant of the m-dimensional Lagrange inversion is then used to enumerate\r\nthese structures. The method of Liskovets for the enumeration of unrooted\r\nplanar maps can also be adapted to m-ary cacti.","comment":"The number of binary cacti is the number of Truchet tiles.","date_added":"2018-10-23","date_published":"1998-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/9804119v2","http:\/\/arxiv.org\/pdf\/math\/9804119v2"],"collections":"Attention-grabbing titles,Combinatorics,Things to make and do","url":"http:\/\/arxiv.org\/abs\/math\/9804119v2 http:\/\/arxiv.org\/pdf\/math\/9804119v2","year":"1998","urldate":"2018-10-23","archivePrefix":"arXiv","eprint":"math\/9804119","primaryClass":"math.CO"},{"key":"SomeFundamentalTheoremsInMathematics","type":"article","title":"Some Fundamental Theorems in Mathematics","author":"Oliver Knill","abstract":"An expository hitchhiker's guide to some theorems in mathematics.","comment":"","date_added":"2018-10-24","date_published":"2018-08-06","urls":["http:\/\/www.math.harvard.edu\/~knill\/graphgeometry\/papers\/fundamental.pdf"],"collections":"Lists and catalogues","url":"http:\/\/www.math.harvard.edu\/~knill\/graphgeometry\/papers\/fundamental.pdf","year":"2018","urldate":"2018-10-24"},{"key":"Noncrossingpartitionsunderrotationandreflection","type":"article","title":"Noncrossing partitions under rotation and reflection","author":"David Callan and Len Smiley","abstract":"We consider noncrossing partitions of [n] under the action of (i) the\r\nreflection group (of order 2), (ii) the rotation group (cyclic of order n) and\r\n(iii) the rotation\/reflection group (dihedral of order 2n). First, we exhibit a\r\nbijection from rotation classes to bicolored plane trees on n edges, and\r\nconsider its implications. Then we count noncrossing partitions of [n]\r\ninvariant under reflection and show that, somewhat surprisingly, they are\r\nequinumerous with rotation classes invariant under reflection. The proof uses a\r\npretty involution originating in work of Germain Kreweras. We conjecture that\r\nthe \"equinumerous\" result also holds for arbitrary partitions of [n].","comment":"","date_added":"2018-10-27","date_published":"2005-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/0510447v3","http:\/\/arxiv.org\/pdf\/math\/0510447v3"],"collections":"Combinatorics,Easily explained,Geometry,Things to make and do","url":"http:\/\/arxiv.org\/abs\/math\/0510447v3 http:\/\/arxiv.org\/pdf\/math\/0510447v3","year":"2005","urldate":"2018-10-27","archivePrefix":"arXiv","eprint":"math\/0510447","primaryClass":"math.CO"},{"key":"Conwaysdoughnuts","type":"article","title":"Conway's doughnuts","author":"Peter Doyle and Shikhin Sethi","abstract":"Morley's Theorem about angle trisectors can be viewed as the statement that a\r\ncertain diagram `exists', meaning that triangles of prescribed shapes meet in a\r\nprescribed pattern. This diagram is the case n=3 of a class of diagrams we call\r\n`Conway's doughnuts'. These diagrams can be proven to exist using John\r\nSmillie's holonomy method, recently championed by Eric Braude: `Guess the\r\nshapes; check the holonomy.' For n = 2, 3, 4 the existence of the doughnut\r\nhappens to be easy to prove because the hole is absent or triangular.","comment":"","date_added":"2018-11-04","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1804.04024v1","http:\/\/arxiv.org\/pdf\/1804.04024v1"],"collections":"Food,Fun maths facts,Geometry","url":"http:\/\/arxiv.org\/abs\/1804.04024v1 http:\/\/arxiv.org\/pdf\/1804.04024v1","year":"2018","urldate":"2018-11-04","archivePrefix":"arXiv","eprint":"1804.04024","primaryClass":"math.HO"},{"key":"Elsholtz2002","type":"misc","title":"A combinatorial approach to sums of two squares and related problems","author":"Elsholtz, Christian","abstract":"Heath-Brown [6] suggested a short proof of the two squares theorem, thereby simplifying ideas of Liouville. Zagier [15] suggested a particularly neat form of this, a \"One sentence proof\". It consists of two suitable involutions on the finite set of the solutions of p = x 2 +4yz in positive integers. A parity argument ensures the existence of a solution with y = z. The proof can be stated in one sentence since elementary calculations (namely to check that the mappings are well defined and are indeed involutory) can be left to the reader. The proof remained somewhat mysterious, since it is not obvious, where these mappings come from. In this paper we reveal this mystery and systematically explore similar proofs that can be given for related problems. We show that the very same method proves results on p = x 2 +2y 2 (see also Jackson [7]), p = x 2 2y 2 , p = 3x 2 + 4y 2 , and p = 3x 2 4y 2 .","comment":"","date_added":"2010-07-16","date_published":"2002-08-06","urls":["https:\/\/link.springer.com\/chapter\/10.1007\/978-0-387-68361-4_8","http:\/\/www.math.tugraz.at\/~elsholtz\/WWW\/papers\/papers.html","http:\/\/www.math.tugraz.at\/~elsholtz\/WWW\/papers\/zagierenglish9thjuly2002.ps"],"collections":"About proof,Fun maths facts","url":"https:\/\/link.springer.com\/chapter\/10.1007\/978-0-387-68361-4_8 http:\/\/www.math.tugraz.at\/~elsholtz\/WWW\/papers\/papers.html http:\/\/www.math.tugraz.at\/~elsholtz\/WWW\/papers\/zagierenglish9thjuly2002.ps","urldate":"2010-07-16","year":"2002"},{"key":"PrimeNumberRaces","type":"article","title":"Prime Number Races","author":"Andrew Granville and Greg Martin","abstract":"This is a survey article on prime number races. Chebyshev noticed in the\r\nfirst half of the nineteenth century that for any given value of x, there\r\nalways seem to be more primes of the form 4n+3 less than x then there are of\r\nthe form 4n+1. Similar observations have been made with primes of the form 3n+2\r\nand 3n+1, with primes of the form 10n+3\/10n+7 and 10n+1\/10n+9, and many others\r\nbesides. More generally, one can consider primes of the form qn+a, qn+b, qn+c,\r\n>... for our favorite constants q, a, b, c, ... and try to figure out which\r\nforms are \"preferred\" over the others. In this paper, we describe these\r\nphenomena in greater detail and explain the efforts that have been made at\r\nunderstanding them.","comment":"","date_added":"2018-11-12","date_published":"2004-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/0408319v1","http:\/\/arxiv.org\/pdf\/math\/0408319v1"],"collections":"Attention-grabbing titles,Easily explained,Fun maths facts,Integerology","url":"http:\/\/arxiv.org\/abs\/math\/0408319v1 http:\/\/arxiv.org\/pdf\/math\/0408319v1","year":"2004","urldate":"2018-11-12","archivePrefix":"arXiv","eprint":"math\/0408319","primaryClass":"math.NT"},{"key":"DuotoneTruchetliketilings","type":"article","title":"Duotone Truchet-like tilings","author":"Cameron Browne","abstract":"This paper explores methods for colouring Truchet-like tiles, with an emphasis on the resulting visual patterns and designs. The methods are extended to non-square tilings that allow Truchet-like patterns of noticeably different character. Underlying parity issues are briefly discussed and solutions presented for parity problems that arise for tiles with odd numbers of sides. A new tile design called the arch tile is introduced and its artistic use demonstrated.","comment":"","date_added":"2018-11-13","date_published":"2009-08-06","urls":["https:\/\/www.tandfonline.com\/doi\/full\/10.1080\/17513470902718252?scroll=top&needAccess=true"],"collections":"Art,Easily explained,Geometry,Things to make and do","url":"https:\/\/www.tandfonline.com\/doi\/full\/10.1080\/17513470902718252?scroll=top&needAccess=true","year":"2009","urldate":"2018-11-13"},{"key":"SevenTreesinOne","type":"article","title":"Seven Trees in One","author":"Andreas Blass","abstract":"Following a remark of Lawvere, we explicitly exhibit a particularly\r\nelementary bijection between the set T of finite binary trees and the set T^7\r\nof seven-tuples of such trees. \"Particularly elementary\" means that the\r\napplication of the bijection to a seven-tuple of trees involves case\r\ndistinctions only down to a fixed depth (namely four) in the given seven-tuple.\r\nWe clarify how this and similar bijections are related to the free commutative\r\nsemiring on one generator X subject to X=1+X^2. Finally, our main theorem is\r\nthat the existence of particularly elementary bijections can be deduced from\r\nthe provable existence, in intuitionistic type theory, of any bijections at\r\nall.","comment":"","date_added":"2018-11-26","date_published":"1994-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/9405205v1","http:\/\/arxiv.org\/pdf\/math\/9405205v1"],"collections":"Fun maths facts,Unusual arithmetic","url":"http:\/\/arxiv.org\/abs\/math\/9405205v1 http:\/\/arxiv.org\/pdf\/math\/9405205v1","year":"1994","urldate":"2018-11-26","archivePrefix":"arXiv","eprint":"math\/9405205","primaryClass":"math.LO"},{"key":"RectangleArithmetic","type":"article","title":"Rectangle Arithmetic","author":"Bill Gosper","abstract":"Another slant on fractions","comment":"","date_added":"2018-11-27","date_published":"2006-08-06","urls":["http:\/\/gosper.org\/rectarith12.pdf"],"collections":"Easily explained,Geometry,Unusual arithmetic","url":"http:\/\/gosper.org\/rectarith12.pdf","urldate":"2018-11-27","year":"2006"},{"key":"Mathematicsappliedtodressmaking","type":"article","title":"Mathematics applied to dressmaking","author":"Christopher Zeeman","abstract":"Dressmaking can raise interesting questions in both geometry and topology. My own involvement began in Bangkok, where I once bought a dress-length of some rather beautiful Thai silk. Unfortunately when I got home all the dress-makers claimed it wasn't long enough to make a dress. It became clear that I either had to abandon the project or make the thing myself.","comment":"","date_added":"2018-11-29","date_published":"1994-08-06","urls":["https:\/\/www.lms.ac.uk\/content\/mathematics-applied-dressmaking","https:\/\/www.lms.ac.uk\/sites\/lms.ac.uk\/files\/1994%20Mathematics%20applied%20to%20dressmaking%20%28preprint%29.pdf"],"collections":"Easily explained,Fun maths facts,Things to make and do","url":"https:\/\/www.lms.ac.uk\/content\/mathematics-applied-dressmaking https:\/\/www.lms.ac.uk\/sites\/lms.ac.uk\/files\/1994%20Mathematics%20applied%20to%20dressmaking%20%28preprint%29.pdf","urldate":"2018-11-29","year":"1994"},{"key":"AmusingPermutationRepresentationsofGroupExtensions","type":"article","title":"Amusing Permutation Representations of Group Extensions","author":"Yongju Bae and J. Scott Carter and Byeorhi Kim","abstract":"Wreath products of finite groups have permutation representations that are\r\nconstructed from the permutation representations of their constituents. One can\r\nenvision these in a metaphoric sense in which a rope is made from a bundle of\r\nthreads. In this way, subgroups and quotients are easily visualized. The\r\ngeneral idea is applied to the finite subgroups of the special unitary group of\r\n$(2\\times 2)$-matrices. Amusing diagrams are developed that describe the unit\r\nquaternions, the binary tetrahedral, octahedral, and icosahedral group as well\r\nas the dicyclic groups. In all cases, the quotients as subgroups of the\r\npermutation group are readily apparent. These permutation representations lead\r\nto injective homomorphisms into wreath products.","comment":"","date_added":"2019-01-01","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1812.08475v1","http:\/\/arxiv.org\/pdf\/1812.08475v1"],"collections":"Attention-grabbing titles,Fun maths facts,The groups group","url":"http:\/\/arxiv.org\/abs\/1812.08475v1 http:\/\/arxiv.org\/pdf\/1812.08475v1","year":"2018","urldate":"2019-01-01","archivePrefix":"arXiv","eprint":"1812.08475","primaryClass":"math.GT"},{"key":"DoingMathinJestReflectionsonUselessMaththeUnreasonableEffectivenessofMathematicsandtheEthicalObligationsofMathematicians","type":"article","title":"Doing Math in Jest: Reflections on Useless Math, the Unreasonable Effectiveness of Mathematics, and the Ethical Obligations of Mathematicians","author":"Gizem Karaali","abstract":"Mathematicians occasionally discover interesting truths even when they are\r\nplaying with mathematical ideas with no thoughts about possible consequences of\r\ntheir actions. This paper describes two specific instances of this phenomenon.\r\nThe discussion touches upon the theme of the unreasonable effectiveness of\r\nmathematics as well as the ethical obligations of mathematicians.","comment":"","date_added":"2019-01-01","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1812.09601v1","http:\/\/arxiv.org\/pdf\/1812.09601v1"],"collections":"The act of doing maths","url":"http:\/\/arxiv.org\/abs\/1812.09601v1 http:\/\/arxiv.org\/pdf\/1812.09601v1","year":"2018","urldate":"2019-01-01","archivePrefix":"arXiv","eprint":"1812.09601","primaryClass":"math.HO"},{"key":"Whatisaclosedformnumber","type":"article","title":"What is a closed-form number?","author":"Timothy Y. Chow","abstract":"If a student asks for an antiderivative of exp(x^2), there is a standard\r\nreply: the answer is not an elementary function. But if a student asks for a\r\nclosed-form expression for the real root of x = cos(x), there is no standard\r\nreply. We propose a definition of a closed-form expression for a number (as\r\nopposed to a *function*) that we hope will become standard. With our\r\ndefinition, the question of whether the root of x = cos(x) has a closed form\r\nis, perhaps surprisingly, still open. We show that Schanuel's conjecture in\r\ntranscendental number theory resolves questions like this, and we also sketch\r\nsome connections with Tarski's problem of the decidability of the first-order\r\ntheory of the reals with exponentiation. Many (hopefully accessible) open\r\nproblems are described.","comment":"","date_added":"2019-01-09","date_published":"1998-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/9805045v1","http:\/\/arxiv.org\/pdf\/math\/9805045v1"],"collections":"Notation and conventions,The act of doing maths","url":"http:\/\/arxiv.org\/abs\/math\/9805045v1 http:\/\/arxiv.org\/pdf\/math\/9805045v1","year":"1998","urldate":"2019-01-09","archivePrefix":"arXiv","eprint":"math\/9805045","primaryClass":"math.NT"},{"key":"Smarandache1991","type":"article","title":"Only problems, not solutions!","author":"Smarandache, Florentin","abstract":"","comment":"","date_added":"2013-07-29","date_published":"1991-08-06","urls":["http:\/\/vixra.org\/abs\/1005.0049","http:\/\/vixra.org\/pdf\/1005.0049v1.pdf"],"collections":"Attention-grabbing titles,Puzzles","url":"http:\/\/vixra.org\/abs\/1005.0049 http:\/\/vixra.org\/pdf\/1005.0049v1.pdf","urldate":"2013-07-29","year":"1991","isbn":"1879585006","volume":"1993"},{"key":"WhatIsanEnvelope","type":"article","title":"What Is an Envelope?","author":"J.W. Bruce and P.J. Giblin","abstract":"","comment":"","date_added":"2019-02-04","date_published":"1981-08-06","urls":["https:\/\/www.jstor.org\/stable\/3617131?seq=1#metadata_info_tab_contents"],"collections":"Easily explained,Geometry","url":"https:\/\/www.jstor.org\/stable\/3617131?seq=1#metadata_info_tab_contents","year":"1981","urldate":"2019-02-04"},{"key":"item16","type":"online","title":"Navigating Hyperbolic Space with Fibonacci Trees","author":"moniker","abstract":"","comment":"","date_added":"2012-04-07","date_published":"2010-08-06","urls":["http:\/\/moniker.name\/worldmaking\/?p=409","https:\/\/web.archive.org\/web\/20161223092813\/http:\/\/moniker.name\/worldmaking\/?p=409"],"collections":"Basically computer science,Fibonaccinalia,Geometry","url":"http:\/\/moniker.name\/worldmaking\/?p=409 https:\/\/web.archive.org\/web\/20161223092813\/http:\/\/moniker.name\/worldmaking\/?p=409","urldate":"2012-04-07","year":"2010"},{"key":"CodesLowerBoundsandPhaseTransitionsintheSymmetricRendezvousProblem","type":"article","title":"Codes, Lower Bounds, and Phase Transitions in the Symmetric Rendezvous Problem","author":"Varsha Dani and Thomas P. Hayes and Cristopher Moore and Alexander Russell","abstract":"In the rendezvous problem, two parties with different labelings of the\r\nvertices of a complete graph are trying to meet at some vertex at the same\r\ntime. It is well-known that if the parties have predetermined roles, then the\r\nstrategy where one of them waits at one vertex, while the other visits all $n$\r\nvertices in random order is optimal, taking at most $n$ steps and averaging\r\nabout $n\/2$. Anderson and Weber considered the symmetric rendezvous problem,\r\nwhere both parties must use the same randomized strategy. They analyzed\r\nstrategies where the parties repeatedly play the optimal asymmetric strategy,\r\ndetermining their role independently each time by a biased coin-flip. By tuning\r\nthe bias, Anderson and Weber achieved an expected meeting time of about $0.829\r\nn$, which they conjectured to be asymptotically optimal.\r\n We change perspective slightly: instead of minimizing the expected meeting\r\ntime, we seek to maximize the probability of meeting within a specified time\r\n$T$. The Anderson-Weber strategy, which fails with constant probability when\r\n$T= \\Theta(n)$, is not asymptotically optimal for large $T$ in this setting.\r\nSpecifically, we exhibit a symmetric strategy that succeeds with probability\r\n$1-o(1)$ in $T=4n$ steps. This is tight: for any $\\alpha < 4$, any symmetric\r\nstrategy with $T = \\alpha n$ fails with constant probability. Our strategy uses\r\na new combinatorial object that we dub a \"rendezvous code,\" which may be of\r\nindependent interest.\r\n When $T \\le n$, we show that the probability of meeting within $T$ steps is\r\nindeed asymptotically maximized by the Anderson-Weber strategy. Our results\r\nimply new lower bounds, showing that the best symmetric strategy takes at least\r\n$0.638 n$ steps in expectation. We also present some partial results for the\r\nsymmetric rendezvous problem on other vertex-transitive graphs.","comment":"","date_added":"2019-03-02","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1609.01582v1","http:\/\/arxiv.org\/pdf\/1609.01582v1"],"collections":"Easily explained,Protocols and strategies","url":"http:\/\/arxiv.org\/abs\/1609.01582v1 http:\/\/arxiv.org\/pdf\/1609.01582v1","year":"2016","urldate":"2019-03-02","archivePrefix":"arXiv","eprint":"1609.01582","primaryClass":"math.CO"},{"key":"YetAnotherSingleLawforLattices","type":"article","title":"Yet Another Single Law for Lattices","author":"William McCune and Ranganathan Padmanabhan and Robert Veroff","abstract":"In this note we show that the equational theory of all lattices is defined by\r\na single absorption law. The identity of length 29 with 8 variables is shorter\r\nthan previously known such equations defining lattices.","comment":"","date_added":"2019-03-02","date_published":"2003-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/0307284v1","http:\/\/arxiv.org\/pdf\/math\/0307284v1"],"collections":"Attention-grabbing titles,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/math\/0307284v1 http:\/\/arxiv.org\/pdf\/math\/0307284v1","year":"2003","urldate":"2019-03-02","archivePrefix":"arXiv","eprint":"math\/0307284","primaryClass":"math.LO"},{"key":"TheInstructorsGuidetoRealInduction","type":"article","title":"The Instructor's Guide to Real Induction","author":"Pete L. Clark","abstract":"We introduce real induction, a proof technique analogous to mathematical\r\ninduction but applicable to statements indexed by an interval on the real line.\r\nMore generally we give an inductive principle applicable in any Dedekind\r\ncomplete linearly ordered set. Real and ordered induction is then applied to\r\ngive streamlined, conceptual proofs of basic results in honors calculus,\r\nelementary real analysis and topology.","comment":"","date_added":"2019-03-02","date_published":"2012-08-06","urls":["http:\/\/arxiv.org\/abs\/1208.0973v1","http:\/\/arxiv.org\/pdf\/1208.0973v1"],"collections":"About proof,Easily explained,Fun maths facts,The act of doing maths","url":"http:\/\/arxiv.org\/abs\/1208.0973v1 http:\/\/arxiv.org\/pdf\/1208.0973v1","year":"2012","urldate":"2019-03-02","archivePrefix":"arXiv","eprint":"1208.0973","primaryClass":"math.HO"},{"key":"Leystatistics","type":"online","title":"Ley statistics","author":"Michael Behrend","abstract":"","comment":"","date_added":"2019-03-02","date_published":"2014-08-06","urls":["https:\/\/www.cantab.net\/users\/michael.behrend\/ley_stats\/index.html"],"collections":"Geometry,Probability and statistics","url":"https:\/\/www.cantab.net\/users\/michael.behrend\/ley_stats\/index.html","year":"2014","urldate":"2019-03-02"},{"key":"TheTakagiFunctionandItsProperties","type":"article","title":"The Takagi Function and Its Properties","author":"Jeffrey C. Lagarias","abstract":"The Takagi function is a continuous non-differentiable function on [0,1]\r\nintroduced by Teiji Takagi in 1903. It has since appeared in a surprising\r\nnumber of different mathematical contexts, including mathematical analysis,\r\nprobability theory and number theory. This paper surveys the known properties\r\nof this function as it relates to these fields.","comment":"","date_added":"2019-03-04","date_published":"2011-08-06","urls":["http:\/\/arxiv.org\/abs\/1112.4205v2","http:\/\/arxiv.org\/pdf\/1112.4205v2"],"collections":"Easily explained,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1112.4205v2 http:\/\/arxiv.org\/pdf\/1112.4205v2","year":"2011","urldate":"2019-03-04","archivePrefix":"arXiv","eprint":"1112.4205","primaryClass":"math.CA"},{"key":"BrazilianPrimesWhichAreAlsoSophieGermainPrimes","type":"article","title":"Brazilian Primes Which Are Also Sophie Germain Primes","author":"Jon Grantham and Hester Graves","abstract":"We disprove a conjecture of Schott that no Brazilian primes are Sophie\r\nGermain primes. We enumerate all counterexamples up to $10^{44}$.","comment":"","date_added":"2019-03-13","date_published":"2019-08-06","urls":["http:\/\/arxiv.org\/abs\/1903.04577v1","http:\/\/arxiv.org\/pdf\/1903.04577v1"],"collections":"Attention-grabbing titles,Easily explained,Fun maths facts,Integerology","url":"http:\/\/arxiv.org\/abs\/1903.04577v1 http:\/\/arxiv.org\/pdf\/1903.04577v1","year":"2019","urldate":"2019-03-13","archivePrefix":"arXiv","eprint":"1903.04577","primaryClass":"math.NT"},{"key":"PerformingMathematicalOperationswithMetamaterials","type":"article","title":"Performing Mathematical Operations with Metamaterials","author":"Alexandre Silva and Francesco Monticone and Giuseppe Castaldi and Vincenzo Galdi and Andrea Al\u00f9 and Nader Engheta","abstract":"We introduce the concept of metamaterial analog computing, based on suitably designed metamaterial blocks that can perform mathematical operations (such as spatial differentiation, integration, or convolution) on the profile of an impinging wave as it propagates through these blocks. Two approaches are presented to achieve such functionality: (i) subwavelength structured metascreens combined with graded-index waveguides and (ii) multilayered slabs designed to achieve a desired spatial Green\u2019s function. Both techniques offer the possibility of miniaturized, potentially integrable, wave-based computing systems that are thinner than conventional lens-based optical signal and data processors by several orders of magnitude.","comment":"","date_added":"2019-03-26","date_published":"2014-08-06","urls":["http:\/\/science.sciencemag.org\/content\/343\/6167\/160"],"collections":"Basically physics,Unusual computers","url":"http:\/\/science.sciencemag.org\/content\/343\/6167\/160","year":"2014","urldate":"2019-03-26"},{"key":"CharlesBabbageNotation","type":"book","title":"Charles Babbage's thoughts on notation","author":"Charles Babbage","abstract":"","comment":"The entry on \"Notation\" from the Edinburgh Encyclopedia, contributed by Charles Babbage. He is not in favour of omitting brackets around trig functions, or the convention of putting the subscript for powers of trig functions after the function name, instead of after the argument.","date_added":"2018-09-04","date_published":"1832-08-06","urls":["https:\/\/books.google.co.uk\/books?id=kQ8bAQAAMAAJ&pg=PA527&lpg=PA527&dq=%22although+a+definition+cannot+be+false+it+may+be+improper%22&source=bl&ots=sUZWWIK67O&sig=MbfxfgGf2LrMYLEgg340SjW8t-Y&hl=en&sa=X&ved=2ahUKEwjx1pK0hqHdAhWCW8AKHfDDCR8Q6AEwAHoECAAQAQ#v=onepage&q=%22although%20a%20definition%20cannot%20be%20false%20it%20may%20be%20improper%22&f=false"],"collections":"Notation and conventions","url":"https:\/\/books.google.co.uk\/books?id=kQ8bAQAAMAAJ&pg=PA527&lpg=PA527&dq=%22although+a+definition+cannot+be+false+it+may+be+improper%22&source=bl&ots=sUZWWIK67O&sig=MbfxfgGf2LrMYLEgg340SjW8t-Y&hl=en&sa=X&ved=2ahUKEwjx1pK0hqHdAhWCW8AKHfDDCR8Q6AEwAHoECAAQAQ#v=onepage&q=%22although%20a%20definition%20cannot%20be%20false%20it%20may%20be%20improper%22&f=false","urldate":"2018-09-04","year":"1832"},{"key":"Glenis2008","type":"article","title":"Comparison of geometric figures","author":"Glenis, Spyros and Kapovich, M. and Brodskiy, N. and Dydak, J. and Lang, U. and Ballinger, B. and Blekherman, G. and Cohn, H. and Giansiracusa, N. and Kelly, E. and Others","abstract":"Although the geometric equality of figures has already been studied thoroughly, little work has been done about the comparison of unequal figures. We are used to compare only similar figures but would it be meaningful to compare non similar ones? In this paper we attempt to build a context where it is possible to compare even non similar figures. Adopting Klein's view for the Euclidean Geometry, we defined a relation \"<=\" as: S<=T whenever there is a rigid motion f so that f(S) is a subset of T. This relation is not an order because there are figures (subsets of the plane) so that S<=T, T<=S and S, T not geometrically equal. Our goal is to avoid this paradox and to track down non-trivial classes of figures where the relation \"<=\" becomes, at least, a partial order. Such a class will be called a good class of figures. A reasonable question is whether the figures forming a good class have certain properties and whether the algebra of these figures is also a good class. Therefore we classified the figures into those that cause the paradox mentioned above and those that never cause it. The last ones are called good figures. Although simple, the definition of the good figure was difficult to handle, therefore we introduced a more technical, but intrinsic and handy definition, that of the strongly good figure. With these tools we constructed a new context, where we expanded our perspective about the geometric comparison not only in the Euclidean but also in the Hyperbolic and in the Elliptic Geometry. Eventually, there are still some open and quite challenging issues, which we present them at the last part of the paper. ","comment":"Once you've said that two shapes aren't similar, you can be more precise and define a partial order that puts nearly-the-same figures closer together.","date_added":"2011-02-03","date_published":"2008-08-06","urls":["https:\/\/arxiv.org\/abs\/math\/0611062","https:\/\/arxiv.org\/pdf\/math\/0611062"],"collections":"Geometry","url":"https:\/\/arxiv.org\/abs\/math\/0611062 https:\/\/arxiv.org\/pdf\/math\/0611062","urldate":"2011-02-03","year":"2008","isbn":"7774553983","issn":"1551-3440","journal":"the montana mathematics enthusiast","keywords":"euclidean geometry,isometries,klein,relations","number":"2&3","pages":"199--214","publisher":"Iap","volume":"5","arxivId":"0611.062"},{"key":"BrainfillingCurvesaFractalBestiary","type":"book","title":"Brainfilling Curves - a Fractal Bestiary","author":"Jeffrey Ventrella","abstract":"This is a full-color, artistic, heavily-illustrated book that introduces an intuitive process of generating plane-filling fractal curves using Koch construction. It also introduces a new way to describe and search for all plane-filling curves, including the classic curves introduced by Mandelbrot. In addition, hundreds of novel fractal curves are shown, many of them in color.\r\n\r\nThis book defines a taxonomy for fractal curves, and shows how all plane-filling curves can be characterized by family-types, each family type having its own characteristic properties, including \"pertiling\" - recursive tiling.\r\n\r\nThis book would be of interest to educated people of all backgrounds, especially geometers, computer scientists, and artists of the Escher ilk. ","comment":"","date_added":"2019-05-09","date_published":"2012-08-06","urls":["https:\/\/archive.org\/details\/BrainfillingCurves-AFractalBestiary"],"collections":"Art,Easily explained,Fun maths facts,Things to make and do","url":"https:\/\/archive.org\/details\/BrainfillingCurves-AFractalBestiary","urldate":"2019-05-09","year":"2012"},{"key":"AContributiontotheMathematicalTheoryofBigGameHunting","type":"article","title":"A Contribution to the Mathematical Theory of Big Game Hunting","author":"Ralph Boas","abstract":"Problem: To Catch a Lion in the Sahara Desert.","comment":"","date_added":"2019-05-09","date_published":"1938-08-06","urls":["https:\/\/stuff.mit.edu\/people\/dpolicar\/writing\/netsam\/lionhunt.html"],"collections":"Animals,Attention-grabbing titles","url":"https:\/\/stuff.mit.edu\/people\/dpolicar\/writing\/netsam\/lionhunt.html","year":"1938","urldate":"2019-05-09"},{"key":"PlanarHypohamiltonianGraphson40Vertices","type":"article","title":"Planar Hypohamiltonian Graphs on 40 Vertices","author":"Mohammadreza Jooyandeh and Brendan D. McKay and Patric R. J. \u00d6sterg\u00e5rd and Ville H. Pettersson and Carol T. Zamfirescu","abstract":"A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any\r\nsingle vertex gives a Hamiltonian graph. Until now, the smallest known planar\r\nhypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That\r\nresult is here improved upon by 25 planar hypohamiltonian graphs of order 40,\r\nwhich are found through computer-aided generation of certain families of planar\r\ngraphs with girth 4 and a fixed number of 4-faces. It is further shown that\r\nplanar hypohamiltonian graphs exist for all orders greater than or equal to 42.\r\nIf Hamiltonian cycles are replaced by Hamiltonian paths throughout the\r\ndefinition of hypohamiltonian graphs, we get the definition of hypotraceable\r\ngraphs. It is shown that there is a planar hypotraceable graph of order 154 and\r\nof all orders greater than or equal to 156. We also show that the smallest\r\nhypohamiltonian planar graph of girth 5 has 45 vertices.","comment":"","date_added":"2019-05-09","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1302.2698v4","http:\/\/arxiv.org\/pdf\/1302.2698v4"],"collections":"Fun maths facts","url":"http:\/\/arxiv.org\/abs\/1302.2698v4 http:\/\/arxiv.org\/pdf\/1302.2698v4","year":"2013","urldate":"2019-05-09","archivePrefix":"arXiv","eprint":"1302.2698","primaryClass":"math.CO"},{"key":"TheSwissCheeseOperad","type":"article","title":"The Swiss-Cheese Operad","author":"Alexander A. Voronov","abstract":"We introduce a new operad, which we call the Swiss-cheese operad. It mixes\r\nnaturally the little disks and the little intervals operads. The Swiss-cheese\r\noperad is related to the configuration spaces of points on the upper half-plane\r\nand points on the real line, considered by Kontsevich for the sake of\r\ndeformation quantization. This relation is similar to the relation between the\r\nlittle disks operad and the configuration spaces of points on the plane. The\r\nSwiss-cheese operad may also be regarded as a finite-dimensional model of the\r\nmoduli space of genus-zero Riemann surfaces appearing in the open-closed string\r\ntheory studied recently by Zwiebach. We describe algebras over the homology of\r\nthe Swiss-cheese operad.","comment":"","date_added":"2019-05-09","date_published":"1998-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/9807037v1","http:\/\/arxiv.org\/pdf\/math\/9807037v1"],"collections":"Attention-grabbing titles,Food","url":"http:\/\/arxiv.org\/abs\/math\/9807037v1 http:\/\/arxiv.org\/pdf\/math\/9807037v1","year":"1998","urldate":"2019-05-09","archivePrefix":"arXiv","eprint":"math\/9807037","primaryClass":"math.QA"},{"key":"ThegraphsbehindReuleauxpolyhedra","type":"article","title":"The graphs behind Reuleaux polyhedra","author":"Luis Montejano and Eric Pauli and Miguel Raggi and Edgardo Rold\u00e1n-Pensado","abstract":"This work is about graphs arising from Reuleaux polyhedra. Such graphs must\r\nnecessarily be planar, $3$-connected and strongly self-dual. We study the\r\nquestion of when these conditions are sufficient.\r\n If $G$ is any such a graph with isomorphism $\\tau : G \\to G^*$ (where $G^*$\r\nis the unique dual graph), a metric mapping is a map $\\eta : V(G) \\to \\mathbb\r\nR^3$ such that the diameter of $\\eta(G)$ is $1$ and for every pair of vertices\r\n$(u,v)$ such that $u\\in \\tau(v)$ we have dist$(\\eta(u),\\eta(v)) = 1$. If $\\eta$\r\nis injective, it is called a metric embedding. Note that a metric embedding\r\ngives rise to a Reuleaux Polyhedra.\r\n Our contributions are twofold: Firstly, we prove that any planar,\r\n$3$-connected, strongly self-dual graph has a metric mapping by proving that\r\nthe chromatic number of the diameter graph (whose vertices are $V(G)$ and whose\r\nedges are pairs $(u,v)$ such that $u\\in \\tau(v)$) is at most $4$, which means\r\nthere exists a metric mapping to the tetrahedron. Furthermore, we use the\r\nLov\\'asz neighborhood-complex theorem in algebraic topology to prove that the\r\nchromatic number of the diameter graph is exactly $4$.\r\n Secondly, we develop algorithms that allow us to obtain every such graph with\r\nup to $14$ vertices. Furthermore, we numerically construct metric embeddings\r\nfor every such graph. From the theorem and this computational evidence we\r\nconjecture that every such graph is realizable as a Reuleaux polyhedron in\r\n$\\mathbb R^3$.\r\n In previous work the first and last authors described a method to construct a\r\nconstant-width body from a Reuleaux polyhedron. So in essence, we also\r\nconstruct hundreds of new examples of constant-width bodies.\r\n This is related to a problem of V\\'azsonyi, and also to a problem of\r\nBlaschke-Lebesgue.","comment":"","date_added":"2019-05-09","date_published":"2019-08-06","urls":["http:\/\/arxiv.org\/abs\/1904.12761v1","http:\/\/arxiv.org\/pdf\/1904.12761v1"],"collections":"Geometry","url":"http:\/\/arxiv.org\/abs\/1904.12761v1 http:\/\/arxiv.org\/pdf\/1904.12761v1","year":"2019","urldate":"2019-05-09","archivePrefix":"arXiv","eprint":"1904.12761","primaryClass":"cs.CG"},{"key":"Auniversaldifferentialequation","type":"article","title":"A universal differential equation","author":"Lee A. Rubel","abstract":"There exists a non trivial fourth-order algebraic differential equation \\[P(y',y'',y''',y'''') = 0,\\] where \\(P\\) is a polynomial in four variables, with integer coefficients, such that for any continuous function \\(\\phi\\) on \\((-\\infty,\\infty)\\) and for any positive continuous function \\(\\varepsilon(t)\\) on \\((-\\infty,\\infty)\\), there exists a \\(C^{\\infty}\\) solution \\(y\\) of \\(P=0\\) such that \\[|y(t)-\\phi(t)| \\lt \\varepsilon(t)\\) \\forall t \\in (-\\infty,\\infty)\\]","comment":"","date_added":"2019-05-09","date_published":"1981-08-06","urls":["https:\/\/projecteuclid.org\/euclid.bams\/1183548125"],"collections":"Fun maths facts,Unusual computers","url":"https:\/\/projecteuclid.org\/euclid.bams\/1183548125","year":"1981","urldate":"2019-05-09"},{"key":"TheSensualquadraticForm","type":"book","title":"The Sensual (quadratic) Form","author":"John Horton Conway","abstract":"John Horton Conway's unique approach to quadratic forms was the subject of the Hedrick Lectures that he gave in August of 1991 at the Joint Meetings of the Mathematical Association of America and the American Mathematical Society in Orono, Maine. This book presents the substance of those lectures.\r\n\r\nThe book should not be thought of as a serious textbook on the theory of quadratic forms. It consists rather of a number of essays on particular aspects of quadratic forms that have interested the author. The lectures are self-contained and will be accessible to the generally informed reader who has no particular background in quadratic form theory. The minor exceptions should not interrupt the flow of ideas. The afterthoughts to the lectures contain discussion of related matters that occasionally presuppose greater knowledge.","comment":"","date_added":"2019-05-09","date_published":"1997-08-06","urls":["https:\/\/bookstore.ams.org\/car-26","https:\/\/www.maths.ed.ac.uk\/~v1ranick\/papers\/conwaysens.pdf"],"collections":"Attention-grabbing titles","url":"https:\/\/bookstore.ams.org\/car-26 https:\/\/www.maths.ed.ac.uk\/~v1ranick\/papers\/conwaysens.pdf","urldate":"2019-05-09","year":"1997"},{"key":"HexAStrategyGuide","type":"book","title":"Hex: A Strategy Guide","author":"Matthew Seymour","abstract":"","comment":"","date_added":"2019-05-31","date_published":"2019-08-06","urls":["http:\/\/www.mseymour.ca\/hex_book\/hexstrat.html"],"collections":"Games to play with friends","url":"http:\/\/www.mseymour.ca\/hex_book\/hexstrat.html","year":"2019","urldate":"2019-05-31"},{"key":"ChordsofanellipseLucaspolynomialsandcubicequations","type":"article","title":"Chords of an ellipse, Lucas polynomials, and cubic equations","author":"Ben Blum-Smith and Japheth Wood","abstract":"A beautiful result of Thomas Price links the Fibonacci numbers and the Lucas\r\npolynomials to the plane geometry of an ellipse. We give a conceptually\r\ntransparent development of this result that provides a tour of several gems of\r\nclassical mathematics: It is inspired by Girolamo Cardano's solution of the\r\ncubic equation, uses Newton's theorem connecting power sums and elementary\r\nsymmetric polynomials, and yields for free an alternative proof of the Binet\r\nformula for the generalized Lucas polynomials.","comment":"","date_added":"2019-06-30","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1810.00492v3","http:\/\/arxiv.org\/pdf\/1810.00492v3"],"collections":"Easily explained,Fun maths facts,Puzzles","url":"http:\/\/arxiv.org\/abs\/1810.00492v3 http:\/\/arxiv.org\/pdf\/1810.00492v3","year":"2018","urldate":"2019-06-30","archivePrefix":"arXiv","eprint":"1810.00492","primaryClass":"math.HO"},{"key":"FiboquadraticSequencesandExtensionsoftheCassiniIdentityRaisedFromtheStudyofRithmomachia","type":"article","title":"Fiboquadratic Sequences and Extensions of the Cassini Identity Raised From the Study of Rithmomachia","author":"Tom\u00e1s Guardia and Douglas Jim\u00e9nez","abstract":"In this paper, we introduce fiboquadratic sequences as an extension to\r\ninfinity of the board of Rithmomachia and we prove that this extension gives\r\nraise to fiboquadratic sequences which we define here. Also, fiboquadratic\r\nsequences provide extensions of Cassini's Identity.","comment":"","date_added":"2019-07-18","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1509.03177v3","http:\/\/arxiv.org\/pdf\/1509.03177v3"],"collections":"Fibonaccinalia,Games to play with friends,History","url":"http:\/\/arxiv.org\/abs\/1509.03177v3 http:\/\/arxiv.org\/pdf\/1509.03177v3","year":"2015","urldate":"2019-07-18","archivePrefix":"arXiv","eprint":"1509.03177","primaryClass":"math.HO"},{"key":"NumbersforMasochistsAGuidetoMentalFactoring","type":"article","title":"Numbers for Masochists: A Guide to Mental Factoring","author":"Hilarie Orman and Richard Schroeppel","abstract":"Many people can multiply large numbers mentally, and there are numerous treatises on how to do it. However, the inverse problem, factoring, is rarely discussed. This paper will show you how to factor numbers up to 100,000 in your head. In fact, you will be able to factor some numbers much larger than that.","comment":"","date_added":"2019-07-31","date_published":"2018-08-06","urls":["http:\/\/www.purplestreak.com\/g4g13\/mentalfactoring.pdf"],"collections":"Easily explained,Integerology","url":"http:\/\/www.purplestreak.com\/g4g13\/mentalfactoring.pdf","year":"2018","urldate":"2019-07-31"},{"key":"CreationofHyperbolicOrnaments","type":"article","title":"Creation of Hyperbolic Ornaments","author":"Martin von Gagern","abstract":"Hyperbolic ornaments are pictures which are invariant under a discrete symmetry group of isometric transformations of the hyperbolic plane. They are the hyperbolic analogue of Euclidean ornaments, including but not limited to those Euclidean ornaments which belong to one of the 17 wallpaper groups. The creation of hyperbolic ornaments has a number of applications. They include artistic goals,communication of mathematical structures and techniques, and experimental research in the hyperbolic plane. Manual creation of hyperbolic ornaments is an arduous task. This work describes two ways in which computers may help with this process. On the one hand, a computer may provide a real-time drawing tool, where any stroke entered by the user will be replicated according to the rules of some previously selected symmetry group. Finding a suitable user interface for the intuitive selection of the symmetry group is a particular challenge in this context. On the other hand, existing Euclidean ornaments can be transported to the hyperbolic plane by changing the orders of their centers of rotation. This requires a deformation of the fundamental domains of the ornament, and one particularlywell suited approach uses conformal deformations for this step, approximated using discrete conformality concepts from discrete differential geometry. Both tools need a way to produce high quality renderings of the hyperbolic ornament, dealing with the fact that in general an infinite number of fundamental domains will be visible in the finite model of the hyperbolic plane. To deal with this problem, an approach similar to ray tracing can be used, variations of which are discussed as well.","comment":"","date_added":"2019-08-08","date_published":"2014-08-06","urls":["http:\/\/martin.von-gagern.net\/publications\/2014-phd\/","http:\/\/mediatum.ub.tum.de\/doc\/1210572\/document.pdf"],"collections":"Art,Basically computer science,Geometry","url":"http:\/\/martin.von-gagern.net\/publications\/2014-phd\/ http:\/\/mediatum.ub.tum.de\/doc\/1210572\/document.pdf","year":"2014","urldate":"2019-08-08"},{"key":"FractalSequences","type":"online","title":"Fractal Sequences","author":"Clark Kimberling","abstract":" Fractal sequences have in common with the more familiar geometric fractals the property of self-containment. An example of a fractal sequence is\r\n\r\n1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, . . .\r\n\r\nIf you delete the first occurrence of each positive integer, you'll see that the remaining sequence is the same as the original. (So, if you do it again and again, you always get the same sequence.)","comment":"","date_added":"2019-08-21","date_published":"1999-08-06","urls":["https:\/\/faculty.evansville.edu\/ck6\/integer\/fractals.html"],"collections":"Easily explained,Fun maths facts,Integerology,Puzzles","url":"https:\/\/faculty.evansville.edu\/ck6\/integer\/fractals.html","year":"1999","urldate":"2019-08-21"},{"key":"PalindromesinDifferentBasesAConjectureofJErnestWilkins","type":"article","title":"Palindromes in Different Bases: A Conjecture of J. Ernest Wilkins","author":"Edray Herber Goins","abstract":"We show that there exist exactly 203 positive integers $N$ such that for some\r\ninteger $d \\geq 2$ this number is a $d$-digit palindrome base 10 as well as a\r\n$d$-digit palindrome for some base $b$ different from 10. To be more precise,\r\nsuch $N$ range from 22 to 9986831781362631871386899.","comment":"","date_added":"2019-09-14","date_published":"2009-08-06","urls":["http:\/\/arxiv.org\/abs\/0909.5452v1","http:\/\/arxiv.org\/pdf\/0909.5452v1"],"collections":"Easily explained,Fun maths facts,Integerology","url":"http:\/\/arxiv.org\/abs\/0909.5452v1 http:\/\/arxiv.org\/pdf\/0909.5452v1","year":"2009","urldate":"2019-09-14","archivePrefix":"arXiv","eprint":"0909.5452","primaryClass":"math.NT"},{"key":"PercolationisOdd","type":"article","title":"Percolation is Odd","author":"Stephan Mertens and Cristopher Moore","abstract":"We discuss the number of spanning configurations in site percolation. We show\r\nthat for a large class of lattices, the number of spanning configrations is odd\r\nfor all lattice sizes. This class includes site percolation on the square\r\nlattice and on the hypercubic lattice in any dimension.","comment":"","date_added":"2019-09-14","date_published":"2019-08-06","urls":["http:\/\/arxiv.org\/abs\/1909.01484v1","http:\/\/arxiv.org\/pdf\/1909.01484v1"],"collections":"Attention-grabbing titles,Combinatorics","url":"http:\/\/arxiv.org\/abs\/1909.01484v1 http:\/\/arxiv.org\/pdf\/1909.01484v1","year":"2019","urldate":"2019-09-14","archivePrefix":"arXiv","eprint":"1909.01484","primaryClass":"cond-mat.stat-mech"},{"key":"FindingaprincessinapalaceApursuitevasionproblem","type":"article","title":"Finding a princess in a palace: A pursuit-evasion problem","author":"John R. Britnell and Mark Wildon","abstract":"This paper solves a pursuit-evasion problem in which a prince must find a\r\nprincess who is constrained to move on each day from one vertex of a finite\r\ngraph to another. Unlike the related and much studied `Cops and Robbers Game',\r\nthe prince has no knowledge of the position of the princess; he may, however,\r\nvisit any single room he wishes on each day. We characterize the graphs for\r\nwhich the prince has a winning strategy, and determine, for each such graph,\r\nthe minimum number of days the prince requires to guarantee to find the\r\nprincess.","comment":"","date_added":"2019-10-08","date_published":"2012-08-06","urls":["http:\/\/arxiv.org\/abs\/1204.5490v1","http:\/\/arxiv.org\/pdf\/1204.5490v1"],"collections":"Combinatorics,Easily explained,Protocols and strategies,Puzzles","url":"http:\/\/arxiv.org\/abs\/1204.5490v1 http:\/\/arxiv.org\/pdf\/1204.5490v1","year":"2012","urldate":"2019-10-08","archivePrefix":"arXiv","eprint":"1204.5490","primaryClass":"math.CO"},{"key":"Catchingamouseonatree","type":"article","title":"Catching a mouse on a tree","author":"Vytautas Gruslys and Ar\u00e8s M\u00e9roueh","abstract":"In this paper we consider a pursuit-evasion game on a graph. A team of cats,\r\nwhich may choose any vertex of the graph at any turn, tries to catch an\r\ninvisible mouse, which is constrained to moving along the vertices of the\r\ngraph. Our main focus shall be on trees. We prove that $\\lceil\r\n(1\/2)\\log_2(n)\\rceil$ cats can always catch a mouse on a tree of order $n$ and\r\ngive a collection of trees where the mouse can avoid being caught by $ (1\/4 -\r\no(1))\\log_2(n)$ cats.","comment":"","date_added":"2019-10-08","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1502.06591v1","http:\/\/arxiv.org\/pdf\/1502.06591v1"],"collections":"Animals,Attention-grabbing titles,Combinatorics","url":"http:\/\/arxiv.org\/abs\/1502.06591v1 http:\/\/arxiv.org\/pdf\/1502.06591v1","year":"2015","urldate":"2019-10-08","archivePrefix":"arXiv","eprint":"1502.06591","primaryClass":"math.CO"},{"key":"HowtoHuntanInvisibleRabbitonaGraph","type":"article","title":"How to Hunt an Invisible Rabbit on a Graph","author":"Tatjana V. Abramovskaya and Fedor V. Fomin and Petr A. Golovach and Micha\u0142 Pilipczuk","abstract":"We investigate Hunters & Rabbit game, where a set of hunters tries to catch\r\nan invisible rabbit that slides along the edges of a graph. We show that the\r\nminimum number of hunters required to win on an (n\\times m)-grid is \\lfloor\r\nmin{n,m}\/2\\rfloor+1. We also show that the extremal value of this number on\r\nn-vertex trees is between \\Omega(log n\/log log n) and O(log n).","comment":"","date_added":"2019-10-08","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1502.05614v2","http:\/\/arxiv.org\/pdf\/1502.05614v2"],"collections":"Animals,Attention-grabbing titles,Combinatorics,Easily explained,Protocols and strategies,Puzzles","url":"http:\/\/arxiv.org\/abs\/1502.05614v2 http:\/\/arxiv.org\/pdf\/1502.05614v2","year":"2015","urldate":"2019-10-08","archivePrefix":"arXiv","eprint":"1502.05614","primaryClass":"math.CO"},{"key":"portandsweepsolitaire","type":"article","title":"Port-and-Sweep Solitaire","author":"Jacob Siehler","abstract":"How does this happen? I just wanted a nice game where I didn\u2019t have to count higher than two, and I ended up dealing with imaginary numbers. But let me back up: I\u2019ve been a little obsessed with a puzzle lately, and I would like to explain what\u2019s puzzling me and how the square root of \u20131 can sneak in where you least expect it. ","comment":"","date_added":"2019-10-12","date_published":"2010-08-06","urls":["http:\/\/homepages.gac.edu\/~jsiehler\/Articles\/port-and-sweep-article.pdf"],"collections":"Easily explained,Games to play with friends,Puzzles","url":"http:\/\/homepages.gac.edu\/~jsiehler\/Articles\/port-and-sweep-article.pdf","year":"2010","urldate":"2019-10-12"},{"key":"TheGraphMenagerieAbstractAlgebraAndTheMadVeterinarian","type":"article","title":"The Graph Menagerie: Abstract Algebra and the Mad Veterinarian","author":"Gene Abrams and Jessica K. Sklar","abstract":"This article begins with a fanciful concept from recreational mathematics: a machine that can transmogrify a single animal of a given species into a finite nonempty collection of animals from any number of species. Given this premise, a natural question arises: if a Mad Veterinarian has a finite slate of such machines, then which animal menageries are equivalent? To answer this question, the authors associate to the slate of machines a directed \"Mad Vet\" graph. They then show that the corresponding collection of equivalence classes of animal menageries forms a semigroup and use the structure of the Mad Vet graph to determine when this collection is actually a group. In addition, the authors show that the Mad Vet groups can be identified explicitly using the Smith normal form of a matrix closely related to the incidence matrix of the Mad Vet graph.","comment":"","date_added":"2019-11-06","date_published":"2011-08-06","urls":["https:\/\/www.maa.org\/programs\/maa-awards\/writing-awards\/the-graph-menagerie-abstract-algebra-and-the-mad-veterinarian","https:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Allendoerfer\/Abrams2011.pdf"],"collections":"Animals,Attention-grabbing titles,Easily explained,Puzzles","url":"https:\/\/www.maa.org\/programs\/maa-awards\/writing-awards\/the-graph-menagerie-abstract-algebra-and-the-mad-veterinarian https:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Allendoerfer\/Abrams2011.pdf","urldate":"2019-11-06","year":"2011"},{"key":"LightsOutandVariants","type":"article","title":"\u201cLights Out\u201d and Variants","author":"Martin Kreh","abstract":"In this article, we investigate the puzzle \u201cLights Out\u201d as well as some variants of it (in particular, varying board size and number of colors). We discuss the complete solvability of such games, i.e., we are interested in the cases such that all starting boards can be solved. We will model the problem with basic linear algebra and develop a criterion for the unsolvability depending on the board size modulo 30. Further, we will discuss two ways of handling the solvability that will rely on algebraic number theory.","comment":"","date_added":"2019-12-09","date_published":"2017-08-06","urls":["https:\/\/www.tandfonline.com\/doi\/abs\/10.4169\/amer.math.monthly.124.10.937"],"collections":"Puzzles","url":"https:\/\/www.tandfonline.com\/doi\/abs\/10.4169\/amer.math.monthly.124.10.937","urldate":"2019-12-09","year":"2017","journal":"The American Mathematical Monthly: Vol 124","volume":"124","number":"10"},{"key":"Thenothreeinlineproblemonatorus","type":"article","title":"The no-three-in-line problem on a torus","author":"Jim Fowler and Andrew Groot and Deven Pandya and Bart Snapp","abstract":"Let $T(\\mathbb{Z}_m \\times \\mathbb{Z}_n)$ denote the maximal number of points that can be\r\nplaced on an $m \\times n$ discrete torus with \"no three in a line,\" meaning no\r\nthree in a coset of a cyclic subgroup of $\\mathbb{Z}_m \\times \\mathbb{Z}_n$. By proving upper\r\nbounds and providing explicit constructions, for distinct primes $p$ and $q$,\r\nwe show that $T(\\mathbb{Z}_p \\times \\mathbb{Z}_{p^2}) = 2p$ and $T(\\mathbb{Z}_p \\times \\mathbb{Z}_{pq}) = p+1$.\r\nVia Grobner bases, we compute $T(\\mathbb{Z}_m \\times \\mathbb{Z}_n)$ for $2 \\leq m \\leq 7$ and\r\n$2 \\leq n \\leq 19$.","comment":"","date_added":"2019-12-10","date_published":"2012-08-06","urls":["http:\/\/arxiv.org\/abs\/1203.6604v1","http:\/\/arxiv.org\/pdf\/1203.6604v1"],"collections":"Fun maths facts,Geometry,Puzzles","url":"http:\/\/arxiv.org\/abs\/1203.6604v1 http:\/\/arxiv.org\/pdf\/1203.6604v1","urldate":"2019-12-10","year":"2012","archivePrefix":"arXiv","eprint":"1203.6604","primaryClass":"math.CO"},{"key":"Auniquepairoftriangles","type":"article","title":"A unique pair of triangles","author":"Yoshinosuke Hirakawa and Hideki Matsumura","abstract":"A rational triangle is a triangle with sides of rational lengths. In this\r\nshort note, we prove that there exists a unique pair of a rational right\r\ntriangle and a rational isosceles triangle which have the same perimeter and\r\nthe same area. In the proof, we determine the set of rational points on a\r\ncertain hyperelliptic curve by a standard but sophisticated argument which is\r\nbased on the 2-descent on its Jacobian variety and Coleman's theory of $p$-adic\r\nabelian integrals.","comment":"","date_added":"2019-12-10","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1809.09936v1","http:\/\/arxiv.org\/pdf\/1809.09936v1"],"collections":"Easily explained,Fun maths facts,Geometry","url":"http:\/\/arxiv.org\/abs\/1809.09936v1 http:\/\/arxiv.org\/pdf\/1809.09936v1","year":"2018","urldate":"2019-12-10","archivePrefix":"arXiv","eprint":"1809.09936","primaryClass":"math.NT"},{"key":"Smalldatacomputingcorrectcalculatorarithmetic","type":"article","title":"Small-data computing: correct calculator arithmetic","author":"Hans-J. Boehm","abstract":"Rounding errors are usually avoidable, and sometimes we can afford to avoid them.","comment":"Description of how the built-in Android calculator uses constructive real numbers to avoid rounding errors.","date_added":"2020-01-13","date_published":"2017-08-06","urls":["https:\/\/dl.acm.org\/doi\/10.1145\/2911981"],"collections":"Basically computer science,Unusual arithmetic","url":"https:\/\/dl.acm.org\/doi\/10.1145\/2911981","year":"2017","urldate":"2020-01-13"},{"key":"Theoryandapplicationsofthedoublebasenumbersystem","type":"article","title":"Theory and applications of the double-base number system","author":" V.S. Dimitrov and G.A. Jullien and W.C. Miller ","abstract":"In this paper, we analyze some of the main properties of a double base number system, using bases 2 and 3; in particular, we emphasize the sparseness of the representation. A simple geometric interpretation allows an efficient implementation of the basic arithmetic operations and we introduce an index calculus for logarithmic-like arithmetic with considerable hardware reductions in lookup table size. We discuss the application of this number system in the area of digital signal processing; we illustrate the discussion with examples of finite impulse response filtering.","comment":"","date_added":"2020-02-03","date_published":"1999-08-06","urls":["https:\/\/ieeexplore.ieee.org\/document\/805158"],"collections":"Basically computer science,Easily explained,Integerology,Unusual arithmetic","url":"https:\/\/ieeexplore.ieee.org\/document\/805158","year":"1999","urldate":"2020-02-03"},{"key":"Sandwichsemigroupsindiagramcategories","type":"article","title":"Sandwich semigroups in diagram categories","author":"Ivana \u0110ur\u0111ev and Igor Dolinka and James East","abstract":"This paper concerns a number of diagram categories, namely the partition,\r\nplanar partition, Brauer, partial Brauer, Motzkin and Temperley-Lieb\r\ncategories. If $\\mathcal K$ denotes any of these categories, and if\r\n$\\sigma\\in\\mathcal K_{nm}$ is a fixed morphism, then an associative operation\r\n$\\star_\\sigma$ may be defined on $\\mathcal K_{mn}$ by\r\n$\\alpha\\star_\\sigma\\beta=\\alpha\\sigma\\beta$. The resulting semigroup $\\mathcal\r\nK_{mn}^\\sigma=(\\mathcal K_{mn},\\star_\\sigma)$ is called a sandwich semigroup.\r\nWe conduct a thorough investigation of these sandwich semigroups, with an\r\nemphasis on structural and combinatorial properties such as Green's relations\r\nand preorders, regularity, stability, mid-identities, ideal structure,\r\n(products of) idempotents, and minimal generation. It turns out that the Brauer\r\ncategory has many remarkable properties not shared by any of the other diagram\r\ncategories we study. Because of these unique properties, we may completely\r\nclassify isomorphism classes of sandwich semigroups in the Brauer category,\r\ncalculate the rank (smallest size of a generating set) of an arbitrary sandwich\r\nsemigroup, enumerate Green's classes and idempotents, and calculate ranks (and\r\nidempotent ranks, where appropriate) of the regular subsemigroup and its\r\nideals, as well as the idempotent-generated subsemigroup. Several illustrative\r\nexamples are considered throughout, partly to demonstrate the sometimes-subtle\r\ndifferences between the various diagram categories.","comment":"","date_added":"2020-02-03","date_published":"2019-08-06","urls":["http:\/\/arxiv.org\/abs\/1910.10286v1","http:\/\/arxiv.org\/pdf\/1910.10286v1"],"collections":"Food,The groups group","url":"http:\/\/arxiv.org\/abs\/1910.10286v1 http:\/\/arxiv.org\/pdf\/1910.10286v1","year":"2019","urldate":"2020-02-03","archivePrefix":"arXiv","eprint":"1910.10286","primaryClass":"math.GR"},{"key":"Henwood1980","type":"article","title":"Eponymy in Mathematical Nomenclature: What's in a Name, and What Should Be?","author":"Henwood, Mervyn R. and Rival, Ivan","abstract":"","comment":"","date_added":"2014-02-11","date_published":"1980-12-01","urls":["http:\/\/ivanrival.com\/docs\/eponymy.pdf","http:\/\/link.springer.com\/10.1007\/BF03028604"],"collections":"The act of doing maths","url":"http:\/\/ivanrival.com\/docs\/eponymy.pdf http:\/\/link.springer.com\/10.1007\/BF03028604","urldate":"2014-02-11","year":"1980","doi":"10.1007\/BF03028604","issn":"0343-6993","journal":"The Mathematical Intelligencer","month":"dec","number":"4","pages":"204--205","volume":"2"},{"key":"OnSometwowayClassificationsofIntegers","type":"article","title":"On Some two way Classifications of Integers","author":"J. Lambek and L. Moser","abstract":"In this note we use the method of generating functions to show that there is a unique way of splitting the non-negative integers into two classes in such a way that the sums of pairs of distinct integers will be the same (with same multiplicities) for both classes. We prove a similar theorem for products of positive integers and consider some related problems.","comment":"","date_added":"2020-03-13","date_published":"1959-08-06","urls":["https:\/\/www.cambridge.org\/core\/journals\/canadian-mathematical-bulletin\/article\/on-some-two-way-classifications-of-integers\/C36087ADE965C59BE843D73B9FA11A87#","https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/C36087ADE965C59BE843D73B9FA11A87\/S0008439559051116a.pdf\/div-class-title-on-some-two-way-classifications-of-integers-div.pdf"],"collections":"Fun maths facts,Integerology","url":"https:\/\/www.cambridge.org\/core\/journals\/canadian-mathematical-bulletin\/article\/on-some-two-way-classifications-of-integers\/C36087ADE965C59BE843D73B9FA11A87# https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/C36087ADE965C59BE843D73B9FA11A87\/S0008439559051116a.pdf\/div-class-title-on-some-two-way-classifications-of-integers-div.pdf","year":"1959","urldate":"2020-03-13","identifier":"doi:10.4153\/CMB-1959-013-x","journal":"Canadian Mathematical Bulletin","publisher":"Cambridge University Press","volume":"2","issue":"2","issn":"1496-4287","doi":"10.4153\/CMB-1959-013-x","pages":"85-89"},{"key":"TheRoleofNumberNotationSignValueNotationNumberProcessingisEasierthanPlaceValue","type":"article","title":"The Role of Number Notation: Sign-Value Notation Number Processing is Easier than Place-Value","author":"Krajcsi, Attila and Szab\u00f3, Eszter","abstract":"Number notations can influence the way numbers are handled in computations; however, the role of notation itself in mental processing has not been examined directly. From a mathematical point of view, it is believed that place-value number notation systems, such as the Indo-Arabic numbers, are superior to sign-value systems, such as the Roman numbers. However, sign-value notation might have sufficient efficiency; for example, sign-value notations were common in flourishing cultures, such as in ancient Egypt. Herein we compared artificial sign-value and place-value notations in simple numerical tasks. We found that, contrary to the dominant view, sign-value notation can be applied more easily than place-value notation for multi-power comparison and addition tasks. Our results are consistent with the popularity of sign-value notations that prevailed for centuries. To explain the notation effect, we propose a natural multi-power number representation based on the numerical representation of objects.","comment":"","date_added":"2020-03-19","date_published":"2012-08-06","urls":["https:\/\/www.frontiersin.org\/articles\/10.3389\/fpsyg.2012.00463\/full","https:\/\/www.frontiersin.org\/articles\/10.3389\/fpsyg.2012.00463\/pdf"],"collections":"Education,Notation and conventions","url":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fpsyg.2012.00463\/full https:\/\/www.frontiersin.org\/articles\/10.3389\/fpsyg.2012.00463\/pdf","year":"2012","urldate":"2020-03-19","volume":"3","journal":"Frontiers in Psychology","publisher":"Frontiers","issn":"1664-1078","doi":"10.3389\/fpsyg.2012.00463","pages":"463"},{"key":"item46","type":"book","title":"Music: a Mathematical Offering","author":"Dave Benson","abstract":"","comment":"","date_added":"2014-12-02","date_published":"2006-08-06","urls":["https:\/\/homepages.abdn.ac.uk\/d.j.benson\/pages\/html\/maths-music.html"],"collections":"Music","url":"https:\/\/homepages.abdn.ac.uk\/d.j.benson\/pages\/html\/maths-music.html","urldate":"2014-12-02","year":"2006"},{"key":"ACatalogueOfMathematicalFormulasInvolvingPiWithAnalysis","type":"article","title":"A catalogue of mathematical formulas involving \u03c0, with analysis","author":"David H. Bailey","abstract":"This paper presents a catalogue of mathematical formulas and iterative algorithms for evaluating the mathematical constant \u03c0, ranging from Archimedes\u2019 2200-year-old iteration to some formulas that were discovered only in the past few decades. Computer implementations and timing results for these formulas and algorithms are also included. In particular, timings are presented for evaluations of various infinite series formulas to approximately 10,000-digit precision, for evaluations of various integral formulas to approximately 4,000-digit precision, and for evaluations of several iterative algorithms to approximately 100,000-digit precision, all based on carefully designed comparative computer runs.","comment":"","date_added":"2020-03-27","date_published":"2020-08-06","urls":["https:\/\/www.davidhbailey.com\/dhbpapers\/pi-formulas.pdf"],"collections":"Basically computer science,History,Lists and catalogues","url":"https:\/\/www.davidhbailey.com\/dhbpapers\/pi-formulas.pdf","year":"2020","urldate":"2020-03-27"},{"key":"FusiblenumbersandPeanoArithmetic","type":"article","title":"Fusible numbers and Peano Arithmetic","author":"Jeff Erickson and Gabriel Nivasch and Junyan Xu","abstract":"Inspired by a mathematical riddle involving fuses, we define the \"fusible\r\nnumbers\" as follows: $0$ is fusible, and whenever $x,y$ are fusible with\r\n$|y-x|<1$, the number $(x+y+1)\/2$ is also fusible. We prove that the set of\r\nfusible numbers, ordered by the usual order on $\\mathbb R$, is well-ordered,\r\nwith order type $\\varepsilon_0$. Furthermore, we prove that the density of the\r\nfusible numbers along the real line grows at an incredibly fast rate: Letting\r\n$g(n)$ be the largest gap between consecutive fusible numbers in the interval\r\n$[n,\\infty)$, we have $g(n)^{-1} \\ge F_{\\varepsilon_0}(n-c)$ for some constant\r\n$c$, where $F_\\alpha$ denotes the fast-growing hierarchy. Finally, we derive\r\nsome true statements that can be formulated but not proven in Peano Arithmetic,\r\nof a different flavor than previously known such statements. For example, PA\r\ncannot prove the true statement \"For every natural number $n$ there exists a\r\nsmallest fusible number larger than $n$.\"","comment":"","date_added":"2020-04-01","date_published":"2020-08-06","urls":["https:\/\/arxiv.org\/abs\/2003.14342v1","https:\/\/arxiv.org\/pdf\/2003.14342v1"],"collections":"Unusual arithmetic","url":"https:\/\/arxiv.org\/abs\/2003.14342v1 https:\/\/arxiv.org\/pdf\/2003.14342v1","urldate":"2020-04-01","year":"2020","archivePrefix":"arXiv","eprint":"2003.14342","primaryClass":"cs.LO"},{"key":"Ufnarovski2003","type":"article","title":"How to differentiate a number","author":"Ufnarovski, Victor and \u00c5hlander, B","abstract":"","comment":"","date_added":"2013-10-31","date_published":"2003-08-06","urls":["https:\/\/www.emis.de\/journals\/JIS\/VOL6\/Ufnarovski\/ufnarovski.html","https:\/\/www.emis.de\/journals\/JIS\/VOL6\/Ufnarovski\/ufnarovski.pdf"],"collections":"Easily explained,Integerology,Unusual arithmetic","url":"https:\/\/www.emis.de\/journals\/JIS\/VOL6\/Ufnarovski\/ufnarovski.html https:\/\/www.emis.de\/journals\/JIS\/VOL6\/Ufnarovski\/ufnarovski.pdf","urldate":"2013-10-31","year":"2003"},{"key":"HowLongIsMyToiletRoll","type":"article","title":"How long is my toilet roll? \u2013 a simple exercise in mathematical modelling","author":"Peter R. Johnston","abstract":"The simple question of how much paper is left on my toilet roll is studied from a mathematical modelling perspective. As is typical with applied mathematics, models of increasing complexity are introduced and solved. Solutions produced at each step are compared with the solution from the previous step. This process exposes students to the typical stages of mathematical modelling via an example from everyday life. Two activities are suggested for students to complete, as well as several extensions to stimulate class discussion.","comment":"","date_added":"2020-04-09","date_published":"2013-08-06","urls":["https:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/0020739X.2013.790502?journalCode=tmes20"],"collections":"Attention-grabbing titles,Easily explained,The act of doing maths,Modelling","url":"https:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/0020739X.2013.790502?journalCode=tmes20","year":"2013","urldate":"2020-04-09"},{"key":"KnotsAndLinksInSpatialGraphs","type":"article","title":"Knots and links in spatial graphs","author":"J. H. Conway and C. McA. Gordon","abstract":"The main purpose of this paper is to show that any embedding of \\(K_7\\) in three\u2010dimensional euclidean space contains a knotted cycle. By a similar but simpler argument, it is also shown that any embedding of \\(K_6\\) contains a pair of disjoint cycles which are homologically linked.","comment":"","date_added":"2020-04-12","date_published":"1983-08-06","urls":["https:\/\/onlinelibrary.wiley.com\/doi\/epdf\/10.1002\/jgt.3190070410","http:\/\/people.reed.edu\/~ormsbyk\/138\/ConwayGordon.pdf"],"collections":"Easily explained,Fun maths facts,Geometry","url":"https:\/\/onlinelibrary.wiley.com\/doi\/epdf\/10.1002\/jgt.3190070410 http:\/\/people.reed.edu\/~ormsbyk\/138\/ConwayGordon.pdf","urldate":"2020-04-12","year":"1983"},{"key":"ExtremeProofsITheIrrationalityof2","type":"article","title":"Extreme Proofs I: The Irrationality of \u221a2","author":"John H. Conway and Joseph Shipman and John H. Conway and Joseph Shipman","abstract":"Mathematicians often ask, ``what is the best proof'' of something, and indeed Erd\\\"{o}s used to speak of ``Proofs from the Book,'' meaning, of course, God\u2019s book. Aigner and Ziegler (1998) have attempted to reconstruct some of this Book. Here we take a different, and more tolerant approach.","comment":"","date_added":"2020-04-13","date_published":"2013-08-06","urls":["https:\/\/link.springer.com\/article\/10.1007\/s00283-013-9373-9","https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00283-013-9373-9.pdf"],"collections":"About proof","url":"https:\/\/link.springer.com\/article\/10.1007\/s00283-013-9373-9 https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00283-013-9373-9.pdf","year":"2013","urldate":"2020-04-13","publisher":"Springer US","fulltext_html_url":"https:\/\/link.springer.com\/article\/10.1007\/s00283-013-9373-9","journal":"The Mathematical Intelligencer","issn":"1866-7414","volume":"35","issue":"3","identifier":"doi:10.1007\/s00283-013-9373-9","doi":"10.1007\/s00283-013-9373-9","pages":"2-7"},{"key":"ComputingLinkages","type":"article","title":"Computing Linkages","author":"Andries de Man","abstract":"Analog calculating machines usually contain lots of gears (differentials), cams, ball-and-disc integrators and rack-and-pinions. But would it be possible to construct such calculating machines only using hinged rods? In the first instance, one would think only linear functions could be represented by such a mechanism but that is not true. This presentation describes \u201ccomputing linkages\u201d and the work of Antonin Svoboda on their systematic development.","comment":"","date_added":"2020-04-06","date_published":"2010-08-06","urls":["https:\/\/sites.google.com\/site\/calculatinghistory\/home\/computing-linkages"],"collections":"Easily explained,History,Things to make and do,Unusual computers","url":"https:\/\/sites.google.com\/site\/calculatinghistory\/home\/computing-linkages","urldate":"2020-04-06","year":"2010"},{"key":"ConwaysTilingGroupsonJSTOR","type":"article","title":"Conway's Tiling Groups on JSTOR","author":"William P. Thurston","abstract":"John Conway disovered a technique using infinite, finitely presented groups that in a number of interesting cases resolves the question of whether a region in the plane can be tessellated by given tiles. The idea is that the tiles can be interpreted as describing relators in a group, in such a way that the plane region can be tiled, only if the group element which describes the boundary of the region is the trivial element 1.","comment":"","date_added":"2020-04-15","date_published":"1990-08-06","urls":["https:\/\/www.jstor.org\/stable\/2324578"],"collections":"Fun maths facts,Geometry,The groups group","url":"https:\/\/www.jstor.org\/stable\/2324578","year":"1990","urldate":"2020-04-15"},{"key":"ProjectivizingSet","type":"article","title":"Projectivizing Set","author":"Cathy Hsu and Jonah Ostroff and Lucas Van Meter","abstract":"You might know the popular game SET. On the surface, this card game appears to be a contest of pattern recognition; however, it is also connected to many deep mathematical ideas, some of which have appeared in previous Math Horizons issues (for instance, see February 2007 and April 2017). In this article, we explore how we can change the mathematics behind SET to create a new variation of this classic game.","comment":"","date_added":"2020-04-19","date_published":"2020-08-06","urls":["https:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/10724117.2020.1714377?journalCode=umho20"],"collections":"Easily explained,Games to play with friends","url":"https:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/10724117.2020.1714377?journalCode=umho20","year":"2020","urldate":"2020-04-19"},{"key":"FourPagesAreIndeedNecessaryforPlanarGraphs","type":"article","title":"Four Pages Are Indeed Necessary for Planar Graphs","author":"Michael A. Bekos and Michael Kaufmann and Fabian Klute and Sergey Pupyrev and Chrysanthi Raftopoulou and Torsten Ueckerdt","abstract":"An embedding of a graph in a book consists of a linear order of its vertices\r\nalong the spine of the book and of an assignment of its edges to the pages of\r\nthe book, so that no two edges on the same page cross. The book thickness of a\r\ngraph is the minimum number of pages over all its book embeddings. Accordingly,\r\nthe book thickness of a class of graphs is the maximum book thickness over all\r\nits members. In this paper, we address a long-standing open problem regarding\r\nthe exact book thickness of the class of planar graphs, which previously was\r\nknown to be either three or four. We settle this problem by demonstrating\r\nplanar graphs that require four pages in any of their book embeddings, thus\r\nestablishing that the book thickness of the class of planar graphs is four.","comment":"","date_added":"2020-04-19","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2004.07630v1","http:\/\/arxiv.org\/pdf\/2004.07630v1"],"collections":"Attention-grabbing titles,Fun maths facts","url":"http:\/\/arxiv.org\/abs\/2004.07630v1 http:\/\/arxiv.org\/pdf\/2004.07630v1","year":"2020","urldate":"2020-04-19","archivePrefix":"arXiv","eprint":"2004.07630","primaryClass":"cs.DS"},{"key":"Gradwohl2007","type":"article","title":"Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles","author":"Gradwohl, Ronen and Naor, M. and Pinkas, Benny and Rothblum, G.","abstract":"","comment":"","date_added":"2012-02-07","date_published":"2007-08-06","urls":["https:\/\/link.springer.com\/chapter\/10.1007\/978-3-540-72914-3_16","http:\/\/www.wisdom.weizmann.ac.il\/~naor\/PAPERS\/sudoku.pdf"],"collections":"About proof,Easily explained,Protocols and strategies","url":"https:\/\/link.springer.com\/chapter\/10.1007\/978-3-540-72914-3_16 http:\/\/www.wisdom.weizmann.ac.il\/~naor\/PAPERS\/sudoku.pdf","urldate":"2012-02-07","year":"2007","booktitle":"Fun with Algorithms","number":"860","pages":"166--182","publisher":"Springer"},{"key":"Shadowmoviesnotarisingfromknots","type":"article","title":"Shadow movies not arising from knots","author":"Daniel Denton and Peter Doyle","abstract":"A shadow diagram is a knot diagram with under-over information omitted; a\r\nshadow movie is a sequence of shadow diagrams related by shadow Reidemeister\r\nmoves. We show that not every shadow movie arises as the shadow of a\r\nReidemeister movie, meaning a sequence of classical knot diagrams related by\r\nclassical Reidemeister moves. This means that in Kaufman's theory of virtual\r\nknots, virtual crossings cannot simply be viewed as classical crossings where\r\nwhich strand is over has been left `to be determined'.","comment":"","date_added":"2020-04-21","date_published":"2011-08-06","urls":["http:\/\/arxiv.org\/abs\/1106.3545v1","http:\/\/arxiv.org\/pdf\/1106.3545v1"],"collections":"Attention-grabbing titles","url":"http:\/\/arxiv.org\/abs\/1106.3545v1 http:\/\/arxiv.org\/pdf\/1106.3545v1","year":"2011","urldate":"2020-04-21","archivePrefix":"arXiv","eprint":"1106.3545","primaryClass":"math.GT"},{"key":"ThedistanceofapermutationfromasubgroupofSn","type":"article","title":"The distance of a permutation from a subgroup of \\(S_n\\)","author":"Richard G. E. Pinch","abstract":"We show that the problem of computing the distance of a given permutation\r\nfrom a subgroup $H$ of $S_n$ is in general NP-complete, even under the\r\nrestriction that $H$ is elementary Abelian of exponent 2. The problem is shown\r\nto be polynomial-time equivalent to a problem related to finding a maximal\r\npartition of the edges of an Eulerian directed graph into cycles and this\r\nproblem is in turn equivalent to the standard NP-complete problem of Boolean\r\nsatisfiability.","comment":"","date_added":"2020-04-30","date_published":"2005-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/0511501v1","http:\/\/arxiv.org\/pdf\/math\/0511501v1"],"collections":"basically-computer-science,computational-complexity-of-games,the-groups-group","url":"http:\/\/arxiv.org\/abs\/math\/0511501v1 http:\/\/arxiv.org\/pdf\/math\/0511501v1","urldate":"2020-04-30","year":"2005","archivePrefix":"arXiv","eprint":"math\/0511501","primaryClass":"math.CO"},{"key":"Demaine2012","type":"article","title":"Picture-Hanging Puzzles","author":"Demaine, Erik D. and Demaine, Martin L. and Minsky, Yair N. and Mitchell, Joseph S. B. and Rivest, Ronald L. and Patrascu, Mihai","abstract":"We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions.","comment":"","date_added":"2012-11-03","date_published":"2012-03-01","urls":["http:\/\/arxiv.org\/abs\/1203.3602","http:\/\/arxiv.org\/pdf\/1203.3602v2"],"collections":"easily-explained,puzzles,the-groups-group","url":"http:\/\/arxiv.org\/abs\/1203.3602 http:\/\/arxiv.org\/pdf\/1203.3602v2","urldate":"2012-11-03","year":"2012","archivePrefix":"arXiv","arxivId":"1203.3602","eprint":"1203.3602","isbn":"9783642303470","month":"mar","pages":"17","primaryClass":"cs.DS"},{"key":"WordcalculusinthefundamentalgroupoftheMengercurve","type":"article","title":"Word calculus in the fundamental group of the Menger curve","author":"Hanspeter Fischer and Andreas Zastrow","abstract":"The fundamental group of the Menger universal curve is uncountable and not\r\nfree, although all of its finitely generated subgroups are free. It contains an\r\nisomorphic copy of the fundamental group of every one-dimensional separable\r\nmetric space and an isomorphic copy of the fundamental group of every planar\r\nPeano continuum. We give an explicit and systematic combinatorial description\r\nof the fundamental group of the Menger universal curve and its generalized\r\nCayley graph in terms of word sequences. The word calculus, which requires only\r\ntwo letters and their inverses, is based on Pasynkov's partial topological\r\nproduct representation and can be expressed in terms of a variation on the\r\nclassical puzzle known as the Towers of Hanoi.","comment":"","date_added":"2020-05-02","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1310.7968v1","http:\/\/arxiv.org\/pdf\/1310.7968v1"],"collections":"combinatorics,fun-maths-facts,games-to-play-with-friends,the-groups-group","url":"http:\/\/arxiv.org\/abs\/1310.7968v1 http:\/\/arxiv.org\/pdf\/1310.7968v1","year":"2013","urldate":"2020-05-02","archivePrefix":"arXiv","eprint":"1310.7968","primaryClass":"math.GT"},{"key":"38406501359372282063949allthatMonodromyofFanoProblems","type":"article","title":"38406501359372282063949 & all that: Monodromy of Fano Problems","author":"Sachi Hashimoto and Borys Kadets","abstract":"A Fano problem is an enumerative problem of counting $r$-dimensional linear\r\nsubspaces on a complete intersection in $\\mathbb{P}^n$ over a field of\r\narbitrary characteristic, whenever the corresponding Fano scheme is finite. A\r\nclassical example is enumerating lines on a cubic surface. We study the\r\nmonodromy of finite Fano schemes $F_{r}(X)$ as the complete intersection $X$\r\nvaries. We prove that the monodromy group is either symmetric or alternating in\r\nmost cases. In the exceptional cases, the monodromy group is one of the Weyl\r\ngroups $W(E_6)$ or $W(D_k)$.","comment":"","date_added":"2020-05-06","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2002.04580v1","http:\/\/arxiv.org\/pdf\/2002.04580v1"],"collections":"attention-grabbing-titles,combinatorics,integerology","url":"http:\/\/arxiv.org\/abs\/2002.04580v1 http:\/\/arxiv.org\/pdf\/2002.04580v1","year":"2020","urldate":"2020-05-06","archivePrefix":"arXiv","eprint":"2002.04580","primaryClass":"math.AG"},{"key":"Crepeau1987","type":"article","title":"A zero-knowledge Poker protocol that achieves confidentiality of the players' strategy or How to achieve an electronic Poker face","author":"Cr\u00e9peau, C.","abstract":"","comment":"","date_added":"2012-01-10","date_published":"1987-08-06","urls":["https:\/\/link.springer.com\/chapter\/10.1007\/3-540-47721-7_18"],"collections":"attention-grabbing-titles,games-to-play-with-friends,protocols-and-strategies","url":"https:\/\/link.springer.com\/chapter\/10.1007\/3-540-47721-7_18","urldate":"2012-01-10","year":"1987","pages":"239--247","publisher":"Springer"},{"key":"InfinitudeofPrimesUsingFormalLanguageTheory","type":"article","title":"Infinitude of Primes Using Formal Language Theory","author":"Aalok Thakkar","abstract":"Formal languages are sets of strings of symbols described by a set of rules\r\nspecific to them. In this note, we discuss a certain class of formal languages,\r\ncalled regular languages, and put forward some elementary results. The\r\nproperties of these languages are then employed to prove that there are\r\ninfinitely many prime numbers.","comment":"","date_added":"2020-05-25","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2005.10372v1","http:\/\/arxiv.org\/pdf\/2005.10372v1"],"collections":"about-proof,basically-computer-science,easily-explained,fun-maths-facts,integerology","url":"http:\/\/arxiv.org\/abs\/2005.10372v1 http:\/\/arxiv.org\/pdf\/2005.10372v1","year":"2020","urldate":"2020-05-25","archivePrefix":"arXiv","eprint":"2005.10372","primaryClass":"cs.FL"},{"key":"GergonnesCardTrickPositionalNotationandRadixSort","type":"article","title":"Gergonne's Card Trick, Positional Notation, and Radix Sort","author":"Ethan D. Bolker","abstract":"Gergonne's three pile card trick has been a favorite of mathematicians for nearly two centuries. This new exposition uses the radix sorting algorithm well known to computer scientists to explain why the trick works, and to explore generalizations. The presentation suggests strategies for introducing the trick and base three arithmetic to elementary school students.","comment":"This is the \"Ace of Base Three\" card trick","date_added":"2020-05-30","date_published":"2017-08-06","urls":["https:\/\/www.tandfonline.com\/doi\/abs\/10.4169\/002557010X479983","https:\/\/www.maa.org\/sites\/default\/files\/Bolker-MMz-201053228.pdf"],"collections":"easily-explained,fun-maths-facts","url":"https:\/\/www.tandfonline.com\/doi\/abs\/10.4169\/002557010X479983 https:\/\/www.maa.org\/sites\/default\/files\/Bolker-MMz-201053228.pdf","urldate":"2020-05-30","year":"2017"},{"key":"IndigenousperspectivesinmathsUnderstandingGurruu","type":"article","title":"Indigenous perspectives in maths: Understanding Gurru\u1e6fu","author":"Chris Matthews","abstract":"Discusses Yol\u014bu mathematics and the interconnected relationships of Gurru\u1e6fu, and shares an activity for teachers and students to explore the connections and patterns in family trees.","comment":"","date_added":"2020-06-08","date_published":"2020-08-06","urls":["https:\/\/www.teachermagazine.com.au\/articles\/indigenous-perspectives-in-maths-understanding-gurruu"],"collections":"easily-explained,fun-maths-facts","url":"https:\/\/www.teachermagazine.com.au\/articles\/indigenous-perspectives-in-maths-understanding-gurruu","year":"2020","urldate":"2020-06-08"},{"key":"GoldbugVariations","type":"article","title":"Goldbug Variations","author":"Michael Kleber","abstract":"This \"Mathematical Entertainments\" column from the Intelligencer is an\r\nexposition of current investigations, rooted in recent work of Jim Propp, into\r\n\"quasirandom\" analogues of random walk and random aggregation processes.\r\nFeatured are the \"Goldbugs\" and the \"Rotor-router\". These are deterministic\r\nprocesses which simulate the random ones, for example having the same limiting\r\nstates, but with faster convergence.\r\n The paper includes three large illustrations, which appear twice in the\r\nsubmission, as both raster image (.png) and postscript (.eps) files. The latter\r\nare much larger but needed for latex inclusion; the former are smaller, used by\r\npdflatex, and better for pixel-level viewing.","comment":"","date_added":"2020-06-12","date_published":"2005-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/0501497v1","http:\/\/arxiv.org\/pdf\/math\/0501497v1"],"collections":"animals,attention-grabbing-titles,easily-explained,fun-maths-facts,puzzles","url":"http:\/\/arxiv.org\/abs\/math\/0501497v1 http:\/\/arxiv.org\/pdf\/math\/0501497v1","year":"2005","urldate":"2020-06-12","archivePrefix":"arXiv","eprint":"math\/0501497","primaryClass":"math.CO"},{"key":"SomeDoublyExponentialSequences","type":"article","title":"Some Doubly Exponential Sequences","author":"A. V. Aho and N. J. A. Sloane","abstract":"Let \\(x_0, x_1, x_2, \\cdots\\) be a sequence of natural numbers satisfying a nonlinear recurrence of the form \\(x_{n+1} = x_n^2 + g_n\\), where \\(|g_n| \\lt \\frac{1}{4}x_n\\) for \\(n \\geq n_0\\). Numerous example of such sequences are given, arising from Boolean functions, graph theory, language theory, automata theory, and number theory. By an elementary method it is shown that the solution is \\(x_n =\\) nearest integer to \\(k^{2^n}\\), for \\(n \\geq n_0\\), where \\(k\\) is a constant. That is, these are doubly exponential sequences. In some cases \\(k\\) is a \"known\" constant (such as \\(\\frac{1}{2}(1+\\sqrt{5})\\), but in general the formula for \\(k\\) involves \\(x_0,x_1,x_2,\\cdots\\)!","comment":"","date_added":"2020-07-13","date_published":"1970-08-06","urls":["http:\/\/neilsloane.com\/doc\/doubly.html"],"collections":"fun-maths-facts,integerology","url":"http:\/\/neilsloane.com\/doc\/doubly.html","urldate":"2020-07-13","year":"1970","journal":"Fibonacci Quarterly"},{"key":"AnOptimalSolutionfortheMuffinProblem","type":"article","title":"An Optimal Solution for the Muffin Problem","author":"Richard E. Chatwin","abstract":"The muffin problem asks us to divide $m$ muffins into pieces and assign each\r\nof those pieces to one of $s$ students so that the sizes of the pieces assigned\r\nto each student total $m\/s$, with the objective being to maximize the size of\r\nthe smallest piece in the solution. We present a recursive algorithm for\r\nsolving any muffin problem and demonstrate that it always produces an optimal\r\nsolution.","comment":"","date_added":"2020-08-17","date_published":"2019-08-06","urls":["http:\/\/arxiv.org\/abs\/1907.08726v2","http:\/\/arxiv.org\/pdf\/1907.08726v2"],"collections":"attention-grabbing-titles,food,fun-maths-facts,protocols-and-strategies","url":"http:\/\/arxiv.org\/abs\/1907.08726v2 http:\/\/arxiv.org\/pdf\/1907.08726v2","year":"2019","urldate":"2020-08-17","archivePrefix":"arXiv","eprint":"1907.08726","primaryClass":"math.CO"},{"key":"Ptak","type":"online","title":"A Do-It-Yourself Paper Digital Computer, 1959.","author":"Ptak, John F.","abstract":"This wonderful cut-away and paste-up template for a digital computer comes to us from the Communications of the Association for Computing Machinery, volume 2, issue 9 for September 1959. The PAPAC-00 is a \u201c2-register, 1-bit, fixed-instruction binary digital computer\u201d and was submitted to the journal by Rollin P. Mayer (of the MIT Lincoln Lab).","comment":"","date_added":"2013-05-05","date_published":"2010-08-06","urls":["https:\/\/longstreet.typepad.com\/thesciencebookstore\/2010\/11\/a-do-it-yourself-paper-digital-computer-1959.html"],"collections":"easily-explained,things-to-make-and-do","url":"https:\/\/longstreet.typepad.com\/thesciencebookstore\/2010\/11\/a-do-it-yourself-paper-digital-computer-1959.html","urldate":"2013-05-05","year":"2010"},{"key":"Akritas1986","type":"article","title":"There is no \"Uspensky's method\"","author":"Akritas, AG","abstract":"In this paper an attempt is made to correct the misconception of several authors that there exists a method by Upensky (based on Vincent's theorem) for the isolation of the real roots of a polynomial equation with rational coefficients. Despite Uspensky's claim, in the preface of his book, that he invented this method, we show that what Uspensky actually did was to take Vincent's method and double its computing time. Uspensky must not have understood Vincent's method probably because he was not aware of Budan's theorem. In view of the above, it is historically incorrect to attribute Vincent's method to Uspensky.","comment":"","date_added":"2011-09-15","date_published":"1986-08-06","urls":["https:\/\/dl.acm.org\/doi\/10.1145\/32439.32457"],"collections":"attention-grabbing-titles,drama","url":"https:\/\/dl.acm.org\/doi\/10.1145\/32439.32457","urldate":"2011-09-15","year":"1986"},{"key":"AstonishingNumbers","type":"article","title":"Astonishing Numbers","author":"Richard Hoshino","abstract":"We say that an ordered pair of positive integers \\(a,b\\) with \\(a \\lt b\\) is astonishing if the sum of the integers from \\(a\\) to \\(b\\), inclusive, is equal to the digits of \\(a\\) followed by the digits of \\(b\\). Determine all astonishing ordered pairs.","comment":"","date_added":"2020-08-28","date_published":"2001-08-06","urls":["https:\/\/cms.math.ca\/publications\/crux\/issue\/?volume=27&issue=1","https:\/\/cms.math.ca\/wp-content\/uploads\/crux-pdfs\/CRUXv27n1.pdf"],"collections":"attention-grabbing-titles,easily-explained,integerology","url":"https:\/\/cms.math.ca\/publications\/crux\/issue\/?volume=27&issue=1 https:\/\/cms.math.ca\/wp-content\/uploads\/crux-pdfs\/CRUXv27n1.pdf","year":"2001","urldate":"2020-08-28"},{"key":"AnInterestingSerendipitousRealNumber","type":"article","title":"An Interesting Serendipitous Real Number","author":"John Ewing and Ciprian Foias","abstract":"This is the story of a remarkable real number, the discovery of which was due to a misprint. Namely, in the mid-seventies, while Ciprian was at the University of Bucharest, one of his former students approached him with the following question: \r\n\r\nIf \\(x_1 \\gt 0\\) and \\(x_{n+1} = \\left(1 + \\frac{1}{x_n}\\right)^n\\), can \\(x_n \\to \\infty\\)?","comment":"Known as the Foias constant.","date_added":"2020-09-21","date_published":"2000-08-06","urls":["https:\/\/link.springer.com\/chapter\/10.1007\/978-1-4471-0751-4_8","https:\/\/link.springer.com\/content\/pdf\/10.1007%2F978-1-4471-0751-4_8.pdf"],"collections":"fun-maths-facts","url":"https:\/\/link.springer.com\/chapter\/10.1007\/978-1-4471-0751-4_8 https:\/\/link.springer.com\/content\/pdf\/10.1007%2F978-1-4471-0751-4_8.pdf","year":"2000","urldate":"2020-09-21","publisher":"Springer, London","doi":"10.1007\/978-1-4471-0751-4_8","fulltext_html_url":"https:\/\/link.springer.com\/chapter\/10.1007\/978-1-4471-0751-4_8","identifier":"10.1007\/978-1-4471-0751-4_8","pages":"119-126"},{"key":"TheNoFlippancyGame","type":"article","title":"The No-Flippancy Game","author":"Isha Agarwal and Matvey Borodin and Aidan Duncan and Kaylee Ji and Tanya Khovanova and Shane Lee and Boyan Litchev and Anshul Rastogi and Garima Rastogi and Andrew Zhao","abstract":"We analyze a coin-based game with two players where, before starting the\r\ngame, each player selects a string of length $n$ comprised of coin tosses. They\r\nalternate turns, choosing the outcome of a coin toss according to specific\r\nrules. As a result, the game is deterministic. The player whose string appears\r\nfirst wins. If neither player's string occurs, then the game must be infinite.\r\n We study several aspects of this game. We show that if, after $4n-4$ turns,\r\nthe game fails to cease, it must be infinite. Furthermore, we examine how a\r\nplayer may select their string to force a desired outcome. Finally, we describe\r\nthe result of the game for particular cases.","comment":"","date_added":"2020-09-21","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2006.09588v1","http:\/\/arxiv.org\/pdf\/2006.09588v1"],"collections":"easily-explained,games-to-play-with-friends","url":"http:\/\/arxiv.org\/abs\/2006.09588v1 http:\/\/arxiv.org\/pdf\/2006.09588v1","year":"2020","urldate":"2020-09-21","archivePrefix":"arXiv","eprint":"2006.09588","primaryClass":"math.CO"},{"key":"AminussignthatusedtoannoymebutnowIknowwhyitisthere","type":"article","title":"A minus sign that used to annoy me but now I know why it is there","author":"Peter Tingley","abstract":"We consider two well known constructions of link invariants. One uses skein\r\ntheory: you resolve each crossing of the link as a linear combination of things\r\nthat don't cross, until you eventually get a linear combination of links with\r\nno crossings, which you turn into a polynomial. The other uses quantum groups:\r\nyou construct a functor from a topological category to some category of\r\nrepresentations in such a way that (directed framed) links get sent to\r\nendomorphisms of the trivial representation, which are just rational functions.\r\nCertain instances of these two constructions give rise to essentially the same\r\ninvariants, but when one carefully matches them there is a minus sign that\r\nseems out of place. We discuss exactly how the constructions match up in the\r\ncase of the Jones polynomial, and where the minus sign comes from. On the\r\nquantum group side, one is led to use a non-standard ribbon element, which then\r\nallows one to consider a larger topological category.","comment":"","date_added":"2020-09-21","date_published":"2010-08-06","urls":["http:\/\/arxiv.org\/abs\/1002.0555v2","http:\/\/arxiv.org\/pdf\/1002.0555v2"],"collections":"attention-grabbing-titles","url":"http:\/\/arxiv.org\/abs\/1002.0555v2 http:\/\/arxiv.org\/pdf\/1002.0555v2","year":"2010","urldate":"2020-09-21","archivePrefix":"arXiv","eprint":"1002.0555","primaryClass":"math.GT"},{"key":"YouCouldHaveInventedSpectralSequences","type":"article","title":"You Could Have Invented Spectral Sequences","author":"Timothy Y. Chow","abstract":"The subject of spectral sequences has a reputation for being difficult for the beginner. Even G. W. Whitehead (quoted in John McCleary) once remarked, \u201cThe machinery of spectral sequences, stemming from the algebraic work of Lyndon and Koszul, seemed complicated and obscure to many topologists.\u201d","comment":"","date_added":"2020-09-21","date_published":"2006-08-06","urls":["http:\/\/www.ams.org\/notices\/200601\/fea-chow.pdf"],"collections":"attention-grabbing-titles","url":"http:\/\/www.ams.org\/notices\/200601\/fea-chow.pdf","year":"2006","urldate":"2020-09-21"},{"key":"MangoesandBlueberries","type":"article","title":"Mangoes and Blueberries","author":"Bruce Reed","abstract":"We prove the following conjecture of Erd\u0151s and Hajnal:\r\n\r\nFor every integer \\(k\\) there is an \\(f(k)\\) such that if for a graph \\(G\\), every subgraph \\(H\\) of \\(G\\) has a stable set containing vertices, then \\(G\\) contains a set \\(X\\) of at most \\(f(k)\\) vertices such that \\(G\u2212X\\) is bipartite.\r\n\r\nThis conjecture was related to me by Paul Erd\u0151s at a conference held in Annecy during July of 1996. I regret not being able to share the answer with him.","comment":"","date_added":"2020-09-21","date_published":"1999-08-06","urls":["https:\/\/link.springer.com\/article\/10.1007\/s004930050056","https:\/\/link.springer.com\/content\/pdf\/10.1007\/s004930050056.pdf"],"collections":"attention-grabbing-titles,food","url":"https:\/\/link.springer.com\/article\/10.1007\/s004930050056 https:\/\/link.springer.com\/content\/pdf\/10.1007\/s004930050056.pdf","year":"1999","urldate":"2020-09-21","publisher":"Bolyai Society \u2013 Springer-Verlag","fulltext_html_url":"https:\/\/link.springer.com\/article\/10.1007\/s004930050056","journal":"Combinatorica","issn":"1439-6912","volume":"19","issue":"2","identifier":"doi:10.1007\/s004930050056","doi":"10.1007\/s004930050056","pages":"267-296"},{"key":"Paperfoldingmorphismsplanefillingcurvesandfractaltiles","type":"article","title":"Paperfolding morphisms, planefilling curves, and fractal tiles","author":"Michel Dekking","abstract":"An interesting class of automatic sequences emerges from iterated\r\npaperfolding. The sequences generate curves in the plane with an almost\r\nperiodic structure. We generalize the results obtained by Davis and Knuth on\r\nthe self-avoiding and planefilling properties of these curves, giving simple\r\ngeometric criteria for a complete classification. Finally, we show how the\r\nautomatic structure of the sequences leads to self-similarity of the curves,\r\nwhich turns the planefilling curves in a scaling limit into fractal tiles. For\r\nsome of these tiles we give a particularly simple formula for the Hausdorff\r\ndimension of their boundary.","comment":"","date_added":"2020-09-21","date_published":"2010-08-06","urls":["http:\/\/arxiv.org\/abs\/1011.5788v2","http:\/\/arxiv.org\/pdf\/1011.5788v2"],"collections":"geometry,things-to-make-and-do","url":"http:\/\/arxiv.org\/abs\/1011.5788v2 http:\/\/arxiv.org\/pdf\/1011.5788v2","year":"2010","urldate":"2020-09-21","archivePrefix":"arXiv","eprint":"1011.5788","primaryClass":"math.CO"},{"key":"Planefillingcurvesonalluniformgrids","type":"article","title":"Plane-filling curves on all uniform grids","author":"J\u00f6rg Arndt","abstract":"We describe a search for plane-filling curves traversing all edges of a grid\r\nonce. The curves are given by Lindenmayer systems with only one non-constant\r\nletter. All such curves for small orders on three grids have been found. For\r\nall uniform grids we show how curves traversing all points once can be obtained\r\nfrom the curves found. Curves traversing all edges once are described for the\r\nfour uniform grids where they exist.","comment":"","date_added":"2020-09-21","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1607.02433v2","http:\/\/arxiv.org\/pdf\/1607.02433v2"],"collections":"fun-maths-facts,geometry,things-to-make-and-do","url":"http:\/\/arxiv.org\/abs\/1607.02433v2 http:\/\/arxiv.org\/pdf\/1607.02433v2","year":"2016","urldate":"2020-09-21","archivePrefix":"arXiv","eprint":"1607.02433","primaryClass":"math.CO"},{"key":"ParkingFunctionsChooseYourOwnAdventure","type":"article","title":"Parking Functions: Choose Your Own Adventure","author":"Joshua Carlson and Alex Christensen and Pamela E. Harris and Zakiya Jones and Andr\u00e9s Ramos Rodr\u00edguez","abstract":"Warning. The reading of this paper will send you down many winding roads\r\ntoward new and exciting research topics enumerating generalized parking\r\nfunctions. Buckle up!","comment":"","date_added":"2020-09-21","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2001.04817v3","http:\/\/arxiv.org\/pdf\/2001.04817v3"],"collections":"attention-grabbing-titles,combinatorics,easily-explained,puzzles,the-act-of-doing-maths","url":"http:\/\/arxiv.org\/abs\/2001.04817v3 http:\/\/arxiv.org\/pdf\/2001.04817v3","year":"2020","urldate":"2020-09-21","archivePrefix":"arXiv","eprint":"2001.04817","primaryClass":"math.CO"},{"key":"TherearenotExactlyFiveObjects","type":"article","title":"There are not Exactly Five Objects","author":"Andreas Blass","abstract":"The purpose of this note is to present a solution to a problem posed informally by A. Wilkie at the 1977 ASL meeting in Wroc\u0142aw: Formulate \"the cardinality of the universe is not exactly five\" as a Horn sentence. Although this solution is not new -- I found it and M. Morley found another solution during the same meeting -- continuing sporadic inquiries about it suggest that its publication may be appropriate, even at this late date.","comment":"","date_added":"2020-09-21","date_published":"1984-08-06","urls":["https:\/\/www.jstor.org\/stable\/2274177?seq=1#metadata_info_tab_contents"],"collections":"attention-grabbing-titles","url":"https:\/\/www.jstor.org\/stable\/2274177?seq=1#metadata_info_tab_contents","urldate":"2020-09-21","year":"1984","journal":"The Journal of Symbolic Logic"},{"key":"ThePenneysGamewithGroupAction","type":"article","title":"The Penney's Game with Group Action","author":"Tanya Khovanova and Sean Li","abstract":"We generalize word avoidance theory by equipping the alphabet $\\mathcal{A}$\r\nwith a group action. We call equivalence classes of words patterns. We extend\r\nthe notion of word correlation to patterns using group stabilizers. We extend\r\nknown word avoidance results to patterns. We use these results to answer\r\nstandard questions for the Penney's game on patterns and show non-transitivity\r\nfor the game on patterns as the length of the pattern tends to infinity. We\r\nalso analyze bounds on the pattern-based Conway leading number and expected\r\nwait time, and further explore the game under the cyclic and symmetric group\r\nactions.","comment":"","date_added":"2020-10-16","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2009.06080v1","http:\/\/arxiv.org\/pdf\/2009.06080v1"],"collections":"games-to-play-with-friends,the-groups-group","url":"http:\/\/arxiv.org\/abs\/2009.06080v1 http:\/\/arxiv.org\/pdf\/2009.06080v1","year":"2020","urldate":"2020-10-16","archivePrefix":"arXiv","eprint":"2009.06080","primaryClass":"math.CO"},{"key":"OntheDreadedRightBousfieldLocalization","type":"article","title":"On the Dreaded Right Bousfield Localization","author":"Clark Barwick","abstract":"I verify the existence of right Bousfield localizations of right semimodel\r\ncategories, and I apply this to construct a model of the homotopy limit of a\r\nleft Quillen presheaf as a right semimodel category.","comment":"","date_added":"2020-10-16","date_published":"2007-08-06","urls":["http:\/\/arxiv.org\/abs\/0708.3435v2","http:\/\/arxiv.org\/pdf\/0708.3435v2"],"collections":"attention-grabbing-titles","url":"http:\/\/arxiv.org\/abs\/0708.3435v2 http:\/\/arxiv.org\/pdf\/0708.3435v2","year":"2007","urldate":"2020-10-16","archivePrefix":"arXiv","eprint":"0708.3435","primaryClass":"math.AT"},{"key":"WorldsshortestexplanationofGdelstheorem","type":"article","title":"World's shortest explanation of G\u00f6del's theorem","author":"Mark Dominus","abstract":"A while back I started writing up an article titled \"World's shortest explanation of G\u00f6del's theorem\". But I didn't finish it, and later I encountered Raymond Smullyan's version, which is much shorter anyway. So here, shamelessly stolen from Smullyan, is the World's shortest explanation of G\u00f6del's theorem.","comment":"","date_added":"2020-10-16","date_published":"2009-08-06","urls":["https:\/\/blog.plover.com\/math\/Gdl-Smullyan.html"],"collections":"about-proof,attention-grabbing-titles,easily-explained,fun-maths-facts","url":"https:\/\/blog.plover.com\/math\/Gdl-Smullyan.html","year":"2009","urldate":"2020-10-16"},{"key":"IcecreamandorbifoldRiemannRoch","type":"article","title":"Ice cream and orbifold Riemann-Roch","author":"Anita Buckley and Miles Reid and Shengtian Zhou","abstract":"We give an orbifold Riemann-Roch formula in closed form for the Hilbert\r\nseries of a quasismooth polarized n-fold X,D, under the assumption that X is\r\nprojectively Gorenstein with only isolated orbifold points. Our formula is a\r\nsum of parts each of which is integral and Gorenstein symmetric of the same\r\ncanonical weight; the orbifold parts are called \"ice cream functions\". This\r\nform of the Hilbert series is particularly useful for computer algebra, and we\r\nillustrate it on examples of K3 surfaces and Calabi-Yau 3-folds.\r\n These results apply also with higher dimensional orbifold strata (see [A.\r\nBuckley and B. Szendroi, Orbifold Riemann-Roch for 3-folds with an application\r\nto Calabi-Yau geometry, J. Algebraic Geometry 14 (2005) 601--622] and\r\n[Shengtian Zhou, Orbifold Riemann-Roch and Hilbert series, University of\r\nWarwick PhD thesis, March 2011, 91+vii pp.], although the correct statements\r\nare considerably trickier. We expect to return to this in future publications.","comment":"","date_added":"2020-10-16","date_published":"2012-08-06","urls":["http:\/\/arxiv.org\/abs\/1208.0457v1","http:\/\/arxiv.org\/pdf\/1208.0457v1"],"collections":"food","url":"http:\/\/arxiv.org\/abs\/1208.0457v1 http:\/\/arxiv.org\/pdf\/1208.0457v1","year":"2012","urldate":"2020-10-16","archivePrefix":"arXiv","eprint":"1208.0457","primaryClass":"math.AG"},{"key":"FertilityNumbers","type":"article","title":"Fertility Numbers","author":"Colin Defant","abstract":"A nonnegative integer is called a fertility number if it is equal to the\r\nnumber of preimages of a permutation under West's stack-sorting map. We prove\r\nstructural results concerning permutations, allowing us to deduce information\r\nabout the set of fertility numbers. In particular, the set of fertility numbers\r\nis closed under multiplication and contains every nonnegative integer that is\r\nnot congruent to $3$ modulo $4$. We show that the lower asymptotic density of\r\nthe set of fertility numbers is at least $1954\/2565\\approx 0.7618$. We also\r\nexhibit some positive integers that are not fertility numbers and conjecture\r\nthat there are infinitely many such numbers.","comment":"","date_added":"2020-10-16","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1809.04421v3","http:\/\/arxiv.org\/pdf\/1809.04421v3"],"collections":"attention-grabbing-titles,combinatorics,integerology","url":"http:\/\/arxiv.org\/abs\/1809.04421v3 http:\/\/arxiv.org\/pdf\/1809.04421v3","year":"2018","urldate":"2020-10-16","archivePrefix":"arXiv","eprint":"1809.04421","primaryClass":"math.CO"},{"key":"ThePhillipIslandpenguinparadeamathematicaltreatment","type":"article","title":"The Phillip Island penguin parade (a mathematical treatment)","author":"Serena Dipierro and Luca Lombardini and Pietro Miraglio and Enrico Valdinoci","abstract":"Penguins are flightless, so they are forced to walk while on land. In\r\nparticular, they show rather specific behaviors in their homecoming, which are\r\ninteresting to observe and to describe analytically. In this paper, we present\r\na simple mathematical formulation to describe the little penguins parade in\r\nPhillip Island. We observed that penguins have the tendency to waddle back and\r\nforth on the shore to create a sufficiently large group and then walk home\r\ncompactly together. The mathematical framework that we introduce describes this\r\nphenomenon, by taking into account \"natural parameters\" such as the eye-sight\r\nof the penguins, their cruising speed and the possible \"fear\" of animals. On\r\nthe one hand, this favors the formation of conglomerates of penguins that\r\ngather together, but, on the other hand, this may lead to the \"panic\" of\r\nisolated and exposed individuals. The model that we propose is based on a set\r\nof ordinary differential equations. Due to the discontinuous behavior of the\r\nspeed of the penguins, the mathematical treatment (to get existence and\r\nuniqueness of the solution) is based on a \"stop-and-go\" procedure. We use this\r\nsetting to provide rigorous examples in which at least some penguins manage to\r\nsafely return home (there are also cases in which some penguins freeze due to\r\npanic). To facilitate the intuition of the model, we also present some simple\r\nnumerical simulations that can be compared with the actual movement of the\r\npenguins parade.","comment":"","date_added":"2020-10-16","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1611.08715v2","http:\/\/arxiv.org\/pdf\/1611.08715v2"],"collections":"animals,modelling","url":"http:\/\/arxiv.org\/abs\/1611.08715v2 http:\/\/arxiv.org\/pdf\/1611.08715v2","year":"2016","urldate":"2020-10-16","archivePrefix":"arXiv","eprint":"1611.08715","primaryClass":"math.CA"},{"key":"MathematicsWithAMetamathematicalFlavour","type":"article","title":"Mathematics with a metamathematical flavour. ","author":"Timothy Gowers","abstract":"Among the most fascinating results of mathematics are unprovability theorems, that is, rigorous proofs that certain statements cannot be deduced from certain axioms. A very famous example is Paul Cohen's demonstration that the continuum hypothesis cannot be deduced from the ZFC axioms . For this, Cohen invented a technique known as forcing, which is far too advanced for a page like this. (Indeed, I am incapable of presenting it anyway - if you are curious you could try visiting this site for some notes on forcing. They seem all right, but I don't know enough to be able to judge with any confidence.) Instead, I shall present here a few examples of low-level unprovability theorems, by which I mean purely mathematical results that, in one way or another, tell us that proofs of certain theorems must necessarily have certain properties. Such conclusions I shall loosely refer to as metamathematics. ","comment":"","date_added":"2020-10-16","date_published":null,"urls":["https:\/\/www.dpmms.cam.ac.uk\/~wtg10\/metamathematics.html"],"collections":"about-proof,the-act-of-doing-maths","url":"https:\/\/www.dpmms.cam.ac.uk\/~wtg10\/metamathematics.html","year":"","urldate":"2020-10-16"},{"key":"MathematicsMorally","type":"article","title":"Mathematics, morally","author":"Eugenia Cheng","abstract":"A source of tension between Philosophers of Mathematics and Mathematicians is the fact that each group feels ignored by the other; daily mathematical practie seems barely affected by the questions the Philosophers are considering. In this talk I will describe an issue that does have an impact on mathematical practice, and a philosophical stance on mathematics that is detectable in the work of practising mathematicians. No doubt controversially, I will call this issue `morality', but the term is not of my coining: there are mathematicians across the world who use the word `morally' to greate effect in private, and I propose that there should be a public theory of what they mean by this.","comment":"","date_added":"2020-10-16","date_published":"2004-08-06","urls":["http:\/\/eugeniacheng.com\/wp-content\/uploads\/2017\/02\/cheng-morality.pdf"],"collections":"the-act-of-doing-maths","url":"http:\/\/eugeniacheng.com\/wp-content\/uploads\/2017\/02\/cheng-morality.pdf","year":"2004","urldate":"2020-10-16"},{"key":"LeastSignificantNonZeroDigitofn","type":"online","title":"Least Significant Non-Zero Digit of n!","author":"Kevin S. Brown","abstract":"Let \\(p(k)\\) be the least significant non-zero decimal digit of \\(k!\\) Can we directly determine the \\(k\\)th term for any given \\(k\\)? ","comment":"","date_added":"2020-10-22","date_published":"2000-08-06","urls":["https:\/\/www.mathpages.com\/home\/kmath489.htm"],"collections":"easily-explained,fun-maths-facts,integerology","url":"https:\/\/www.mathpages.com\/home\/kmath489.htm","year":"2000","urldate":"2020-10-22"},{"key":"Generatingfunctionsforgeneratingtrees","type":"article","title":"Generating functions for generating trees","author":"Cyril Banderier and Mireille Bousquet-M\u00e9lou and Alain Denise and Philippe Flajolet and Dani\u00e8le Gardy and Dominique Gouyou-Beauchamps","abstract":"Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.","comment":"","date_added":"2020-10-24","date_published":"2002-08-06","urls":["https:\/\/www.sciencedirect.com\/science\/article\/abs\/pii\/S0012365X01002503"],"collections":"integerology","url":"https:\/\/www.sciencedirect.com\/science\/article\/abs\/pii\/S0012365X01002503","year":"2002","urldate":"2020-10-24"},{"key":"APrimeRepresentingConstant","type":"article","title":"A Prime-Representing Constant","author":"Dylan Fridman and Juli Garbulsky and Bruno Glecer and James Grime and Massi Tron Florentin","abstract":"We present a constant and a recursive relation to define a sequence $f_n$\r\nsuch that the floor of $f_n$ is the $n$th prime. Therefore, this constant\r\ngenerates the complete sequence of primes. We also show this constant is\r\nirrational and consider other sequences that can be generated using the same\r\nmethod.","comment":"","date_added":"2020-11-06","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2010.15882v1","http:\/\/arxiv.org\/pdf\/2010.15882v1"],"collections":"easily-explained,fun-maths-facts,integerology","url":"http:\/\/arxiv.org\/abs\/2010.15882v1 http:\/\/arxiv.org\/pdf\/2010.15882v1","year":"2020","urldate":"2020-11-06","archivePrefix":"arXiv","eprint":"2010.15882","primaryClass":"math.NT"},{"key":"Configurationspacesofhardsquaresinarectangle","type":"article","title":"Configuration spaces of hard squares in a rectangle","author":"Hannah Alpert and Ulrich Bauer and Matthew Kahle and Robert MacPherson and Kelly Spendlove","abstract":"We study the configuration spaces C(n;p,q) of n labeled hard squares in a p\r\nby q rectangle, a generalization of the well-known \"15 Puzzle\". Our main\r\ninterest is in the topology of these spaces. Our first result is to describe a\r\ncubical cell complex and prove that is homotopy equivalent to the configuration\r\nspace. We then focus on determining for which n, j, p, and q the homology group\r\n$H_j [ C(n;p,q) ]$ is nontrivial. We prove three homology-vanishing theorems,\r\nbased on discrete Morse theory on the cell complex. Then we describe several\r\nexplicit families of nontrivial cycles, and a method for interpolating between\r\nparameters to fill in most of the picture for \"large-scale\" nontrivial\r\nhomology.","comment":"","date_added":"2020-11-06","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2010.14480v1","http:\/\/arxiv.org\/pdf\/2010.14480v1"],"collections":"puzzles","url":"http:\/\/arxiv.org\/abs\/2010.14480v1 http:\/\/arxiv.org\/pdf\/2010.14480v1","year":"2020","urldate":"2020-11-06","archivePrefix":"arXiv","eprint":"2010.14480","primaryClass":"math.AT"},{"key":"Bigfieldsthatarenotlarge","type":"article","title":"Big fields that are not large","author":"Mazur, Barry and Rubin, Karl","abstract":"A subfield \\(K\\) of \\(\\bar {\\mathbb{Q}}\\) is large if every smooth curve \\(C\\) over \\(K\\) with a \\(K\\)-rational point has infinitely many \\(K\\)-rational points. A subfield \\(K\\) of \\(\\bar {\\mathbb{Q}}\\) is big if for every positive integer \\(n\\), \\(K\\) contains a number field \\(F\\) with \\([F:\\mathbb{Q}]\\) divisible by \\(n\\). The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large. ","comment":"","date_added":"2020-11-17","date_published":"2020-08-06","urls":["https:\/\/www.ams.org\/journals\/bproc\/2020-07-14\/S2330-1511-2020-00057-8\/","https:\/\/www.ams.org\/bproc\/2020-07-14\/S2330-1511-2020-00057-8\/S2330-1511-2020-00057-8.pdf"],"collections":"attention-grabbing-titles","url":"https:\/\/www.ams.org\/journals\/bproc\/2020-07-14\/S2330-1511-2020-00057-8\/ https:\/\/www.ams.org\/bproc\/2020-07-14\/S2330-1511-2020-00057-8\/S2330-1511-2020-00057-8.pdf","year":"2020","urldate":"2020-11-17","journal":"Proceedings of the American Mathematical Society, Series B","issn":"2330-1511","volume":"7","issue":"14","doi":"10.1090\/bproc\/57","pages":"159-169"},{"key":"Graphsfriendsandacquaintances","type":"article","title":"Graphs, friends and acquaintances","author":"C. Dalf\u00f3 and M. A. Fiol","abstract":"As is well known, a graph is a mathematical object modeling the existence of\r\na certain relation between pairs of elements of a given set. Therefore, it is\r\nnot surprising that many of the first results concerning graphs made reference\r\nto relationships between people or groups of people. In this article, we\r\ncomment on four results of this kind, which are related to various general\r\ntheories on graphs and their applications: the Handshake lemma (related to\r\ngraph colorings and Boolean algebra), a lemma on known and unknown people at a\r\ncocktail party (to Ramsey theory), a theorem on friends in common (to\r\ndistance-regularity and coding theory), and Hall's Marriage theorem (to the\r\ntheory of networks). These four areas of graph theory, often with problems\r\nwhich are easy to state but difficult to solve, are extensively developed and\r\ncurrently give rise to much research work. As examples of representative\r\nproblems and results of these areas, which are discussed in this paper, we may\r\ncite the following: the Four Colors Theorem (4CTC), the Ramsey numbers,\r\nproblems of the existence of distance-regular graphs and completely regular\r\ncodes, and finally the study of topological proprieties of interconnection\r\nnetworks.","comment":"","date_added":"2020-11-25","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1611.07462v2","http:\/\/arxiv.org\/pdf\/1611.07462v2"],"collections":"attention-grabbing-titles,easily-explained","url":"http:\/\/arxiv.org\/abs\/1611.07462v2 http:\/\/arxiv.org\/pdf\/1611.07462v2","year":"2016","urldate":"2020-11-25","archivePrefix":"arXiv","eprint":"1611.07462","primaryClass":"math.HO"},{"key":"Warrington2009","type":"article","title":"Juggling Probabilities","author":"Warrington, Gregory S.","abstract":"The act of a person juggling can be viewed as a Markov process if we assume that the juggler throws to random heights. I make this association for the simplest reasonable model of random juggling and compute the steady state probabilities in terms of the Stirling numbers of the second kind. I also explore several alternate models of juggling. ","comment":"","date_added":"2011-03-20","date_published":"2009-01-01","urls":["https:\/\/arxiv.org\/abs\/math\/0302257v1","https:\/\/arxiv.org\/pdf\/math\/0302257v1","http:\/\/www.jstor.org\/pss\/30037409"],"collections":"easily-explained","url":"https:\/\/arxiv.org\/abs\/math\/0302257v1 https:\/\/arxiv.org\/pdf\/math\/0302257v1 http:\/\/www.jstor.org\/pss\/30037409","urldate":"2011-03-20","year":"2009","language":"EN","month":"jan","publisher":"Mathematical Association of America"},{"key":"MatchStickGeometry","type":"article","title":"\"Match-Stick\" Geometry","author":"T.R. Dawson","abstract":"The following paper arose out of an attempt to improve upon the familiar \"match puzzles\" one sees in the popular press and which are generally too puerile to be interesting. Following some desultory experiments, it developed that the methods of constructions postulated are capable of determining all points obtainable with ruler and compass, but no others.","comment":"","date_added":"2020-12-20","date_published":"1939-08-06","urls":["https:\/\/www.jstor.org\/stable\/3607531?seq=1#metadata_info_tab_contents","https:\/\/www.cambridge.org\/core\/journals\/mathematical-gazette\/article\/abs\/matchstick-geometry\/7E1F1089E56A1C1EDA907BC95BF59129"],"collections":"about-proof,easily-explained,fun-maths-facts,geometry,things-to-make-and-do","url":"https:\/\/www.jstor.org\/stable\/3607531?seq=1#metadata_info_tab_contents https:\/\/www.cambridge.org\/core\/journals\/mathematical-gazette\/article\/abs\/matchstick-geometry\/7E1F1089E56A1C1EDA907BC95BF59129","year":"1939","urldate":"2020-12-20","doi":"10.2307\/3607531"},{"key":"AreThereMoreFiniteRingsthanFiniteGroups","type":"article","title":"Are There More Finite Rings than Finite Groups?","author":"Desmond MacHale","abstract":"We compare the number of finite groups of order \\(n\\) with the number of finite rings of order \\(n\\), with some surprising results.","comment":"","date_added":"2021-01-06","date_published":"2020-08-06","urls":["https:\/\/www.tandfonline.com\/doi\/full\/10.1080\/00029890.2020.1820790","https:\/\/doi.org\/10.1080\/00029890.2020.1820790"],"collections":"fun-maths-facts,the-groups-group","url":"https:\/\/www.tandfonline.com\/doi\/full\/10.1080\/00029890.2020.1820790 https:\/\/doi.org\/10.1080\/00029890.2020.1820790","year":"2020","urldate":"2021-01-06"},{"key":"CountingCandyCrushConfigurations","type":"article","title":"Counting Candy Crush Configurations","author":"Adam Hamilton and Giang T. Nguyen and Matthew Roughan","abstract":"A k-stable c-coloured Candy Crush grid is a weak proper c-colouring of a\r\nparticular type of k-uniform hypergraph. In this paper we introduce a fully\r\npolynomial randomised approximation scheme (FPRAS) which counts the number of\r\nk-stable c-coloured Candy Crush grids of a given size (m, n) for certain values\r\nof c and k. We implemented this algorithm on Matlab, and found that in a Candy\r\nCrush grid with7 available colours there are approximately 4.3*10^61 3-stable\r\ncolourings. (Note that, typical Candy Crush games are played with 6 colours and\r\nour FPRAS is not guaranteed to work in expected polynomial time with k= 3 and\r\nc= 6.) We also discuss the applicability of this FPRAS to the problem of\r\ncounting the number of weak c-colourings of other, more general hypergraphs.","comment":"","date_added":"2021-01-06","date_published":"2019-08-06","urls":["http:\/\/arxiv.org\/abs\/1908.09996v1","http:\/\/arxiv.org\/pdf\/1908.09996v1"],"collections":"combinatorics,games-to-play-with-friends","url":"http:\/\/arxiv.org\/abs\/1908.09996v1 http:\/\/arxiv.org\/pdf\/1908.09996v1","year":"2019","urldate":"2021-01-06","archivePrefix":"arXiv","eprint":"1908.09996","primaryClass":"math.CO"},{"key":"Envelopesaresolvingmachinesforquadraticsandcubicsandcertainpolynomialsofarbitrarydegree","type":"article","title":"Envelopes are solving machines for quadratics and cubics and certain polynomials of arbitrary degree","author":"Michael Schmitz and Andr\u00e9 Streicher","abstract":"Everybody knows from school how to solve a quadratic equation of the form\r\n$x^2-px+q=0$ graphically. But this method can become tedious if several\r\nequations ought to be solved, as for each pair $(p,q)$ a new parabola has to be\r\ndrawn. Stunningly, there is one single curve that can be used to solve every\r\nquadratic equation via drawing tangent lines through a given point $(p,q)$ to\r\nthis curve.\r\n In this article we derive this method in an elementary way and generalize it\r\nto equations of the form $x^n-px+q=0$ for arbitrary $n \\ge 2$. Moreover, the\r\nnumber of solutions of a specific equation of this form can be seen immediately\r\nwith this technique. Concluding the article we point out connections to the\r\nduality of points and lines in the plane and to the the concept of Legendre\r\ntransformation.","comment":"","date_added":"2021-01-06","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2012.06821v1","http:\/\/arxiv.org\/pdf\/2012.06821v1"],"collections":"easily-explained,fun-maths-facts,geometry,things-to-make-and-do,unusual-computers","url":"http:\/\/arxiv.org\/abs\/2012.06821v1 http:\/\/arxiv.org\/pdf\/2012.06821v1","year":"2020","urldate":"2021-01-06","archivePrefix":"arXiv","eprint":"2012.06821","primaryClass":"math.HO"},{"key":"Therealnumbersasurveyofconstructions","type":"article","title":"The real numbers - a survey of constructions","author":"Ittay Weiss","abstract":"We present a comprehensive survey of constructions of the real numbers (from\r\neither the rationals or the integers) in a unified fashion, thus providing an\r\noverview of most (if not all) known constructions ranging from the earliest\r\nattempts to recent results, and allowing for a simple comparison-at-a-glance\r\nbetween different constructions.","comment":"","date_added":"2021-01-06","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1506.03467v1","http:\/\/arxiv.org\/pdf\/1506.03467v1"],"collections":"about-proof,history","url":"http:\/\/arxiv.org\/abs\/1506.03467v1 http:\/\/arxiv.org\/pdf\/1506.03467v1","year":"2015","urldate":"2021-01-06","archivePrefix":"arXiv","eprint":"1506.03467","primaryClass":"math.HO"},{"key":"Hilbert13Arethereanygenuinecontinuousmultivariaterealvaluedfunctions","type":"article","title":"Hilbert 13: Are there any genuine continuous multivariate real-valued functions?","author":"Morris, Sidney","abstract":"This article begins with a provocative question: Are there any genuine continuous multivariate real-valued functions? This may seem to be a silly question, but it is in essence what David Hilbert asked as one of the 23 problems he posed at the second International Congress of Mathematicians, held in Paris in 1900. These problems guided a large portion of the research in mathematics of the 20th century. Hilbert's 13th problem conjectured that there exists a continuous function $ f:\\mathbb{I}^3\\to \\mathbb{R}$, where $ {\\mathbb{I}=[0,1]}$, which cannot be expressed in terms of composition and addition of continuous functions from $ \\mathbb{R}^2 \\to \\mathbb{R}$, that is, as composition and addition of continuous real-valued functions of two variables. It took over 50 years to prove that Hilbert's conjecture is false. This article discusses the solution. ","comment":"","date_added":"2021-02-02","date_published":"2021-08-06","urls":["https:\/\/www.ams.org\/journals\/bull\/2021-58-01\/S0273-0979-2020-01698-8\/","https:\/\/www.ams.org\/bull\/2021-58-01\/S0273-0979-2020-01698-8\/S0273-0979-2020-01698-8.pdf"],"collections":"about-proof,fun-maths-facts","url":"https:\/\/www.ams.org\/journals\/bull\/2021-58-01\/S0273-0979-2020-01698-8\/ https:\/\/www.ams.org\/bull\/2021-58-01\/S0273-0979-2020-01698-8\/S0273-0979-2020-01698-8.pdf","year":"2021","urldate":"2021-02-02","journal":"Bulletin of the American Mathematical Society","issn":"1088-9485","volume":"58","issue":"1","doi":"10.1090\/bull\/1698","pages":"107-118"},{"key":"Maymin2011","type":"article","title":"Markets are efficient if and only if P = NP","author":"Maymin, PZ","abstract":" I prove that if markets are weak-form efficient, meaning current prices fully reflect all information available in past prices, then P = NP, meaning every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. I also prove the converse by showing how we can \"program\" the market to solve NP-complete problems. Since P probably does not equal NP, markets are probably not efficient. Specifically, markets become increasingly inefficient as the time series lengthens or becomes more frequent. An illustration by way of partitioning the excess returns to momentum strategies based on data availability confirms this prediction. ","comment":"","date_added":"2013-02-11","date_published":"2011-08-06","urls":["https:\/\/arxiv.org\/abs\/1002.2284","https:\/\/arxiv.org\/pdf\/1002.2284"],"collections":"computational-complexity-of-games","url":"https:\/\/arxiv.org\/abs\/1002.2284 https:\/\/arxiv.org\/pdf\/1002.2284","urldate":"2013-02-11","year":"2011","journal":"Algorithmic Finance","volume":"2010"},{"key":"ProgrammingtheHilbertcurve","type":"article","title":"Programming the Hilbert curve","author":"John Skilling","abstract":"The Hilbert curve has previously been constructed recursively, using \\(p\\) levels of recursion of \\(n\\)\u2010bit Gray codes to attain a precision of \\(p\\) bits in \\(n\\) dimensions. Implementations have reflected the awkwardness of aligning the recursive steps to preserve geometrical adjacency. We point out that a single global Gray code can instead be applied to all \\(np\\) bits of a Hilbert length. Although this \u201cover\u2010transforms\u201d the length, the excess work can be undone in a single pass over the bits, leading to compact and efficient computer code.","comment":"","date_added":"2021-02-12","date_published":"2004-08-06","urls":["https:\/\/aip.scitation.org\/doi\/abs\/10.1063\/1.1751381"],"collections":"basically-computer-science,things-to-make-and-do","url":"https:\/\/aip.scitation.org\/doi\/abs\/10.1063\/1.1751381","year":"2004","urldate":"2021-02-12"},{"key":"PythagoreanTriplesComplexNumbersAbelianGroupsandPrimeNumbers","type":"article","title":"Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers","author":"Amnon Yekutieli","abstract":"It is well-known that pythagorean triples can be represented by points of the\r\nunit circle with rational coordinates. These points form an abelian group, and\r\nwe describe its structure. This structural description yields, almost\r\nimmediately, an enumeration of the normalized pythagorean triples with a given\r\nhypotenuse, and also to an effective method for producing all such triples.\r\nThis effective method seems to be new.\r\n This paper is intended for the general mathematical audience, including\r\nundergraduate mathematics students, and therefore it contains plenty of\r\nbackground material, some history and several examples and exercises.","comment":"","date_added":"2021-02-12","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2101.12166v1","http:\/\/arxiv.org\/pdf\/2101.12166v1"],"collections":"fun-maths-facts,integerology,the-groups-group","url":"http:\/\/arxiv.org\/abs\/2101.12166v1 http:\/\/arxiv.org\/pdf\/2101.12166v1","year":"2021","urldate":"2021-02-12","archivePrefix":"arXiv","eprint":"2101.12166","primaryClass":"math.NT"},{"key":"ItislikeeggPaulLorenzenandthecollapseofproofsofconsistency","type":"article","title":"\"It is like egg\": Paul Lorenzen and the collapse of proofs of consistency","author":"Stefan Neuwirth","abstract":"Paul Lorenzen, mathematician and philosopher of the 20th century, mentions\r\nOctober 1947 as the date of a crisis in his mathematical and philosophical\r\ninvestigations. An autograph dated 15 October 1947 documents this crisis. This\r\narticle proposes a traduction and a commentary of it and sketches the\r\ncircumstances of its writing on the base of his correspondence with Paul\r\nBernays. A lettter from Lorenzen to Carl Friedrich Gethmann dated 14 January\r\n1988 carves out the story of this crisis by showing how he soaks up the\r\nindications of his correspondents and transmutes them into an absolutely\r\noriginal research.","comment":"","date_added":"2021-02-12","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2101.04381v1","http:\/\/arxiv.org\/pdf\/2101.04381v1"],"collections":"about-proof,food,history","url":"http:\/\/arxiv.org\/abs\/2101.04381v1 http:\/\/arxiv.org\/pdf\/2101.04381v1","year":"2021","urldate":"2021-02-12","archivePrefix":"arXiv","eprint":"2101.04381","primaryClass":"math.HO"},{"key":"Braidswhichcanbeplaitedwiththeirthreadstiedtogetherateachend","type":"article","title":"Braids which can be plaited with their threads tied together at each end","author":"J.A.H. Shepperd","abstract":"The group of braids, which can be plaited from n untwisted threads tied together at each end, is examined and its structure is determined. An algorithm is derived for deciding whether or not a given braid can be so plaited and a calculation procedure is described. The problem arises from a process of manufacturing braids by a machine which plaits by passing a shuttle, on which the constructed braid is wound, between the threads, which are supplied from bobbins effectively fixed and inaccessible. Every plait on three threads can be constructed in this way. For more than three threads, examples are given both of plaits which can be so constructed and of plaits which cannot.","comment":"","date_added":"2021-02-12","date_published":"1962-08-06","urls":["https:\/\/royalsocietypublishing.org\/doi\/10.1098\/rspa.1962.0006"],"collections":"fun-maths-facts,the-groups-group,things-to-make-and-do","url":"https:\/\/royalsocietypublishing.org\/doi\/10.1098\/rspa.1962.0006","year":"1962","urldate":"2021-02-12"},{"key":"Maximumoverhang","type":"article","title":"Maximum overhang","author":"Mike Paterson and Yuval Peres and Mikkel Thorup and Peter Winkler and Uri Zwick","abstract":"How far can a stack of $n$ identical blocks be made to hang over the edge of\r\na table? The question dates back to at least the middle of the 19th century and\r\nthe answer to it was widely believed to be of order $\\log n$. Recently,\r\nPaterson and Zwick constructed $n$-block stacks with overhangs of order\r\n$n^{1\/3}$, exponentially better than previously thought possible. We show here\r\nthat order $n^{1\/3}$ is indeed best possible, resolving the long-standing\r\noverhang problem up to a constant factor.","comment":"","date_added":"2021-03-15","date_published":"2007-08-06","urls":["http:\/\/arxiv.org\/abs\/0707.0093v1","http:\/\/arxiv.org\/pdf\/0707.0093v1"],"collections":"basically-physics,easily-explained,fun-maths-facts,puzzles","url":"http:\/\/arxiv.org\/abs\/0707.0093v1 http:\/\/arxiv.org\/pdf\/0707.0093v1","year":"2007","urldate":"2021-03-15","archivePrefix":"arXiv","eprint":"0707.0093","primaryClass":"math.HO"},{"key":"TheAmazing3nTheoremanditsevenmoreAmazingProofDiscoveredbyXavierGViennotandhiscoleBordelaisegang","type":"article","title":"The Amazing $3^n$ Theorem and its even more Amazing Proof [Discovered by Xavier G. Viennot and his \u00c9cole Bordelaise gang]","author":"Doron Zeilberger","abstract":"The most amazing (at least to me) result in Enumerative Combinatorics is\r\nDominique Gouyou-Beauchamps and Xavier Viennot's theorem that states that the\r\nnumber of so-called directed animals with compact source (that are equivalent,\r\nvia Viennot's beautiful concept of heaps, to towers of dominoes, that I take\r\nthe liberty of renaming xaviers) with n+1 points equals 3^n. This amazing\r\nresult received an even more amazing proof by Jean B\\'etrema and Jean-Guy\r\nPenaud. Both theorem and proof deserve to be better known! Hence this article,\r\nthat is also accompanied by a comprehensive Maple package\r\nhttp:\/\/www.math.rutgers.edu\/~zeilberg\/tokhniot\/BORDELAISE that implements\r\neverything (and much more)","comment":"","date_added":"2021-03-15","date_published":"2012-08-06","urls":["http:\/\/arxiv.org\/abs\/1208.2258v1","http:\/\/arxiv.org\/pdf\/1208.2258v1"],"collections":"attention-grabbing-titles,combinatorics,fun-maths-facts,things-to-make-and-do","url":"http:\/\/arxiv.org\/abs\/1208.2258v1 http:\/\/arxiv.org\/pdf\/1208.2258v1","year":"2012","urldate":"2021-03-15","archivePrefix":"arXiv","eprint":"1208.2258","primaryClass":"math.CO"},{"key":"Thetermangleintheinternationalsystemofunits","type":"article","title":"The term `angle' in the international system of units","author":"Michael P. Krystek","abstract":"The concept of an angle is one that often causes difficulties in metrology.\r\nThese are partly caused by a confusing mixture of several mathematical terms,\r\npartly by real mathematical difficulties and finally by imprecise terminology.\r\nThe purpose of this publication is to clarify misunderstandings and to explain\r\nwhy strict terminology is important. It will also be shown that most\r\nmisunderstandings regarding the `radian' can be avoided if some simple rules\r\nare obeyed.","comment":"","date_added":"2021-03-15","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2101.01023v2","http:\/\/arxiv.org\/pdf\/2101.01023v2"],"collections":"history,notation-and-conventions","url":"http:\/\/arxiv.org\/abs\/2101.01023v2 http:\/\/arxiv.org\/pdf\/2101.01023v2","year":"2021","urldate":"2021-03-15","archivePrefix":"arXiv","eprint":"2101.01023","primaryClass":"math.HO"},{"key":"Threefriendlywalkers","type":"article","title":"Three friendly walkers","author":"Iwan Jensen","abstract":"More than 15 years ago Guttmann and V\u00f6ge (2002 J. Stat. Plan. Inference 101 107), introduced a model of friendly walkers. Since then it has remained unsolved. In this paper we provide the exact solution to a closely allied model which essentially only differs in the boundary conditions. The exact solution is expressed in terms of the reciprocal of the generating function for vicious walkers which is a D-finite function. However, ratios of D-finite functions are inherently not D-finite and in this case we prove that the friendly walkers generating function is the solution to a non-linear differential equation with polynomial coefficients, it is in other words D-algebraic. We find using numerically exact calculations a conjectured expression for the generating function of the original model as a ratio of a D-finite function and the generating function for vicious walkers. We obtain an expression for this D-finite function in terms of a \\({{}_{2}}{{F}_{1}}\\) hypergeometric function with a rational pullback and its first and second derivatives.","comment":"Contains objects called vicious, friendly and super-friendly watermelons. I have no idea why.","date_added":"2021-03-22","date_published":"2016-08-06","urls":["https:\/\/iopscience.iop.org\/article\/10.1088\/1751-8121\/50\/2\/024003"],"collections":"attention-grabbing-titles,food","url":"https:\/\/iopscience.iop.org\/article\/10.1088\/1751-8121\/50\/2\/024003","year":"2016","urldate":"2021-03-22"},{"key":"Thecockedhat","type":"article","title":"The cocked hat","author":"Imre B\u00e1r\u00e1ny and William Steiger and Sivan Toledo","abstract":"We revisit the cocked hat -- an old problem from navigation -- and examine\r\nunder what conditions its old solution is valid.","comment":"","date_added":"2021-03-23","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2007.06838v1","http:\/\/arxiv.org\/pdf\/2007.06838v1"],"collections":"fun-maths-facts,geometry,modelling,probability-and-statistics","url":"http:\/\/arxiv.org\/abs\/2007.06838v1 http:\/\/arxiv.org\/pdf\/2007.06838v1","year":"2020","urldate":"2021-03-23","archivePrefix":"arXiv","eprint":"2007.06838","primaryClass":"math.MG"},{"key":"ASelfReferentialPropertyofZiminWords","type":"article","title":"A Self-Referential Property of Zimin Words","author":"John Connor","abstract":"This paper gives a short overview of Zimin words, and proves an interesting\r\nproperty of their distribution. Let $L_q^m$ to be the lexically ordered\r\nsequence of $q$-ary words of length $m$, and let $T_n(L_q^m)$ to be the binary\r\nsequence where the $i$-th term is $1$ if and only if the $i$-th word of $L_q^m$\r\nencounters the $n$-th Zimin word, $Z_n$. We show that the sequence $T_n(L_q^m)$\r\nis an instance of $Z_{n+1}$ when $1 < n$ and $m=2^n-1$.","comment":"","date_added":"2021-03-31","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1611.01061v1","http:\/\/arxiv.org\/pdf\/1611.01061v1"],"collections":"combinatorics,fun-maths-facts","url":"http:\/\/arxiv.org\/abs\/1611.01061v1 http:\/\/arxiv.org\/pdf\/1611.01061v1","year":"2016","urldate":"2021-03-31","archivePrefix":"arXiv","eprint":"1611.01061","primaryClass":"math.CO"},{"key":"TheHandbookofMathematicalDiscourse","type":"book","title":"A Handbook of Mathematical Discourse","author":"Charles Wells","abstract":"This handbook is an intensive description of many aspects of the vocabulary and forms of the English language used to communicate mathematics. It is designed to be read and consulted by anyone who teaches or writes about mathematics, as a guide to what possible meanings the students or readers will extract (or fail to extract) from what is said or written. Students should also find it useful, especially upper-level undergraduate students and graduate students studying subjects that make substantial use of mathematical reasoning.\r\n\r\nThis handbook is written from a personal point of view by a mathematician. I have been particularly interested in and observant of the use of language from before the time I knew abstract mathematics existed, and I have taught mathematics for 37 years. During most of that time I kept a file of notes on language usages that students find difficult. Many of those observations may be found in this volume. However, a much larger part of this dictionary is based on the works of others (acknowledged in the individual entries), and the reports of usage are based, incompletely in this early version, citations from the literature.\r\n\r\nSomeday, I hope, there will be a complete dictionary based on extensive scientific observation of written and spoken mathematical English, created by a collaborative team of mathematicians, linguists and lexicographers. This handbook points the way to such an endeavor. However, its primary reason for being is to provide information about the language to instructors and students that will make it easier for them to explain, learn and use mathematics.\r\n\r\nThe earliest dictionaries of the English language listed only \"difficult\" words. Dictionaries such as Dr. Johnson's that attempted completeness came later. This handbook is more like the earlier dictionaries, with a focus on usages that cause problems for those who are just beginning to learn how to do abstract mathematics.","comment":"","date_added":"2017-06-19","date_published":"2003-08-06","urls":["https:\/\/abstractmath.org\/","https:\/\/abstractmath.org\/Handbook\/handbook.pdf"],"collections":"education,notation-and-conventions,the-act-of-doing-maths","url":"https:\/\/abstractmath.org\/ https:\/\/abstractmath.org\/Handbook\/handbook.pdf","urldate":"2017-06-19","year":"2003"},{"key":"Generationofrealalgebraiclociviacomplexdetours","type":"article","title":"Generation of real algebraic loci via complex detours","author":"Stefan Kranich","abstract":"We discuss the locus generation algorithm used by the dynamic geometry\r\nsoftware Cinderella, and how it uses complex detours to resolve singularities.\r\nWe show that the algorithm is independent of the orientation of its complex\r\ndetours. We conjecture that the algorithm terminates if it takes small enough\r\ncomplex detours and small enough steps on every complex detour. Moreover, we\r\nintroduce a variant of the algorithm that possibly generates entire real\r\nconnected components of real algebraic loci. Several examples illustrate its\r\nuse for organic generation of real algebraic loci. Another example shows how we\r\ncan apply the algorithm to simulate mechanical linkages. Apparently, the use of\r\ncomplex detours produces physically reasonable motion of such linkages.","comment":"","date_added":"2021-04-10","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1510.05464v3","http:\/\/arxiv.org\/pdf\/1510.05464v3"],"collections":"basically-computer-science,geometry","url":"http:\/\/arxiv.org\/abs\/1510.05464v3 http:\/\/arxiv.org\/pdf\/1510.05464v3","year":"2015","urldate":"2021-04-10","archivePrefix":"arXiv","eprint":"1510.05464","primaryClass":"math.AG"},{"key":"GdelforGoldilocksARigorousStreamlinedProofofavariantofGdelsFirstIncompletenessTheorem","type":"article","title":"G\u00f6del for Goldilocks: A Rigorous, Streamlined Proof of (a variant of) G\u00f6del's First Incompleteness Theorem","author":"Dan Gusfield","abstract":"Most discussions of G\\\"odel's theorems fall into one of two types: either\r\nthey emphasize perceived philosophical, cultural \"meanings\" of the theorems,\r\nand perhaps sketch some of the ideas of the proofs, usually relating G\\\"odel's\r\nproofs to riddles and paradoxes, but do not attempt to present rigorous,\r\ncomplete proofs; or they do present rigorous proofs, but in the traditional\r\nstyle of mathematical logic, with all of its heavy notation and difficult\r\ndefinitions, and technical issues which reflect G\\\"odel's original approach and\r\nbroader logical issues. Many non-specialists are frustrated by these two\r\nextreme types of expositions and want a complete, rigorous proof that they can\r\nunderstand. Such an exposition is possible, because many people have realized\r\nthat variants of G\\\"odel's first incompleteness theorem can be rigorously\r\nproved by a simpler middle approach, avoiding philosophical discussions and\r\nhand-waiving at one extreme; and also avoiding the heavy machinery of\r\ntraditional mathematical logic, and many of the harder detail's of G\\\"odel's\r\noriginal proof, at the other extreme. This is the just-right Goldilocks\r\napproach. In this exposition we give a short, self-contained Goldilocks\r\nexposition of G\\\"odel's first theorem, aimed at a broad, undergraduate\r\naudience.","comment":"","date_added":"2021-04-10","date_published":"2014-08-06","urls":["http:\/\/arxiv.org\/abs\/1409.5944v3","http:\/\/arxiv.org\/pdf\/1409.5944v3"],"collections":"about-proof","url":"http:\/\/arxiv.org\/abs\/1409.5944v3 http:\/\/arxiv.org\/pdf\/1409.5944v3","year":"2014","urldate":"2021-04-10","archivePrefix":"arXiv","eprint":"1409.5944","primaryClass":"math.LO"},{"key":"DependencyGraphOfPropositionsInEuclidsElements","type":"article","title":"Dependency Graph of Propositions in Euclid\u2019s Elements","author":"Thomson Nguyen","abstract":"This is a dependency graph of propositions from the first book of Euclid\u2019s Elements. We say that a proposition \\(A\\) depends on \\(B\\) iff proposition \\(B\\) is necessary in the proof of proposition \\(A\\). In the dependency graph below,this will be denoted by an arrow starting at \\(A\\) and pointing at \\(B\\). \r\n\r\nFigure 1 is a dependency graph of all propositions in the first book. Figure 2 is a dependency graph of all propositions that state a relation between two objects, while Figure 3 is a dependency graph of all propositions that state the existence of an unmarked straightedge and compass construction of something.The dependencies were gratefully extracted from Richard Fitzpatrick\u2019s edition of Euclid\u2019s Elements. The graph itself was written in DOT and converted to pslatex with dot2tex. The motivation for this graph was from Mariusz Wodzicki\u2019s Spring 2007 History of Mathematics course at the University of California, Berkeley. Corrections and comments are always appreciated at thomson@ocf.berkeley.edu.","comment":"","date_added":"2021-04-14","date_published":"2007-08-06","urls":["https:\/\/www.ocf.berkeley.edu\/~thomson\/euclid\/euclid.pdf"],"collections":"history","url":"https:\/\/www.ocf.berkeley.edu\/~thomson\/euclid\/euclid.pdf","year":"2007","urldate":"2021-04-14"},{"key":"EqWorld","type":"article","title":"EqWorld - The World of Mathematical Equations","author":"Alexei I. Zhurov and Alexander L. Levitin and Dmitry A. Polyanin","abstract":"Equations play a crucial role in modern mathematics and form the basis for mathematical modelling of numerous phenomena and processes in science and engineering.\r\n\r\nThe international scientific-educational website EqWorld presents extensive information on solutions to various classes of ordinary differential, partial differential, integral, functional, and other mathematical equations. It also outlines some methods for solving equations, includes interesting articles, gives links to mathematical websites and software packages, lists useful handbooks and monographs, and refers to scientific publishers, journals, etc. The website includes a dynamic section Equation Archive which allows authors to quickly publish their equations (differential, integral, and other) and also exact solutions, first integrals, and transformations.\r\n\r\nThe EqWorld website is intended for researchers, university teachers, engineers, and students all over the world. It contains about 2000 webpages and is visited by over 3000 users a day (coming from 200 countries worldwide). All resources presented on this site are free to its users.","comment":"","date_added":"2021-04-15","date_published":"2004-08-06","urls":["http:\/\/eqworld.ipmnet.ru\/"],"collections":"education,lists-and-catalogues","url":"http:\/\/eqworld.ipmnet.ru\/","year":"2004","urldate":"2021-04-15"},{"key":"TheUsefulnessOfUselessKnowledge","type":"article","title":"The usefulness of useless knowledge","author":"Flexner, Abraham","abstract":"","comment":"","date_added":"2012-07-16","date_published":"1939-10-01","urls":["https:\/\/www.ias.edu\/sites\/default\/files\/library\/UsefulnessHarpers.pdf"],"collections":"history","url":"https:\/\/www.ias.edu\/sites\/default\/files\/library\/UsefulnessHarpers.pdf","urldate":"2012-07-16","year":"1939","journal":"Harper's Magazine","keywords":"Research,Science","month":"oct","number":"117","pages":"544--552","pmid":"13024481","volume":"17"},{"key":"Gould2011","type":"article","title":"Table for Fundamentals of Series : Part I : Basic Properties of Series and Products","author":"Gould, Henry W.","abstract":"","comment":"","date_added":"2014-02-11","date_published":"2011-08-06","urls":["https:\/\/web.archive.org\/web\/20180218233224\/http:\/\/math.wvu.edu\/~gould\/Vol.1.PDF"],"collections":"lists-and-catalogues","url":"https:\/\/web.archive.org\/web\/20180218233224\/http:\/\/math.wvu.edu\/~gould\/Vol.1.PDF","urldate":"2014-02-11","year":"2011"},{"key":"Thefluidmechanicsofpoohsticks","type":"article","title":"The fluid mechanics of poohsticks","author":"Julyan H. E. Cartwright and Oreste Piro","abstract":"2019 is the bicentenary of George Gabriel Stokes, who in 1851 described the\r\ndrag - Stokes drag - on a body moving immersed in a fluid, and 2020 is the\r\ncentenary of Christopher Robin Milne, for whom the game of poohsticks was\r\ninvented; his father A. A. Milne's \"The House at Pooh Corner\", in which it was\r\nfirst described in print, appeared in 1928. So this is an apt moment to review\r\nthe state of the art of the fluid mechanics of a solid body in a complex fluid\r\nflow, and one floating at the interface between two fluids in motion.\r\nPoohsticks pertains to the latter category, when the two fluids are water and\r\nair.","comment":"","date_added":"2021-04-20","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2104.03056v1","http:\/\/arxiv.org\/pdf\/2104.03056v1"],"collections":"attention-grabbing-titles,basically-physics,modelling","url":"http:\/\/arxiv.org\/abs\/2104.03056v1 http:\/\/arxiv.org\/pdf\/2104.03056v1","urldate":"2021-04-20","year":"2021","archivePrefix":"arXiv","eprint":"2104.03056","primaryClass":"physics.flu-dyn"},{"key":"OnMathematicalSymbolsinChina","type":"article","title":"On Mathematical Symbols in China","author":"Fang Li and Yong Zhang","abstract":"When studying the history of mathematical symbols, one finds that the\r\ndevelopment of mathematical symbols in China is a significant piece of Chinese\r\nhistory; however, between the beginning of mathematics and modern day\r\nmathematics in China, there exists a long blank period. Let us focus on the\r\ndevelopment of Chinese mathematical symbols, and find out the significance of\r\ntheir origin, evolution, rise and fall within Chinese mathematics.","comment":"","date_added":"2021-04-20","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1511.08033v1","http:\/\/arxiv.org\/pdf\/1511.08033v1"],"collections":"history,notation-and-conventions","url":"http:\/\/arxiv.org\/abs\/1511.08033v1 http:\/\/arxiv.org\/pdf\/1511.08033v1","year":"2015","urldate":"2021-04-20","archivePrefix":"arXiv","eprint":"1511.08033","primaryClass":"math.HO"},{"key":"Anintegralsjourneyovertherealline","type":"article","title":"An integral's journey over the real line","author":"Robert Reynolds and Allan Stauffer","abstract":"In 1826 Cauchy presented an Integral over the real line. Al and I thought a\r\nderivation would be mighty fine. So we packed our contour integral bags that\r\nday, and we now present an analytic continuation this time.","comment":"","date_added":"2021-04-20","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2103.03355v1","http:\/\/arxiv.org\/pdf\/2103.03355v1"],"collections":"the-act-of-doing-maths","url":"http:\/\/arxiv.org\/abs\/2103.03355v1 http:\/\/arxiv.org\/pdf\/2103.03355v1","urldate":"2021-04-20","year":"2021","archivePrefix":"arXiv","eprint":"2103.03355","primaryClass":"math.GM"},{"key":"Ordner","type":"online","title":"Ordner: index of real numbers","author":"Fredrik Johansson","abstract":"Ordner is indexed by 30-digit floating-point decimal keys such as 0.707106781186547524400844362105. For each key, Ordner lists constant symbolic expressions (for example Div(1, Sqrt(2))) with numerical value within \u00b11 ulp of the key. For each expression x, Ordner also links to the Fungrim entries where x appears.\r\n\r\nOrdner is generated automatically by searching all Fungrim formulas for constant subexpressions that Arb can evaluate numerically. Only expressions that appear explicitly in Fungrim are covered, with the following exceptions. For tables with numerical data, all the listed instances of the ground expression are included. All decimal keys in Ordner are normalized to be nonnegative, so expressions x representing negative values are indexed as Neg(x) in Ordner. Complex numbers are indexed by the real and imaginary parts (Re(x), Im(x)), as well as the absolute value and complex argument (Abs(x), Arg(x)) when both the real and imaginary parts are nonzero. The number 0 is a special case: a vanishing expression is only included when the numerical evaluation code can prove that the expression exactly represents 0. Some trivially zero-valued expressions are excluded to prevent bloat. Finally, since the Fungrim formula language normally uses Exp(x) instead of Pow(ConstE, x) to represent the exponential function, formulas containing Exp(...) are listed under 2.71828182845904523536028747135 as a special case, so as to represent this fundamental constant fairly!\r\n\r\nORDNER stands for Online Real Decimal Number Encyclopedia Reference.","comment":"","date_added":"2021-04-20","date_published":"2019-08-06","urls":["https:\/\/fungrim.org\/ordner\/"],"collections":"lists-and-catalogues","url":"https:\/\/fungrim.org\/ordner\/","year":"2019","urldate":"2021-04-20"},{"key":"Thebestknownpackingsofequalcirclesinasquare","type":"online","title":"The best known packings of equal circles in a square","author":"Eckard Specht","abstract":"","comment":"","date_added":"2021-04-20","date_published":"1999-08-06","urls":["http:\/\/hydra.nat.uni-magdeburg.de\/packing\/csq\/csq.html"],"collections":"geometry,lists-and-catalogues","url":"http:\/\/hydra.nat.uni-magdeburg.de\/packing\/csq\/csq.html","urldate":"2021-04-20","year":"1999"},{"key":"FungrimTheMathematicalFunctionsGrimoire","type":"online","title":"Fungrim: The Mathematical Functions Grimoire","author":"Fredrik Johansson","abstract":"The Mathematical Functions Grimoire (Fungrim) is an open source library of formulas and data for special functions.","comment":"","date_added":"2021-04-20","date_published":"2019-08-06","urls":["https:\/\/fungrim.org\/"],"collections":"lists-and-catalogues","url":"https:\/\/fungrim.org\/","urldate":"2021-04-20","year":"2019"},{"key":"Perimeterminimizingpentagonaltilings","type":"article","title":"Perimeter-minimizing pentagonal tilings","author":"Chung, Ping Ngai and Fernandez, Miguel and Shah, Niralee and Sordo Vieira, Luis and Wikner, Elena","abstract":"We provide examples of perimeter-minimizing tilings of the plane by convex pentagons and examples of perimeter-minimizing tilings of certain small flat tori. ","comment":"","date_added":"2021-05-04","date_published":"2014-08-06","urls":["https:\/\/msp.org\/involve\/2014\/7-4\/p02.xhtml","http:\/\/msp.org\/involve\/2014\/7-4\/involve-v7-n4-p02-s.pdf"],"collections":"easily-explained,geometry,things-to-make-and-do","url":"https:\/\/msp.org\/involve\/2014\/7-4\/p02.xhtml http:\/\/msp.org\/involve\/2014\/7-4\/involve-v7-n4-p02-s.pdf","year":"2014","urldate":"2021-05-04","publisher":"Mathematical Sciences Publishers","journal":"Involve, a Journal of Mathematics","volume":"7","issue":"4","doi":"10.2140\/involve.2014.7.453","issn":"1944-4184","pages":"453-478"},{"key":"NoThisisnotaCircle","type":"article","title":"No, This is not a Circle","author":"Zolt\u00e1n Kov\u00e1cs","abstract":"A curve, also shown in introductory maths textbooks, seems like a circle. But\r\nit is actually a different curve. This paper discusses some easy approaches to\r\nclassify the result, including a GeoGebra applet construction.","comment":"","date_added":"2017-05-02","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1704.08483","http:\/\/arxiv.org\/pdf\/1704.08483"],"collections":"attention-grabbing-titles,drama,easily-explained,geometry","url":"http:\/\/arxiv.org\/abs\/1704.08483 http:\/\/arxiv.org\/pdf\/1704.08483","urldate":"2017-05-02","year":"2017","archivePrefix":"arXiv","eprint":"1704.08483","primaryClass":"math.HO"},{"key":"TilingsEncyclopedia","type":"online","title":"Tilings Encyclopedia","author":"Dirk Frettl\u00f6h and Edmund Harriss and Franz G\u00e4hler","abstract":"The tilings encyclopedia shows a wealth of examples of nonperiodic substitution tilings. ","comment":"","date_added":"2021-05-18","date_published":null,"urls":["https:\/\/tilings.math.uni-bielefeld.de\/"],"collections":"geometry,lists-and-catalogues","url":"https:\/\/tilings.math.uni-bielefeld.de\/","year":"","urldate":"2021-05-18"},{"key":"EverySalamihastwoends","type":"article","title":"Every Salami has two ends","author":"Bobo Hua and Florentin M\u00fcnch","abstract":"A salami is a connected, locally finite, weighted graph with non-negative\r\nOllivier Ricci curvature and at least two ends of infinite volume. We show that\r\nevery salami has exactly two ends and no vertices with positive curvature. We\r\nmoreover show that every salami is recurrent and admits harmonic functions with\r\nconstant gradient. The proofs are based on extremal Lipschitz extensions, a\r\nvariational principle and the study of harmonic functions. Assuming a lower\r\nbound on the edge weight, we prove that salamis are quasi-isometric to the\r\nline, that the space of all harmonic functions has finite dimension, and that\r\nthe space of subexponentially growing harmonic functions is two-dimensional.\r\nMoreover, we give a Cheng-Yau gradient estimate for harmonic functions on\r\nballs.","comment":"","date_added":"2021-05-26","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2105.11887v1","http:\/\/arxiv.org\/pdf\/2105.11887v1"],"collections":"attention-grabbing-titles,food","url":"http:\/\/arxiv.org\/abs\/2105.11887v1 http:\/\/arxiv.org\/pdf\/2105.11887v1","year":"2021","urldate":"2021-05-26","archivePrefix":"arXiv","eprint":"2105.11887","primaryClass":"math.DG"},{"key":"item59","type":"article","title":"A Dozen Hat Problems","author":"Ezra Brown and James Tanton","abstract":"Hat problems are all the rage these days, proliferating on various web sites and generating a great deal of conversation\u2014and research\u2014among mathematicians\r\nand students. But they have been around for quite a while in different forms.","comment":"Also known as prisoner puzzles","date_added":"2016-04-12","date_published":"2009-08-06","urls":["http:\/\/intranet.math.vt.edu\/people\/brown\/doc\/dozen_hats.pdf"],"collections":"games-to-play-with-friends,protocols-and-strategies,puzzles","url":"http:\/\/intranet.math.vt.edu\/people\/brown\/doc\/dozen_hats.pdf","urldate":"2016-04-12","year":"2009"},{"key":"AGamblerthatBetsForeverandtheStrongLawofLargeNumbers","type":"article","title":"A Gambler that Bets Forever and the Strong Law of Large Numbers","author":"Calvin Wooyoung Chin","abstract":"In this expository note, we give a simple proof that a gambler repeating a\r\ngame with positive expected value never goes broke with a positive probability.\r\nThis does not immediately follow from the strong law of large numbers or other\r\nbasic facts on random walks. Using this result, we provide an elementary proof\r\nof the strong law of large numbers. The ideas of the proofs come from the\r\nmaximal ergodic theorem and Birkhoff's ergodic theorem.","comment":"","date_added":"2021-06-07","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2105.03803v1","http:\/\/arxiv.org\/pdf\/2105.03803v1"],"collections":"probability-and-statistics","url":"http:\/\/arxiv.org\/abs\/2105.03803v1 http:\/\/arxiv.org\/pdf\/2105.03803v1","year":"2021","urldate":"2021-06-07","archivePrefix":"arXiv","eprint":"2105.03803","primaryClass":"math.PR"},{"key":"Gamilaraaykinshiprevisitedincidenceofrecessivediseaseisdynamicallytradedoffagainstbenefitsofcooperativebehaviours","type":"article","title":"Gamilaraay kinship revisited: incidence of recessive disease is dynamically traded-off against benefits of cooperative behaviours","author":"Jared M. Field","abstract":"Traditional Indigenous marriage rules have been studied extensively since the\r\nmid 1800s. Despite this, they have historically been cast aside as having very\r\nlittle utility. This is, in large part, due to a focus on trying to understand\r\nbroad-stroke marriage restrictions or how they may evolve. Here, taking the\r\nGamilaraay system as a case study, we instead ask how relatedness may be\r\ndistributed under such a system. We show, remarkably, that this system\r\ndynamically trades off kin avoidance to minimise incidence of recessive\r\ndiseases against expected levels of cooperation, as understood formally through\r\nHamilton's rule.","comment":"","date_added":"2021-06-07","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2104.14779v1","http:\/\/arxiv.org\/pdf\/2104.14779v1"],"collections":"modelling","url":"http:\/\/arxiv.org\/abs\/2104.14779v1 http:\/\/arxiv.org\/pdf\/2104.14779v1","year":"2021","urldate":"2021-06-07","archivePrefix":"arXiv","eprint":"2104.14779","primaryClass":"q-bio.PE"},{"key":"RealAnalysisinReverse","type":"article","title":"Real Analysis in Reverse","author":"James Propp","abstract":"Many of the theorems of real analysis, against the background of the ordered\r\nfield axioms, are equivalent to Dedekind completeness, and hence can serve as\r\ncompleteness axioms for the reals. In the course of demonstrating this, the\r\narticle offers a tour of some less-familiar ordered fields, provides some of\r\nthe relevant history, and considers pedagogical implications.","comment":"","date_added":"2021-06-07","date_published":"2012-08-06","urls":["http:\/\/arxiv.org\/abs\/1204.4483v4","http:\/\/arxiv.org\/pdf\/1204.4483v4"],"collections":"about-proof,attention-grabbing-titles,the-act-of-doing-maths","url":"http:\/\/arxiv.org\/abs\/1204.4483v4 http:\/\/arxiv.org\/pdf\/1204.4483v4","year":"2012","urldate":"2021-06-07","archivePrefix":"arXiv","eprint":"1204.4483","primaryClass":"math.HO"},{"key":"Knutson2012","type":"article","title":"A stratification of the space of all $k$-planes in $\\mathbb{C}_n$","author":"Allen Knutson","abstract":"To each $k \\times n$ matrix $\\mathrm{M}$ of rank $k$, we associate a juggling pattern of periodicity $n$ with $k$ balls. The juggling pattern actually only depends\r\non the $k$-plane spanned by the rows, so gives a decomposition of the \u201cGrassmannian\u201d of all $k$-planes in $n$-space.\r\n\r\nThere are many connections between the geometry and the juggling. For example, the natural topology on the space of matrices induces a partial order on the space of juggling patterns, which indicates whether one pattern is \u201cmore excited\u201d than another. This same decomposition turns out to naturally arise from totally\r\npositive geometry, characteristic $p$ geometry, and noncommutative geometry. It also arises by projection from the manifold of full flags in $n$-space, where there is no cyclic symmetry","comment":"","date_added":"2012-11-19","date_published":"2012-08-06","urls":["https:\/\/pi.math.cornell.edu\/~allenk\/joint.pdf"],"collections":"geometry","url":"https:\/\/pi.math.cornell.edu\/~allenk\/joint.pdf","urldate":"2012-11-19","year":"2012"},{"key":"OuroborosFunctionalsFamiliesofOuroborosFunctionsandTheirRelationshiptoPartialDifferentialEquationsandProbabilityTheory","type":"article","title":"Ouroboros Functionals, Families of Ouroboros Functions, and Their Relationship to Partial Differential Equations and Probability Theory","author":"Nathan Thomas Provost","abstract":"Previously, we have introduced a very small number of examples of what we\r\ncall Ouroboros functions. Using our already established theory of Ouroboros\r\nspaces and their functions, we will provide a set of families of Ouroboros\r\nfunctions that bolster our overall understanding of the Ouroboros spaces. From\r\nhere, we extend the theory of Ouroboros functions by introducing Ouroboros\r\nfunctionals and Ouroboros functional spaces. Furthermore, we re-frame the\r\nexpected value of a random variable as an Ouroboros functional, which proves to\r\nbe more intuitive in view of probabilistic measure theory. We then show that\r\nthese Ouroboros functions have additional applications, as they are general\r\nsolutions to certain elementary linear first order partial differential\r\nequations (PDEs). We conclude by elaborating upon this connection and\r\ndiscussing future endeavors, which will be centered on answering a given\r\nhypothesis.","comment":"","date_added":"2021-08-29","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2106.04680v1","http:\/\/arxiv.org\/pdf\/2106.04680v1"],"collections":"attention-grabbing-titles","url":"http:\/\/arxiv.org\/abs\/2106.04680v1 http:\/\/arxiv.org\/pdf\/2106.04680v1","year":"2021","urldate":"2021-08-29","archivePrefix":"arXiv","eprint":"2106.04680","primaryClass":"math.FA"},{"key":"Someinstructivemathematicalerrors","type":"article","title":"Some instructive mathematical errors","author":"Richard P. Brent","abstract":"We describe various errors in the mathematical literature, and consider how\r\nsome of them might have been avoided, or at least detected at an earlier stage,\r\nusing tools such as Maple or Sage. Our examples are drawn from three broad\r\ncategories of errors. First, we consider some significant errors made by\r\nhighly-regarded mathematicians. In some cases these errors were not detected\r\nuntil many years after their publication. Second, we consider in some detail an\r\nerror that was recently detected by the author. This error in a refereed\r\njournal led to further errors by at least one author who relied on the\r\n(incorrect) result. Finally, we mention some instructive errors that have been\r\ndetected in the author's own published papers.","comment":"","date_added":"2021-08-29","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2106.07269v2","http:\/\/arxiv.org\/pdf\/2106.07269v2"],"collections":"history,the-act-of-doing-maths","url":"http:\/\/arxiv.org\/abs\/2106.07269v2 http:\/\/arxiv.org\/pdf\/2106.07269v2","year":"2021","urldate":"2021-08-29","archivePrefix":"arXiv","eprint":"2106.07269","primaryClass":"math.NT"},{"key":"SETwithaTwist","type":"article","title":"SET with a Twist","author":"Cathy Hsu, Jonah Ostroff and Lucas Van Meter","abstract":"If you can\u2019t get enough of the card game SET and enjoyed our version of Projective SET (see \u201cProjectivizing Set\u201d in the April 2020 issue of this magazine), then get excited, because we are back with another round of variations on the game. This time, we\u2019ll explore the game from a purely algebraic perspective. This article is self-contained\u2014so feel free to just keep reading\u2014but we encourage you to check out our last article for a more geometric take on game modifications.","comment":"","date_added":"2021-08-29","date_published":"2021-08-06","urls":["https:\/\/maa.tandfonline.com\/doi\/full\/10.1080\/10724117.2021.1881376?needAccess=true"],"collections":"easily-explained,games-to-play-with-friends,the-groups-group","url":"https:\/\/maa.tandfonline.com\/doi\/full\/10.1080\/10724117.2021.1881376?needAccess=true","year":"2021","urldate":"2021-08-29"},{"key":"FruitDiophantineEquation","type":"article","title":"Fruit Diophantine Equation","author":"Dipramit Majumdar and B. Sury","abstract":"We show that the Diophantine equation given by X^3+ XYZ = Y^2+Z^2+5 has no\r\nintegral solution. As a consequence, we show that the family of elliptic curve\r\ngiven by the Weierstrass equations Y^2-kXY = X^3 - (k^2+5) has no integral\r\npoint.","comment":"","date_added":"2021-08-29","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2108.02640v2","http:\/\/arxiv.org\/pdf\/2108.02640v2"],"collections":"attention-grabbing-titles,easily-explained,food,integerology","url":"http:\/\/arxiv.org\/abs\/2108.02640v2 http:\/\/arxiv.org\/pdf\/2108.02640v2","year":"2021","urldate":"2021-08-29","archivePrefix":"arXiv","eprint":"2108.02640","primaryClass":"math.NT"},{"key":"YulesNonsenseCorrelationSolved","type":"article","title":"Yule's \"Nonsense Correlation\" Solved!","author":"Philip Ernst and Larry Shepp and Abraham Wyner","abstract":"In this paper, we resolve a longstanding open statistical problem. The\r\nproblem is to mathematically confirm Yule's 1926 empirical finding of \"nonsense\r\ncorrelation\" (\\cite{Yule}). We do so by analytically determining the second\r\nmoment of the empirical correlation coefficient\r\n \\beqn \\theta := \\frac{\\int_0^1W_1(t)W_2(t) dt - \\int_0^1W_1(t) dt \\int_0^1\r\nW_2(t) dt}{\\sqrt{\\int_0^1 W^2_1(t) dt - \\parens{\\int_0^1W_1(t) dt}^2}\r\n\\sqrt{\\int_0^1 W^2_2(t) dt - \\parens{\\int_0^1W_2(t) dt}^2}}, \\eeqn of two {\\em\r\nindependent} Wiener processes, $W_1,W_2$. Using tools from Fred- holm integral\r\nequation theory, we successfully calculate the second moment of $\\theta$ to\r\nobtain a value for the standard deviation of $\\theta$ of nearly .5. The\r\n\"nonsense\" correlation, which we call \"volatile\" correlation, is volatile in\r\nthe sense that its distribution is heavily dispersed and is frequently large in\r\nabsolute value. It is induced because each Wiener process is \"self-correlated\"\r\nin time. This is because a Wiener process is an integral of pure noise and thus\r\nits values at different time points are correlated. In addition to providing an\r\nexplicit formula for the second moment of $\\theta$, we offer implicit formulas\r\nfor higher moments of $\\theta$.","comment":"","date_added":"2021-08-29","date_published":"2016-08-06","urls":["http:\/\/arxiv.org\/abs\/1608.04120v2","http:\/\/arxiv.org\/pdf\/1608.04120v2"],"collections":"drama,probability-and-statistics","url":"http:\/\/arxiv.org\/abs\/1608.04120v2 http:\/\/arxiv.org\/pdf\/1608.04120v2","year":"2016","urldate":"2021-08-29","archivePrefix":"arXiv","eprint":"1608.04120","primaryClass":"math.ST"},{"key":"SkateboardTricksandTopologicalFlips","type":"article","title":"Skateboard Tricks and Topological Flips","author":"Justus Carlisle and Kyle Hammer and Robert Hingtgen and Gabriel Martins","abstract":"We study the motion of skateboard flip tricks by modeling them as continuous\r\ncurves in the group \\(SO(3)\\) of special orthogonal matrices. We show that up to\r\ncontinuous deformation there are only four flip tricks. The proof relies on an\r\nanalysis of the lift of such curves to the unit 3-sphere. We also derive\r\nexplicit formulas for a number of tricks and continuous deformations between\r\nthem.","comment":"","date_added":"2021-08-29","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2108.06307v1","http:\/\/arxiv.org\/pdf\/2108.06307v1"],"collections":"attention-grabbing-titles,easily-explained,games-to-play-with-friends,geometry,the-groups-group","url":"http:\/\/arxiv.org\/abs\/2108.06307v1 http:\/\/arxiv.org\/pdf\/2108.06307v1","year":"2021","urldate":"2021-08-29","archivePrefix":"arXiv","eprint":"2108.06307","primaryClass":"math-ph"},{"key":"HowNottoComputeaFourierTransform","type":"article","title":"How Not to Compute a Fourier Transform","author":"J. A. Grzesik","abstract":"We revisit the Fourier transform of a Hankel function, of considerable\r\nimportance in the theory of knife edge diffraction. Our approach is based\r\ndirectly upon the underlying Bessel equation, which admits manipulation into an\r\nalternate second order differential equation, one of whose solutions is\r\nprecisely the desired transform, apart from an {\\em{a priori}} unknown\r\nconstant, and a second, undesired solution of logarithmic type. A modest amount\r\nof analysis is then required to exhibit that constant as having its proper\r\nvalue, and to purge the logarithmic accompaniment. The intervention of this\r\nanalysis, which relies upon an interplay of asymptotic and close-in functional\r\nbehaviors, prompts our somewhat ironic, mildly puckish caveat, our negation\r\n{\\em{\"not\"}} in the title. In a concluding section we show that this same\r\ntransform is still more readily exhibited as an easy by-product of the\r\ninhomogeneous wave equation in two dimensions satisfied by the Green's function\r\n$G,$ itself proportional to a Hankel function. This latter discussion lapses of\r\ncourse into the argot of physicists.","comment":"","date_added":"2021-11-20","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2001.11987v3","http:\/\/arxiv.org\/pdf\/2001.11987v3"],"collections":"attention-grabbing-titles,the-act-of-doing-maths","url":"http:\/\/arxiv.org\/abs\/2001.11987v3 http:\/\/arxiv.org\/pdf\/2001.11987v3","year":"2020","urldate":"2021-11-20","archivePrefix":"arXiv","eprint":"2001.11987","primaryClass":"math.GM"},{"key":"Bellringingmethodsaspolyhedra","type":"online","title":"Bell-ringing methods as polyhedra","author":"Hugh C. Pumphrey","abstract":" In his popular book \"The Bob Caller's companion\" Mr. Steve Coleman notes how any bell-ringing method can be represented as a directed graph, with a node for each lead head and an edge for each plain lead or bob lead. He further notes that if the graph can be drawn without any of the edges crossing, then it can be made into a polyhedron, with the edges and nodes of the graph being the edges and nodes of the polyhedron. He gives Grandsire Doubles and Plain Bob Doubles as examples of this.\r\n\r\nThis left me asking myself the question: how many different polyhedra are there whose nodes and edges map onto the graph of the lead-heads of a ringing method, popular or otherwise? I enumerated the possible cases for lead heads of plain doubles and minor methods and found the following cases, some of which have polyhedral graphs and some of which do not.","comment":"","date_added":"2021-11-20","date_published":"2015-08-06","urls":["https:\/\/www.geos.ed.ac.uk\/~hcp\/bells\/"],"collections":"combinatorics,easily-explained,fun-maths-facts,the-groups-group,things-to-make-and-do","url":"https:\/\/www.geos.ed.ac.uk\/~hcp\/bells\/","year":"2015","urldate":"2021-11-20"},{"key":"YetanotherProofofanoldHat","type":"article","title":"Yet another Proof of an old Hat","author":"Roland Bacher","abstract":"Every odd prime number \\(p\\) can be written in exactly \\((p + 1)\/2\\) ways as a sum \\(ab + cd\\) with \\(\\min(a, b) > \\max(c, d)\\) of two ordered products. This gives a new proof of Fermat's Theorem expressing primes of the form \\(1 + 4\\mathbb{N}\\) as sums of two squares.","comment":"","date_added":"2021-11-20","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2111.02788v1","http:\/\/arxiv.org\/pdf\/2111.02788v1"],"collections":"about-proof,attention-grabbing-titles,fun-maths-facts,integerology","url":"http:\/\/arxiv.org\/abs\/2111.02788v1 http:\/\/arxiv.org\/pdf\/2111.02788v1","urldate":"2021-11-20","year":"2021","archivePrefix":"arXiv","eprint":"2111.02788","primaryClass":"math.HO"},{"key":"DatabaseOfRingTheory","type":"article","title":"Database of Ring Theory","author":"Ryan C. Schwiebert","abstract":"A repository of rings, their properties, and more ring theory stuff.","comment":"","date_added":"2021-11-20","date_published":"2014-08-06","urls":["http:\/\/ringtheory.herokuapp.com\/"],"collections":"lists-and-catalogues","url":"http:\/\/ringtheory.herokuapp.com\/","year":"2014","urldate":"2021-11-20"},{"key":"IconicityinMathematicalNotationCommutativityandSymmetry","type":"article","title":"Iconicity in Mathematical Notation: Commutativity and Symmetry","author":"Theresa Elise Wege and Sophie Batchelor and Matthew Inglis and Honali Mistry and Dirk Schlimm","abstract":"Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects \u2013 those which visually resemble in some way the concepts they represent \u2013 offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd\u2019s hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance. ","comment":"","date_added":"2021-11-20","date_published":"2020-08-06","urls":["https:\/\/jnc.psychopen.eu\/index.php\/jnc\/article\/view\/5923","https:\/\/jnc.psychopen.eu\/index.php\/jnc\/article\/download\/5923\/5923.pdf"],"collections":"notation-and-conventions,the-act-of-doing-maths","url":"https:\/\/jnc.psychopen.eu\/index.php\/jnc\/article\/view\/5923 https:\/\/jnc.psychopen.eu\/index.php\/jnc\/article\/download\/5923\/5923.pdf","year":"2020","urldate":"2021-11-20","journal":"Journal of Numerical Cognition","issn":"2363-8761","volume":"6","issue":"3","doi":"10.5964\/jnc.v6i3.314","fulltext_html_url":"https:\/\/jnc.psychopen.eu\/index.php\/jnc\/article\/view\/5923\/5923.html","pages":"378-392"},{"key":"Thedesignofmathematicallanguage","type":"article","title":"The design of mathematical language","author":"Jeremy Avigad","abstract":"As idealized descriptions of mathematical language, there is a sense in which formal systems specify too little, and there is a sense in which they specify too much. They are silent with respect to a number of features of mathematical language that are essential to the communicative and inferential goals of the subject, while many of these features are independent of a specific choice of foundation. This chapter begins to map out the design features of mathematical language without descending to the level of formal implementation, drawing on examples from the mathematical literature and insights from the design of computational proof assistants.","comment":"","date_added":"2021-11-20","date_published":"2021-08-06","urls":["http:\/\/philsci-archive.pitt.edu\/19508\/"],"collections":"easily-explained,notation-and-conventions","url":"http:\/\/philsci-archive.pitt.edu\/19508\/","year":"2021","urldate":"2021-11-20"},{"key":"TheMathematicalMovieDatabase","type":"article","title":"The Mathematical Movie Database","author":"Burkard Polster and Marty Ross","abstract":"This page, along with our TV Database, complements our book Mathematics goes to the movies. It contains a list of titles and short descriptions of about 800 movies that contain mathematics. Also included are links to the respective entries in the International Movie Database (IMDB) and links to clips from some of the movies.","comment":"","date_added":"2021-11-20","date_published":"2004-08-06","urls":["https:\/\/www.qedcat.com\/moviemath\/"],"collections":"lists-and-catalogues","url":"https:\/\/www.qedcat.com\/moviemath\/","year":"2004","urldate":"2021-11-20"},{"key":"RustanLeinoPuzzles","type":"article","title":"Rustan Leino's Puzzles","author":"Rustan Leino","abstract":"Here are some mathematical puzzles that I have enjoyed. Most of them are of the kind that you can discuss and solve at a dinner table, usually without pen and paper. So as not to spoil your fun, no solutions are given on this page, but for some problems I have provided some hints.","comment":"Contains the princess in a castle puzzle, as \"Finding a hermit\".","date_added":"2021-11-23","date_published":"2009-08-06","urls":["https:\/\/web.archive.org\/web\/20161120223815\/http:\/\/research.microsoft.com:80\/en-us\/um\/people\/leino\/puzzles.html"],"collections":"puzzles","url":"https:\/\/web.archive.org\/web\/20161120223815\/http:\/\/research.microsoft.com:80\/en-us\/um\/people\/leino\/puzzles.html","year":"2009","urldate":"2021-11-23"},{"key":"FibbinaryZippers","type":"article","title":"Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-Style Square-Filling Curves","author":"Douglas M. McKenna","abstract":"Within the recursive subdivision of the \\(n \\times n\\) square, what characterizes a Hilbert-style space-filling curve motif of length \\(n^2\\) when\u2014under iterated, self-similar, pure edge-replacement\u2014a sequence of always self-avoiding lattice paths results? How many motifs are there and what do they look like? Such motifs are composable elements of a monoid, where all such motifs map to a particular subset of Hamiltonian cycles on the \\(n \\times n\\) toroidal grid-graph. We prove that for any odd \\(n \\geq 1\\) each motif has a shape that falls into exactly one of \\(F_{(n\u22123)\/2}\\) boundary \u201czipping\u201d modes, where \\(F_i\\) is the \\(i\\)th Fibonacci number; for even \\(n\\) no solution motifs exist. Each mode is governed by a special palindromic Fibbinary bit sequence (i.e., having no adjacent 1 bits). To varying degrees, each zipping mode emanates further combinatorial constraint inward from the square\u2019s\r\nboundary, especially at the corners. The zipping mode whose Fibbinary bits have the most consecutive 0s freezes over half of the \\(n^2\\) edges of an order-\\(n\\) motif into only one distinct (either left- or right-handed) configuration. Manual and machine enumeration for small \\(n\\) is significantly enhanced by these results. For \\(n = 1, 3, 5, 7, 9, 11\\) there are 1, 0, 1, 7, 10101, 20305328 distinct, globally self-avoiding motifs, falling into \\(F_{(n\u22123)\/2} = 1, 0, 1, 1, 2, 3\\) zipping modes, respectively. For \\(n \\geq 5\\), each such motif, when infinitely exponentiated within its monoid, converges to an open-ended, square-filling, continuous curve.","comment":"","date_added":"2021-12-08","date_published":"2022-08-06","urls":["http:\/\/ecajournal.haifa.ac.il\/Volume2022\/ECA2022_S2A13.pdf"],"collections":"art,combinatorics,fibonaccinalia,things-to-make-and-do","url":"http:\/\/ecajournal.haifa.ac.il\/Volume2022\/ECA2022_S2A13.pdf","year":"2022","urldate":"2021-12-08"},{"key":"Benjamin2007","type":"article","title":"Fibonacci determinants-a combinatorial approach","author":"Benjamin, A.T. and Cameron, N.T. and Quinn, J.J.","abstract":"","comment":"","date_added":"2012-02-08","date_published":"2007-08-06","urls":["https:\/\/scholarship.claremont.edu\/cgi\/viewcontent.cgi?referer=&httpsredir=1&article=2055&context=hmc_fac_pub"],"collections":"fibonaccinalia,fun-maths-facts","url":"https:\/\/scholarship.claremont.edu\/cgi\/viewcontent.cgi?referer=&httpsredir=1&article=2055&context=hmc_fac_pub","urldate":"2012-02-08","year":"2007","journal":"Fibonacci Quarterly","number":"1","pages":"39","publisher":"THE FIBONACCI ASSOCIATION","volume":"45"},{"key":"OnKaprekarsJunctionNumbers","type":"article","title":"On Kaprekar's Junction Numbers","author":"Max A. Alekseyev and Donovan Johnson and N. J. A. Sloane","abstract":"A base b junction number u has the property that there are at least two ways\r\nto write it as u = v + s(v), where s(v) is the sum of the digits in the\r\nexpansion of the number v in base b. For the base 10 case, Kaprekar in the\r\n1950's and 1960's studied the problem of finding K(n), the smallest u such that\r\nthe equation u=v+s(v) has exactly n solutions. He gave the values K(2)=101,\r\nK(3)=10^13+1, and conjectured that K(4)=10^24+102. In 1966 Narasinga Rao gave\r\nthe upper bound 10^1111111111124+102 for K(5), as well as upper bounds for\r\nK(6), K(7), K(8), and K(16). In the present work, we derive a set of\r\nrecurrences, which determine K(n) for any base b and in particular imply that\r\nthese conjectured values of K(n) are correct. The key to our approach is an\r\napparently new recurrence for F(u), the number of solutions to u=v+s(v). We\r\nhave applied our method to compute K(n) for n <= 16 and bases b <= 10. These\r\nsequences grow extremely rapidly. Rather surprisingly, the solution to the base\r\n5 problem is determined by the classical Thue-Morse sequence. For a fixed b, it\r\nappears that K(n) grows as a tower of height about log_2(n).","comment":"","date_added":"2022-01-10","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2112.14365v1","http:\/\/arxiv.org\/pdf\/2112.14365v1"],"collections":"fun-maths-facts,integerology","url":"http:\/\/arxiv.org\/abs\/2112.14365v1 http:\/\/arxiv.org\/pdf\/2112.14365v1","year":"2021","urldate":"2022-01-10","archivePrefix":"arXiv","eprint":"2112.14365","primaryClass":"math.NT"},{"key":"MathematicsClipArt","type":"online","title":"Mathematics ClipArt","author":"Florida Center for Instructional Technology","abstract":"The Mathematics ClipArt collection includes 9,820 images for algebra, geometry, trigonometry, probability, money, number sense, and more., conveniently arranged in 222 galleries. This comprehensive set of illustrations for teachers and students consists of ClipArt for all levels of K-12 math classes. Included are coordinate and polar grids, graphs, number lines, clocks, patterns, flashcards, protractors, thermometers, fractions, geometric shapes & solids, angles & lines, bags of marbles, spinners, constructions, theorems & proofs, and dice. From the common place to the hard-to-find ClipArt, everything an educator needs for math activities, assessments, and presentations can be found here.","comment":"","date_added":"2022-01-18","date_published":"2004-08-06","urls":["https:\/\/etc.usf.edu\/clipart\/galleries\/722-mathematics"],"collections":"art,lists-and-catalogues","url":"https:\/\/etc.usf.edu\/clipart\/galleries\/722-mathematics","year":"2004","urldate":"2022-01-18"},{"key":"CheckDigits","type":"online","title":"Check Digits","author":"Jonathan Mohr","abstract":"A decimal (or alphanumeric) digit added to a number for the purpose of detecting the sorts of errors humans typically make on data entry.","comment":"Notes about check digits, including Verhoeff's algorithm for a decimal check digit.","date_added":"2022-01-18","date_published":"1999-08-06","urls":["http:\/\/www.augustana.ualberta.ca\/~mohrj\/algorithms\/checkdigit.html"],"collections":"basically-computer-science,easily-explained,fun-maths-facts,unusual-arithmetic","url":"http:\/\/www.augustana.ualberta.ca\/~mohrj\/algorithms\/checkdigit.html","year":"1999","urldate":"2022-01-18"},{"key":"LessMundaneApplicationsoftheMostMundaneFunctions","type":"article","title":"Less Mundane Applications of the Most Mundane Functions","author":"Pisheng Ding","abstract":"Linear functions are arguably the most mundane among all functions. However,\r\nthe basic fact that a multi-variable linear function has a constant gradient\r\nfield can provide simple geometric insights into several familiar results such\r\nas the Cauchy-Schwarz inequality, the GM-AM inequality, and some distance\r\nformulae, as we shall show.","comment":"","date_added":"2022-02-16","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2201.08798v2","http:\/\/arxiv.org\/pdf\/2201.08798v2"],"collections":"attention-grabbing-titles,easily-explained","url":"http:\/\/arxiv.org\/abs\/2201.08798v2 http:\/\/arxiv.org\/pdf\/2201.08798v2","year":"2022","urldate":"2022-02-16","archivePrefix":"arXiv","eprint":"2201.08798","primaryClass":"math.GM"},{"key":"MATHREPO","type":"online","title":"MATHREPO - Mathematical Data and Software","author":"Claudia Fevola and Christiane G\u00f6rgen","abstract":"This is the repository website of the Max Planck Institute for Mathematics in the Sciences in Leipzig, dedicated to software, computations, and research data in mathematics.\r\n\r\nThe purpose of this webpage is to collect and explain mathematical software developed in various projects among members and collaborators of this institute and its respective applications. The website also contains supplementary material to publications and materials created at events held at MPI MiS.","comment":"","date_added":"2022-02-16","date_published":"2021-08-06","urls":["https:\/\/mathrepo.mis.mpg.de\/"],"collections":"lists-and-catalogues","url":"https:\/\/mathrepo.mis.mpg.de\/","year":"2021","urldate":"2022-02-16"},{"key":"HowtoSolveTheHardestLogicPuzzleEverandItsGeneralization","type":"article","title":"How to Solve \"The Hardest Logic Puzzle Ever\" and Its Generalization","author":"Daniel Vallstrom","abstract":"Raymond Smullyan came up with a puzzle that George Boolos called \"The Hardest\r\nLogic Puzzle Ever\".[1] The puzzle has truthful, lying, and random gods who\r\nanswer yes or no questions with words that we don't know the meaning of. The\r\nchallenge is to figure out which type each god is. Various \"top-down\" solutions\r\nto the puzzle have been developed.[1,2] Here a systematic bottom-up approach to\r\nthe puzzle and its generalization is presented. We prove that an n gods puzzle\r\nis solvable if and only if the random gods are less than the non-random gods.\r\nWe develop a solution using 4.13 questions to the 5 gods variant with 2 random\r\nand 3 lying gods. There is also an aside on mathematical vs. computational\r\nthinking.","comment":"","date_added":"2022-02-16","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2201.09801v2","http:\/\/arxiv.org\/pdf\/2201.09801v2"],"collections":"protocols-and-strategies,puzzles,the-act-of-doing-maths","url":"http:\/\/arxiv.org\/abs\/2201.09801v2 http:\/\/arxiv.org\/pdf\/2201.09801v2","year":"2022","urldate":"2022-02-16","archivePrefix":"arXiv","eprint":"2201.09801","primaryClass":"math.GM"},{"key":"Generatinggraphsrandomly","type":"article","title":"Generating graphs randomly","author":"Catherine Greenhill","abstract":"Graphs are used in many disciplines to model the relationships that exist\r\nbetween objects in a complex discrete system. Researchers may wish to compare a\r\nnetwork of interest to a \"typical\" graph from a family (or ensemble) of graphs\r\nwhich are similar in some way. One way to do this is to take a sample of\r\nseveral random graphs from the family, to gather information about what is\r\n\"typical\". Hence there is a need for algorithms which can generate graphs\r\nuniformly (or approximately uniformly) at random from the given family. Since a\r\nlarge sample may be required, the algorithm should also be computationally\r\nefficient.\r\n Rigorous analysis of such algorithms is often challenging, involving both\r\ncombinatorial and probabilistic arguments. We will focus mainly on the set of\r\nall simple graphs with a particular degree sequence, and describe several\r\ndifferent algorithms for sampling graphs from this family uniformly, or almost\r\nuniformly.","comment":"","date_added":"2022-02-16","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2201.04888v1","http:\/\/arxiv.org\/pdf\/2201.04888v1"],"collections":"basically-computer-science","url":"http:\/\/arxiv.org\/abs\/2201.04888v1 http:\/\/arxiv.org\/pdf\/2201.04888v1","year":"2022","urldate":"2022-02-16","archivePrefix":"arXiv","eprint":"2201.04888","primaryClass":"math.CO"},{"key":"MaxMinPuzzlesinGeometry","type":"article","title":"Max\/Min Puzzles in Geometry","author":"James M Parks","abstract":"The objective here is to find the maximum polygon, in area, which can be\r\nenclosed in a given triangle, for the polygons: parallelograms, rectangles and\r\nsquares. It will initially be assumed that the choices are inscribed polygons,\r\nthat is all vertices of the polygon are on the sides of the triangle. This\r\nconcept will be generalized later to include wedged polygons.","comment":"","date_added":"2022-02-16","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2201.02050v4","http:\/\/arxiv.org\/pdf\/2201.02050v4"],"collections":"geometry,puzzles","url":"http:\/\/arxiv.org\/abs\/2201.02050v4 http:\/\/arxiv.org\/pdf\/2201.02050v4","year":"2022","urldate":"2022-02-16","archivePrefix":"arXiv","eprint":"2201.02050","primaryClass":"math.HO"},{"key":"PopUpGeometry","type":"article","title":"Pop-Up Geometry","author":"Joseph O'Rourke","abstract":"Anyone browsing at the stationery store will see an incredible array of pop-up cards available for any occasion. The workings of pop-up cards and pop-up books can be remarkably intricate. Behind such designs lies beautiful geometry involving the intersection of circles, cones, and spheres, the movements of linkages, and other constructions. The geometry can be modelled by algebraic equations, whose solutions explain the dynamics. For example, several pop-up motions rely on the intersection of three spheres, a computation made every second for GPS location. Connecting the motions of the card structures with the algebra and geometry reveals abstract mathematics performing tangible calculations. Beginning with the nephroid in the 19th-century, the mathematics of pop-up design is now at the frontiers of rigid origami and algorithmic computational complexity. All topics are accessible to those familiar with high-school mathematics; no calculus required. Explanations are supplemented by 140+ figures and 20 animations.","comment":"","date_added":"2022-02-16","date_published":"2022-08-06","urls":["http:\/\/www.science.smith.edu\/~jorourke\/PopUps\/","https:\/\/www.cambridge.org\/us\/academic\/subjects\/mathematics\/recreational-mathematics\/pop-geometry-mathematics-behind-pop-cards"],"collections":"easily-explained,geometry,things-to-make-and-do","url":"http:\/\/www.science.smith.edu\/~jorourke\/PopUps\/ https:\/\/www.cambridge.org\/us\/academic\/subjects\/mathematics\/recreational-mathematics\/pop-geometry-mathematics-behind-pop-cards","year":"2022","urldate":"2022-02-16"},{"key":"EncyclopediaOfTriangleCenters","type":"article","title":"Clark Kimberling's Encyclopedia of Triangle Centers","author":"Clark Kimberling","abstract":"Long ago, someone drew a triangle and three segments across it. Each segment started at a vertex and stopped at the midpoint of the opposite side. The segments met in a point. The person was impressed and repeated the experiment on a different shape of triangle. Again the segments met in a point. The person drew yet a third triangle, very carefully, with the same result. He told his friends. To their surprise and delight, the coincidence worked for them, too. Word spread, and the magic of the three segments was regarded as the work of a higher power.\r\n\r\nCenturies passed, and someone proved that the three medians do indeed concur in a point, now called the centroid. The ancients found other points, too, now called the incenter, circumcenter, and orthocenter. More centuries passed, more special points were discovered, and a definition of triangle center emerged. Like the definition of continuous function, this definition is satisfied by infinitely many objects, of which only finitely many will ever be published. The Encyclopedia of Triangle Centers (ETC) extends a list of 400 triangle centers published in the 1998 book Triangle Centers and Central Triangles.","comment":"","date_added":"2022-02-16","date_published":"1999-08-06","urls":["https:\/\/faculty.evansville.edu\/ck6\/encyclopedia\/ETC.html"],"collections":"geometry,lists-and-catalogues","url":"https:\/\/faculty.evansville.edu\/ck6\/encyclopedia\/ETC.html","year":"1999","urldate":"2022-02-16"},{"key":"ATilingDatabase","type":"online","title":"A Tiling Database","author":"Brian Wichmann and Tony Lee","abstract":"This database has three aims:\r\n\r\n1. Provide a comprehensive collection of high quality images of geometric tiling patterns;\r\n\r\n2. Provide a means of locating images by means of their geometric properties;\r\n\r\n3. Provide an authoritative source for such patterns. ","comment":"","date_added":"2018-10-24","date_published":"2011-08-06","urls":["http:\/\/www.tilingsearch.org"],"collections":"geometry,lists-and-catalogues","url":"http:\/\/www.tilingsearch.org","urldate":"2018-10-24","year":"2011"},{"key":"Graphlopedia","type":"article","title":"Graphlopedia","author":"Sara Billey and Kimberly Bautista and Aaron Bode and Riley Casper and Dien Dang and Nicholas Farn and Graham Kelley and Stanley Lai and Adharsh Ranganathan and Michael Trinh and Alex Tsun and Katrina Warner","abstract":"A database of graphs for the use of mathematicians and other graph lovers. The graphs are ordered by degree sequence.","comment":"","date_added":"2017-06-21","date_published":"2017-08-06","urls":["http:\/\/graphlopedia.org","http:\/\/www.math.washington.edu\/~billey\/graphlopediaPDF.pdf"],"collections":"lists-and-catalogues","url":"http:\/\/graphlopedia.org http:\/\/www.math.washington.edu\/~billey\/graphlopediaPDF.pdf","urldate":"2017-06-21","year":"2017"},{"key":"DatabaseofPermutationPatternAvoidance","type":"online","title":"Database of Permutation Pattern Avoidance","author":"Bridget Tenner","abstract":"The aim of this database is to provide a resource of phenomena characterized by avoiding a finite number of permutation patterns.","comment":"","date_added":"2022-02-16","date_published":"2011-08-06","urls":["https:\/\/math.depaul.edu\/bridget\/patterns.html"],"collections":"combinatorics,lists-and-catalogues","url":"https:\/\/math.depaul.edu\/bridget\/patterns.html","urldate":"2022-02-16","year":"2011"},{"key":"FindStat","type":"online","title":"FindStat","author":"Martin Rubey and Christian Stump","abstract":"This collaborative project is a database of combinatorial statistics and maps on combinatorial collections and a search engine, identifying your data as the composition of known maps and statistics.","comment":"","date_added":"2022-02-16","date_published":"2019-08-06","urls":["http:\/\/www.findstat.org\/"],"collections":"combinatorics,lists-and-catalogues","url":"http:\/\/www.findstat.org\/","year":"2019","urldate":"2022-02-16"},{"key":"HouseofGraphs","type":"article","title":"House of Graphs","author":"Gunnar Brinkmann and Kris Coolsaet and Jan Goedgebeur and Hadrien M\u00e9lot","abstract":"Most graph theorists will agree that among the vast number of graphs that exist there are only a few that can be considered really interesting.\r\n\r\nIt is the aim of this House of Graphs project to find a workable definition of 'interesting' and provide a searchable database of graphs that conform to this definition. And to allow users to add additional graphs which they find interesting. In order to avoid abuse, only registered users can add new graphs.\r\n\r\nWe would also like to serve as a repository for lists of graphs (which can be downloaded in several formats) and graph generators. Currently we only provide a small selection. ","comment":"","date_added":"2022-02-16","date_published":"2010-08-06","urls":["https:\/\/hog.grinvin.org\/"],"collections":"lists-and-catalogues","url":"https:\/\/hog.grinvin.org\/","year":"2010","urldate":"2022-02-16"},{"key":"Acatalogofmatchstickgraphs","type":"article","title":"A catalog of matchstick graphs","author":"Raffaele Salvia","abstract":"Classification of planar unit-distance graphs with up to 9 edges, by\r\nhomeomorphism and isomorphism classes. With exactly nine edges, there are 633\r\nnonisomorphic connected matchstick graphs, of which 196 are topologically\r\ndistinct from each other. Increasing edges' number, their quantities rise more\r\nthan exponentially, in a still unclear way.","comment":"","date_added":"2022-02-16","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1303.5965v5","http:\/\/arxiv.org\/pdf\/1303.5965v5"],"collections":"lists-and-catalogues","url":"http:\/\/arxiv.org\/abs\/1303.5965v5 http:\/\/arxiv.org\/pdf\/1303.5965v5","year":"2013","urldate":"2022-02-16","archivePrefix":"arXiv","eprint":"1303.5965","primaryClass":"math.CO"},{"key":"Tilingwitharbitrarytiles","type":"article","title":"Tiling with arbitrary tiles","author":"Vytautas Gruslys and Imre Leader and Ta Sheng Tan","abstract":"Let $T$ be a tile in $\\mathbb{Z}^n$, meaning a finite subset of\r\n$\\mathbb{Z}^n$. It may or may not tile $\\mathbb{Z}^n$, in the sense of\r\n$\\mathbb{Z}^n$ having a partition into copies of $T$. However, we prove that\r\n$T$ does tile $\\mathbb{Z}^d$ for some $d$. This resolves a conjecture of\r\nChalcraft.","comment":"","date_added":"2022-02-23","date_published":"2015-08-06","urls":["http:\/\/arxiv.org\/abs\/1505.03697v2","http:\/\/arxiv.org\/pdf\/1505.03697v2"],"collections":"easily-explained,fun-maths-facts,geometry","url":"http:\/\/arxiv.org\/abs\/1505.03697v2 http:\/\/arxiv.org\/pdf\/1505.03697v2","year":"2015","urldate":"2022-02-23","archivePrefix":"arXiv","eprint":"1505.03697","primaryClass":"math.CO"},{"key":"EasyProofofThreeRecursiveAlgorithmsEinfacherBeweisdreierrekursiverAlgorithmen","type":"article","title":"Easy Proof of Three Recursive $\u03c0$-Algorithms -- Einfacher Beweis dreier rekursiver $\u03c0$-Algorithmen","author":"Lorenz Milla","abstract":"This paper consists of three independent parts: First we use only elementary\r\nalgebra to prove that the quartic algorithm of the Borwein brothers has exactly\r\nthe same output as the Brent-Salamin algorithm, but that the latter needs twice\r\nas many iterations. Second we use integral calculus to prove that the\r\nBrent-Salamin algorithm approximates $\\pi$. Combining these results proves that\r\nthe Borwein brothers' quartic algorithm also approximates $\\pi$. Third, we\r\nprove the quadratic convergence of the Brent-Salamin algorithm, which also\r\nproves the quartic convergence of Borwein's algorithm.\r\n -----\r\n Dieses Paper besteht aus drei unabh\\\"angigen Teilen: Erstens beweisen wir mit\r\nelementarer Algebra, dass der Borwein-Algorithmus vierter Ordnung die gleichen\r\nErgebnisse liefert wie der Brent-Salamin-Algorithmus, wobei letzterer doppelt\r\nso viele Iterationen ben\\\"otigt. Zweitens beweisen wir mit Integralrechnung,\r\ndass der Brent-Salamin-Algorithmus gegen $\\pi$ konvergiert. Hieraus folgt, dass\r\nder Borwein-Algorithmus vierter Ordnung ebenfalls gegen $\\pi$ konvergiert.\r\nDrittens beweisen wir die quadratische Konvergenz des Brent-Salamin-Algorithmus\r\nund somit auch die quartische Konvergenz des Borwein-Algorithmus.","comment":"","date_added":"2022-02-25","date_published":"2019-08-06","urls":["http:\/\/arxiv.org\/abs\/1907.04110v2","http:\/\/arxiv.org\/pdf\/1907.04110v2"],"collections":"about-proof,basically-computer-science,fun-maths-facts","url":"http:\/\/arxiv.org\/abs\/1907.04110v2 http:\/\/arxiv.org\/pdf\/1907.04110v2","year":"2019","urldate":"2022-02-25","archivePrefix":"arXiv","eprint":"1907.04110","primaryClass":"math.NT"},{"key":"DescendingDungeonsandIteratedBaseChanging","type":"article","title":"Descending Dungeons and Iterated Base-Changing","author":"David Applegate and Marc LeBrun and N. J. A. Sloane","abstract":"For real numbers a, b> 1, let as a_b denote the result of interpreting a in\r\nbase b instead of base 10. We define ``dungeons'' (as opposed to ``towers'') to\r\nbe numbers of the form a_b_c_d_..._e, parenthesized either from the bottom\r\nupwards (preferred) or from the top downwards. Among other things, we show that\r\nthe sequences of dungeons with n-th terms 10_11_12_..._(n-1)_n or\r\nn_(n-1)_..._12_11_10 grow roughly like 10^{10^{n log log n}}, where the\r\nlogarithms are to the base 10. We also investigate the behavior as n increases\r\nof the sequence a_a_a_..._a, with n a's, parenthesized from the bottom upwards.\r\nThis converges either to a single number (e.g. to the golden ratio if a = 1.1),\r\nto a two-term limit cycle (e.g. if a = 1.05) or else diverges (e.g. if a =\r\nfrac{100{99).","comment":"","date_added":"2022-02-28","date_published":"2006-08-06","urls":["http:\/\/arxiv.org\/abs\/math\/0611293v3","http:\/\/arxiv.org\/pdf\/math\/0611293v3"],"collections":"attention-grabbing-titles,easily-explained,fun-maths-facts,integerology","url":"http:\/\/arxiv.org\/abs\/math\/0611293v3 http:\/\/arxiv.org\/pdf\/math\/0611293v3","year":"2006","urldate":"2022-02-28","archivePrefix":"arXiv","eprint":"math\/0611293","primaryClass":"math.NT"},{"key":"CharacteristicPolynomialDatabase","type":"online","title":"Characteristic Polynomial Database","author":"Steven E. Thornton","abstract":"The characteristic polynomial database contains characteristic polynomials, minimal polynomials and properties for a variety of families of Bohemian matrices.","comment":"","date_added":"2022-04-24","date_published":"2019-08-06","urls":["http:\/\/www.bohemianmatrices.com\/cpdb\/"],"collections":"lists-and-catalogues","url":"http:\/\/www.bohemianmatrices.com\/cpdb\/","year":"2019","urldate":"2022-04-24"},{"key":"Asimplemnemonictocomputesumsofpowers","type":"article","title":"A simple mnemonic to compute sums of powers","author":"Alessandro Mariani","abstract":"We give a simple recursive formula to obtain the general sum of the first $N$\r\nnatural numbers to the $r$th power. Our method allows one to obtain the general\r\nformula for the $(r+1)$th power once one knows the general formula for the\r\n$r$th power. The method is very simple to remember owing to an analogy with\r\ndifferentiation and integration. Unlike previously known methods, no knowledge\r\nof additional specific constants (such as the Bernoulli numbers) is needed.\r\nThis makes it particularly suitable for applications in cases when one cannot\r\nconsult external references, for example mathematics competitions.","comment":"","date_added":"2022-04-24","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2203.13870v1","http:\/\/arxiv.org\/pdf\/2203.13870v1"],"collections":"easily-explained,fun-maths-facts,integerology","url":"http:\/\/arxiv.org\/abs\/2203.13870v1 http:\/\/arxiv.org\/pdf\/2203.13870v1","year":"2022","urldate":"2022-04-24","archivePrefix":"arXiv","eprint":"2203.13870","primaryClass":"math.GM"},{"key":"AnoteontheScreamingToesgame","type":"article","title":"A note on the Screaming Toes game","author":"Simon Tavar\u00e9","abstract":"We investigate properties of random mappings whose core is composed of\r\nderangements as opposed to permutations. Such mappings arise as the natural\r\nframework to study the Screaming Toes game described, for example, by Peter\r\nCameron. This mapping differs from the classical case primarily in the\r\nbehaviour of the small components, and a number of explicit results are\r\nprovided to illustrate these differences.","comment":"","date_added":"2022-04-24","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2006.04805v1","http:\/\/arxiv.org\/pdf\/2006.04805v1"],"collections":"attention-grabbing-titles,combinatorics,games-to-play-with-friends,probability-and-statistics,the-groups-group","url":"http:\/\/arxiv.org\/abs\/2006.04805v1 http:\/\/arxiv.org\/pdf\/2006.04805v1","urldate":"2022-04-24","year":"2020","archivePrefix":"arXiv","eprint":"2006.04805","primaryClass":"math.PR"},{"key":"ConvexEquipartitionsTheSpicyChickenTheorem","type":"article","title":"Convex Equipartitions: The Spicy Chicken Theorem","author":"Roman Karasev and Alfredo Hubard and Boris Aronov","abstract":"We show that, for any prime power n and any convex body K (i.e., a compact\r\nconvex set with interior) in Rd, there exists a partition of K into n convex\r\nsets with equal volumes and equal surface areas. Similar results regarding\r\nequipartitions with respect to continuous functionals and absolutely continuous\r\nmeasures on convex bodies are also proven. These include a generalization of\r\nthe ham-sandwich theorem to arbitrary number of convex pieces confirming a\r\nconjecture of Kaneko and Kano, a similar generalization of perfect partitions\r\nof a cake and its icing, and a generalization of the Gromov-Borsuk-Ulam theorem\r\nfor convex sets in the model spaces of constant curvature.\r\n Most of the results in this paper appear in arxiv:1011.4762 and in\r\narxiv:1010.4611. Since the main results and techniques there are essentially\r\nthe same, we have merged the papers for journal publication. In this version we\r\nalso provide a technical alternative to a part of the proof of the main\r\ntopological result that avoids the use of compactly supported homology.","comment":"","date_added":"2022-04-24","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1306.2741v2","http:\/\/arxiv.org\/pdf\/1306.2741v2"],"collections":"food,fun-maths-facts","url":"http:\/\/arxiv.org\/abs\/1306.2741v2 http:\/\/arxiv.org\/pdf\/1306.2741v2","year":"2013","urldate":"2022-04-24","archivePrefix":"arXiv","eprint":"1306.2741","primaryClass":"math.MG"},{"key":"item35","type":"article","title":"WHAT IS Lehmer's number?","author":"Eriko Hironaka","abstract":"Lehmer's number \\(\\lambda \\approx 1.17628\\) is the largest real root of the polynomial \\(f_\\lambda(x) = x^{10} + x^9 - x^7 - x^6 -x^5 -x^4 - x^3 + x + 1\\).\r\n\r\nThis number appears in various contexts in number theory and topology as the (sometimes conjectural) answer to natural questions involving ``minimality'' and ``small complexity''.","comment":"","date_added":"2013-12-03","date_published":"2009-08-06","urls":["http:\/\/www.math.fsu.edu\/~aluffi\/archive\/paper355.pdf"],"collections":"attention-grabbing-titles,integerology","url":"http:\/\/www.math.fsu.edu\/~aluffi\/archive\/paper355.pdf","urldate":"2013-12-03","year":"2009"},{"key":"FillingspacewithhypercubesoftwosizesThepythagoreantilinginhigherdimensions","type":"article","title":"Filling space with hypercubes of two sizes -- The pythagorean tiling in higher dimensions","author":"Jakob F\u00fchrer","abstract":"We construct a unilateral lattice tiling of $\\mathbb{R}^n$ into hypercubes of\r\ntwo differnet side lengths $p$ or $q$. This generalizes the Pythagorean tiling\r\nin $\\mathbb{R}^2$. We also show that this tiling is unique up to symmetries,\r\nwhich proves a variation of a conjecture by B\\\"olcskei from 2001. For positive\r\nintegers $p$ and $q$ this tiling also provides a tiling of\r\n$(\\mathbb{Z}\/(p^n+q^n)\\mathbb{Z})^n$.","comment":"","date_added":"2022-05-13","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2204.11529v2","http:\/\/arxiv.org\/pdf\/2204.11529v2"],"collections":"easily-explained,fun-maths-facts,geometry","url":"http:\/\/arxiv.org\/abs\/2204.11529v2 http:\/\/arxiv.org\/pdf\/2204.11529v2","year":"2022","urldate":"2022-05-13","archivePrefix":"arXiv","eprint":"2204.11529","primaryClass":"math.CO"},{"key":"WhosAfraidofMathematicalDiagrams","type":"article","title":"Who's Afraid of Mathematical Diagrams?","author":"Silvia De Toffoli","abstract":"Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability. I then consider whether diagrams can be essential to the proofs in which they appear. ","comment":"","date_added":"2022-05-13","date_published":false,"urls":["https:\/\/philpapers.org\/rec\/DETWAO","https:\/\/philpapers.org\/archive\/DETWAO.pdf"],"collections":"attention-grabbing-titles,the-act-of-doing-maths","url":"https:\/\/philpapers.org\/rec\/DETWAO https:\/\/philpapers.org\/archive\/DETWAO.pdf","year":"forthcoming","urldate":"2022-05-13","doi":"","journal":"Philosophers","volume":"0","issue":""},{"key":"Barany2011","type":"article","title":"Chalk: Materials and Concepts in Mathematics Research","author":"Barany, Michael J and Mackenzie, Donald","abstract":"","comment":"","date_added":"2011-11-01","date_published":"2011-08-06","urls":["https:\/\/www.sps.ed.ac.uk\/sites\/default\/files\/assets\/pdf\/Chalk_0.pdf","https:\/\/mitpress.universitypressscholarship.com\/view\/10.7551\/mitpress\/9780262525381.001.0001\/upso-9780262525381-chapter-6"],"collections":"notation-and-conventions,the-act-of-doing-maths","url":"https:\/\/www.sps.ed.ac.uk\/sites\/default\/files\/assets\/pdf\/Chalk_0.pdf https:\/\/mitpress.universitypressscholarship.com\/view\/10.7551\/mitpress\/9780262525381.001.0001\/upso-9780262525381-chapter-6","urldate":"2011-11-01","year":"2011","pages":"1--30"},{"key":"EarliestUsesofVariousMathematicalSymbols","type":"online","title":"Earliest Uses of Various Mathematical Symbols","author":"Jeff Miller","abstract":"These pages show the names of the individuals who first used various common mathematical symbols, and the dates the symbols first appeared. The most important written source is the definitive A History of Mathematical Notations by Florian Cajori.","comment":"","date_added":"2017-09-12","date_published":"2008-08-06","urls":["https:\/\/mathshistory.st-andrews.ac.uk\/Miller\/mathsym\/"],"collections":"history,notation-and-conventions","url":"https:\/\/mathshistory.st-andrews.ac.uk\/Miller\/mathsym\/","urldate":"2017-09-12","year":"2008"},{"key":"TheStrongLawOfSmallNumbers","type":"article","title":"The Strong Law of Small Numbers","author":"Richard K. Guy","abstract":"This article is in two parts, the first of which is a do-it-yourself operation, in which I'll show you 35 examples of patterns that seem to appear when we look at several small values of n, in various problems whose answers depend on n. The question will be, in each case: do you think that the pattern persists for all n, or do you believe that it is a figment of the smallness of the values of n that are worked out in the examples?\r\n\r\nCaution: examples of both kinds appear; they are not all figments!\r\n\r\nIn the second part I'll give you the answers, insofar as I know them, together with references.","comment":"","date_added":"2022-07-05","date_published":"1988-08-06","urls":["https:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Ford\/Guy697-712.pdf"],"collections":"easily-explained,fun-maths-facts,the-act-of-doing-maths","url":"https:\/\/www.maa.org\/sites\/default\/files\/pdf\/upload_library\/22\/Ford\/Guy697-712.pdf","year":"1988","urldate":"2022-07-05"},{"key":"StutteringlookandsaysequencesandachallengertoConwaysmostcomplicatedalgebraicnumberfromthesilliestsource","type":"article","title":"Stuttering look and say sequences and a challenger to Conway's most complicated algebraic number from the silliest source","author":"Jonathan Comes","abstract":"We introduce stuttering look and say sequences and describe their chemical\r\nstructure in the spirit of Conway's work on audioactive decay. We show the\r\ngrowth rate of a stuttering look and say sequence is an algebraic integer of\r\ndegree 415.","comment":"","date_added":"2022-07-08","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2206.11991v1","http:\/\/arxiv.org\/pdf\/2206.11991v1"],"collections":"attention-grabbing-titles,easily-explained,fun-maths-facts,integerology","url":"http:\/\/arxiv.org\/abs\/2206.11991v1 http:\/\/arxiv.org\/pdf\/2206.11991v1","year":"2022","urldate":"2022-07-08","archivePrefix":"arXiv","eprint":"2206.11991","primaryClass":"math.HO"},{"key":"OnTheEuclideanAlgorithmRhythmWithoutRecursion","type":"article","title":"On The Euclidean Algorithm: Rhythm Without Recursion","author":"Thomas Morrill","abstract":"A modified form of Euclid's algorithm has gained popularity among musical\r\ncomposers following Toussaint's 2005 survey of so-called Euclidean rhythms in\r\nworld music. We offer a method to easily calculate Euclid's algorithm by hand\r\nas a modification of Bresenham's line-drawing algorithm. Notably, this modified\r\nalgorithm is a non-recursive matrix construction, using only modular arithmetic\r\nand combinatorics. This construction does not outperform the traditional\r\ndivide-with-remainder method; it is presented for combinatorial interest and\r\nease of hand computation.","comment":"","date_added":"2022-07-08","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2206.12421v1","http:\/\/arxiv.org\/pdf\/2206.12421v1"],"collections":"easily-explained,fun-maths-facts,music","url":"http:\/\/arxiv.org\/abs\/2206.12421v1 http:\/\/arxiv.org\/pdf\/2206.12421v1","urldate":"2022-07-08","year":"2022","archivePrefix":"arXiv","eprint":"2206.12421","primaryClass":"math.HO"},{"key":"Themathematicsofburgerflipping","type":"article","title":"The mathematics of burger flipping","author":"Jean-Luc Thiffeault","abstract":"What is the most effective way to grill food? Timing is everything, since\r\nonly one surface is exposed to heat at a given time. Should we flip only once,\r\nor many times? We present a simple model of cooking by flipping, and some\r\ninteresting observations emerge. The rate of cooking depends on the spectrum of\r\na linear operator, and on the fixed point of a map. If the system has symmetric\r\nthermal properties, the rate of cooking becomes independent of the sequence of\r\nflips, as long as the last point to be cooked is the midpoint. After numerical\r\noptimization, the flipping intervals become roughly equal in duration as their\r\nnumber is increased, though the final interval is significantly longer. We find\r\nthat the optimal improvement in cooking time, given an arbitrary number of\r\nflips, is about 29% over a single flip. This toy problem has some\r\ncharacteristics reminiscent of turbulent thermal convection, such as a uniform\r\naverage interior temperature with boundary layers.","comment":"","date_added":"2022-07-08","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2206.13900v2","http:\/\/arxiv.org\/pdf\/2206.13900v2"],"collections":"basically-physics,food,modelling","url":"http:\/\/arxiv.org\/abs\/2206.13900v2 http:\/\/arxiv.org\/pdf\/2206.13900v2","year":"2022","urldate":"2022-07-08","archivePrefix":"arXiv","eprint":"2206.13900","primaryClass":"physics.pop-ph"},{"key":"MathematicalMarbling","type":"online","title":"Mathematical Marbling","author":"Aubrey Jaffer","abstract":"Marbling refers to painting techniques for creating a stone-like appearance or intricate flowing designs.\r\n\r\nMarbling originated in Asia more than 800 years ago and spread to Europe in the 1500s, where it was used for endpapers and book covers.\r\n\r\nMy web-pages are about generating marbling designs mathematically. ","comment":"","date_added":"2022-07-12","date_published":"2003-08-06","urls":["https:\/\/people.csail.mit.edu\/jaffer\/Marbling\/"],"collections":"art,modelling,things-to-make-and-do","url":"https:\/\/people.csail.mit.edu\/jaffer\/Marbling\/","year":"2003","urldate":"2022-07-12"},{"key":"ListofnumbersGoogologyWiki","type":"online","title":"List of numbers - Googology Wiki","author":"","abstract":"This is a list of googolisms (names for numbers) in ascending order.\r\n\r\nThis list contains ill-defined large numbers, e.g. BEAF numbers beyond tetrational arrays, BIG FOOT, Little Bigeddon, Sasquatch, and large numbers whose well-definedness is not known, e.g. large numbers defined by Taranovsky's ordinal notation and Bashicu matrix number with respect to Bashicu matrix system version 2.3. ","comment":"","date_added":"2022-08-09","date_published":"2013-08-06","urls":["https:\/\/googology.miraheze.org\/wiki\/List_of_numbers"],"collections":"integerology,lists-and-catalogues","url":"https:\/\/googology.miraheze.org\/wiki\/List_of_numbers","year":"2013","urldate":"2022-08-09"},{"key":"Battisti2006","type":"article","title":"A Generalized Fibonacci LSB Data Hiding Technique","author":"Battisti, F and Carli, M and Neri, A and Egiaziarian, K","abstract":"","comment":"","date_added":"2012-01-10","date_published":"2006-08-06","urls":["https:\/\/web.archive.org\/web\/20181123041419\/http:\/\/www.comlab.uniroma3.it\/Marco\/Articoli%20Battisti\/A%20Generalized%20Fibonacci%20LSB%20Data%20Hiding%20Technique.pdf","http:\/\/www.comlab.uniroma3.it\/Marco\/Articoli%20Battisti\/A%20Generalized%20Fibonacci%20LSB%20Data%20Hiding%20Technique.pdf"],"collections":"basically-computer-science,fibonaccinalia","url":"https:\/\/web.archive.org\/web\/20181123041419\/http:\/\/www.comlab.uniroma3.it\/Marco\/Articoli%20Battisti\/A%20Generalized%20Fibonacci%20LSB%20Data%20Hiding%20Technique.pdf http:\/\/www.comlab.uniroma3.it\/Marco\/Articoli%20Battisti\/A%20Generalized%20Fibonacci%20LSB%20Data%20Hiding%20Technique.pdf","urldate":"2012-01-10","year":"2006","booktitle":"3rd International Conference on Computers and Devices for Communication (CODEC-06) TEA, Institute of Radio Physics and Electronics, University of Calcutta","keywords":"data hiding,fibonacci,image,security","pages":"2--5"},{"key":"StrangeExpectationsandtheWinniethePoohProblem","type":"article","title":"Strange Expectations and the Winnie-the-Pooh Problem","author":"Marko Thiel and Nathan Williams","abstract":"Motivated by the study of simultaneous cores, we give three proofs (in\r\nvarying levels of generality) for the expected norm of a weight in a highest\r\nweight representation of a complex simple Lie algebra. First, we argue directly\r\nusing the polynomial method and the Weyl character formula. Second, we use the\r\ncombinatorics of semistandard tableaux to obtain the result in type A. Third,\r\nand most interestingly, we relate this problem to the \"Winnie-the-Pooh problem\"\r\nregarding orthogonal decompositions of Lie algebras; although this approach\r\noffers the most explanatory power, it applies only to Cartan types other than A\r\nand C. We conclude with computations of many combinatorial cumulants.","comment":"","date_added":"2022-08-22","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1811.02550v1","http:\/\/arxiv.org\/pdf\/1811.02550v1"],"collections":"attention-grabbing-titles","url":"http:\/\/arxiv.org\/abs\/1811.02550v1 http:\/\/arxiv.org\/pdf\/1811.02550v1","year":"2018","urldate":"2022-08-22","archivePrefix":"arXiv","eprint":"1811.02550","primaryClass":"math.CO"},{"key":"DistanceRegular","type":"article","title":"DistanceRegular.org","author":"Robert Bailey","abstract":"An online repository of distance-regular graphs. Here you will find distance-regular graphs available for download in a variety of formats.","comment":"","date_added":"2022-08-22","date_published":"2017-08-06","urls":["https:\/\/www.distanceregular.org\/"],"collections":"lists-and-catalogues","url":"https:\/\/www.distanceregular.org\/","year":"2017","urldate":"2022-08-22"},{"key":"GroupNames","type":"online","title":"GroupNames","author":"Tim Dokchitser","abstract":"GroupNames.org is a database, under construction, of names, extensions, properties and character tables of finite groups of small order.","comment":"","date_added":"2022-09-09","date_published":"2022-08-06","urls":["https:\/\/people.maths.bris.ac.uk\/~matyd\/GroupNames\/"],"collections":"lists-and-catalogues,the-groups-group","url":"https:\/\/people.maths.bris.ac.uk\/~matyd\/GroupNames\/","year":"2022","urldate":"2022-09-09"},{"key":"Euclideantravellerinhyperbolicworlds","type":"article","title":"Euclidean traveller in hyperbolic worlds","author":"Hee Oh","abstract":"We will discuss all possible closures of a Euclidean line in various\r\ngeometric spaces. Imagine the Euclidean traveller, who travels only along a\r\nEuclidean line. She will be travelling to many different geometric worlds, and\r\nour question will be \"what places does she get to see in each world?\". Here is\r\nthe itinerary of our Euclidean traveller: In 1884, she travels to the torus of\r\nany dimension, guided by Kronecker. In 1936, she travels to the world, called a\r\nclosed hyperbolic surface, guided by Hedlund. In 1991, she then travels to a\r\nclosed hyperbolic manifold of higher dimension $n\\ge 3$ guided by Ratner.\r\nFinally, she adventures into hyperbolic manifolds of infinite volume guided by\r\nDal'bo in dimension $2$ in 2000, by McMullen-Mohammadi-Oh in dimension $3$ in\r\n2016 and by Lee-Oh in all higher dimensions in 2019.","comment":"","date_added":"2022-09-09","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2209.01306v1","http:\/\/arxiv.org\/pdf\/2209.01306v1"],"collections":"easily-explained,fun-maths-facts,geometry","url":"http:\/\/arxiv.org\/abs\/2209.01306v1 http:\/\/arxiv.org\/pdf\/2209.01306v1","year":"2022","urldate":"2022-09-09","archivePrefix":"arXiv","eprint":"2209.01306","primaryClass":"math.GT"},{"key":"Moller2009","type":"article","title":"Das 2: 3-Ei-ein praktikables Eimodell","author":"M\u00f6ller, H","abstract":"","comment":"","date_added":"2013-03-19","date_published":"2009-08-06","urls":["https:\/\/web.archive.org\/web\/20130905000237\/http:\/\/www.math.uni-muenster.de\/u\/mollerh\/data\/ZweiDreiEi.pdf","http:\/\/www.math.uni-muenster.de\/u\/mollerh\/data\/ZweiDreiEi.pdf"],"collections":"animals,art,easily-explained,food,geometry,lists-and-catalogues,modelling","url":"https:\/\/web.archive.org\/web\/20130905000237\/http:\/\/www.math.uni-muenster.de\/u\/mollerh\/data\/ZweiDreiEi.pdf http:\/\/www.math.uni-muenster.de\/u\/mollerh\/data\/ZweiDreiEi.pdf","urldate":"2013-03-19","year":"2009","journal":"math.uni-muenster.de","pages":"1--32"},{"key":"MaclaurinIntegrationAWeaponAgainstInfamousIntegrals","type":"article","title":"Maclaurin Integration: A Weapon Against Infamous Integrals","author":"Glenn Bruda","abstract":"Maclaurin Integration is a new series-based technique for solving infamously\r\ndifficult integrals in terms of elementary functions. It has fairly liberal\r\nconditions for sound use, making it one of the most versatile integration\r\ntechniques. Additionally, there is essentially zero human labor involved in\r\ncalculating integrals using this technique, making it one of the easiest\r\nintegration techniques to use. Its scope is mainly in pure mathematics.","comment":"","date_added":"2022-10-08","date_published":"2022-08-06","urls":["http:\/\/arxiv.org\/abs\/2201.12717v1","http:\/\/arxiv.org\/pdf\/2201.12717v1"],"collections":"fun-maths-facts","url":"http:\/\/arxiv.org\/abs\/2201.12717v1 http:\/\/arxiv.org\/pdf\/2201.12717v1","year":"2022","urldate":"2022-10-08","archivePrefix":"arXiv","eprint":"2201.12717","primaryClass":"math.GM"},{"key":"ArbitrarilyClose","type":"book","title":"Arbitrarily Close","author":"John A. Rock","abstract":"Mathematicians tend to use the phrase \"arbitrarily close\" to mean something\r\nalong the lines of \"every neighborhood of a point intersects a set\". Taking the\r\nlatter statement as a technical definition for arbitrarily close leads to an\r\nalternative development of classic concepts in real analysis such as supremum,\r\nclosure, convergence and limits of sequences, closure, connectedness,\r\ncompactness, and continuity. The goal of this text is to provide readers with\r\nan introduction to real analysis by taking deliberate steps to parse these\r\ndifficult concepts using arbitrarily close as the kernel.","comment":"","date_added":"2022-10-08","date_published":"2019-08-06","urls":["http:\/\/arxiv.org\/abs\/1912.13159v2","http:\/\/arxiv.org\/pdf\/1912.13159v2"],"collections":"the-act-of-doing-maths","url":"http:\/\/arxiv.org\/abs\/1912.13159v2 http:\/\/arxiv.org\/pdf\/1912.13159v2","year":"2019","urldate":"2022-10-08","archivePrefix":"arXiv","eprint":"1912.13159","primaryClass":"math.HO"},{"key":"PIGTIKALpuzzlesingeometrythatIknowandlove","type":"book","title":"PIGTIKAL (puzzles in geometry that I know and love)","author":"Anton Petrunin","abstract":"Problems for the graduate students who want to improve problem-solving skills\r\nin geometry. Every problem has a short elegant solution -- this gives a hint\r\nwhich was not available when the problem was discovered.","comment":"","date_added":"2022-10-14","date_published":"2009-08-06","urls":["http:\/\/arxiv.org\/abs\/0906.0290v16","http:\/\/arxiv.org\/pdf\/0906.0290v16"],"collections":"geometry,lists-and-catalogues,puzzles","url":"http:\/\/arxiv.org\/abs\/0906.0290v16 http:\/\/arxiv.org\/pdf\/0906.0290v16","year":"2009","urldate":"2022-10-14","archivePrefix":"arXiv","eprint":"0906.0290","primaryClass":"math.HO"},{"key":"Raymer2007","type":"article","title":"Spontaneous knotting of an agitated string.","author":"Raymer, Dorian M and Smith, Douglas E","abstract":"It is well known that a jostled string tends to become knotted; yet the factors governing the \"spontaneous\" formation of various knots are unclear. We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds. We used mathematical knot theory to analyze the knots. Above a critical string length, the probability P of knotting at first increased sharply with length but then saturated below 100%. This behavior differs from that of mathematical self-avoiding random walks, where P has been proven to approach 100%. Finite agitation time and jamming of the string due to its stiffness result in lower probability, but P approaches 100% with long, flexible strings. We analyzed the knots by calculating their Jones polynomials via computer analysis of digital photos of the string. Remarkably, almost all were identified as prime knots: 120 different types, having minimum crossing numbers up to 11, were observed in 3,415 trials. All prime knots with up to seven crossings were observed. The relative probability of forming a knot decreased exponentially with minimum crossing number and M\u00f6bius energy, mathematical measures of knot complexity. Based on the observation that long, stiff strings tend to form a coiled structure when confined, we propose a simple model to describe the knot formation based on random \"braid moves\" of the string end. Our model can qualitatively account for the observed distribution of knots and dependence on agitation time and string length.","comment":"If you jiggle a string about for a while, it will get a knot in it. The authors look at which kinds of knots are formed most often, from a knot theory perspective, and find a surprising variety. They give a mathematical model that matches the observed distribution of knots, useful if you ever want to pretend you've wobbled a rope for longer than you really have.","date_added":"2011-01-12","date_published":"2007-10-01","urls":["https:\/\/www.pnas.org\/doi\/10.1073\/pnas.0611320104"],"collections":"easily-explained,modelling","url":"https:\/\/www.pnas.org\/doi\/10.1073\/pnas.0611320104","urldate":"2011-01-12","year":"2007","doi":"10.1073\/pnas.0611320104","issn":"0027-8424","journal":"Proceedings of the National Academy of Sciences of the United States of America","month":"oct","number":"42","pages":"16432--7","pmid":"17911269","volume":"104"},{"key":"LucyandLilyAGameofGeometryandNumberTheoryTheAmericanMathematicalMonthlyVol109No1","type":"article","title":"Lucy and Lily: A Game of Geometry and Number Theory: The American Mathematical Monthly: Vol 109, No 1","author":"Richard Evan Schwartz","abstract":"The purpose of this article is to describe a computer game I created. I named the game \u201cLucy and Lily,\u201d after my two daughters. Many people have expressed enthusiasm for the game, and their enthusiasm has encouraged me to write this article. At some point, Daniel Allcock and Brian Conrad worked out an informal but careful\r\nanalysis of \u201cLucy and Lily.\u201d The several challenges I issue to the reader, during the\r\ncourse of the article, derive from facts that one or both of them established","comment":"","date_added":"2022-11-14","date_published":false,"urls":["https:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/00029890.2002.11919835","https:\/\/www.math.brown.edu\/reschwar\/Papers\/ll.pdf"],"collections":"games-to-play-with-friends,geometry","url":"https:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/00029890.2002.11919835 https:\/\/www.math.brown.edu\/reschwar\/Papers\/ll.pdf","year":"1 Feb 2018","urldate":"2022-11-14","journal":"The American Mathematical Monthly","publisher":"Taylor & Francis","identifier":"10.1080\/00029890.2002.11919835"},{"key":"MathMagic","type":"online","title":"Math Magic","author":"Erich Friedman","abstract":"Math Magic is a web site devoted to original mathematical recreations.","comment":"A monthly recreational maths problem, stretching back to 1998.","date_added":"2017-06-21","date_published":"1998-08-06","urls":["https:\/\/web.archive.org\/web\/20200706040018\/http:\/\/www2.stetson.edu\/~efriedma\/mathmagic\/","http:\/\/www2.stetson.edu\/~efriedma\/mathmagic\/"],"collections":"puzzles","url":"https:\/\/web.archive.org\/web\/20200706040018\/http:\/\/www2.stetson.edu\/~efriedma\/mathmagic\/ http:\/\/www2.stetson.edu\/~efriedma\/mathmagic\/","urldate":"2017-06-21","year":"1998"},{"key":"Subshoot","type":"article","title":"Sub shoot!","author":"Kleber, Michael","abstract":"By day, Leonidas Kontothanassis works as my colleague at Google. By night, he runs a gaudy carnival booth on the boardwalk outside of\r\ntown. \"Step right up and try your luck! Shoot the sub and win a prize,\" Leonidas was calling out one fine Fall evening. He was standing in the middle of the booth, surrounded by a moat packed with plastic toy submarines. The subs were of every shape and size, and they circled the moat at all different speeds, propelled by a complex system of currents.","comment":"A puzzle similar to the \"Princess in a castle\" puzzle.","date_added":"2023-01-08","date_published":"2008-08-06","urls":["https:\/\/link.springer.com\/article\/10.1007\/BF03038093","https:\/\/link.springer.com\/content\/pdf\/10.1007\/BF03038093.pdf"],"collections":"puzzles","url":"https:\/\/link.springer.com\/article\/10.1007\/BF03038093 https:\/\/link.springer.com\/content\/pdf\/10.1007\/BF03038093.pdf","year":"2008","urldate":"2023-01-08","publisher":"Springer-Verlag","fulltext_html_url":"https:\/\/link.springer.com\/article\/10.1007\/BF03038093","journal":"The Mathematical Intelligencer","volume":"30","issue":"4","identifier":"doi:10.1007\/BF03038093","doi":"10.1007\/BF03038093","pages":"26-30"},{"key":"ThePackingChromaticNumberoftheInfiniteSquareGridis15","type":"article","title":"The Packing Chromatic Number of the Infinite Square Grid is 15","author":"Bernardo Subercaseaux and Marijn J. H. Heule","abstract":"A packing $k$-coloring is a natural variation on the standard notion of graph\r\n$k$-coloring, where vertices are assigned numbers from $\\{1, \\ldots, k\\}$, and\r\nany two vertices assigned a common color $c \\in \\{1, \\ldots, k\\}$ need to be at\r\na distance greater than $c$ (as opposed to $1$, in standard graph colorings).\r\nDespite a sequence of incremental work, determining the packing chromatic\r\nnumber of the infinite square grid has remained an open problem since its\r\nintroduction in 2002. We culminate the search by proving this number to be 15.\r\nWe achieve this result by improving the best-known method for this problem by\r\nroughly two orders of magnitude. The most important technique to boost\r\nperformance is a novel, surprisingly effective propositional encoding for\r\npacking colorings. Additionally, we developed an alternative symmetry-breaking\r\nmethod. Since both new techniques are more complex than existing techniques for\r\nthis problem, a verified approach is required to trust them. We include both\r\ntechniques in a proof of unsatisfiability, reducing the trusted core to the\r\ncorrectness of the direct encoding.","comment":"","date_added":"2023-01-26","date_published":"2023-08-06","urls":["http:\/\/arxiv.org\/abs\/2301.09757v1","http:\/\/arxiv.org\/pdf\/2301.09757v1"],"collections":"combinatorics,fun-maths-facts","url":"http:\/\/arxiv.org\/abs\/2301.09757v1 http:\/\/arxiv.org\/pdf\/2301.09757v1","year":"2023","urldate":"2023-01-26","archivePrefix":"arXiv","eprint":"2301.09757","primaryClass":"cs.DM"},{"key":"HedraZoo","type":"online","title":"Hedra Zoo","author":"Stefan Forcey","abstract":"Encyclopedia of Combinatorial Polytope Sequences","comment":"","date_added":"2023-02-02","date_published":"2022-08-06","urls":["https:\/\/sforcey.github.io\/sf34\/hedra.htm"],"collections":"geometry,lists-and-catalogues","url":"https:\/\/sforcey.github.io\/sf34\/hedra.htm","year":"2022","urldate":"2023-02-02"},{"key":"item31","type":"online","title":"Pascal's Pyramid Or Pascal's Tetrahedron","author":"Jim Nugent","abstract":"A lattice of octahedra and tetrahedra (oct-tet lattice) is a useful paradigm for understanding the structure of Pascal's pyramid, the 3-D analog of Pascal's triangle. Notation for levels and coordinates of elements, a standard algorithm for generating the values of various elements, and a ratio method that is not dependent on the calculation of previous levels are discussed. Figures show a bell curve in 3 dimensions, the association of elements to primes and twin primes, and the values of elements mod(x) through patterns arranged in triangular plots. It is conjectured that the largest factor of any element is less than the level index.","comment":"","date_added":"2013-06-25","date_published":"1990-08-06","urls":["https:\/\/web.archive.org\/web\/20160410142410\/http:\/\/buckydome.com\/math\/Article2.htm","http:\/\/buckydome.com\/math\/Article2.htm"],"collections":"easily-explained","url":"https:\/\/web.archive.org\/web\/20160410142410\/http:\/\/buckydome.com\/math\/Article2.htm http:\/\/buckydome.com\/math\/Article2.htm","urldate":"2013-06-25","year":"1990","keywords":"3-D,Galton Board,JimNugent,Pascal,Pascal's tetrahedron,Pascal's triangle,Sierpinski,Stephen Mueller,True BASIC,bell curve,binomial,expansion,geometry,mathematics,oct-tet,octahedron,prime numbers,taxicab,taxicab geometry,tetrahedron,three dimensional,trinomial,twin prime,twin primes"},{"key":"almanachoudictionnairedesnombrescuriositsetproprits","type":"online","title":"almanach ou dictionnaire des nombres - curiosit\u00e9s et propri\u00e9t\u00e9s","author":"G\u00e9rard Villemin","abstract":"","comment":"","date_added":"2018-03-05","date_published":null,"urls":["http:\/\/web.archive.org\/web\/20190510154629\/http:\/\/yoda.guillaume.pagesperso-orange.fr\/index.html","http:\/\/yoda.guillaume.pagesperso-orange.fr\/index.htm"],"collections":"integerology,lists-and-catalogues","url":"http:\/\/web.archive.org\/web\/20190510154629\/http:\/\/yoda.guillaume.pagesperso-orange.fr\/index.html http:\/\/yoda.guillaume.pagesperso-orange.fr\/index.htm","urldate":"2018-03-05","year":""},{"key":"AcuriousresultrelatedtoKempnersseries","type":"article","title":"A curious result related to Kempner's series","author":"Bakir Farhi","abstract":"It is well known since A. J. Kempner's work that the series of the\r\nreciprocals of the positive integers whose the decimal representation does not\r\ncontain any digit 9, is convergent. This result was extended by F. Irwin and\r\nothers to deal with the series of the reciprocals of the positive integers\r\nwhose the decimal representation contains only a limited quantity of each digit\r\nof a given nonempty set of digits. Actually, such series are known to be all\r\nconvergent. Here, letting $S^{(r)}$ $(r \\in \\mathbb{N})$ denote the series of\r\nthe reciprocal of the positive integers whose the decimal representation\r\ncontains the digit 9 exactly $r$ times, the impressive obtained result is that\r\n$S^{(r)}$ tends to $10 \\log{10}$ as $r$ tends to infinity!","comment":"","date_added":"2023-02-27","date_published":"2008-08-06","urls":["http:\/\/arxiv.org\/abs\/0807.3518v1","http:\/\/arxiv.org\/pdf\/0807.3518v1"],"collections":"","url":"http:\/\/arxiv.org\/abs\/0807.3518v1 http:\/\/arxiv.org\/pdf\/0807.3518v1","year":"2008","urldate":"2023-02-27","archivePrefix":"arXiv","eprint":"0807.3518","primaryClass":"math.NT"},{"key":"ErichsPackingCenter","type":"online","title":"Erich's Packing Center","author":"Erich Friedman","abstract":"","comment":"A collection of best-known solutions to packing problems.","date_added":"2023-03-20","date_published":"2020-08-06","urls":["https:\/\/erich-friedman.github.io\/packing\/index.html"],"collections":"geometry,lists-and-catalogues","url":"https:\/\/erich-friedman.github.io\/packing\/index.html","year":"2020","urldate":"2023-03-20"},{"key":"SolvingRushHour","type":"article","title":"Solving Rush Hour, the Puzzle","author":"Michael Fogleman","abstract":"\r\n\r\nRush Hour is a 6x6 sliding block puzzle invented by Nob Yoshigahara in the 1970s. It was first sold in the United States in 1996.\r\n\r\nI played a clone of this game on my first iPhone several years ago. Recently, I stumbled on the physical incarnation of it and instantly bought it on Amazon for my kids to play. We've been having fun with it, but naturally I was most interested in writing some code to solve the puzzles.\r\n\r\nAfter writing a solver, I wrote a puzzle generator that would create more starting positions for us to try (the game comes with 40 levels printed on playing cards). The generator used simulated annealing to try to maximize the number of moves required to solve the puzzle.\r\n\r\nUnsatisfied with the results, I then decided to try generating all possible puzzles. Ultimately I ended up with a complete database of every \"interesting\" starting position. It was quite challenging (and exciting!) and that's what I want to talk about in this article. My code is open source with a permissive license and the resulting database is available for download.","comment":"","date_added":"2023-03-22","date_published":"2018-08-06","urls":["https:\/\/www.michaelfogleman.com\/rush\/"],"collections":"basically-computer-science,combinatorics,puzzles","url":"https:\/\/www.michaelfogleman.com\/rush\/","year":"2018","urldate":"2023-03-22"},{"key":"FoldingsandMeanders","type":"article","title":"Foldings and Meanders","author":"St\u00e9phane Legendre","abstract":"We review the stamp folding problem, the number of ways to fold a strip of\r\n$n$ stamps, and the related problem of enumerating meander configurations. The\r\nstudy of equivalence classes of foldings and meanders under symmetries allows\r\nto characterize and enumerate folding and meander shapes. Symmetric foldings\r\nand meanders are described, and relations between folding and meandric\r\nsequences are given. Extended tables for these sequences are provided.","comment":"","date_added":"2023-03-22","date_published":"2013-08-06","urls":["http:\/\/arxiv.org\/abs\/1302.2025v1","http:\/\/arxiv.org\/pdf\/1302.2025v1"],"collections":"combinatorics,easily-explained,puzzles","url":"http:\/\/arxiv.org\/abs\/1302.2025v1 http:\/\/arxiv.org\/pdf\/1302.2025v1","year":"2013","urldate":"2023-03-22","archivePrefix":"arXiv","eprint":"1302.2025","primaryClass":"math.CO"},{"key":"MetaNumbers","type":"online","title":"MetaNumbers - Number encyclopedia","author":"MetaNumbers","abstract":"MetaNumbers is a free math tool providing information about any positive integer (up to 9223372036854775807), such as its factorized form, its divisors, its classification, or its arithmetic properties (widely used in the field of number theory). ","comment":"","date_added":"2023-03-22","date_published":"2019-08-06","urls":["https:\/\/metanumbers.com\/"],"collections":"integerology,lists-and-catalogues","url":"https:\/\/metanumbers.com\/","year":"2019","urldate":"2023-03-22"},{"key":"FuzzyGeometry","type":"article","title":"Fuzzy plane geometry I: Points and lines ","author":"J.J. Buckley and E. Aslami","abstract":"We introduce a comprehensive study of fuzzy geometry in this paper by first defining a fuzzy point and a fuzzy line\r\nin fuzzy plane geometry. We consider the fuzzy distance between fuzzy points and show it is a (weak) fuzzy metric.\r\nWe study various definitions of a fuzzy line, develop their basic properties, and investigate parallel fuzzy lines. ","comment":"","date_added":"2016-06-17","date_published":"1997-08-06","urls":["https:\/\/www.sciencedirect.com\/science\/article\/abs\/pii\/0165011495003428"],"collections":"geometry,probability-and-statistics","url":"https:\/\/www.sciencedirect.com\/science\/article\/abs\/pii\/0165011495003428","urldate":"2016-06-17","year":"1997"},{"key":"WallpaperFunctions","type":"article","title":"Wallpaper Functions","author":"Frank A. Farris and Rima Lanning","abstract":"Instead of making wallpaper by repeating copies of a motif, we construct wallpaper functions. These are functions on \\(\\mathbb{R}^2\\) that are invariant under the action of one of the 17 planar crystallographic groups. We also construct functions with antisymmetries, and offer a complete analysis of types. Techniques include exhibiting bases for various spaces of wallpaper functions, and an algebraic definition of equivalence of pattern type.","comment":"","date_added":"2023-06-20","date_published":"2002-08-06","urls":["https:\/\/core.ac.uk\/download\/pdf\/82036994.pdf","https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0723086902800196"],"collections":"art,easily-explained,fun-maths-facts,geometry,the-groups-group","url":"https:\/\/core.ac.uk\/download\/pdf\/82036994.pdf https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0723086902800196","urldate":"2023-06-20","year":"2002"},{"key":"Lang2000","type":"article","title":"Origami Burrs and Woven Polyhedra","author":"Lang, Robert J","abstract":"","comment":"","date_added":"2011-12-14","date_published":"2000-08-06","urls":["https:\/\/www.langorigami.com\/wp-content\/uploads\/2016\/02\/Polypolyhedra_part_1.pdf"],"collections":"things-to-make-and-do","url":"https:\/\/www.langorigami.com\/wp-content\/uploads\/2016\/02\/Polypolyhedra_part_1.pdf","urldate":"2011-12-14","year":"2000","pages":"1--26","volume":"4"},{"key":"Aludaat2008","type":"article","title":"A Note on Approximating the Normal Distribution Function","author":"Aludaat, K M and Alodat, M T","abstract":"","comment":"","date_added":"2011-03-28","date_published":"2008-08-06","urls":["http:\/\/www.m-hikari.com\/ams\/ams-password-2008\/ams-password9-12-2008\/alodatAMS9-12-2008-2.pdf"],"collections":"","url":"http:\/\/www.m-hikari.com\/ams\/ams-password-2008\/ams-password9-12-2008\/alodatAMS9-12-2008-2.pdf","urldate":"2011-03-28","year":"2008","keywords":"cumulative distribution function,normal distribution","number":"9","pages":"425 -- 429","volume":"2"},{"key":"Maximummutationalrobustnessingenotypephenotypemapsfollowsaselfsimilarblancmangelikecurve","type":"article","title":"Maximum mutational robustness in genotype\u2013phenotype maps follows a self-similar blancmange-like curve","author":"Vaibhav Mohanty , Sam F. Greenbury , Tasmin Sarkany , Shyam Narayanan , Kamaludin Dingle , Sebastian E. Ahnert, Ard A. Louis","abstract":"Phenotype robustness, defined as the average mutational robustness of all the genotypes that map to a given phenotype, plays a key role in facilitating neutral exploration of novel phenotypic variation by an evolving population. By applying results from coding theory, we prove that the maximum phenotype robustness occurs when genotypes are organized as bricklayer\u2019s graphs, so-called because they resemble the way in which a bricklayer would fill in a Hamming graph. The value of the maximal robustness is given by a fractal continuous everywhere but differentiable nowhere sums-of-digits function from number theory. Interestingly, genotype\u2013phenotype maps for RNA secondary structure and the hydrophobic-polar (HP) model for protein folding can exhibit phenotype robustness that exactly attains this upper bound. By exploiting properties of the sums-of-digits function, we prove a lower bound on the deviation of the maximum robustness of phenotypes with multiple neutral components from the bricklayer\u2019s graph bound, and show that RNA secondary structure phenotypes obey this bound. Finally, we show how robustness changes when phenotypes are coarse-grained and derive a formula and associated bounds for the transition probabilities between such phenotypes.","comment":"","date_added":"2023-08-02","date_published":"2023-08-06","urls":["https:\/\/royalsocietypublishing.org\/doi\/10.1098\/rsif.2023.0169"],"collections":"food,unusual-arithmetic","url":"https:\/\/royalsocietypublishing.org\/doi\/10.1098\/rsif.2023.0169","urldate":"2023-08-02","year":"2023"},{"key":"DistantdecimalsofPi","type":"article","title":"Distant decimals of $\u03c0$","author":"Yves Bertot and Laurence Rideau and Laurent Th\u00e9ry","abstract":"We describe how to compute very far decimals of \\(\\pi\\) and how to provide\r\nformal guarantees that the decimals we compute are correct. In particular, we\r\nreport on an experiment where 1 million decimals of \\(\\pi\\) and the billionth\r\nhexadecimal (without the preceding ones) have been computed in a formally\r\nverified way. Three methods have been studied, the first one relying on a\r\nspigot formula to obtain at a reasonable cost only one distant digit (more\r\nprecisely a hexadecimal digit, because the numeration basis is 16) and the\r\nother two relying on arithmetic-geometric means. All proofs and computations\r\ncan be made inside the Coq system. We detail the new formalized material that\r\nwas necessary for this achievement and the techniques employed to guarantee the\r\naccuracy of the computed digits, in spite of the necessity to work with fixed\r\nprecision numerical computation.","comment":"","date_added":"2023-08-14","date_published":"2017-08-06","urls":["http:\/\/arxiv.org\/abs\/1709.01743v2","http:\/\/arxiv.org\/pdf\/1709.01743v2"],"collections":"basically-computer-science,fun-maths-facts","url":"http:\/\/arxiv.org\/abs\/1709.01743v2 http:\/\/arxiv.org\/pdf\/1709.01743v2","year":"2017","urldate":"2023-08-14","archivePrefix":"arXiv","eprint":"1709.01743","primaryClass":"cs.LO"},{"key":"HyperbolicMinesweeperisinP","type":"article","title":"Hyperbolic Minesweeper is in P","author":"Eryk Kopczy\u0144ski","abstract":"We show that, while Minesweeper is NP-complete, its hyperbolic variant is in\r\nP. Our proof does not rely on the rules of Minesweeper, but is valid for any\r\npuzzle based on satisfying local constraints on a graph embedded in the\r\nhyperbolic plane.","comment":"","date_added":"2023-08-14","date_published":"2020-08-06","urls":["http:\/\/arxiv.org\/abs\/2002.09534v2","http:\/\/arxiv.org\/pdf\/2002.09534v2"],"collections":"computational-complexity-of-games","url":"http:\/\/arxiv.org\/abs\/2002.09534v2 http:\/\/arxiv.org\/pdf\/2002.09534v2","year":"2020","urldate":"2023-08-14","archivePrefix":"arXiv","eprint":"2002.09534","primaryClass":"cs.CC"},{"key":"Asignthatusedtoannoymeandstilldoes","type":"article","title":"A sign that used to annoy me, and still does","author":"Andrea T. Ricolfi","abstract":"We provide a proof of the following fact: if a complex scheme $Y$ has Behrend\r\nfunction constantly equal to a sign $\\sigma \\in \\{\\pm 1\\}$, then all of its\r\ncomponents $Z \\subset Y$ are generically reduced and satisfy\r\n$(-1)^{\\mathrm{dim}_{\\mathbb C} T_pY} = \\sigma = (-1)^{\\mathrm{dim}Z}$ for $p\r\n\\in Z$ a general point. Given the recent counterexamples to the parity\r\nconjecture for the Hilbert scheme of points $\\mathrm{Hilb}^n(\\mathbb A^3)$, our\r\nargument suggests a possible path to disprove the constancy of the Behrend\r\nfunction of $\\mathrm{Hilb}^n(\\mathbb A^3)$.","comment":"","date_added":"2023-08-14","date_published":"2023-08-06","urls":["http:\/\/arxiv.org\/abs\/2306.08457v1","http:\/\/arxiv.org\/pdf\/2306.08457v1"],"collections":"attention-grabbing-titles","url":"http:\/\/arxiv.org\/abs\/2306.08457v1 http:\/\/arxiv.org\/pdf\/2306.08457v1","year":"2023","urldate":"2023-08-14","archivePrefix":"arXiv","eprint":"2306.08457","primaryClass":"math.AG"},{"key":"LyonsTaming","type":"article","title":"Lyons Taming","author":"Wolfram Neutsch","abstract":"Based on Kantor's geometry, we give a new Highly symmetric construction of\r\nLyons' sporadic simple group $Ly$ via its minimal representation over $\\mathbb\r\nF_5^{111}$, thus obtaining elementary existence proofs for both the group and\r\nthe representation at one stroke.","comment":"","date_added":"2023-08-14","date_published":"2023-08-06","urls":["http:\/\/arxiv.org\/abs\/2307.11399v1","http:\/\/arxiv.org\/pdf\/2307.11399v1"],"collections":"animals,the-groups-group","url":"http:\/\/arxiv.org\/abs\/2307.11399v1 http:\/\/arxiv.org\/pdf\/2307.11399v1","year":"2023","urldate":"2023-08-14","archivePrefix":"arXiv","eprint":"2307.11399","primaryClass":"math.GR"},{"key":"asgunziMathproofscardboardpaperMathematicalartwithpaperandcardboard","type":"article","title":"Mathematical proofs with cardboard and paper","author":"Arnaldo Gunzi and Ern\u00e9e Kozyroff Filho","abstract":"In this tutorial, we'll show visual mathematical proofs using cardboard and paper. This ludic technique is especially good for children. These math proofs are like puzzles, and, best of all, you can do it yourself, at home.","comment":"","date_added":"2023-08-24","date_published":"2023-08-06","urls":["https:\/\/github.com\/asgunzi\/Math_proofs_cardboard_paper"],"collections":"about-proof,things-to-make-and-do","url":"https:\/\/github.com\/asgunzi\/Math_proofs_cardboard_paper","year":"2023","urldate":"2023-08-24"},{"key":"item2","type":"article","title":"Digital halftoning with space filling curves","author":"Luiz Velho and Jonas de Miranda Gomes","abstract":"This paper introduces a new digital halftoning technique that uses space filling curves to generate aperiodic patterns of clustered dots. This method allows the parameterization of the size of pixel clusters, which can vary in one pixel steps. The algorithm unifies, in this way, the dispersed and clustered-dot dithering techniques.","comment":"","date_added":"2010-08-31","date_published":"1991-08-06","urls":["https:\/\/dl.acm.org\/doi\/abs\/10.1145\/122718.122727"],"collections":"basically-computer-science","url":"https:\/\/dl.acm.org\/doi\/abs\/10.1145\/122718.122727","urldate":"2010-08-31","year":"1991"},{"key":"ConwayandDoyleCanDividebyThreeButICant","type":"article","title":"Conway and Doyle Can Divide by Three, But I Can't","author":"Patrick Lutz","abstract":"Conway and Doyle have claimed to be able to divide by three. We attempt to\r\nreplicate their achievement and fail. In the process, we get tangled up in some\r\nshoes and socks and forget how to multiply.","comment":"","date_added":"2023-09-29","date_published":"2023-08-06","urls":["http:\/\/arxiv.org\/abs\/2309.11634v1","http:\/\/arxiv.org\/pdf\/2309.11634v1"],"collections":"attention-grabbing-titles,fun-maths-facts","url":"http:\/\/arxiv.org\/abs\/2309.11634v1 http:\/\/arxiv.org\/pdf\/2309.11634v1","year":"2023","urldate":"2023-09-29","archivePrefix":"arXiv","eprint":"2309.11634","primaryClass":"math.LO"},{"key":"Ropesfractionsandmodulispaces","type":"article","title":"Ropes, fractions, and moduli spaces","author":"Nick Salter","abstract":"This is an exposition of John H. Conway's tangle trick. We discuss what the\r\ntrick is, how to perform it, why it works mathematically, and finally offer a\r\nconceptual explanation for why a trick like this should exist in the first\r\nplace. The mathematical centerpiece is the relationship between braids on three\r\nstrands and elliptic curves, and we a draw a line from the tangle trick back to\r\nwork of Weierstrass, Abel, and Jacobi in the 19th century. For the most part we\r\nassume only a familiarity with the language of group actions, but some prior\r\nexposure to the fundamental group is beneficial in places.","comment":"","date_added":"2023-09-29","date_published":"2023-08-06","urls":["http:\/\/arxiv.org\/abs\/2309.11311v2","http:\/\/arxiv.org\/pdf\/2309.11311v2"],"collections":"games-to-play-with-friends,the-groups-group,things-to-make-and-do","url":"http:\/\/arxiv.org\/abs\/2309.11311v2 http:\/\/arxiv.org\/pdf\/2309.11311v2","year":"2023","urldate":"2023-09-29","archivePrefix":"arXiv","eprint":"2309.11311","primaryClass":"math.HO"},{"key":"FlatorigamiisTuringComplete","type":"article","title":"Flat origami is Turing Complete","author":"Thomas C. Hull and Inna Zakharevich","abstract":"Flat origami refers to the folding of flat, zero-curvature paper such that\r\nthe finished object lies in a plane. Mathematically, flat origami consists of a\r\ncontinuous, piecewise isometric map $f:P\\subseteq\\mathbb{R}^2\\to\\mathbb{R}^2$\r\nalong with a layer ordering $\\lambda_f:P\\times P\\to \\{-1,1\\}$ that tracks which\r\npoints of $P$ are above\/below others when folded. The set of crease lines that\r\na flat origami makes (i.e., the set on which the mapping $f$ is\r\nnon-differentiable) is called its \\textit{crease pattern}. Flat origami\r\nmappings and their layer orderings can possess surprisingly intricate\r\nstructure. For instance, determining whether or not a given straight-line\r\nplanar graph drawn on $P$ is the crease pattern for some flat origami has been\r\nshown to be an NP-complete problem, and this result from 1996 led to numerous\r\nexplorations in computational aspects of flat origami. In this paper we prove\r\nthat flat origami, when viewed as a computational device, is Turing complete.\r\nWe do this by showing that flat origami crease patterns with \\textit{optional\r\ncreases} (creases that might be folded or remain unfolded depending on\r\nconstraints imposed by other creases or inputs) can be constructed to simulate\r\nRule 110, a one-dimensional cellular automaton that was proven to be Turing\r\ncomplete by Matthew Cook in 2004.","comment":"","date_added":"2023-09-29","date_published":"2023-08-06","urls":["http:\/\/arxiv.org\/abs\/2309.07932v1","http:\/\/arxiv.org\/pdf\/2309.07932v1"],"collections":"basically-computer-science,computational-complexity-of-games,unusual-computers","url":"http:\/\/arxiv.org\/abs\/2309.07932v1 http:\/\/arxiv.org\/pdf\/2309.07932v1","year":"2023","urldate":"2023-09-29","archivePrefix":"arXiv","eprint":"2309.07932","primaryClass":"math.CO"},{"key":"WhenCanYouTileanIntegerRectanglewithIntegerSquares","type":"article","title":"When Can You Tile an Integer Rectangle with Integer Squares?","author":"MIT CompGeom Group and Zachary Abel and Hugo A. Akitaya and Erik D. Demaine and Adam C. Hesterberg and Jayson Lynch","abstract":"This paper characterizes when an $m \\times n$ rectangle, where $m$ and $n$\r\nare integers, can be tiled (exactly packed) by squares where each has an\r\ninteger side length of at least 2. In particular, we prove that tiling is\r\nalways possible when both $m$ and $n$ are sufficiently large (at least 10).\r\nWhen one dimension $m$ is small, the behavior is eventually periodic in $n$\r\nwith period 1, 2, or 3. When both dimensions $m,n$ are small, the behavior is\r\ndetermined computationally by an exhaustive search.","comment":"","date_added":"2023-10-09","date_published":"2023-08-06","urls":["http:\/\/arxiv.org\/abs\/2308.15317v1","http:\/\/arxiv.org\/pdf\/2308.15317v1"],"collections":"easily-explained,fun-maths-facts,geometry,integerology","url":"http:\/\/arxiv.org\/abs\/2308.15317v1 http:\/\/arxiv.org\/pdf\/2308.15317v1","year":"2023","urldate":"2023-10-09","archivePrefix":"arXiv","eprint":"2308.15317","primaryClass":"cs.CG"},{"key":"DubiousIdentitiesAVisittotheBorweinZoo","type":"article","title":"Dubious Identities: A Visit to the Borwein Zoo","author":"Zachary P. Bradshaw and Christophe Vignat","abstract":"We contribute to the zoo of dubious identities established by J.M. and P.B.\r\nBorwein in their 1992 paper, \"Strange Series and High Precision Fraud\" with\r\nfive new entries, each of a different variety than the last. Some of these\r\nidentities are again a high precision fraud and picking out the true from the\r\nbogus can be a challenging task with many unexpected twists along the way.","comment":"","date_added":"2023-10-09","date_published":"2023-08-06","urls":["http:\/\/arxiv.org\/abs\/2307.05565v1","http:\/\/arxiv.org\/pdf\/2307.05565v1"],"collections":"attention-grabbing-titles","url":"http:\/\/arxiv.org\/abs\/2307.05565v1 http:\/\/arxiv.org\/pdf\/2307.05565v1","year":"2023","urldate":"2023-10-09","archivePrefix":"arXiv","eprint":"2307.05565","primaryClass":"math.HO"},{"key":"TheEuclideanAlgorithmGeneratesTraditionalMusicalRhythms","type":"article","title":"The Euclidean Algorithm Generates Traditional Musical Rhythms","author":"Godfried Toussaint","abstract":"The Euclidean algorithm (which comes down to us from Euclid\u2019s Elements) computes the greatest common divisor of two given integers. It is shown here that the structure of the Euclidean algorithm may be used to automatically generate, very efficiently, a large family of rhythms used as timelines (rhythmic ostinatos), in traditional world music. These rhythms, here dubbed Euclidean rhythms, have the property that their onset patterns are distributed as evenly as possible in a mathematically precise sense, and optimal manner. Euclidean rhythms are closely related to the family of Aksak rhythms studied by ethnomusicologists, and occur in a wide variety of other disciplines as well. For example they characterize algorithms for drawing digital straight lines in computer graphics, as well as algorithms for calculating\r\nleap years in calendar design. Euclidean rhythms also find application in nuclear physics accelerators and in computer science, and are closely related to several families of words and sequences of interest\r\nin the study of the combinatorics of words, such as mechanical words, Sturmian words, two-distance\r\nsequences, and Euclidean strings, to which the Euclidean rhythms are compared.","comment":"","date_added":"2023-10-25","date_published":"2005-08-06","urls":["http:\/\/cgm.cs.mcgill.ca\/~godfried\/publications\/banff-extended.pdf"],"collections":"integerology,music","url":"http:\/\/cgm.cs.mcgill.ca\/~godfried\/publications\/banff-extended.pdf","year":"2005","urldate":"2023-10-25"},{"key":"Sorrythenilpotentsareinthecenter","type":"article","title":"Sorry, the nilpotents are in the center","author":"Vineeth Chintala","abstract":"The behavior of nilpotents can reveal valuable information about the algebra.\r\nWe give a simple proof of a classic result that a finite ring is commutative if\r\nall its nilpotents lie in the center.","comment":"","date_added":"2023-11-17","date_published":"2018-08-06","urls":["http:\/\/arxiv.org\/abs\/1805.11451v4","http:\/\/arxiv.org\/pdf\/1805.11451v4"],"collections":"attention-grabbing-titles","url":"http:\/\/arxiv.org\/abs\/1805.11451v4 http:\/\/arxiv.org\/pdf\/1805.11451v4","year":"2018","urldate":"2023-11-17","archivePrefix":"arXiv","eprint":"1805.11451","primaryClass":"math.RA"},{"key":"ContinuedLogarithms","type":"article","title":"Continued Logarithms And Associated Continued Fractions","author":"Jonathan M. Borwein and Neil J. Calkin and Scott B. Lindstrom and Andrew Mattingly","abstract":"We investigate some of the connections between continued fractions and continued logarithms. We study the binary continued logarithms as introduced by Bill Gosper and explore two generalizations of the continued logarithm to base $b$. We show convergence for them using equivalent forms of their corresponding continued fractions. Through numerical experimentation we discover that, for one such formulation, the exponent terms have finite arithmetic means for almost all real numbers. This set of means, which we call the logarithmic Khintchine numbers, has a pleasing relationship with the geometric means of the corresponding continued fraction terms. While the classical Khintchine\u2019s constant is believed not to be related to any naturally occurring number, we find surprisingly that the logarithmic Khintchine numbers are elementary.","comment":"","date_added":"2016-06-15","date_published":"2016-08-06","urls":["https:\/\/carmamaths.org\/resources\/jon\/clogs.pdf","https:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/10586458.2016.1195307"],"collections":"","url":"https:\/\/carmamaths.org\/resources\/jon\/clogs.pdf https:\/\/www.tandfonline.com\/doi\/abs\/10.1080\/10586458.2016.1195307","urldate":"2016-06-15","year":"2016"},{"key":"MathBases","type":"online","title":"MathBases","author":"Katja Ber\u010di\u010d","abstract":"MathBases.org brings together a searchable index of existing mathematical databases and resources for mathematicians interested in creating new mathematical databases.","comment":"","date_added":"2023-12-09","date_published":"2023-08-06","urls":["https:\/\/mathbases.org\/"],"collections":"lists-and-catalogues","url":"https:\/\/mathbases.org\/","urldate":"2023-12-09","year":"2023"},{"key":"TheSineOfASingleDegree","type":"article","title":"The Sine of a Single Degree","author":"Travis Kowalski","abstract":"Ostensibly a derivation of an algebraically exact formula for the value of the sine of 1 degree, we present this calculation as a \u201chistorical romp\u201d looking at the problem through the tools of geometry, then algebra, and finally complex analysis. Each one of these approaches gets the reader nearer to the correct value, but also serves to frame a vignette of surprising or beautiful mathematics.","comment":"","date_added":"2024-01-15","date_published":"2017-08-06","urls":["https:\/\/maa.org\/sites\/default\/files\/pdf\/awards\/college.math.j.47.5.322.pdf","https:\/\/www.tandfonline.com\/doi\/abs\/10.4169\/college.math.j.47.5.322"],"collections":"easily-explained,fun-maths-facts,geometry","url":"https:\/\/maa.org\/sites\/default\/files\/pdf\/awards\/college.math.j.47.5.322.pdf https:\/\/www.tandfonline.com\/doi\/abs\/10.4169\/college.math.j.47.5.322","year":"2017","urldate":"2024-01-15"},{"key":"Howbigatabledoyouneedforyourjigsawpuzzle","type":"article","title":"How big a table do you need for your jigsaw puzzle?","author":"Madeleine Bonsma-Fisher and Kent Bonsma-Fisher","abstract":"Jigsaw puzzles are typically labeled with their finished area and number of\r\npieces. With this information, is it possible to estimate the area required to\r\nlay each piece flat before assembly? We derive a simple formula based on\r\ntwo-dimensional circular packing and show that the unassembled puzzle area is\r\n$\\sqrt{3}$ times the assembled puzzle area, independent of the number of\r\npieces. We perform measurements on 9 puzzles ranging from 333 cm$^2$ (9 pieces)\r\nto 6798 cm$^2$ (2000 pieces) and show that the formula accurately predicts\r\nrealistic assembly scenarios.","comment":"","date_added":"2024-01-15","date_published":"2023-08-06","urls":["http:\/\/arxiv.org\/abs\/2312.04588v1","http:\/\/arxiv.org\/pdf\/2312.04588v1"],"collections":"easily-explained,fun-maths-facts,geometry","url":"http:\/\/arxiv.org\/abs\/2312.04588v1 http:\/\/arxiv.org\/pdf\/2312.04588v1","year":"2023","urldate":"2024-01-15","archivePrefix":"arXiv","eprint":"2312.04588","primaryClass":"math.HO"},{"key":"MartinGardnercolumntobookmappingproject","type":"online","title":"Martin Gardner column to book mapping project","author":"Peter Rowlett","abstract":"A table mapping each column written by Martin Gardner in Scientific American to the book it's reproduced in.","comment":"","date_added":"2022-09-08","date_published":"2022-08-06","urls":["https:\/\/peterrowlett.net\/gardner-index\/"],"collections":"history,lists-and-catalogues","url":"https:\/\/peterrowlett.net\/gardner-index\/","urldate":"2022-09-08","year":"2022"},{"key":"MyFavoriteMathJokes","type":"article","title":"My Favorite Math Jokes","author":"Tanya Khovanova","abstract":"For many years, I have been collecting math jokes and posting them on my\r\nwebsite. I have more than 400 jokes there. In this paper, which is an extended\r\nversion of my talk at the G4G15, I would like to present 66 of them.","comment":"","date_added":"2024-03-06","date_published":"2024-08-06","urls":["http:\/\/arxiv.org\/abs\/2403.01010v1","http:\/\/arxiv.org\/pdf\/2403.01010v1"],"collections":"lists-and-catalogues","url":"http:\/\/arxiv.org\/abs\/2403.01010v1 http:\/\/arxiv.org\/pdf\/2403.01010v1","year":"2024","urldate":"2024-03-06","archivePrefix":"arXiv","eprint":"2403.01010","primaryClass":"math.HO"},{"key":"TheFlowerCalculus","type":"article","title":"The Flower Calculus","author":"Pablo Donato","abstract":"We introduce the flower calculus, a deep inference proof system for intuitionistic first-order logic inspired by Peirce's existential graphs. It works as a rewriting system over inductive objects called \"flowers\", that enjoy both a graphical interpretation as topological diagrams, and a textual presentation as nested sequents akin to coherent formulas. Importantly, the calculus dispenses completely with the traditional notion of symbolic connective, operating solely on nested flowers containing atomic predicates. We prove both the soundness of the full calculus and the completeness of an analytic fragment with respect to Kripke semantics. This provides to our knowledge the first analyticity result for a proof system based on existential graphs, adapting semantic cut-elimination techniques to a deep inference setting. Furthermore, the kernel of rules targetted by completeness is fully invertible, a desirable property for both automated and interactive proof search.","comment":"","date_added":"2024-03-11","date_published":"2024-08-06","urls":["https:\/\/hal.science\/hal-04472717","https:\/\/hal.science\/hal-04472717\/document"],"collections":"attention-grabbing-titles","url":"https:\/\/hal.science\/hal-04472717 https:\/\/hal.science\/hal-04472717\/document","year":"2024","urldate":"2024-03-11"},{"key":"RandomFormulaGenerators","type":"article","title":"Random Formula Generators","author":"Ariel J. Roffe and Joaquin S. Toranzo Calderon","abstract":"In this article, we provide three generators of propositional formulae for\r\narbitrary languages, which uniformly sample three different formulae spaces.\r\nThey take the same three parameters as input, namely, a desired depth, a set of\r\natomics and a set of logical constants (with specified arities). The first\r\ngenerator returns formulae of exactly the given depth, using all or some of the\r\npropositional letters. The second does the same but samples up-to the given\r\ndepth. The third generator outputs formulae with exactly the desired depth and\r\nall the atomics in the set. To make the generators uniform (i.e. to make them\r\nreturn every formula in their space with the same probability), we will prove\r\nvarious cardinality results about those spaces.","comment":"","date_added":"2024-03-11","date_published":"2021-08-06","urls":["http:\/\/arxiv.org\/abs\/2110.09228v1","http:\/\/arxiv.org\/pdf\/2110.09228v1"],"collections":"basically-computer-science","url":"http:\/\/arxiv.org\/abs\/2110.09228v1 http:\/\/arxiv.org\/pdf\/2110.09228v1","year":"2021","urldate":"2024-03-11","archivePrefix":"arXiv","eprint":"2110.09228","primaryClass":"cs.LO"},{"key":"TheFlappingBirdsinthePentagramZoo","type":"article","title":"The Flapping Birds in the Pentagram Zoo","author":"Richard Evan Schwartz","abstract":"We study the $(k+1,k)$ diagonal map for $k=2,3,4,...$. We call this map\r\n$\\Delta_k$. The map $\\Delta_1$ is the pentagram map and $\\Delta_k$ is a\r\ngeneralization. $\\Delta_k$ does not preserve convexity, but we prove that\r\n$\\Delta_k$ preserves a subset $B_k$ of certain star-shaped polygons which we\r\ncall $k$-birds. The action of $\\Delta_k$ on $B_k$ seems similar to the action\r\nof $\\Delta_1$ on the space of convex polygons. We show that some classic\r\ngeometric results about $\\Delta_1$ generalize to this setting.","comment":"","date_added":"2024-03-14","date_published":"2024-08-06","urls":["http:\/\/arxiv.org\/abs\/2403.05735v1","http:\/\/arxiv.org\/pdf\/2403.05735v1"],"collections":"animals","url":"http:\/\/arxiv.org\/abs\/2403.05735v1 http:\/\/arxiv.org\/pdf\/2403.05735v1","year":"2024","urldate":"2024-03-14","archivePrefix":"arXiv","eprint":"2403.05735","primaryClass":"math.DS"},{"key":"Asimplegroupoforder44352000","type":"article","title":"A simple group of order 44,352,000","author":"Higman, Donald G. and Sims, Charles C.","abstract":"The group \\(G\\) of the title is obtained as a primitive permutation group of degree 100 in which the stabilizer of a point has orbits of lengths 1, 22 and 77 and is isomorphic to the Mathieu group \\(M_{22}\\). Thus \\(G\\) has rank 3 in the sense\r\nof [1]. \\(G\\) is an automorphism group of a graph constructed from the Steiner system \\(\\mathfrak{S}(3, 6, 22)\\).","comment":"","date_added":"2024-04-02","date_published":"1968-08-06","urls":["https:\/\/deepblue.lib.umich.edu\/bitstream\/handle\/2027.42\/46258\/209_2005_Article_BF01110435.pdf","https:\/\/link.springer.com\/article\/10.1007\/BF01110435","https:\/\/link.springer.com\/content\/pdf\/10.1007\/BF01110435.pdf"],"collections":"the-groups-group","url":"https:\/\/deepblue.lib.umich.edu\/bitstream\/handle\/2027.42\/46258\/209_2005_Article_BF01110435.pdf https:\/\/link.springer.com\/article\/10.1007\/BF01110435 https:\/\/link.springer.com\/content\/pdf\/10.1007\/BF01110435.pdf","urldate":"2024-04-02","year":"1968","publisher":"Springer-Verlag","fulltext_html_url":"https:\/\/link.springer.com\/article\/10.1007\/BF01110435","journal":"Mathematische Zeitschrift","issn":"1432-1823","volume":"105","issue":"2","identifier":"doi:10.1007\/BF01110435","doi":"10.1007\/BF01110435","pages":"110-113"},{"key":"SquigonometryTheStudyofImperfectCircles","type":"book","title":"Squigonometry: The Study of Imperfect Circles","author":" Robert D. Poodiack and William E. Wood","abstract":"This textbook introduces generalized trigonometric functions through the exploration of imperfect circles: curves defined by \\(|x|^p + |y|^p = 1\\) where \\(p \\geq 1\\). Grounded in visualization and computations, this accessible, modern perspective encompasses new and old results, casting a fresh light on duality, special functions, geometric curves, and differential equations. Projects and opportunities for research abound, as we explore how similar (or different) the trigonometric and squigonometric worlds might be.\r\n\r\nComprised of many short chapters, the book begins with core definitions and techniques. Successive chapters cover inverse squigonometric functions, the many possible re-interpretations of \u03c0, two deeper dives into parameterizing the squigonometric functions, and integration. Applications include a celebration of Piet Hein\u2019s work in design. From here, more technical pathways offer further exploration. Topics include infinite series; hyperbolic, exponential, and logarithmic functions; metrics and norms; and lemniscatic and elliptic functions. Illuminating illustrations accompany the text throughout, along with historical anecdotes, engaging exercises, and wry humor.\r\n\r\nSquigonometry: The Study of Imperfect Circles invites readers to extend familiar notions from trigonometry into a new setting. Ideal for an undergraduate reading course in mathematics or a senior capstone, this book offers scaffolding for active discovery. Knowledge of the trigonometric functions, single-variable calculus, and initial-value problems is assumed, while familiarity with multivariable calculus and linear algebra will allow additional insights into certain later material.","comment":"","date_added":"2024-04-10","date_published":"2022-08-06","urls":["https:\/\/link.springer.com\/book\/10.1007\/978-3-031-13783-9"],"collections":"attention-grabbing-titles,easily-explained,geometry","url":"https:\/\/link.springer.com\/book\/10.1007\/978-3-031-13783-9","urldate":"2024-04-10","year":"2022"},{"key":"TheCardboardComputer","type":"online","title":"The Cardboard Computer","author":"David Megginson","abstract":"The Cardboard Computer is a free (Public Domain) do-it-yourself circular slide rule that you can print out and put together at home or in a classroom. It's available in two versions: a basic version that can do basic ratios, multiplication, and division, and an advanced version that adds more features, including square and cube roots. ","comment":"","date_added":"2024-04-10","date_published":"2024-08-06","urls":["https:\/\/cardboard-computer.org\/"],"collections":"things-to-make-and-do,unusual-computers","url":"https:\/\/cardboard-computer.org\/","year":"2024","urldate":"2024-04-10"},{"key":"PolyamorousScheduling","type":"article","title":"Polyamorous Scheduling","author":"Leszek G\u0105sieniec and Benjamin Smith and Sebastian Wild","abstract":"Finding schedules for pairwise meetings between the members of a complex\r\nsocial group without creating interpersonal conflict is challenging, especially\r\nwhen different relationships have different needs. We formally define and study\r\nthe underlying optimisation problem: Polyamorous Scheduling.\r\n In Polyamorous Scheduling, we are given an edge-weighted graph and try to\r\nfind a periodic schedule of matchings in this graph such that the maximal\r\nweighted waiting time between consecutive occurrences of the same edge is\r\nminimised. We show that the problem is NP-hard and that there is no efficient\r\napproximation algorithm with a better ratio than 4\/3 unless P = NP. On the\r\npositive side, we obtain an $O(\\log n)$-approximation algorithm; indeed, a\r\n$O(\\log \\Delta)$-approximation for $\\Delta$ the maximum degree, i.e., the\r\nlargest number of relationships of any individual. We also define a\r\ngeneralisation of density from the Pinwheel Scheduling Problem, \"poly density\",\r\nand ask whether there exists a poly-density threshold similar to the\r\n5\/6-density threshold for Pinwheel Scheduling [Kawamura, STOC 2024].\r\nPolyamorous Scheduling is a natural generalisation of Pinwheel Scheduling with\r\nrespect to its optimisation variant, Bamboo Garden Trimming.\r\n Our work contributes the first nontrivial hardness-of-approximation reduction\r\nfor any periodic scheduling problem, and opens up numerous avenues for further\r\nstudy of Polyamorous Scheduling.","comment":"","date_added":"2024-04-10","date_published":"2024-08-06","urls":["http:\/\/arxiv.org\/abs\/2403.00465v2","http:\/\/arxiv.org\/pdf\/2403.00465v2"],"collections":"basically-computer-science","url":"http:\/\/arxiv.org\/abs\/2403.00465v2 http:\/\/arxiv.org\/pdf\/2403.00465v2","year":"2024","urldate":"2024-04-10","archivePrefix":"arXiv","eprint":"2403.00465","primaryClass":"cs.DS"},{"key":"Devlin","type":"article","title":"Good stories, pity they're not true","author":"Devlin, Keith","abstract":"The enormous success of Dan Brown\u2019s novel The Da Vinci Code has introduced the famous Golden Ratio (henceforth GR) to a whole new audience. Regular readers of this column will surely be familiar with the story. The ancient Greeks believed that there is a rectangle that the human eye finds the most pleasing, and that its aspect ratio is the positive root of the quadratic equation \\(x^2 \u2013 x \u2013 1 = 0\\).","comment":"","date_added":"2012-02-08","date_published":"2004-08-06","urls":["https:\/\/profkeithdevlin.org\/devlins-angle\/2004-posts\/#jun04"],"collections":"attention-grabbing-titles,drama,history","url":"https:\/\/profkeithdevlin.org\/devlins-angle\/2004-posts\/#jun04","urldate":"2012-02-08","year":"2004"},{"key":"OnlineMatchingPennies","type":"article","title":"Online Matching Pennies","author":"Olivier Gossner, Penelope Hernandez, Abraham Neyman","abstract":"We study a repeated game in which one player, the prophet, acquires more information than another player, the follower, about the play that is going to be played. We characterize the optimal amount of information that can be transmitted online by the prophet to the follower, and provide applications to repeated games played by finite automata, and by players with bounded recall.","comment":"","date_added":"2024-05-20","date_published":"2003-08-06","urls":["https:\/\/ratio.huji.ac.il\/publications\/online-matching-pennies","https:\/\/ratio.huji.ac.il\/files\/dp316.pdf"],"collections":"games-to-play-with-friends,protocols-and-strategies","url":"https:\/\/ratio.huji.ac.il\/publications\/online-matching-pennies https:\/\/ratio.huji.ac.il\/files\/dp316.pdf","year":"2003","urldate":"2024-05-20"},{"key":"DONUTDatabaseofOriginalNonTheoreticalUsesofTopology","type":"online","title":"DONUT: Database of Original & Non-Theoretical Uses of Topology","author":"Giunti, Barbara and Lazovskis, J\u0101nis and Rieck, Bastian","abstract":"This is a database of applications of Topological Data Analysis, an emerging mathematical paradigm for performing multi-scale analyses of complex data sets. ","comment":"","date_added":"2024-05-20","date_published":"2022-08-06","urls":["https:\/\/donut.topology.rocks\/"],"collections":"lists-and-catalogues","url":"https:\/\/donut.topology.rocks\/","year":"2022","urldate":"2024-05-20"},{"key":"ErdosProblems","type":"online","title":"Erd\u0151s Problems","author":"Thomas Bloom","abstract":"A collection of problems or conjectures posed by Paul Erd\u0151s.","comment":"","date_added":"2024-05-20","date_published":"2024-08-06","urls":["http:\/\/www.erdosproblems.com\/"],"collections":"lists-and-catalogues","url":"http:\/\/www.erdosproblems.com\/","year":"2024","urldate":"2024-05-20"},{"key":"Kamel1994","type":"article","title":"Hilbert R-tree: An improved R-tree using fractals","author":"Kamel, Ibrahim and Faloutsos, Christos","abstract":"We propose a new \\(\\mathbb{R}\\)-tree structure that outperforms all the older ones. The heart of the idea is to facilitate the deferred splitting approach in \\(\\mathbb{R}\\)-trees. The is done by proposing an ordering on the \\(\\mathbb{R}\\)-tree nodes. This ordering has to be `good', in the sense that it should group `similar' data rectangles together, to minimize the area and perimeter of the resulting minimum bounding rectangles (MBRs).\r\n\r\nFollowing [19], we have chosen the so-called `\"D-c' method, which sorts rectangles according to the Hilbert value of the center of the rectangles. Given the ordering, every node has a well defined set of sibling nodes; thus, we can use deferred splitting. By adjusting the split policy, the Hilbert \\(\\mathbb{R}\\)-tree can achieve as high utilization as desired. To the contrary, the \\(\\mathbb{R}^{\\ast}\\)-tree has no control over the space utilization, typically achieving up to 70%. We designed the manipulation algorithms in detail, and we did a full implementation of the the Hilbert \\(\\mathbb{R}\\)-tree. Our experiments show that the `2-to-3' split policy provides a compromise between the insertion complexity and the search cost, giving up to 28% savings over the \\(\\mathbb{R}^{\\ast}\\)-tree on real data.","comment":"An R-tree is a way of storing data corresponding to points in Euclidean space in a way that makes it easy to find all the elements in a given rectangle. This paper claims to give a version of an R-tree that's better than all the others.","date_added":"2010-08-31","date_published":"1994-08-06","urls":["https:\/\/cis.temple.edu\/~vasilis\/Courses\/CIS750\/Papers\/HilbertRtree-Kamel.pdf","https:\/\/drum.lib.umd.edu\/items\/72ba2d12-9e7b-4689-9be6-ba19a58c5736"],"collections":"basically-computer-science","url":"https:\/\/cis.temple.edu\/~vasilis\/Courses\/CIS750\/Papers\/HilbertRtree-Kamel.pdf https:\/\/drum.lib.umd.edu\/items\/72ba2d12-9e7b-4689-9be6-ba19a58c5736","urldate":"2010-08-31","year":"1994","booktitle":"PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON VERY LARGE DATA BASES","pages":"500--500","publisher":"Citeseer","volume":"8958546"},{"key":"Surlaloiderpartitiondukimefacteurpremierdunentier","type":"article","title":"Sur la loi de r\u00e9partition du k-i\u00e8me facteur premier d'un entier","author":"J.-M. DE KONINCK and G. TENENBAUM","abstract":"Soit \\({p_k(n)}^{w(n)}_{k=1}\\) la suite croissante des facteurs premiers distincts d'un entier \\(n\\). Nous donnons, lorsque \\(k \\to \\infty\\), une approximation uniforme de la loi de r\u00e9partition limite de la fonction arithm\u00e9tique \\(n \\mapsto pk(n)\\), pr\u00e9cisant ainsi un r\u00e9sultat classique d'Erd\u0151s. Deux applications en sont d\u00e9duites, relatives \u00e0 la m\u00e9diane de cette loi et \u00e0 celle de la fonction \u201c nombre de facteurs premiers \u201d.","comment":"37 is the median value for the second prime factor of an integer. The probability that the second prime factor of an integer chosen at random is smaller than 37 is approximately 1 in 2.\r\n","date_added":"2024-05-24","date_published":"2002-08-06","urls":["https:\/\/www.cambridge.org\/core\/journals\/mathematical-proceedings-of-the-cambridge-philosophical-society\/article\/abs\/sur-la-loi-de-repartition-du-kieme-facteur-premier-dun-entier\/8A74855F1EE7A0A0986858CEB3D89FD2","https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/8A74855F1EE7A0A0986858CEB3D89FD2\/S0305004102005972a.pdf\/div-class-title-sur-la-loi-de-repartition-du-span-class-italic-k-span-ieme-facteur-premier-d-un-entier-div.pdf"],"collections":"fun-maths-facts,integerology","url":"https:\/\/www.cambridge.org\/core\/journals\/mathematical-proceedings-of-the-cambridge-philosophical-society\/article\/abs\/sur-la-loi-de-repartition-du-kieme-facteur-premier-dun-entier\/8A74855F1EE7A0A0986858CEB3D89FD2 https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/8A74855F1EE7A0A0986858CEB3D89FD2\/S0305004102005972a.pdf\/div-class-title-sur-la-loi-de-repartition-du-span-class-italic-k-span-ieme-facteur-premier-d-un-entier-div.pdf","urldate":"2024-05-24","year":"2002","identifier":"doi:10.1017\/S0305004102005972","journal":"Mathematical Proceedings of the Cambridge Philosophical Society","publisher":"Cambridge University Press","volume":"133","issue":"2","issn":"0305-0041","doi":"10.1017\/S0305004102005972","pages":"191-204"},{"key":"TheSleepingBeautyControversy","type":"article","title":"The Sleeping Beauty Controversy","author":"Peter Winkler","abstract":"In 2000, Adam Elga posed the following problem:\r\n\r\nSome researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?\r\n\r\nThis may seem like a simple question about conditional probability, but 100 or so articles (including thousands of pages in major philosophy journals) have been devoted to it. Herein is an attempt to summarize the main arguments and to determine what, if anything, has been learned.","comment":"","date_added":"2017-08-14","date_published":"2017-08-06","urls":["https:\/\/math.dartmouth.edu\/~pw\/sb.pdf","http:\/\/www.jstor.org\/stable\/10.4169\/amer.math.monthly.124.7.579"],"collections":"drama,easily-explained,probability-and-statistics","url":"https:\/\/math.dartmouth.edu\/~pw\/sb.pdf http:\/\/www.jstor.org\/stable\/10.4169\/amer.math.monthly.124.7.579","urldate":"2017-08-14","year":"2017"},{"key":"Iteratedfailuresofchoice","type":"article","title":"Iterated failures of choice","author":"Asaf Karagila","abstract":"We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of counterexamples. For example, the rational numbers have a proper class of non-isomorphic algebraic closures, every partial order embeds into the cardinals of the model, every set is the image of a Dedekind-finite set, every weak choice axiom of the form $\\mathsf{AC}_X^Y$ fails with a proper class of counterexamples, every field has a vector space with two linearly independent vectors but without endomorphisms that are not scalar multiplication, etc.","comment":"","date_added":"2024-06-10","date_published":"2019-08-06","urls":["http:\/\/arxiv.org\/abs\/1911.09285v3","http:\/\/arxiv.org\/pdf\/1911.09285v3"],"collections":"fun-maths-facts","url":"http:\/\/arxiv.org\/abs\/1911.09285v3 http:\/\/arxiv.org\/pdf\/1911.09285v3","year":"2019","urldate":"2024-06-10","archivePrefix":"arXiv","eprint":"1911.09285","primaryClass":"math.LO"},{"key":"Integersthatarenotthesumofpositivepowers","type":"article","title":"Integers that are not the sum of positive powers","author":"Brennan Benfield and Oliver Lippard","abstract":"Exactly which positive integers cannot be expressed as the sum of \\(j\\)\r\npositive \\(k\\)-th powers? This paper utilizes theoretical and computational\r\ntechniques to answer this question for \\(k\\leq9\\). Results from Waring's problem\r\nare used throughout to catalogue the sets of such integers. These sets are then\r\nconsidered in a general setting, and several curious properties are\r\nestablished.","comment":"1072 is the sum of 2, 3, 4, 5, 6, 7 and 8 positive cubes.","date_added":"2024-06-21","date_published":"2024-08-06","urls":["http:\/\/arxiv.org\/abs\/2404.08193v1","http:\/\/arxiv.org\/pdf\/2404.08193v1"],"collections":"easily-explained,fun-maths-facts,integerology","url":"http:\/\/arxiv.org\/abs\/2404.08193v1 http:\/\/arxiv.org\/pdf\/2404.08193v1","urldate":"2024-06-21","year":"2024","archivePrefix":"arXiv","eprint":"2404.08193","primaryClass":"math.NT"},{"key":"Masuda2005","type":"article","title":"VIP-club phenomenon: emergence of elites and masterminds in social networks","author":"Masuda, Naoki and Konno, Norio","abstract":"Hubs, or vertices with large degrees, play massive roles in, for example, epidemic dynamics, innovation diffusion, and synchronization on networks. However, costs of owning edges can motivate agents to decrease their degrees and avoid becoming hubs, whereas they would somehow like to keep access to a major part of the network. By analyzing a model and tennis players' partnership networks, we show that combination of vertex fitness and homophily yields a VIP club made of elite vertices that are influential but not easily accessed from the majority. Intentionally formed VIP members can even serve as masterminds, which manipulate hubs to control the entire network without exposing themselves to a large mass. From conventional viewpoints based on network topology and edge direction, elites are not distinguished from many other vertices. Understanding network data is far from sufficient; individualistic factors greatly affect network structure and functions per se.","comment":"","date_added":"2012-09-02","date_published":"2005-01-01","urls":["http:\/\/arxiv.org\/abs\/cond-mat\/0501129","http:\/\/arxiv.org\/pdf\/cond-mat\/0501129","http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0378873305000535"],"collections":"modelling","url":"http:\/\/arxiv.org\/abs\/cond-mat\/0501129 http:\/\/arxiv.org\/pdf\/cond-mat\/0501129 http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0378873305000535","urldate":"2012-09-02","year":"2005","archivePrefix":"arXiv","arxivId":"cond-mat\/0501129","doi":"10.1016\/j.socnet.2005.07.005","eprint":"0501129","journal":"Social networks","month":"jan","pages":"15","primaryClass":"cond-mat"},{"key":"DoHaresEatLynx","type":"article","title":"Do Hares Eat Lynx?","author":"Michael E. Gilpin","abstract":"To test a recently developed predator-prey model against reality, I chose the well-known Canadian hare-lynx system. A measure of the state of this system for the last 200-odd years is available in the fur catch records of the Hudson Bay Company (MacLulich 1937; Elton and Nicholson 1942). Although the accuracy of these data is questionable (see Elton and Nicholson for a full discussion), they represent the only long-term population record available to ecologists.\r\n\r\nThe model I tested is\r\n\r\n\\[ \\begin{align}\r\ndH\/dt &= H(r_H + C_{HL} L + S_H H + I_H H^2), \\\r\ndL\/dt &= L(R_L + C_{LH} H + S_L L + I_L L^2),\r\n\\end{align} \\]\r\n","comment":"","date_added":"2024-06-29","date_published":"1973-08-06","urls":["https:\/\/www.journals.uchicago.edu\/doi\/abs\/10.1086\/282870?journalCode=an"],"collections":"animals,modelling","url":"https:\/\/www.journals.uchicago.edu\/doi\/abs\/10.1086\/282870?journalCode=an","urldate":"2024-06-29","year":"1973","journal":"The American Naturalist","volume":"107","number":"957"},{"key":"Marasco2011","type":"article","title":"Doc, What Are My Chances?","author":"Marasco, Joe and Doerfler, Ron and Roschier, Leif","abstract":"","comment":"","date_added":"2012-03-11","date_published":"2011-08-06","urls":["https:\/\/www.comap.com\/membership\/member-resources\/item\/doc-what-are-my-chances","http:\/\/www.myreckonings.com\/modernnomogramsTMP\/Doc_What_Are_My_Chances_UMAP_32-4-2011.pdf"],"collections":"attention-grabbing-titles,probability-and-statistics","url":"https:\/\/www.comap.com\/membership\/member-resources\/item\/doc-what-are-my-chances http:\/\/www.myreckonings.com\/modernnomogramsTMP\/Doc_What_Are_My_Chances_UMAP_32-4-2011.pdf","urldate":"2012-03-11","year":"2011","number":"4","pages":"279--298","volume":"32"},{"key":"NormalNumbersAreNormal","type":"article","title":"Normal Numbers are Normal","author":"Davar Khoshnevisan","abstract":"A number is normal in base \\(b\\) if every sequence of \\(k\\) symbols in the letters \\(0, 1, \\ldots, b \u2212 1\\)\r\noccurs in the base-\\(b\\) expansion of the given number with the expected frequency \\(b\u2212k\\) . From an informal\r\npoint of view, we can think of numbers normal in base 2 as those produced by flipping a fair coin,\r\nrecording 1 for heads and 0 for tails. Normal numbers are those which are normal in every base.\r\nIn this expository article, we recall Borel\u2019s result that almost all numbers are normal. Despite the\r\nabundance of such numbers, it is exceedingly difficult to find specific exemplars. While it is known\r\nthat the Champernowne number \\(0.123456789101112131415\\ldots\\) is normal in base 10, it is (for example)\r\nunknown whether \\(\\sqrt{2}\\) is normal in any base. We sketch a bit of what is known and what is not known\r\nof this peculiar class of numbers, and we discuss connections with areas such as computability theory.","comment":"","date_added":"2024-07-11","date_published":"2006-08-06","urls":["https:\/\/www.claymath.org\/library\/annual_report\/ar2006\/06report_normalnumbers.pdf","https:\/\/www.math.utah.edu\/~davar\/ps-pdf-files\/NN2.pdf"],"collections":"attention-grabbing-titles,fun-maths-facts","url":"https:\/\/www.claymath.org\/library\/annual_report\/ar2006\/06report_normalnumbers.pdf https:\/\/www.math.utah.edu\/~davar\/ps-pdf-files\/NN2.pdf","urldate":"2024-07-11","year":"2006"},{"key":"TumblingDownhillalongaGivenCurve","type":"article","title":"Tumbling Downhill along a Given Curve","author":"Jean-Pierre Eckmann and Yaroslav I. Sobolev and Tsvi Tlusty","abstract":"A cylinder will roll down an inclined plane in a straight line. A cone will\r\nroll around a circle on that plane and then will stop rolling. We ask the\r\ninverse question: For which curves drawn on the inclined plane $\\mathbb{R}^2$\r\ncan one carve a shape that will roll downhill following precisely this\r\nprescribed curve and its translationally repeated copies? This simple question\r\nhas a solution essentially always, but it turns out that for most curves, the\r\nshape will return to its initial orientation only after crossing a few copies\r\nof the curve - most often two copies will suffice, but some curves require an\r\narbitrarily large number of copies.","comment":"","date_added":"2024-07-24","date_published":"2024-08-06","urls":["http:\/\/arxiv.org\/abs\/2406.16336v1","http:\/\/arxiv.org\/pdf\/2406.16336v1","https:\/\/www.ams.org\/journals\/notices\/202406\/rnoti-p740.pdf"],"collections":"basically-physics,easily-explained,fun-maths-facts,geometry,things-to-make-and-do","url":"http:\/\/arxiv.org\/abs\/2406.16336v1 http:\/\/arxiv.org\/pdf\/2406.16336v1 https:\/\/www.ams.org\/journals\/notices\/202406\/rnoti-p740.pdf","urldate":"2024-07-24","year":"2024","archivePrefix":"arXiv","eprint":"2406.16336","primaryClass":"math-ph"}]