@article{Gibbons2006,
author = {Gibbons, J.},
journal = {American Mathematical Monthly},
number = 4,
pages = {318--328},
publisher = {Citeseer},
title = {Unbounded spigot algorithms for the digits of pi},
url = {http://www.cs.ox.ac.uk/jeremy.gibbons/publications/spigot.pdf},
volume = 113,
year = 2006,
urldate = {2010-01-05},
collections = {Basically computer science,Fun maths facts},
comment = {There are several algorithms which produce digits of π 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 π as the composition of infinitely many Mobius transforms, which are implemented as integer matrix multiplications.}
}
@article{Quisquater,
author = {Quisquater, JJ and Quisquater, M},
journal = {Advances in Cryptology— \ldots},
title = {How to explain zero-knowledge protocols to your children},
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},
collections = {Attention-grabbing titles,Easily explained,Protocols and strategies},
year = 1989,
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.}
}
@article{Socolar2010,
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.},
author = {Socolar, Joshua E. S. and Taylor, Joan M.},
keywords = {Combinatorics,Other Condensed Matter},
month = {mar},
pages = 21,
title = {An aperiodic hexagonal tile},
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},
collections = {Geometry},
comment = {A single shape which tiles the plane, but not periodically. The catch is that it's not a single connected component.}
}
@article{Avigad2008,
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.},
author = {Avigad, Jeremy and Dean, Edward and Mumma, John},
keywords = {Logic},
month = {oct},
title = {A formal system for Euclid's Elements},
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},
collections = {Geometry,About proof}
}
@article{Hall2001,
author = {Hall, Rachel W. and Josic, Kresimir},
journal = {The American Mathematical Monthly},
month = {apr},
number = 4,
pages = 347,
title = {The Mathematics of Musical Instruments},
url = {http://www.jstor.org/stable/2695241?origin=crossref},
volume = 108,
year = 2001,
urldate = {2010-07-24},
collections = {Music}
}
@book{Moritz2008,
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...},
author = {Moritz, Robert Edoward},
pages = 448,
publisher = {READ BOOKS},
title = {On Mathematics and Mathematicians},
url = {http://books.google.com/books?id=2sML4rpqucEC{\&}pgis=1},
year = 2008,
urldate = {2010-08-24},
collections = {The act of doing maths},
comment = {A collection of quotes about mathematics and mathematicians, from 1914.}
}
@article{Alpher1948,
author = {Alpher, R. and Bethe, H. and Gamow, G.},
journal = {Physical Review},
month = {apr},
number = 7,
pages = {803--804},
title = {The Origin of Chemical Elements},
url = {http://en.wikipedia.org/wiki/Alpher{\%}E2{\%}80{\%}93Bethe{\%}E2{\%}80{\%}93Gamow{\_}paper},
volume = 73,
year = 1948,
urldate = {2010-09-03},
collections = {Basically physics},
comment = {The "Alpher-Bethe-Gamow" paper.}
}
@inproceedings{Weber2002,
author = {Weber, K.},
booktitle = {Proceedings from the 2nd International Conference for the Teaching of Mathematics},
title = {The role of instrumental and relational understanding in proofs about group isomorphisms},
url = {http://www.math.uoc.gr/{\~{}}ictm2/Proceedings/pap86.pdf},
year = 2002,
urldate = {2010-09-03},
collections = {The act of doing maths,About proof,The groups group},
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.}
}
@article{Grunbaum2006,
author = {Gr{\"{u}}nbaum, Branko},
journal = {Notices of the American Mathematical Society},
number = {ICM},
pages = {2--5},
title = {What symmetry groups are present in the Alhambra?},
url = {http://www.ams.org/notices/200606/comm-grunbaum.pdf},
year = 2006,
urldate = {2010-09-06},
collections = {History,Easily explained,Geometry},
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.}
}
@article{Brunvand1963,
author = {Brunvand, J.H.},
journal = {Journal of American Folklore},
number = 299,
pages = {42--68},
publisher = {JSTOR},
title = {A classification for shaggy dog stories},
url = {http://www.jstor.org/stable/538078},
volume = 76,
year = 1963,
urldate = {2011-01-12},
collections = {Lists and catalogues,Animals},
comment = {Over 300 shaggy dog stories, organised into six sections and over 200 types.}
}
@article{Raymer2007,
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{\"{o}}bius 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.},
author = {Raymer, Dorian M and Smith, Douglas E},
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,
title = {Spontaneous knotting of an agitated string.},
url = {http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2034230{\&}tool=pmcentrez{\&}rendertype=abstract},
volume = 104,
year = 2007,
urldate = {2011-01-12},
collections = {Easily explained,Modelling},
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.}
}
@article{Ord2002,
author = {Ord, Toby},
journal = {Arxiv preprint math/0209332},
title = {Hypercomputation: computing more than the Turing machine},
url = {http://arxiv.org/abs/math/0209332 http://arxiv.org/pdf/math/0209332v1},
year = 2002,
archivePrefix = {arXiv},
eprint = {math/0209332},
primaryClass = {math.LO},
abstract = {Due to common misconceptions about the Church-Turing thesis, it has been
widely assumed that the Turing machine provides an upper bound on what is
computable. This is not so. The new field of hypercomputation studies models of
computation that can compute more than the Turing machine and addresses their
implications. In this report, I survey much of the work that has been done on
hypercomputation, explaining how such non-classical models fit into the
classical theory of computation and comparing their relative powers. I also
examine the physical requirements for such machines to be constructible and the
kinds of hypercomputation that may be possible within the universe. Finally, I
show how the possibility of hypercomputation weakens the impact of Godel's
Incompleteness Theorem and Chaitin's discovery of 'randomness' within
arithmetic.},
urldate = {2011-02-09},
collections = {Basically computer science,Unusual computers},
comment = {A probably overblown argument for a model of computation that does more than Turing machines.}
}
@article{Duarte2005,
author = {Duarte, J},
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},
title = {A discursive grammar for customizing mass housing: the case of Siza's houses at Malagueira},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0926580504000810},
volume = 14,
year = 2005,
urldate = {2011-02-15},
collections = {Basically computer science,Art}
}
@article{Blasjo,
author = {Blasjo, Viktor},
isbn = 0821835874,
issn = {1534-6455},
journal = {The Mathematical Association of America Montly},
number = 112,
pages = {526--566},
publisher = {Amer Mathematical Society},
title = {The isoperimetric problem},
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},
collections = {Puzzles,Geometry},
year = 2001
}
@article{Warrington2009,
author = {Warrington, Gregory S.},
language = {EN},
month = {jan},
publisher = {Mathematical Association of America},
title = {Juggling Probabilities},
url = {http://www.jstor.org/pss/30037409},
year = 2009,
urldate = {2011-03-20},
collections = {Easily explained}
}
@article{Comon1997,
author = {Comon, Hubert and Dauchet, M and Gilleron, R},
title = {Tree automata techniques and applications},
url = {http://en.scientificcommons.org/42494218},
year = 1997,
urldate = {2011-03-25},
collections = {Basically computer science}
}
@article{Nollenburg2010,
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.},
author = {N{\"{o}}llenburg, Martin and Wolff, Alexander},
doi = {10.1109/TVCG.2010.81},
issn = {1077-2626},
journal = {IEEE transactions on visualization and computer graphics},
month = {may},
pages = {1--25},
pmid = 20498505,
title = {Drawing and Labeling High-Quality Metro Maps by Mixed-Integer Programming.},
url = {http://www.ncbi.nlm.nih.gov/pubmed/20498505},
year = 2010,
urldate = {2011-04-07},
collections = {Basically computer science}
}
@article{Khosla2002,
author = {Khosla, D},
journal = {Intelligent Vehicle Symposium, 2002. IEEE},
title = {Accurate estimation of forward path geometry using two-clothoid road model},
url = {http://ieeexplore.ieee.org/xpls/abs{\_}all.jsp?arnumber=1187944},
year = 2002,
urldate = {2011-04-12},
collections = {Basically computer science,Geometry,Modelling}
}
@article{Boolos1994,
author = {Boolos, George},
doi = {10.1093/mind/103.409.1},
issn = {0026-4423},
journal = {Mind},
number = 409,
pages = {1--3},
title = {G{\"{o}}del's Second Incompleteness Theorem Explained in Words of One Syllable},
url = {http://mind.oxfordjournals.org/cgi/doi/10.1093/mind/103.409.1},
volume = 103,
year = 1994,
urldate = {2011-09-24},
collections = {Easily explained,Fun maths facts}
}
@article{Hardy1971,
author = {Hardy, C. and Baronet, C. N. and Tordion, G. V.},
issn = {0029-5981},
journal = {International Journal for Numerical Methods in Engineering},
month = {oct},
number = 4,
pages = {451--462},
title = {The elasto-plastic indentation of a half-space by a rigid sphere},
url = {http://doi.wiley.com/10.1002/nme.1620030402},
volume = 3,
year = 1971,
urldate = {2011-10-02}
}
@article{Calogero2003,
author = {Calogero, Francesco},
issn = {0343-6993},
journal = {The Mathematical Intelligencer},
keywords = {Mathematics},
month = {dec},
number = 4,
pages = {72--76},
publisher = {Springer New York},
title = {Cool irrational numbers and their rather cool rational approximations},
url = {https://link.springer.com/article/10.1007{\%}2FBF02984865},
volume = 25,
year = 2003,
urldate = {2011-10-13},
collections = {Attention-grabbing titles,Fun maths facts}
}
@article{Ball2009,
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.},
author = {Ball, Rafael},
issn = {0138-9130},
journal = {Scientometrics},
keywords = {Computer Science},
month = {jan},
number = 3,
pages = {667--679},
publisher = {Akad{\'{e}}miai Kiad{\'{o}}, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic Publishers B.V.},
title = {Scholarly communication in transition: The use of question marks in the titles of scientific articles in medicine, life sciences and physics 1966–2005},
url = {https://link.springer.com/article/10.1007/s11192-007-1984-5},
volume = 79,
year = 2009,
urldate = {2011-10-14},
collections = {History,The act of doing maths}
}
@article{Hilhorst2005,
archivePrefix = {arXiv},
arxivId = {arXiv:cond-mat/0507567v1},
author = {Hilhorst, H.J.},
eprint = {0507567v1},
journal = {Journal of Statistical Mechanics: Theory and Experiment},
keywords = {exact results,random graphs},
pages = {P09005},
primaryClass = {arXiv:cond-mat},
publisher = {IOP Publishing},
title = {Asymptotic statistics of the n-sided planar Poisson–Voronoi cell: I. Exact results},
url = {http://iopscience.iop.org/1742-5468/2005/09/P09005},
volume = 2005,
year = 2005,
urldate = {2011-12-08},
collections = {Probability and statistics,Geometry}
}
@article{Fink2000,
author = {Fink, T and Mao, Y},
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},
title = {Tie knots, random walks and topology},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0378437199002265},
volume = 276,
year = 2000,
urldate = {2011-12-09},
collections = {Probability and statistics}
}
@article{Ekhad2011,
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).},
archivePrefix = {arXiv},
arxivId = {1112.1645},
author = {Ekhad, Shalosh B and Georgiadis, Evangelos and Zeilberger, Doron},
eprint = {1112.1645},
journal = {Strategy},
keywords = {Computer Science and Game Theory,Probability},
month = {dec},
pages = 6,
title = {How to Gamble If You're In a Hurry},
url = {http://arxiv.org/abs/1112.1645 http://arxiv.org/pdf/1112.1645v2},
year = 2011,
primaryClass = {math.PR},
urldate = {2011-12-15},
collections = {Attention-grabbing titles,Games to play with friends,Probability and statistics}
}
@article{Petersen2010,
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.},
archivePrefix = {arXiv},
arxivId = {1003.3947},
author = {Petersen, Carsten Lunde and Roesch, Pascale},
eprint = {1003.3947},
keywords = {Dynamical Systems},
month = {mar},
pages = 21,
title = {Carrots for dessert},
url = {http://arxiv.org/abs/1003.3947 http://arxiv.org/pdf/1003.3947v1},
volume = 0,
year = 2010,
primaryClass = {math.DS},
urldate = {2012-01-02},
collections = {Attention-grabbing titles,Food}
}
@article{Gajardo2002,
author = {Gajardo, A and Moreira, A and Goles, E},
journal = {Discrete Applied Mathematics},
keywords = {arti{\"{y}}cial life,complexity,universality,virtual ant},
number = 1,
pages = {41--50},
publisher = {Elsevier},
title = {Complexity of Langton's ant},
url = {http://www.sciencedirect.com/science/article/pii/S0166218X00003346},
volume = 117,
year = 2002,
urldate = {2012-01-04},
collections = {Basically computer science,Easily explained,Animals}
}
@article{Schmidt1966,
author = {Schmidt, WM},
journal = {Trans. Amer. Math. Soc},
number = {C},
pages = {178--199},
title = {On badly approximable numbers and certain games},
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},
collections = {Games to play with friends}
}
@article{Fisher2007,
author = {Fisher, GL and Mellor, B.},
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},
title = {Three-dimensional finite point groups and the symmetry of beaded beads},
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},
collections = {Easily explained,Things to make and do,The groups group}
}
@article{Funes1999,
author = {Funes, Pablo and Pollack, Jordan},
journal = {Evolutionary design by computers},
pages = {387--403},
publisher = {Morgan Kaufmann, San Francisco},
title = {Computer evolution of buildable objects},
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},
collections = {Basically computer science}
}
@article{Diaconis2011,
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.},
archivePrefix = {arXiv},
arxivId = {1107.2961},
author = {Diaconis, Persi and Fulman, Jason and Holmes, Susan},
eprint = {1107.2961},
journal = {Analysis},
keywords = {Combinatorics,Probability},
month = {jul},
pages = 23,
title = {Analysis of Casino Shelf Shuffling Machines},
url = {http://arxiv.org/abs/1107.2961 http://arxiv.org/pdf/1107.2961v2},
year = 2011,
primaryClass = {math.CO},
urldate = {2012-01-11},
collections = {Basically physics,Probability and statistics}
}
@article{Viglietta2012,
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.},
archivePrefix = {arXiv},
arxivId = {1201.4995},
author = {Viglietta, Giovanni},
eprint = {1201.4995},
keywords = {Computational Complexity,Computer Science and Game Theory},
month = {jan},
pages = 12,
title = {Gaming is a hard job, but someone has to do it!},
url = {http://arxiv.org/abs/1201.4995 http://arxiv.org/pdf/1201.4995v5},
year = 2012,
primaryClass = {cs.CC},
urldate = {2012-01-27},
collections = {Basically computer science,Games to play with friends,Computational complexity of games}
}
@article{Moszkowski2011,
author = {Moszkowski, Ben},
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},
title = {Compositional Reasoning Using Intervals and Time Reversal},
url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6065235},
year = 2011,
urldate = {2012-01-27}
}
@article{Wolf2010,
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.},
archivePrefix = {arXiv},
arxivId = {1003.4015},
author = {Wolf, Marek},
eprint = {1003.4015},
isbn = 9999637051,
journal = {Arxiv preprint arXiv:1003.4015},
keywords = {History and Overview,Number Theory},
month = {mar},
number = 1,
pages = 35,
title = {Continued fractions constructed from prime numbers},
url = {http://arxiv.org/abs/1003.4015 http://arxiv.org/pdf/1003.4015v2},
volume = 1,
year = 2010,
primaryClass = {math.NT},
urldate = {2012-02-01},
collections = {Easily explained}
}
@article{Izhakian2005,
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.},
archivePrefix = {arXiv},
arxivId = {math/0505458},
author = {Izhakian, Zur},
eprint = {math/0505458},
journal = {ReCALL},
keywords = {Algebraic Geometry,Combinatorics},
month = {may},
pages = 17,
primaryClass = {math.AG},
title = {Tropical Arithmetic and Tropical Matrix Algebra},
url = {http://arxiv.org/abs/math/0505458 http://arxiv.org/pdf/math/0505458v3},
year = 2005,
urldate = {2012-02-03},
collections = {Unusual arithmetic}
}
@article{Hodge2010,
author = {Hodge, Jonathan K. and Marshall, Emily and Patterson, Geoff},
doi = {10.4169/074683410X510317},
issn = 07468342,
journal = {The College Mathematics Journal},
month = {sep},
number = 4,
pages = {312--324},
title = {Gerrymandering and Convexity},
url = {http://www.jstor.org/stable/10.4169/074683410X510317},
volume = 41,
year = 2010,
urldate = {2012-02-03},
collections = {Easily explained,Geometry}
}
@article{Ball2011,
author = {Ball, Philip},
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},
title = {In retrospect: On the Six-Cornered Snowflake},
url = {http://dx.doi.org/10.1038/480455a},
volume = 480,
year = 2011,
urldate = {2012-02-06}
}
@article{Meier1980,
author = {Meier, Paul and Zabell, Sandy},
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},
title = {Benjamin Peirce and the Howland will},
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},
collections = {History}
}
@article{Benjamin2007,
author = {Benjamin, A.T. and Cameron, N.T. and Quinn, J.J.},
journal = {Fibonacci Quarterly},
number = 1,
pages = 39,
publisher = {THE FIBONACCI ASSOCIATION},
title = {Fibonacci determinants-a combinatorial approach},
url = {http://www.math.hmc.edu/{\~{}}benjamin/papers/FiboDet.pdf},
volume = 45,
year = 2007,
urldate = {2012-02-08},
collections = {Fibonaccinalia,Fun maths facts}
}
@article{Speyer2004,
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.},
archivePrefix = {arXiv},
arxivId = {math/0408099},
author = {Speyer, David and Sturmfels, Bernd},
eprint = {math/0408099},
keywords = {Algebraic Geometry,Combinatorics},
month = {aug},
number = {July},
pages = 15,
primaryClass = {math.CO},
title = {Tropical Mathematics},
url = {http://arxiv.org/abs/math/0408099 http://arxiv.org/pdf/math/0408099v1},
year = 2004,
urldate = {2012-02-12},
collections = {Unusual arithmetic,Easily explained}
}
@article{Erlichson2003,
author = {Erlichson, Herman},
doi = {10.1016/S0315-0860(02)00008-3},
issn = 03150860,
journal = {Historia Mathematica},
month = {nov},
number = 4,
pages = {432--440},
title = {Passage to the limit in Proposition I, Book I of Newton's Principia},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0315086002000083},
volume = 30,
year = 2003,
urldate = {2012-02-15},
collections = {History}
}
@article{Bartholdi2012,
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.},
archivePrefix = {arXiv},
arxivId = {1202.3033},
author = {Bartholdi, Laurent and Henriques, Andr{\'{e}} G.},
eprint = {1202.3033},
journal = {Time},
month = {feb},
pages = {1--3},
title = {Orange Peels and Fresnel Integrals},
url = {http://arxiv.org/abs/1202.3033 http://arxiv.org/pdf/1202.3033v1},
year = 2012,
primaryClass = {math.HO},
urldate = {2012-02-15},
collections = {Attention-grabbing titles,Easily explained,Things to make and do,Food,Geometry,Fun maths facts}
}
@article{Pepperberg2012,
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 0–6 items (up to 6), and to identify and serially order Arabic numerals (1–8), all by using English labels (Pepperberg in J Comp Psychol 108:36–44, 1994 ; J Comp Psychol 120:1–11, 2006a ; J Comp Psychol 120:205–216, 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, “How many total?” 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 “What color number (is the) total?” Although his death precluded testing on all possible arrays, his accuracy was statistically significant and suggested addition abilities comparable with those of nonhuman primates.},
author = {Pepperberg, Irene M.},
issn = {1435-9448},
journal = {Animal Cognition},
keywords = {Biomedical and Life Sciences},
month = {feb},
pages = {1--7},
publisher = {Springer Berlin / Heidelberg},
title = {Further evidence for addition and numerical competence by a Grey parrot (Psittacus erithacus)},
url = {https://www.ncbi.nlm.nih.gov/pubmed/22402776},
year = 2012,
urldate = {2012-02-21},
collections = {Animals}
}
@article{Ghrist2008,
author = {Ghrist, Robert},
journal = {Bulletin-American Mathematical Society},
number = 3,
pages = {1--15},
title = {Barcodes: the persistent topology of data},
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},
collections = {Probability and statistics}
}
@article{Washington1986,
author = {Washington, Lawrence and Zagier, Don},
title = {Quotients Homophones des Groupes Libres Homophonic Quotients of Free Groups},
url = {http://people.mpim-bonn.mpg.de/zagier/files/exp-math-2/fulltext.pdf},
year = 1986,
urldate = {2012-02-27},
collections = {Easily explained,The groups group}
}
@article{VanLeeuwaarden2011,
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.},
archivePrefix = {arXiv},
arxivId = {1104.3034},
author = {van Leeuwaarden, Johan S. H. and Raschel, Kilian},
eprint = {1104.3034},
journal = {Plays},
month = {apr},
pages = 17,
title = {Random walks reaching against all odds the other side of the quarter plane},
url = {http://arxiv.org/abs/1104.3034 http://arxiv.org/pdf/1104.3034v2},
year = 2011,
primaryClass = {math.PR},
urldate = {2012-03-01},
collections = {Probability and statistics}
}
@misc{Ferguson,
author = {Ferguson, D. F. and Wrench, John W},
title = {A New Approximation to {\$}\pi{\$} (Conclusion)},
url = {http://www.jstor.org/discover/10.2307/2002657?uid=3738032{\&}uid=2{\&}uid=4{\&}sid=47698756219137},
urldate = {2012-03-14},
collections = {Easily explained},
year = 1948
}
@article{Giblin1990,
author = {Giblin, PJ},
journal = {American Mathematical Monthly},
title = {The bitangent sphere problem},
url = {http://www.jstor.org/stable/10.2307/2323998 http://dl.acm.org/citation.cfm?id=87129},
year = 1990,
urldate = {2012-03-24},
collections = {Geometry}
}
@article{Petersen2011,
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.},
archivePrefix = {arXiv},
arxivId = {1107.3707},
author = {Petersen, Alexander M and Tenenbaum, Joel and Havlin, Shlomo and Stanley, H Eugene},
eprint = {1107.3707},
journal = {Methods},
month = {jul},
pages = 31,
title = {Statistical Laws Governing Fluctuations in Word Use from Word Birth to Word Death},
url = {http://arxiv.org/abs/1107.3707 http://arxiv.org/pdf/1107.3707v2},
year = 2011,
primaryClass = {physics.soc-ph},
urldate = {2012-03-27},
collections = {Probability and statistics}
}
@article{Ianovski2012,
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.},
archivePrefix = {arXiv},
arxivId = {1203.0100},
author = {Ianovski, Egor},
eprint = {1203.0100},
journal = {October},
month = {mar},
number = {March},
title = {Cake Cutting Mechanisms},
url = {http://arxiv.org/abs/1203.0100 http://arxiv.org/pdf/1203.0100v2},
year = 2012,
primaryClass = {cs.GT},
urldate = {2012-04-07},
collections = {Puzzles,Easily explained,Protocols and strategies,Food}
}
@article{Chelghoum2004,
author = {Chelghoum, Kamel and Margenstern, Maurice and Martin, Beno\^it},
journal = {Cellular Automata},
pages = {1--11},
title = {Cellular automata in the hyperbolic plane: proposal for a new environment},
url = {https://link.springer.com/chapter/10.1007/978-3-540-30479-1{\_}70},
year = 2004,
urldate = {2012-04-08},
collections = {Basically computer science}
}
@article{Lemoine2010,
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.},
archivePrefix = {arXiv},
arxivId = {1008.2320},
author = {Lemoine, Julien and Viennot, Simon},
eprint = {1008.2320},
journal = {Analysis},
month = {aug},
pages = 17,
title = {Computer analysis of Sprouts with nimbers},
url = {http://arxiv.org/abs/1008.2320 http://arxiv.org/pdf/1008.2320v1},
year = 2010,
primaryClass = {math.CO},
urldate = {2012-04-21},
collections = {Unusual arithmetic,Computational complexity of games}
}
@book{Thompson1914,
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},
author = {Thompson, Silvanus P},
title = {Calculus Made Easy},
url = {http://www.gutenberg.org/ebooks/33283},
year = 1914,
urldate = {2012-04-26},
collections = {Education}
}
@article{Vecer2009,
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.},
author = {Vecer, Jan and Kopriva, Frantisek},
journal = {Journal of Quantitative Analysis in},
number = 1992,
pages = {1--13},
title = {Estimating the Effect of the Red Card in Soccer},
url = {http://www.stat.columbia.edu/{\~{}}vecer/redcard.pdf},
year = 2009,
urldate = {2012-05-09},
collections = {Easily explained}
}
@article{Baez2012,
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.},
archivePrefix = {arXiv},
arxivId = {1205.2447},
author = {Baez, John C and Huerta, John},
eprint = {1205.2447},
month = {may},
pages = 28,
title = {G2 and the Rolling Ball},
url = {http://arxiv.org/abs/1205.2447 http://arxiv.org/pdf/1205.2447v4},
year = 2012,
primaryClass = {math.DG},
urldate = {2012-05-14},
collections = {Basically physics}
}
@article{Geiges2011,
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.},
author = {Geiges, Hansj{\"{o}}rg and Zehmisch, Kai},
month = {apr},
pages = 26,
title = {How to recognise a 4-ball when you see one},
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},
collections = {Attention-grabbing titles,Geometry}
}
@article{Ravsky2012,
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.},
archivePrefix = {arXiv},
arxivId = {1205.0747},
author = {Ravsky, Alex},
eprint = {1205.0747},
journal = {Star},
month = {may},
pages = 2,
title = {On an error in the star puzzle by Henry E. Dudeney},
url = {http://arxiv.org/abs/1205.0747 http://arxiv.org/pdf/1205.0747v2},
year = 2012,
primaryClass = {math.HO},
urldate = {2012-05-22},
collections = {Puzzles}
}
@article{Feist2012,
abstract = {We use braids and linking number to explain why automobile shades fold into an odd number of loops.},
archivePrefix = {arXiv},
arxivId = {1205.4797},
author = {Feist, Curtis and Naimi, Ramin},
eprint = {1205.4797},
month = {may},
pages = 8,
title = {Topology Explains Why Automobile Sunshades Fold Oddly},
url = {http://arxiv.org/abs/1205.4797 http://arxiv.org/pdf/1205.4797v1},
year = 2012,
primaryClass = {math.GT},
urldate = {2012-05-23},
collections = {Easily explained,Basically physics,Fun maths facts}
}
@article{Kindler2004,
author = {Kindler, G},
journal = {of the fifteenth annual ACM-SIAM symposium on},
title = {On distributions computable by random walks on graphs},
url = {http://dl.acm.org/citation.cfm?id=982809},
year = 2004,
urldate = {2012-06-07}
}
@article{Isaksen2002,
author = {Isaksen, DC},
journal = {The American mathematical monthly},
number = 9,
pages = {796--805},
title = {A cohomological viewpoint on elementary school arithmetic},
url = {http://www.jstor.org/stable/10.2307/3072368},
volume = 109,
year = 2002,
urldate = {2012-06-14},
collections = {Unusual arithmetic}
}
@article{Kraft2001,
author = {Kraft, Hanspeter},
journal = {J. Biol. Phys. Chem},
number = {December},
pages = {95--96},
title = {The wobbly garden table},
url = {http://www.amsi.ge/jbpc/020101/full/9{\_}18KR01F.pdf},
volume = 1,
year = 2001,
urldate = {2012-06-17},
collections = {Easily explained,Basically physics,Fun maths facts}
}
@article{Baek2010,
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.},
archivePrefix = {arXiv},
arxivId = {1001.1065},
author = {Baek, Seung Ki and Bernhardsson, Sebastian},
doi = {10.1142/S0219477510000071},
eprint = {1001.1065},
keywords = {game,lowest unique positive integer,nash equilibrium,projection operator},
month = {jan},
number = 0,
pages = 8,
title = {Equilibrium solution to the lowest unique positive integer game},
url = {http://arxiv.org/abs/1001.1065 http://arxiv.org/pdf/1001.1065v1},
volume = 0,
year = 2010,
primaryClass = {math.CO},
urldate = {2012-06-22},
collections = {Protocols and strategies,Fun maths facts}
}
@misc{Chalcraft,
author = {Chalcraft, Adam and Greene, Michael},
title = {Train Sets},
url = {http://www.monochrom.at/turingtrainterminal/Chalcraft.pdf},
urldate = {2012-06-25},
collections = {Easily explained,Things to make and do},
year = 1994
}
@article{Dubner2001,
author = {Dubner, H},
title = {The top ten prime numbers},
url = {http://primes.utm.edu/lists/top{\_}ten/topten.pdf},
year = 2001,
urldate = {2012-06-28},
collections = {Easily explained,Lists and catalogues,Integerology}
}
@article{Sloane2006,
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.},
author = {Sloane, N J A},
pages = {1--12},
title = {Seven Staggering Sequences},
url = {http://neilsloane.com/doc/g4g7.pdf},
year = 2006,
urldate = {2012-07-14},
collections = {Easily explained,Integerology}
}
@article{Flexner1952,
author = {Flexner, Abraham},
journal = {Harper's Magazine},
keywords = {Research,Science},
month = {oct},
number = 117,
pages = {544--552},
pmid = 13024481,
title = {The usefulness of useless knowledge},
url = {http://library.ias.edu/files/UsefulnessHarpers.pdf},
volume = 17,
year = 1952,
urldate = {2012-07-16},
collections = {History}
}
@article{Mamakani2012,
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.},
archivePrefix = {arXiv},
arxivId = {1207.6452},
author = {Mamakani, Khalegh and Ruskey, Frank},
eprint = {1207.6452},
keywords = {crosscut symmetry,hypercube,symmetric graphs,venn diagram},
month = {jul},
title = {A New Rose : The First Simple Symmetric 11-Venn Diagram},
url = {http://arxiv.org/abs/1207.6452 http://arxiv.org/pdf/1207.6452v1},
year = 2012,
primaryClass = {cs.CG},
urldate = {2012-08-09},
collections = {Art,Easily explained}
}
@article{Cui2012,
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.},
archivePrefix = {arXiv},
arxivId = {1208.5531},
author = {Cui, Weideng},
eprint = {1208.5531},
keywords = {algebra,canonical basis,the modified quantized enveloping,the quasi-},
month = {aug},
title = {The Canonical Basis of {\$}\dot{\{}\mathbf{\{}U{\}}{\}}{\$} for Type {\$}A{\_}{\{}2{\}}{\$}},
url = {http://arxiv.org/abs/1208.5531 http://arxiv.org/pdf/1208.5531v3},
year = 2012,
primaryClass = {math.RT},
urldate = {2012-08-29}
}
@article{Stuckman2005,
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.},
archivePrefix = {arXiv},
arxivId = {cs/0512049},
author = {Stuckman, Jeff and Zhang, Guo-Qiang},
eprint = {cs/0512049},
journal = {Arxiv preprint cs/0512049},
keywords = {computational complexity,mastermind,theory of computation},
month = {dec},
pages = {1--7},
primaryClass = {cs.CC},
title = {Mastermind is NP-Complete},
url = {http://arxiv.org/abs/cs/0512049 http://arxiv.org/pdf/cs/0512049v1},
year = 2005,
urldate = {2012-09-02},
collections = {Puzzles,Basically computer science,Games to play with friends,Computational complexity of games}
}
@article{Masuda2005,
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.},
archivePrefix = {arXiv},
arxivId = {cond-mat/0501129},
author = {Masuda, Naoki and Konno, Norio},
doi = {10.1016/j.socnet.2005.07.005},
eprint = 0501129,
journal = {Social networks},
month = {jan},
pages = 15,
primaryClass = {cond-mat},
title = {VIP-club phenomenon: emergence of elites and masterminds in social networks},
url = {http://www.sciencedirect.com/science/article/pii/S0378873305000535 http://arxiv.org/abs/cond-mat/0501129},
year = 2005,
urldate = {2012-09-02},
collections = {Modelling}
}
@article{Kahan1998,
author = {Kahan, W and Darcy, JD},
journal = {\ldots 1998 Workshop on Java \ldots},
pages = {1--81},
title = {How Java's floating-point hurts everyone everywhere},
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},
collections = {Basically computer science}
}
@article{Sunic2011,
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.},
archivePrefix = {arXiv},
arxivId = {1108.4494},
author = {Sunic, Zoran},
eprint = {1108.4494},
month = {aug},
pages = {1--19},
title = {Twin Towers of Hanoi},
url = {http://arxiv.org/abs/1108.4494 http://arxiv.org/pdf/1108.4494v1},
year = 2011,
primaryClass = {math.CO},
urldate = {2012-09-28},
collections = {Puzzles,Easily explained}
}
@article{Coulbois2012,
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.},
archivePrefix = {arXiv},
arxivId = {1210.1518},
author = {Coulbois, Thierry and Pellicer, Daniel and Raggi, Miguel and Ram{\'{i}}rez, Camilo and Valdez, Ferr{\'{a}}n},
eprint = {1210.1518},
month = {oct},
pages = 21,
title = {The topology of the minimal regular cover of the Archimedean tessellations},
url = {http://arxiv.org/abs/1210.1518 http://arxiv.org/pdf/1210.1518v1},
year = 2012,
primaryClass = {math.GT},
urldate = {2012-10-05},
collections = {Geometry}
}
@article{Bruijn1981,
author = {Bruijn, NG De},
journal = {Kon. Nederl. Akad. Wetensch. Proc. Ser. A},
title = {Algebraic theory of Penrose's non-periodic tilings of the plane},
url = {http://alexandria.tue.nl/repository/freearticles/597566.pdf},
year = 1981,
urldate = {2012-10-13},
collections = {Geometry}
}
@article{Gerlach2010,
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.},
archivePrefix = {arXiv},
arxivId = {1005.4609},
author = {Gerlach, Henryk and von der Mosel, Heiko},
eprint = {1005.4609},
journal = {Nature},
month = {may},
pages = 15,
title = {On sphere-filling ropes},
url = {http://arxiv.org/abs/1005.4609 http://arxiv.org/pdf/1005.4609v1},
year = 2010,
primaryClass = {math.GT},
urldate = {2012-10-29},
collections = {Easily explained,Geometry,Fun maths facts}
}
@article{Pasles2001,
author = {Pasles, PC},
journal = {The American Mathematical Monthly},
number = 6,
pages = {489--511},
title = {The lost squares of Dr. Franklin: Ben Franklin's missing squares and the secret of the magic circle},
url = {http://www.jstor.org/stable/10.2307/2695704},
volume = 108,
year = 2001,
urldate = {2012-10-30},
collections = {Puzzles,History}
}
@article{Papy1970,
author = {Papy, F},
journal = {Math Teaching},
title = {Papy's Minicomputer},
url = {http://www.rkennes.be/Papy-Minicomputer/minicomp-anglais.pdf},
year = 1970,
urldate = {2012-10-31},
collections = {Easily explained,Things to make and do,Education}
}
@article{Khovanova2012,
abstract = {I present and discuss a puzzle about wizards invented by John H. Conway.},
archivePrefix = {arXiv},
arxivId = {1210.5460},
author = {Khovanova, Tanya},
eprint = {1210.5460},
month = {oct},
pages = 5,
title = {Conway's Wizards},
url = {http://arxiv.org/abs/1210.5460 http://arxiv.org/pdf/1210.5460v1},
year = 2012,
primaryClass = {math.HO},
urldate = {2012-11-03},
collections = {Puzzles,Easily explained}
}
@article{Alexander2012,
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.},
archivePrefix = {arXiv},
arxivId = {1212.0186},
author = {Alexander, Samuel},
eprint = {1212.0186},
month = {dec},
pages = 9,
title = {Biologically Unavoidable Sequences},
url = {http://arxiv.org/abs/1212.0186 http://arxiv.org/pdf/1212.0186v2},
volume = 63,
year = 2012,
primaryClass = {math.CO},
urldate = {2012-12-16},
collections = {Fun maths facts}
}
@article{Wilcox2006,
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{\'{a}}ndez-D{\'{i}}az 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{\'{a}}ndez-D{\'{i}}az 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.},
author = {Wilcox, Allen J},
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,
title = {Invited commentary: the perils of birth weight--a lesson from directed acyclic graphs.},
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},
collections = {Easily explained,Probability and statistics}
}
@article{Rothemund2004,
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.},
author = {Rothemund, Paul W K and Papadakis, Nick and Winfree, Erik},
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},
title = {Algorithmic self-assembly of DNA Sierpinski triangles.},
url = {http://dx.plos.org/10.1371/journal.pbio.0020424},
volume = 2,
year = 2004,
urldate = {2013-01-18}
}
@article{Bielczyk2012,
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.},
author = {Bielczyk, Natalia and Bodnar, Marek and Fory{\'{s}}, Urszula},
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},
title = {Delay can stabilize: Love affairs dynamics},
url = {http://dx.doi.org/10.1016/j.amc.2012.10.028},
volume = 219,
year = 2012,
urldate = {2013-01-19},
collections = {Easily explained,Modelling}
}
@article{Fiore2013,
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.},
author = {Fiore, Thomas M. and Noll, Thomas and Satyendra, Ramon},
month = {jan},
pages = 15,
title = {Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality},
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},
collections = {Music,The groups group}
}
@article{Davis2012,
author = {Davis, Tom},
pages = {1--21},
title = {Conway's Rational Tangles},
url = {http://www.geometer.org/mathcircles/tangle.pdf},
year = 2012,
urldate = {2013-02-06},
collections = {Easily explained,Things to make and do,Fun maths facts}
}
@article{Maymin2011,
author = {Maymin, PZ},
journal = {Algorithmic Finance},
title = {Markets are efficient if and only if P = NP},
url = {http://iospress.metapress.com/index/0WH736N18J682433.pdf},
volume = 2010,
year = 2011,
urldate = {2013-02-11}
}
@article{Galvin1993,
author = {Galvin, Fred},
journal = {American Mathematical Monthly},
number = 6,
pages = {578--580},
title = {Embedding countable groups in 2-generator groups},
url = {http://www.jstor.org/stable/10.2307/2324618},
volume = 100,
year = 1993,
urldate = {2013-02-19},
collections = {Fun maths facts,The groups group}
}
@article{Hughes2007,
author = {Hughes, Jesse},
title = {The Muddy Children : A logic for public announcement},
url = {http://phiwumbda.org/{\~{}}jesse/slides/MuddyChildren.handouts.pdf},
year = 2007,
urldate = {2013-02-19}
}
@article{Ghang2013,
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.},
archivePrefix = {arXiv},
arxivId = {1303.0885},
author = {Ghang, EK and Zeilberger, Doron},
eprint = {1303.0885},
isbn = 1111111111,
journal = {arXiv preprint arXiv:1303.0885},
month = {mar},
pages = {1--7},
title = {Zeroless Arithmetic: Representing Integers ONLY using ONE},
url = {http://arxiv.org/abs/1303.0885 http://arxiv.org/pdf/1303.0885v2},
year = 2013,
primaryClass = {math.CO},
urldate = {2013-03-07},
collections = {Unusual arithmetic,Easily explained,Integerology}
}
@article{Moller2009,
author = {M{\"{o}}ller, H},
journal = {math.uni-muenster.de},
pages = {1--32},
title = {Das 2: 3-Ei-ein praktikables Eimodell},
url = {http://www.math.uni-muenster.de/u/mollerh/data/ZweiDreiEi.pdf},
year = 2009,
urldate = {2013-03-19},
collections = {Art,Easily explained,Lists and catalogues,Animals,Food,Geometry,Modelling}
}
@article{Cherowitzo2006,
author = {Cherowitzo, WE},
journal = {To appear},
title = {Constructing the Tits ovoid from an elliptic quadric},
url = {http://www-math.ucdenver.edu/{\~{}}wcherowi/research/htmltalks/ovoids/ovoids.pdf},
year = 2006,
urldate = {2013-03-19},
collections = {Geometry}
}
@article{Kranakis2010,
author = {Kranakis, Evangelos and Krizanc, Danny},
journal = {Fun with Algorithms},
number = 1,
title = {The urinal problem},
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},
collections = {Puzzles,Easily explained}
}
@misc{Goucher,
author = {Goucher, Adam},
title = {Circuitry in 3D chess},
url = {http://cp4space.wordpress.com/2013/04/03/circuitry-in-3d-chess/},
urldate = {2013-04-03},
year = 2013
}
@article{Sauerland2013,
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.},
author = {Sauerland, Uli and Gotzner, Nicole},
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},
title = {Familial sinistrals avoid exact numbers.},
url = {http://dx.plos.org/10.1371/journal.pone.0059103},
volume = 8,
year = 2013,
urldate = {2013-04-15},
collections = {Easily explained,Probability and statistics}
}
@article{Vierling-Claassen2013,
author = {Vierling-Claassen, a.},
doi = {10.1177/1043463112473795},
issn = {1043-4631},
journal = {Rationality and Society},
month = {may},
number = 2,
pages = {198--228},
title = {Division of labor in child care: A game-theoretic approach},
url = {http://rss.sagepub.com/cgi/doi/10.1177/1043463112473795},
volume = 25,
year = 2013,
urldate = {2013-05-07},
collections = {Easily explained,Protocols and strategies}
}
@article{Coecke2005,
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.},
archivePrefix = {arXiv},
arxivId = {quant-ph/0510032},
author = {Coecke, Bob},
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},
title = {Kindergarten Quantum Mechanics},
url = {http://arxiv.org/abs/quant-ph/0510032},
year = 2005,
urldate = {2013-05-28}
}
@article{Popoff2013,
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.},
archivePrefix = {arXiv},
arxivId = {1305.7192},
author = {Popoff, Alexandre},
eprint = {1305.7192},
month = {may},
pages = 17,
title = {Using Monoidal Categories in the Transformational Study of Musical Time-Spans and Rhythms},
url = {http://arxiv.org/abs/1305.7192 http://arxiv.org/pdf/1305.7192v3},
year = 2013,
primaryClass = {math.GR},
urldate = {2013-06-02}
}
@article{Khovanova2013,
author = {Khovanova, Tanya},
pages = {1--4},
title = {A Line of Sages},
url = {http://tanyakhovanova.com/publications/ALineOfWizards.pdf},
year = 2013,
urldate = {2013-06-20},
collections = {Puzzles}
}
@article{Conrey2012,
abstract = {We give a new algorithm that quickly finds smooth neighbors.},
archivePrefix = {arXiv},
arxivId = {1212.5161},
author = {Conrey, Brian and Holmstrom, Mark and McLaughlin, Tara},
eprint = {1212.5161},
month = {dec},
pages = {1--14},
title = {Smooth neighbors},
url = {http://arxiv.org/abs/1212.5161 http://arxiv.org/pdf/1212.5161v1},
year = 2012,
primaryClass = {math.NT},
urldate = {2013-07-11},
collections = {Integerology}
}
@article{Lehmer1964,
author = {Lehmer, DH},
journal = {Illinois Journal of Mathematics},
title = {On a problem of St{\"{o}}rmer},
url = {http://projecteuclid.org/euclid.ijm/1256067456},
year = 1964,
urldate = {2013-07-11},
collections = {Easily explained,Fun maths facts}
}
@article{Haralambous1999,
author = {Haralambous, Yannis},
journal = {\ldots 14th International Unicode Conference, available from \ldots},
pages = {1--36},
title = {From Unicode to Typography, a Case Study the Greek Script},
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},
collections = {Basically computer science,Notation and conventions,History}
}
@article{Fouhey2013,
author = {Fouhey, David F and Maturana, Daniel},
title = {On n-Dimensional Polytope Schemes},
url = {http://www.oneweirdkerneltrick.com/polytope.pdf},
year = 2013,
urldate = {2013-08-02},
collections = {Geometry}
}
@article{Braswell2013a,
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.},
author = {Braswell, Leigh Marie and Khovanova, Tanya},
month = {may},
pages = 8,
title = {Cookie Monster Devours Naccis},
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},
collections = {Attention-grabbing titles,Puzzles,Animals,Food,Fibonaccinalia}
}
@article{Speyer2004a,
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.},
author = {Speyer, David E},
month = {feb},
title = {Perfect Matchings and the Octahedron Recurrence},
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}
}
@article{Buchholz2013,
author = {Buchholz, Ralph H and Rathbun, Randall L},
number = 2,
pages = {107--115},
title = {An Infinite Set of Heron Triangles with Two Rational Medians},
url = {http://www.jstor.org/stable/2974977},
volume = 104,
year = 2013,
urldate = {2013-10-09},
collections = {Geometry,Fun maths facts}
}
@article{Fomin2001,
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.},
author = {Fomin, Sergey and Zelevinsky, Andrei},
month = {apr},
pages = 21,
title = {The Laurent phenomenon},
url = {http://arxiv.org/abs/math.CO/0104241},
year = 2001,
urldate = {2013-10-09}
}
@article{Feinstein2010,
author = {Feinstein, JF and Heath, MJ},
journal = {Studia Math},
keywords = {and phrases,rational approximation,swiss cheeses,uniform algebras},
title = {Swiss cheeses, rational approximation and universal plane curves},
url = {http://www.maths.nottingham.ac.uk/personal/jff/Papers/pdf/Swisscheeses2.pdf},
year = 2010,
urldate = {2013-11-14},
collections = {Attention-grabbing titles,Food,Geometry}
}
@article{Bulteau2011,
author = {Bulteau, Laurent and Fertin, Guillaume and Rusu, Irena},
month = {nov},
title = {Pancake Flipping is Hard},
url = {http://arxiv.org/abs/1111.0434 http://arxiv.org/pdf/1111.0434v2},
year = 2011,
archivePrefix = {arXiv},
eprint = {1111.0434},
primaryClass = {cs.CC},
abstract = {Pancake Flipping is the problem of sorting a stack of pancakes of different
sizes (that is, a permutation), when the only allowed operation is to insert a
spatula anywhere in the stack and to flip the pancakes above it (that is, to
perform a prefix reversal). In the burnt variant, one side of each pancake is
marked as burnt, and it is required to finish with all pancakes having the
burnt side down. Computing the optimal scenario for any stack of pancakes and
determining the worst-case stack for any stack size have been challenges over
more than three decades. Beyond being an intriguing combinatorial problem in
itself, it also yields applications, e.g. in parallel computing and
computational biology. In this paper, we show that the Pancake Flipping
problem, in its original (unburnt) variant, is NP-hard, thus answering the
long-standing question of its computational complexity.},
urldate = {2013-11-15},
collections = {Attention-grabbing titles,Puzzles,Basically computer science,Easily explained,Food}
}
@article{Grau2011,
author = {Grau, Jos{\'{e}} Mar{\'{i}}a and Oller-Marc{\'{e}}n, Antonio M.},
month = {mar},
title = {Giuga Numbers and the arithmetic derivative},
url = {http://arxiv.org/abs/1103.2298 http://arxiv.org/pdf/1103.2298v1},
year = 2011,
archivePrefix = {arXiv},
eprint = {1103.2298},
primaryClass = {math.NT},
abstract = {We characterize Giuga Numbers as solutions to the equation {\$}n'=an+1{\$}, with {\$}a
\in \mathbb{\{}N{\}}{\$} and {\$}n'{\$} being the arithmetic derivative. Although this fact
does not refute Lava's conjecture, it brings doubts about its veracity.},
urldate = {2013-11-15},
collections = {Unusual arithmetic,Integerology}
}
@article{Sloane2013,
author = {Sloane, NJA},
journal = {arXiv preprint arXiv:1307.0453},
title = {2178 And All That},
url = {http://arxiv.org/abs/1307.0453 http://arxiv.org/pdf/1307.0453v4},
year = 2013,
archivePrefix = {arXiv},
eprint = {1307.0453},
primaryClass = {math.NT},
abstract = {For integers g >= 3, k >= 2, call a number N a (g,k)-reverse multiple if the
reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are
the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089
and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to
study the problem of finding all (g,k)-reverse multiples. By using modified
versions of her trees, which we call Young graphs, we determine the possible
values of k for bases g = 2 through 100, and then show how to apply the
transfer-matrix method to enumerate the (g,k)-reverse multiples with a given
number of base-g digits. These Young graphs are interesting finite directed
graphs, whose structure is not at all well understood.},
urldate = {2013-11-20},
collections = {Easily explained,Integerology}
}
@article{Mestrovic2011,
author = {Mestrovic, Romeo},
month = {nov},
title = {Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)},
url = {http://arxiv.org/abs/1111.3057 http://arxiv.org/pdf/1111.3057v2},
year = 2011,
archivePrefix = {arXiv},
eprint = {1111.3057},
primaryClass = {math.NT},
abstract = {In 1862 Wolstenholme proved that for any prime {\$}p\ge 5{\$} the numerator of the
fraction {\$}{\$} 1+\frac 12 +\frac 13+...+\frac{\{}1{\}}{\{}p-1{\}}
{\$}{\$} written in reduced form is divisible by {\$}p^2{\$}, {\$}(2){\$} and the numerator of
the fraction
{\$}{\$} 1+\frac{\{}1{\}}{\{}2^2{\}} +\frac{\{}1{\}}{\{}3^2{\}}+...+\frac{\{}1{\}}{\{}(p-1)^2{\}}
{\$}{\$} written in reduced form is divisible by {\$}p{\$}. The first of the above
congruences, the so called {\{}\it Wolstenholme's theorem{\}}, is a fundamental
congruence in combinatorial number theory. In this article, consisting of 11
sections, we provide a historical survey of Wolstenholme's type congruences and
related problems. Namely, we present and compare several generalizations and
extensions of Wolstenholme's theorem obtained in the last hundred and fifty
years. In particular, we present more than 70 variations and generalizations of
this theorem including congruences for Wolstenholme primes. These congruences
are discussed here by 33 remarks.
The Bibliography of this article contains 106 references consisting of 13
textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of
Integer Sequences. In this article, some results of these references are cited
as generalizations of certain Wolstenholme's type congruences, but without the
expositions of related congruences. The total number of citations given here is
189.},
urldate = {2013-11-21},
collections = {History,Integerology}
}
@article{Bell2008,
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.},
author = {Bell, George I.},
month = {jan},
pages = 11,
title = {The mathematics of Septoku},
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},
collections = {Puzzles,Easily explained}
}
@article{Middleton2013,
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.},
author = {Middleton, Chad A. and Langston, Michael},
month = {dec},
pages = 13,
title = {Circular orbits on a warped spandex fabric},
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},
collections = {Basically physics,Things to make and do,Geometry}
}
@article{Benjamin2003,
author = {Benjamin, AT and Hanusa, CRH and Su, FE},
journal = {Utilitas Mathematica},
pages = {3--17},
title = {Linear recurrences through tilings and Markov chains},
url = {http://math.hmc.edu/{\~{}}benjamin/papers/hanusa.pdf},
volume = 64,
year = 2003,
urldate = {2014-01-21}
}
@article{Hirsch2014,
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.},
author = {Hirsch, Dan and Patterson, Meredith L and Sandberg, Anders and Vejdemo-Johansson, Mikael},
month = {jan},
title = {More ties than we thought},
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},
collections = {Attention-grabbing titles,Easily explained}
}
@article{Reifegerste2003,
author = {Reifegerste, Astrid},
doi = {10.1016/S0195-6698(03)00065-9},
issn = 01956698,
journal = {European Journal of Combinatorics},
month = {aug},
number = 6,
pages = {759--776},
title = {On the diagram of 132-avoiding permutations},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0195669803000659},
volume = 24,
year = 2003,
urldate = {2014-03-31},
collections = {Combinatorics,The groups group}
}
@article{Demontigny2013,
author = {Demontigny, Philippe and Do, Thao and Kulkarni, Archit and Miller, Steven J. and Moon, David and Varma, Umang},
month = {sep},
title = {Generalizing Zeckendorf's Theorem to f-decompositions},
url = {http://arxiv.org/abs/1309.5599 http://arxiv.org/pdf/1309.5599v1},
year = 2013,
archivePrefix = {arXiv},
eprint = {1309.5599},
primaryClass = {math.NT},
abstract = {A beautiful theorem of Zeckendorf states that every positive integer can be
uniquely decomposed as a sum of non-consecutive Fibonacci numbers {\$}\{\{}F{\_}n\{\}}{\$},
where {\$}F{\_}1 = 1{\$}, {\$}F{\_}2 = 2{\$} and {\$}F{\_}{\{}n+1{\}} = F{\_}n + F{\_}{\{}n-1{\}}{\$}. For general
recurrences {\$}\{\{}G{\_}n\{\}}{\$} with non-negative coefficients, there is a notion of a
legal decomposition which again leads to a unique representation, and the
number of summands in the representations of uniformly randomly chosen {\$}m \in
[G{\_}n, G{\_}{\{}n+1{\}}){\$} converges to a normal distribution as {\$}n \to \infty{\$}.
We consider the converse question: given a notion of legal decomposition, is
it possible to construct a sequence {\$}\{\{}a{\_}n\{\}}{\$} such that every positive integer
can be decomposed as a sum of terms from the sequence? We encode a notion of
legal decomposition as a function {\$}f:\N{\_}0\to\N{\_}0{\$} and say that if {\$}a{\_}n{\$} is in
an "{\$}f{\$}-decomposition", then the decomposition cannot contain the {\$}f(n){\$} terms
immediately before {\$}a{\_}n{\$} in the sequence; special choices of {\$}f{\$} yield many
well known decompositions (including base-{\$}b{\$}, Zeckendorf and factorial). We
prove that for any {\$}f:\N{\_}0\to\N{\_}0{\$}, there exists a sequence
{\$}\{\{}a{\_}n\{\}}{\_}{\{}n=0{\}}^\infty{\$} such that every positive integer has a unique
{\$}f{\$}-decomposition using {\$}\{\{}a{\_}n\{\}}{\$}. Further, if {\$}f{\$} is periodic, then the unique
increasing sequence {\$}\{\{}a{\_}n\{\}}{\$} that corresponds to {\$}f{\$} satisfies a linear
recurrence relation. Previous research only handled recurrence relations with
no negative coefficients. We find a function {\$}f{\$} that yields a sequence that
cannot be described by such a recurrence relation. Finally, for a class of
functions {\$}f{\$}, we prove that the number of summands in the {\$}f{\$}-decomposition of
integers between two consecutive terms of the sequence converges to a normal
distribution.},
urldate = {2014-04-28},
collections = {Unusual arithmetic,Fibonaccinalia,Integerology}
}
@article{Khovanova2014,
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.},
author = {Khovanova, Tanya and Xiong, Joshua},
month = {may},
pages = 19,
title = {Nim Fractals},
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},
collections = {Games to play with friends,Combinatorics}
}
@article{Clifford2008,
abstract = {Ulam has defined a history-dependent random sequence by the recursion Xn+1=Xn+XU(n), where (U(n); n[{\&}ge;]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.},
author = {Clifford, P. and Stirzaker, D.},
issn = {1364-5021},
journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
month = {may},
number = 2093,
pages = {1105--1124},
title = {History-dependent random processes},
url = {http://rspa.royalsocietypublishing.org/content/464/2093/1105},
volume = 464,
year = 2008,
urldate = {2014-06-30}
}
@article{Petersen2004,
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.},
author = {Petersen, T. K. and Speyer, D.},
month = {jul},
pages = 25,
title = {An arctic circle theorem for groves},
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}
}
@article{Papadopoulos2014,
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.},
author = {Papadopoulos, Athanase},
month = {jul},
title = {Mathematics and group theory in music},
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},
collections = {Music,The groups group}
}
@article{Legendre2013,
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.},
author = {Legendre, St{\'{e}}phane},
month = {feb},
pages = 18,
title = {Foldings and Meanders},
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},
collections = {Easily explained,Geometry}
}
@article{Mahadevan1993,
author = {Mahadevan, L and Keller, JB},
journal = {Proceedings: Mathematical and Physical Sciences},
number = 1908,
pages = {149--162},
title = {The shape of a Mobius band},
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},
collections = {Geometry}
}
@article{Martin2013,
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.},
author = {Martin, Kimball and Shankar, Krishnan},
month = {may},
pages = 28,
title = {How often should you clean your room?},
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},
collections = {Easily explained}
}
@article{Bailey2008,
author = {Bailey, RA},
journal = {American Mathematical \ldots},
pages = {1--31},
title = {Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes},
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},
collections = {Puzzles}
}
@article{Hughes2006,
author = {Hughes, DJD},
journal = {Annals of Mathematics},
number = 3,
pages = {1065--1076},
title = {Proofs without syntax},
url = {http://www.jstor.org/stable/20160016},
volume = 164,
year = 2006,
urldate = {2014-10-09},
collections = {Notation and conventions,About proof}
}
@article{Richardson2014,
author = {Richardson, Thomas M.},
month = {oct},
title = {The Super Patalan Numbers},
url = {http://arxiv.org/abs/1410.5880 http://arxiv.org/pdf/1410.5880v1},
year = 2014,
archivePrefix = {arXiv},
eprint = {1410.5880},
primaryClass = {math.CO},
abstract = {We introduce the super Patalan numbers, a generalization of the super Catalan
numbers in the sense of Gessel, and prove a number of properties analagous to
those of the super Catalan numbers. The super Patalan numbers generalize the
super Catalan numbers similarly to how the Patalan numbers generalize the
Catalan numbers.},
urldate = {2014-10-23},
collections = {Attention-grabbing titles,Easily explained,Integerology}
}
@article{Bell2007,
author = {Bell, George I.},
month = {mar},
title = {Solving Triangular Peg Solitaire},
url = {http://arxiv.org/abs/math/0703865v6 http://arxiv.org/pdf/math/0703865v6},
year = 2007,
archivePrefix = {arXiv},
eprint = {math/0703865},
primaryClass = {math.CO},
abstract = {We consider the one-person game of peg solitaire on a triangular board of
arbitrary size. The basic game begins from a full board with one peg missing
and finishes with one peg at a specified board location. We develop necessary
and sufficient conditions for this game to be solvable. For all solvable
problems, we give an explicit solution algorithm. On the 15-hole board, we
compare three simple solution strategies. We then consider the problem of
finding solutions that minimize the number of moves (where a move is one or
more consecutive jumps by the same peg), and find the shortest solution to the
basic game on all triangular boards with up to 55 holes (10 holes on a side).},
urldate = {2014-11-10},
collections = {Puzzles,Fun maths facts}
}
@article{Miller2009a,
author = {Miller, Steven J. and Montague, David},
month = {sep},
title = {Irrationality From The Book},
url = {http://arxiv.org/abs/0909.4913 http://arxiv.org/pdf/0909.4913v2},
year = 2009,
archivePrefix = {arXiv},
eprint = {0909.4913},
primaryClass = {math.HO},
abstract = {We generalize Tennenbaum's geometric proof of the irrationality of sqrt(2) to
sqrt(n) for n = 3, 5, 6 and 10.},
urldate = {2014-12-01},
collections = {About proof,Fun maths facts}
}
@article{McGuire2012,
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.},
author = {McGuire, Gary},
month = {mar},
title = {Bells, Motels and Permutation Groups},
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},
collections = {Easily explained,Music,Fun maths facts,Combinatorics,The groups group}
}
@article{Arthan2004,
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.},
author = {Arthan, R. D.},
month = {may},
pages = 15,
title = {The Eudoxus Real Numbers},
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},
collections = {Unusual arithmetic}
}
@article{Loregian2015,
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.},
author = {Loregian, Fosco},
month = {jan},
pages = 39,
title = {This is the (co)end, my only (co)friend},
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},
collections = {Attention-grabbing titles}
}
@article{Feldman1993,
author = {Feldman, D and Impagliazzo, R and Naor, M},
journal = {Information and \ldots},
title = {On dice and coins: Models of computation for random generation},
url = {http://www.sciencedirect.com/science/article/pii/S089054018371028X},
year = 1993,
urldate = {2015-01-26},
collections = {Easily explained,Things to make and do,Modelling}
}
@article{Gouyou-Beauchamps2005,
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.},
author = {Gouyou-Beauchamps, Dominique and Leroux, Pierre},
issn = 03043975,
journal = {Theoretical Computer Science},
month = {nov},
number = {2-3},
pages = {307--334},
title = {Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice},
url = {http://www.sciencedirect.com/science/article/pii/S0304397505004950},
volume = 346,
year = 2005,
urldate = {2015-02-02},
collections = {Geometry,Combinatorics}
}
@misc{Calude2006,
author = {Calude, C.S and Dinneen, Michael},
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},
title = {Exact Approximations of Omega Numbers},
url = {https://researchspace.auckland.ac.nz/handle/2292/3800},
year = 2006,
urldate = {2015-02-03}
}
@article{Ahlbach2012,
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.},
author = {Ahlbach, Connor and Usatine, Jeremy and Pippenger, Nicholas},
month = {jul},
title = {Efficient Algorithms for Zeckendorf Arithmetic},
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},
collections = {Easily explained,Fun maths facts,Integerology}
}
@article{Oller2007,
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.},
author = {Oller, Antonio M.},
month = {oct},
pages = 8,
title = {The dying rabbit problem revisited},
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},
collections = {Easily explained,Animals,Fibonaccinalia}
}
@article{Chvatal1975,
author = {Chv{\'{a}}tal, V},
issn = 00958956,
journal = {Journal of Combinatorial Theory, Series B},
month = {feb},
number = 1,
pages = {39--41},
title = {A combinatorial theorem in plane geometry},
url = {http://www.sciencedirect.com/science/article/pii/0095895675900611},
volume = 18,
year = 1975,
urldate = {2015-02-23},
collections = {Easily explained,Geometry,Fun maths facts,Combinatorics}
}
@article{Golumbic1998,
author = {Golumbic, Martin Charles and Wassermann, Amir},
issn = {0911-0119},
journal = {Graphs and Combinatorics},
month = {aug},
number = 3,
pages = {223--239},
title = {Complexity and Algorithms for Graph and Hypergraph Sandwich Problems},
url = {http://link.springer.com/10.1007/s003730050028},
volume = 14,
year = 1998,
urldate = {2015-02-24}
}
@article{Cotogno2009,
author = {Cotogno, Paolo},
issn = {0924-6495},
journal = {Minds and Machines},
month = {oct},
number = 3,
pages = {391--405},
title = {A Brief Critique of Pure Hypercomputation},
url = {http://link.springer.com/10.1007/s11023-009-9161-7},
volume = 19,
year = 2009,
urldate = {2015-03-05},
collections = {Basically computer science}
}
@article{Anderson2015,
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.},
author = {Anderson, Ross and Ashlagi, Itai and Gamarnik, David and Roth, Alvin E.},
issn = {0027-8424},
journal = {Proceedings of the National Academy of Sciences},
month = {jan},
number = 3,
pages = 201421853,
title = {Finding long chains in kidney exchange using the traveling salesman problem},
url = {http://www.pnas.org/content/112/3/663},
volume = 112,
year = 2015,
urldate = {2015-03-07},
collections = {Basically computer science,Protocols and strategies}
}
@article{Mugford2001,
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.},
author = {Mugford, S. T.},
issn = 14657279,
journal = {Behavioral Ecology},
month = {nov},
number = 6,
pages = {655--658},
title = {The accuracy of Buffon's needle: a rule of thumb used by ants to estimate area},
url = {http://beheco.oxfordjournals.org/content/12/6/655.full},
volume = 12,
year = 2001,
urldate = {2015-03-09},
collections = {Animals,Probability and statistics}
}
@article{Ferrie2015,
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.},
author = {Ferrie, Christopher and Kueng, Richard},
month = {mar},
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},
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},
collections = {Attention-grabbing titles,Probability and statistics}
}
@article{Schumacher2004,
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.},
author = {Schumacher, B. and Werner, R. F.},
month = {may},
pages = 20,
title = {Reversible quantum cellular automata},
url = {http://arxiv.org/abs/quant-ph/0405174},
year = 2004,
urldate = {2015-06-28},
collections = {Basically computer science}
}
@article{Gerogiorgakis2015,
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—named CP—whose 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.},
author = {Gerogiorgakis, Stamatios},
issn = {0144-5340},
journal = {History and Philosophy of Logic},
language = {en},
month = {jun},
pages = {1--13},
publisher = {Taylor {\&} Francis},
title = {Mind the Croc! Rationality Gaps vis-{\`{a}}-vis the Crocodile Paradox},
url = {http://www.tandfonline.com/doi/full/10.1080/01445340.2015.1046211{\#}.VegVv{\_}lViHg},
year = 2015,
urldate = {2015-09-03},
collections = {Attention-grabbing titles,Animals}
}
@article{Diaconis1989a,
author = {Diaconis, Persi and Keller, Joseph B},
journal = {The American Mathematical Monthly},
number = 4,
pages = {337--339},
title = {Fair Dice},
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},
collections = {Games to play with friends,Probability and statistics}
}
@article{Muller2013,
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.},
author = {M{\"{u}}ller, Peter},
month = {nov},
pages = 2,
title = {Another Proof of Segre's Theorem about Ovals},
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},
collections = {About proof}
}
@article{Rechnitzer2003,
abstract = {Guttmann and Enting [Phys. Rev. Lett. 76 (1996) 344–347] 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 wide range of solved and unsolved families of bond animals, we show that the coefficients of yn is rational, the degree of its numerator is at most that of its denominator, and the denominator is a product 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.},
author = {Rechnitzer, Andrew},
issn = 01968858,
journal = {Advances in Applied Mathematics},
month = {feb},
number = {1-2},
pages = {228--257},
title = {Haruspicy and anisotropic generating functions},
url = {http://www.sciencedirect.com/science/article/pii/S0196885802005341},
volume = 30,
year = 2003,
urldate = {2015-11-08}
}
@article{Blote1984,
author = {Blote, H W J and Hilhorst, H J},
issn = {0305-4470},
journal = {Journal of Physics A: Mathematical and General},
language = {en},
month = {feb},
number = 3,
pages = {L111--L115},
publisher = {IOP Publishing},
title = {Spiralling self-avoiding walks: an exact solution},
url = {http://iopscience.iop.org/article/10.1088/0305-4470/17/3/004},
volume = 17,
year = 1984,
urldate = {2015-11-08}
}
@article{Katz2010,
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.},
author = {Katz, Karin Usadi and Katz, Mikhail G.},
month = {jul},
pages = 28,
title = {When is .999... less than 1?},
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},
collections = {Unusual arithmetic}
}
@article{Sanders2015,
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.},
author = {Sanders, Sam},
month = {nov},
title = {The effective content of Reverse Nonstandard Mathematics and the nonstandard content of effective Reverse Mathematics},
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},
collections = {Attention-grabbing titles,About proof}
}
@article{Gordinowicz2015,
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.},
author = {Gordinowicz, Przemys{\l}aw},
issn = 03043975,
journal = {Theoretical Computer Science},
month = {aug},
pages = {160--164},
title = {Planar graph is on fire},
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},
collections = {Attention-grabbing titles,Puzzles,Fun maths facts}
}
@article{Diaz2003,
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.},
author = {Diaz, Rodolfo A. and Gonzalez, Diego L. and Marin, Francisco and Martinez, R.},
month = {oct},
pages = 16,
title = {Comparative kinetics of the snowball respect to other dynamical objects},
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},
collections = {Basically physics}
}
@misc{BlancoAbellan2015,
abstract = {Postprint (author's final draft)},
author = {Blanco Abell{\'{a}}n, M{\'{o}}nica},
booktitle = {Bulletin of the Scientific Instrument Society},
issn = {0956-8271},
keywords = {Mathematical instruments,{\`{A}}rees tem{\`{a}}tiques de la UPC::Matem{\`{a}}tiques i estad{\'{i}}s},
language = {eng},
pages = {22--28},
title = {On gardeners, dukes and mathematical instruments},
url = {http://upcommons.upc.edu/handle/2117/78617},
volume = 125,
year = 2015,
urldate = {2016-01-18},
collections = {History}
}
@article{Hung,
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.},
author = {Hung, J.W.Nienhuys (Ling-Ju and Eds.), Ton Kloks},
pages = 192,
title = {De Bruijn's Combinatorics},
url = {http://vixra.org/abs/1208.0223},
urldate = {2016-02-05},
year = 2012,
collections = {Combinatorics}
}
@article{Uspensky1994,
title = {G{\"{o}}del's incompleteness theorem},
author = {Uspensky, V},
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},
collections = {Fun maths facts}
}
@article{FIBONACCIJIGSAWPUZZLE,
title = {Fibonacci Jigsaw Puzzle},
url = {http://starcage.org/fibonacci{\_}puzzle/fibonacci{\_}puzzle.html},
author = {Akio Hizume},
comment = {},
urldate = {2016-05-19},
collections = {Puzzles,Easily explained,Fibonaccinalia},
year = 2016
}
@article{Bernhardsson2011,
title = {A Paradoxical Property of the Monkey Book},
author = {Bernhardsson, Sebastian and Baek, Seung Ki and Minnhagen, Petter},
url = {http://arxiv.org/abs/1103.2681 http://arxiv.org/pdf/1103.2681v1.pdf},
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.},
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},
collections = {Animals}
}
@article{Culbertson2012,
title = {A categorical foundation for Bayesian probability},
author = {Culbertson, Jared and Sturtz, Kirk},
url = {http://arxiv.org/abs/1205.1488 http://arxiv.org/pdf/1205.1488v3},
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.},
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},
collections = {Probability and statistics}
}
@article{DismalArithmetic,
title = {Dismal Arithmetic},
abstract = {Dismal arithmetic is just like the arithmetic you learned in school, only
simpler: there are no carries, when you add digits you just take the largest,
and when you multiply digits you take the smallest. This paper studies basic
number theory in this world, including analogues of the primes, number of
divisors, sum of divisors, and the partition function.},
url = {http://arxiv.org/abs/1107.1130v2 http://arxiv.org/pdf/1107.1130v2},
author = {David Applegate and Marc LeBrun and N. J. A. Sloane},
comment = {},
urldate = {2016-05-19},
archivePrefix = {arXiv},
eprint = {1107.1130},
primaryClass = {math.NT},
collections = {Unusual arithmetic,Easily explained,Integerology},
year = 2011
}
@article{TwoNotesOnNotation,
title = {Two notes on notation},
abstract = {The author advocates two specific mathematical notations from his popular
course and joint textbook, "Concrete Mathematics". The first of these,
extending an idea of Iverson, is the notation "[P]" for the function which is 1
when the Boolean condition P is true and 0 otherwise. This notation can
encourage and clarify the use of characteristic functions and Kronecker deltas
in sums and integrals.
The second notation puts Stirling numbers on the same footing as binomial
coefficients. Since binomial coefficients are written on two lines in
parentheses and read "n choose k", Stirling numbers of the first kind should be
written on two lines in brackets and read "n cycle k", while Stirling numbers
of the second kind should be written in braces and read "n subset k". (I might
say "n partition k".) The written form was first suggested by Imanuel Marx. The
virtues of this notation are that Stirling partition numbers frequently appear
in combinatorics, and that it more clearly presents functional relations
similar to those satisfied by binomial coefficients.},
url = {http://arxiv.org/abs/math/9205211v1 http://arxiv.org/pdf/math/9205211v1},
author = {Donald E. Knuth},
comment = {},
urldate = {2016-05-19},
archivePrefix = {arXiv},
eprint = {math/9205211},
primaryClass = {math.HO},
collections = {Notation and conventions},
year = 1992
}
@article{OntheCookieMonsterProblem,
title = {On the Cookie Monster Problem},
abstract = {The Cookie Monster Problem supposes that the Cookie 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 of a set is the minimum number of moves
the 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 jars
containing cookies in the Fibonacci, Tribonacci, n-nacci, and Super-n-nacci
sequences. We also construct sequences of k jars such that their Cookie Monster
numbers are asymptotically rk, where r is any real number between 0 and 1
inclusive.},
url = {http://arxiv.org/abs/1309.5985v1 http://arxiv.org/pdf/1309.5985v1},
author = {Leigh Marie Braswell and Tanya Khovanova},
comment = {},
urldate = {2016-05-19},
archivePrefix = {arXiv},
eprint = {1309.5985},
primaryClass = {math.HO},
collections = {Puzzles,Animals,Food,Fibonaccinalia,Fun maths facts},
year = 2013
}
@article{Brumleve2012,
title = {The mate-in-n problem of infinite chess is decidable},
author = {Brumleve, Dan and Hamkins, Joel David and Schlicht, Philipp},
url = {http://arxiv.org/abs/1201.5597 http://arxiv.org/pdf/1201.5597v4},
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.},
archivePrefix = {arXiv},
arxivId = {1201.5597},
eprint = {1201.5597},
journal = {New York},
month = {jan},
pages = 10,
year = 2012,
urldate = {2012-03-28},
collections = {Puzzles,Games to play with friends,Fun maths facts}
}
@article{Xu2012,
title = {Survey on fusible numbers},
author = {Xu, Junyan},
url = {http://arxiv.org/abs/1202.5614 http://arxiv.org/pdf/1202.5614v1},
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.},
archivePrefix = {arXiv},
arxivId = {1202.5614},
eprint = {1202.5614},
month = {feb},
pages = 10,
year = 2012,
urldate = {2012-05-02},
collections = {Easily explained}
}
@article{Balbiani2008,
title = {‘Knowable' As ‘Known After an Announcement'},
author = {Balbiani, Philippe and Baltag, Alexandru and Ditmarsch, Hans Van and Herzig, Andreas and Hoshi, Tomohiro and De Lima, Tiago},
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}
}
@article{Galperin2003,
title = {Playing pool with {\$}\pi{\$} (the number {\$}\pi{\$} from a billiard point of view)},
author = {Galperin, G},
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},
collections = {Easily explained,Fun maths facts}
}
@article{Hata1993,
title = {Rational approximations to {\$}\pi{\$} and some other numbers},
author = {Hata, Masayoshi and Mignotte, M and Chudnovsky, G V and Beukers, F},
url = {http://matwbn.icm.edu.pl/ksiazki/aa/aa63/aa6344.pdf},
journal = {Acta Arithmetica},
number = 4,
volume = 63,
year = 1993,
urldate = {2015-02-16},
collections = {Easily explained}
}
@article{Chen2010,
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},
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},
collections = {Basically computer science}
}
@article{Marasco2011,
title = {Doc, What Are My Chances?},
author = {Marasco, Joe and Doerfler, Ron and Roschier, Leif},
url = {http://www.myreckonings.com/modernnomogramsTMP/Doc{\_}What{\_}Are{\_}My{\_}Chances{\_}UMAP{\_}32-4-2011.pdf},
number = 4,
pages = {279--298},
volume = 32,
year = 2011,
urldate = {2012-03-11},
collections = {Attention-grabbing titles,Probability and statistics}
}
@article{Barany2011,
title = {Chalk : Materials and Concepts in Mathematics Chalk in Hand},
author = {Barany, Michael J and Mackenzie, Donald},
url = {http://www.sps.ed.ac.uk/{\_}{\_}data/assets/file/0020/60518/Chalk.pdf},
pages = {1--30},
year = 2011,
urldate = {2011-11-01},
collections = {Notation and conventions,The act of doing maths}
}
@article{Renteln,
title = {Foolproof : A Sampling of Mathematical Folk Humor},
author = {Renteln, Paul and Dundes, Alan},
url = {http://www.ams.org/notices/200501/fea-dundes.pdf},
journal = {Physics},
urldate = {2010-06-18},
collections = {Lists and catalogues},
year = 2005
}
@misc{Scarle2008,
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},
url = {http://www.ncbi.nlm.nih.gov/pubmed/19577519 http://research.microsoft.com/pubs/79271/turing.pdf},
year = 2008,
urldate = {2010-09-30},
collections = {Basically computer science,Basically physics,Unusual computers}
}
@article{Flajolet2011,
title = {On Buffon Machines and Numbers},
author = {Flajolet, Philippe},
url = {http://arxiv.org/abs/0906.5560 http://arxiv.org/pdf/0906.5560v2},
journal = {Bernoulli},
pages = {1--12},
year = 2011,
urldate = {2011-01-12},
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.},
archivePrefix = {arXiv},
eprint = {0906.5560},
collections = {Easily explained,Probability and statistics}
}
@article{West2002,
title = {Better approximations to cumulative normal functions},
author = {West, Graeme},
url = {http://www.codeplanet.eu/files/download/accuratecumnorm.pdf},
journal = {Wilmott},
pages = {70--76},
year = 2002,
urldate = {2011-01-22},
collections = {Probability and statistics}
}
@article{Tondering2005,
title = {Surreal Numbers – An Introduction},
author = {T{\o}ndering, Claus},
url = {http://www.tondering.dk/download/sur16.pdf},
number = {January},
year = 2005,
urldate = {2011-02-16},
collections = {Unusual arithmetic}
}
@article{Ellermeyer2008,
title = {James Garfield's Proof of the Pythagorean Theorem},
author = {Ellermeyer, S F},
url = {http://math.kennesaw.edu/{\~{}}sellerme/sfehtml/classes/math1112/garfieldpro.pdf},
pages = {1--4},
year = 2008,
urldate = {2011-02-18},
collections = {History,About proof}
}
@article{Gerh2010,
title = {Automatic calculation of plane loci using Grobner bases and integration into a Dynamic Geometry System},
author = {Gerh, Michael},
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},
collections = {Basically computer science}
}
@article{Figueira2002,
title = {An example of a computable absolutely normal number},
author = {Figueira, Santiago},
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},
collections = {Fun maths facts}
}
@article{Aludaat2008,
title = {A Note on Approximating the Normal Distribution Function},
author = {Aludaat, K M and Alodat, M T},
url = {http://www.m-hikari.com/ams/ams-password-2008/ams-password9-12-2008/alodatAMS9-12-2008-2},
keywords = {cumulative distribution function,normal distribution},
number = 9,
pages = {425 -- 429},
volume = 2,
year = 2008,
urldate = {2011-03-28}
}
@article{Chandler,
title = {Testing Petri Nets for Mobile Robots Using Gr{\"{o}}bner Bases},
author = {Chandler, Angie and Heyworth, Anne and Blair, Lynne and Seward, Derek},
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},
collections = {Basically computer science},
year = 2000
}
@article{Owens,
title = {Investigations of Game of Life cellular automata rules on Penrose Tilings : lifetime and ash statistics},
author = {Owens, Nick and Stepney, Susan},
url = {https://www-users.cs.york.ac.uk/susan/bib/ss/nonstd/auto08b.pdf},
pages = {1--34},
urldate = {2011-08-14},
collections = {Probability and statistics},
year = 2010
}
@article{Bacchus1995,
title = {Against Conditionalization},
author = {Bacchus, Fahiem},
url = {http://www.cs.toronto.edu/{\~{}}fbacchus/Papers/BKTSYN90.pdf},
year = 1995,
urldate = {2011-08-28},
collections = {Probability and statistics}
}
@article{Miller2009,
title = {Irrationality from the book},
author = {Miller, Steven J and Montague, David},
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},
collections = {Easily explained}
}
@article{Shamos2011,
title = {Shamos's Catalog of the Real Numbers},
author = {Shamos, Michael Ian},
url = {http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf},
booktitle = {Science},
isbn = 0671750615,
year = 2011,
urldate = {2011-09-08},
collections = {Lists and catalogues}
}
@article{Mccarthy2006,
title = {Mad Abel : A card game for 2 + players},
author = {Mccarthy, Sm{\'{a}}ri},
url = {https://www.pagat.com/docs/madabel.pdf},
pages = {3--7},
year = 2006,
urldate = {2011-09-13},
collections = {Unusual arithmetic,Easily explained,Games to play with friends,The groups group}
}
@article{Erickson,
title = {Fusible Numbers},
author = {Erickson, Jeff},
url = {http://www.mathpuzzle.com/fusible.pdf},
urldate = {2011-09-21},
collections = {Unusual arithmetic,Easily explained},
year = 2010
}
@article{Khovanova2010,
title = {Baron Munchhausen Redeems Himself : Bounds for a Coin-Weighing Puzzle Background},
author = {Khovanova, Tanya and Lewis, Joel Brewster},
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},
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.},
collections = {Attention-grabbing titles,Puzzles}
}
@article{Brenton2008,
title = {Remainder Wheels and Group Theory},
author = {Brenton, Lawrence},
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},
collections = {Unusual arithmetic,Easily explained,The groups group}
}
@article{Demaine2008,
title = {Tetris is Hard, Even to Approximate},
author = {Demaine, Erik D and Hohenberger, Susan and Liben-Nowell, David},
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},
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.},
collections = {Attention-grabbing titles,Computational complexity of games}
}
@article{Lane,
title = {What Are the Odds?},
author = {Lane, F.C.},
url = {http://library.la84.org/SportsLibrary/BBM/1919/bbm226k.pdf},
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},
keywords = {1919,Iss. 6,Vol. 22,pgs.337-339},
pages = {337--339},
urldate = {2011-12-09},
collections = {Games to play with friends,Probability and statistics},
year = 1919
}
@article{Appel2002,
title = {Deobfuscation is in NP},
author = {Appel, Andrew W},
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},
collections = {Basically computer science}
}
@article{Gerdes2000,
title = {Ethnomathematics as a new research field , illustrated by studies of mathematical ideas in African history},
author = {Gerdes, Paulus},
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},
collections = {History,Art}
}
@article{Deuber2004,
title = {A note on paradoxical metric spaces},
author = {Deuber, W A and Simonovits, M and Os, V T S},
url = {http://renyi.hu/{\~{}}miki/walter07.pdf},
number = 1995,
pages = {17--23},
volume = 30,
year = 2004,
urldate = {2013-01-26}
}
@article{Dijkstra1981,
title = {Fibonacci numbers and Leonardo numbers},
author = {Dijkstra, E.W.},
url = {https://www.cs.utexas.edu/users/EWD/transcriptions/EWD07xx/EWD797.html},
pages = {1--8},
year = 1981,
urldate = {2013-11-19},
collections = {Easily explained,Fibonaccinalia,Integerology}
}
@article{Hampton2009,
title = {A Mathematical Coloring Book},
author = {Hampton, Marshall},
url = {http://www.d.umn.edu/{\~{}}mhampton/mathcolor17b.pdf},
year = 2009,
urldate = {2014-04-24},
collections = {Art,Easily explained,Things to make and do}
}
@article{Bialostocki1998,
title = {An Application of Elementary Group Theory to Central Solitaire},
author = {Bialostocki, Arie},
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}
}
@article{Casazza,
title = {A mathematician's survival guide},
author = {Casazza, Peter G},
url = {https://faculty.missouri.edu/{\~{}}casazzap/pdf/140-MAA.pdf},
pages = {1--18},
urldate = {2012-04-29},
collections = {The act of doing maths},
year = 2012
}
@article{Demaine2010,
title = {Circle Packing for Origami Design Is Hard},
author = {Demaine, E.D. and Fekete, S.P. and Lang, R.J.},
url = {http://arxiv.org/abs/1008.1224 http://arxiv.org/pdf/1008.1224v2},
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.},
journal = {Arxiv preprint arXiv:1008.1224},
pages = {1--17},
publisher = {arxiv.org},
year = 2010,
urldate = {2010-08-13},
archivePrefix = {arXiv},
eprint = {1008.1224},
collections = {Geometry}
}
@article{CountingGroups,
title = {Counting groups: gnus, moas and other exotica},
abstract = {The number of groups of a given order is a fascinating function. We report on
its known values, discuss some of its properties, and study some related functions.},
url = {https://www.math.auckland.ac.nz/{\~{}}obrien/research/gnu.pdf},
author = {John H. Conway and Heiko Dietrich and E.A. O’Brien},
comment = {},
urldate = {2016-05-20},
collections = {Attention-grabbing titles,Animals,The groups group},
year = 2008
}
@article{LostMathematics,
title = {Lectures on lost mathematics},
author = {Branko Gr{\"{u}}nbaum},
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},
collections = {History,The act of doing maths}
}
@article{Poloblanco2007,
title = {Theory and History of Geometric Models},
author = {Polo-blanco, Irene},
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},
collections = {History,Geometry}
}
@article{OnPellegrinos20CapsinS43,
title = {On Pellegrino's 20-Caps in {\$}S{\_}{\{}4,3{\}}{\$}},
author = {R. Hill},
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},
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 = {},
urldate = {2016-06-01},
year = 1983
}
@article{ChallengingProblems,
title = {Challenging mathematical problems with elementary solutions},
abstract = {},
url = {https://drive.google.com/file/d/0B45juCGJ7U7PNmJhY2M0ZTMtNzU1MC00NGU3LTg3NGItMzE3OTViOThiYmU3/view},
author = {A.M. Yaglom and I.M. Yaglom},
comment = {},
urldate = {2016-06-08},
collections = {Puzzles},
year = 1964
}
@article{DrMitchillandtheMathematicalTetrodonThePublicDomainReview,
title = {Dr Mitchill and the Mathematical Tetrodon},
abstract = {},
url = {http://publicdomainreview.org/2015/09/16/dr-mitchill-and-the-mathematical-tetrodon/},
author = {Kevin Dann},
comment = {},
urldate = {2016-06-13},
collections = {Attention-grabbing titles,Art,Animals},
year = 2015
}
@article{Dudley,
title = {What to do when the trisector comes},
author = {Dudley, Underwood},
url = {http://web.mst.edu/{\~{}}lmhall/WhatToDoWhenTrisectorComes.pdf},
urldate = {2015-11-30},
collections = {Attention-grabbing titles,The act of doing maths},
year = 1983
}
@article{Thesnaillemma,
title = {The snail lemma},
abstract = {The classical snake lemma produces a six terms exact sequence starting from
a commutative square with one of the edge being a regular epimorphism. We establish
a new diagram lemma, that we call snail lemma, removing such a condition. We also
show that the snail lemma subsumes the snake lemma and we give an interpretation of
the snail lemma in terms of strong homotopy kernels. Our results hold in any pointed
regular protomodular category.},
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},
author = {Enrico M. Vitale},
comment = {},
urldate = {2016-06-13},
collections = {Attention-grabbing titles,Animals},
year = 2016
}
@article{ContinuedLogarithms,
title = {Continued Logarithms And Associated Continued Fractions},
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’s constant is believed not to be related to any naturally occurring number, we find surprisingly that the logarithmic Khintchine numbers are elementary.},
url = {https://www.carma.newcastle.edu.au/jon/clogs.pdf},
author = {Jonathan M. Borwein and Neil J. Calkin and Scott B. Lindstrom and Andrew Mattingly},
comment = {},
urldate = {2016-06-15},
year = 2016
}
@article{FuzzyGeometry,
title = {Fuzzy plane geometry I: Points and lines },
abstract = {We introduce a comprehensive study of fuzzy geometry in this paper by first defining a fuzzy point and a fuzzy line
in fuzzy plane geometry. We consider the fuzzy distance between fuzzy points and show it is a (weak) fuzzy metric.
We study various definitions of a fuzzy line, develop their basic properties, and investigate parallel fuzzy lines. },
url = {http://www.caam.rice.edu/{\~{}}yad1/miscellaneous/References/Math/Topology/Fuzzy/Fuzzy{\%}20plane{\%}20geometry{\%}20I{\%}20Points{\%}20and{\%}20lines.pdf?platform=hootsuite},
author = {J.J. Buckley and E. Aslami},
comment = {},
urldate = {2016-06-17},
collections = {Probability and statistics,Geometry},
year = 1997
}
@article{Yato2003,
title = {Complexity and Completeness of Finding Another solution and its Application to Puzzles},
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{\$}
solutions 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{\$}.
In 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
solvable in polynomial time.
As 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.},
url = {http://www-imai.is.s.u-tokyo.ac.jp/{\~{}}yato/data2/MasterThesis.pdf},
author = {Takayushi Yato},
comment = {},
urldate = {2016-06-18},
collections = {Puzzles,Basically computer science,About proof},
year = 2003
}
@article{AnIrrationalityMeasureforRegularPaperfoldingNumbers,
title = {An Irrationality Measure for Regular Paperfolding Numbers},
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
\[ \mu(F(1/b)) \leq 275331112987/137522851840 = 2.002075359 \ldots \]
for all integers {\$}b \geq 2{\$}. This improves upon the previous bound of {\$}\mu(F(1/b)) \leq 5{\$} given by Adamczewski and Rivoal.},
url = {https://cs.uwaterloo.ca/journals/JIS/VOL15/Coons/coons3.html https://cs.uwaterloo.ca/journals/JIS/VOL15/Coons/coons3.pdf},
author = {Michael Coons and Paul Vrbik},
comment = {},
urldate = {2016-07-11},
collections = {Things to make and do,Integerology},
year = 2012
}
@article{RationalPolynomialsThatTakeIntegerValuesattheFibonacciNumbers,
title = {Rational Polynomials That Take Integer Values at the Fibonacci Numbers},
author = {Keith Johnson and Kira Scheibelhut},
url = {http://www.jstor.org/stable/10.4169/amer.math.monthly.123.4.338},
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 = {},
urldate = {2016-08-02},
year = 2016,
collections = {Fibonaccinalia,Fun maths facts,Integerology}
}
@article{TenLessonsRota,
title = {Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations},
abstract = {},
url = {https://web.williams.edu/Mathematics/lg5/Rota.pdf},
author = {Giancarlo Rota},
comment = {An entertainingly indignant rant about the way DEs are taught.},
urldate = {2016-08-03},
collections = {Attention-grabbing titles,The act of doing maths},
year = 1997
}
@book{MattersComputational,
title = {Matters Computational - Ideas, Algorithms, Source Code},
author = {J{\"{o}}rg Arndt},
url = {http://www.jjj.de/fxt/fxtbook.pdf http://www.jjj.de/fxt/{\#}fxtbook},
abstract = {This is the book "Matters Computational" (formerly titled "Algorithms for Programmers"), published with Springer.},
comment = {Lots and lots of algorithms!},
urldate = {2016-08-03},
collections = {Basically computer science},
year = 2010
}
@article{Thedenominatorsofconvergentsforcontinuedfractions,
title = {The denominators of convergents for continued fractions},
abstract = {For any real number {\$}x \in [0,1){\$}, we denote by {\$}q{\_}n(x){\$} the denominator of
the {\$}n{\$}-th convergent of the continued fraction expansion of {\$}x{\$} {\$}(n \in
\mathbb{\{}N{\}}){\$}. It is well-known that the Lebesgue measure of the set of points
{\$}x \in [0,1){\$} for which {\$}\log q{\_}n(x)/n{\$} deviates away from {\$}\pi^2/(12\log2){\$}
decays to zero as {\$}n{\$} tends to infinity. In this paper, we study the rate of
this decay by giving an upper bound and a lower bound. What is interesting is
that the upper bound is closely related to the Hausdorff dimensions of the
level sets for {\$}\log q{\_}n(x)/n{\$}. As a consequence, we obtain a large deviation
type result for {\$}\log q{\_}n(x)/n{\$}, which indicates that the rate of this decay is
exponential.},
url = {http://arxiv.org/abs/1608.01246v1 http://arxiv.org/pdf/1608.01246v1},
author = {Lulu Fang and Min Wu and Bing Li},
comment = {},
urldate = {2016-08-06},
archivePrefix = {arXiv},
eprint = {1608.01246},
primaryClass = {math.NT},
year = 2016
}
@article{BeckettGrayCodes,
title = {Beckett-Gray Codes},
abstract = {In this paper we discuss a natural mathematical structure that is derived
from Samuel Beckett's play "Quad". This structure is called a binary
Beckett-Gray code. Our goal is to formalize the definition of a binary
Beckett-Gray code and to present the work done to date. In addition, we
describe the methodology used to obtain enumeration results for binary
Beckett-Gray codes of order {\$}n = 6{\$} and existence results for binary
Beckett-Gray codes of orders {\$}n = 7,8{\$}. We include an estimate, using Knuth's
method, for the size of the exhaustive search tree for {\$}n=7{\$}. Beckett-Gray
codes can be realized as successive states of a queue data structure. We show
that the binary reflected Gray code can be realized as successive states of two
stack data structures.},
url = {http://arxiv.org/abs/1608.06001v1 http://arxiv.org/pdf/1608.06001v1},
author = {Mark Cooke and Chris North and Megan Dewar and Brett Stevens},
comment = {},
urldate = {2016-08-24},
archivePrefix = {arXiv},
eprint = {1608.06001},
primaryClass = {math.CO},
year = 2016
}
@article{TopologicallyDistinctSetsofNonintersectingCirclesinthePlane,
title = {Topologically Distinct Sets of Non-intersecting Circles in the Plane},
abstract = {Nested parentheses are forms in an algebra which define orders of
evaluations. A class of well-formed sets of associated opening and closing
parentheses is well studied in conjunction with Dyck paths and Catalan numbers.
Nested parentheses also represent cuts through circles on a line. These become
topologies of non-intersecting circles in the plane if the underlying algebra
is commutative.
This paper generalizes the concept and answers quantitatively - as
recurrences and generating functions of matching rooted forests - the
questions: how many different topologies of nested circles exist in the plane
if (i) pairs of circles may intersect, or (ii) even triples of circles may
intersect. That analysis is driven by examining the symmetry properties of the
inner regions of the fundamental type(s) of the intersecting pairs and triples.},
url = {http://arxiv.org/abs/1603.00077v1 http://arxiv.org/pdf/1603.00077v1},
author = {Richard J. Mathar},
comment = {},
urldate = {2016-08-25},
archivePrefix = {arXiv},
eprint = {1603.00077},
primaryClass = {math.CO},
collections = {Easily explained,Geometry},
year = 2016
}
@article{WinklerSevenPuzzles,
title = {Seven Puzzles You Think You Must Not Have Heard Correctly},
abstract = {A typical mathematical puzzle sounds tricky but solvable — if not by you, then perhaps by the
genius down the hall. But sometimes the task at hand is so obviously impossible that you are moved
to ask whether you understood the problem correctly, and other times, the task seems so trivial
that you are sure you must have missed something.
Here, I have compiled seven puzzles which have often been greeted by words similar to “Wait
a minute — I must not have heard that correctly.” Some seem too hard, some too easy; after you've
worked on them for a while, you may find that the hard ones now seem easy and vice versa.},
url = {https://math.dartmouth.edu/{\~{}}pw/solutions.pdf},
author = {Peter Winkler},
comment = {},
urldate = {2016-08-30},
collections = {Attention-grabbing titles,Puzzles},
year = 2006
}
@article{PonytailMotion,
title = {Ponytail Motion},
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.},
url = {http://epubs.siam.org/doi/abs/10.1137/090760477 http://epubs.siam.org/doi/pdf/10.1137/090760477},
author = {Joseph B. Keller},
comment = {},
urldate = {2016-09-19},
collections = {Easily explained,Basically physics},
year = 2010
}
@article{HistoricalMethodsForMultiplication,
title = {Historical methods for multiplication},
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.},
url = {https://www.researchgate.net/publication/263733700{\_}Historical{\_}methods{\_}for{\_}multiplication},
author = {Bj{\o}rn Smestad and Konstantinos Nikolantonakis},
comment = {},
urldate = {2016-09-22},
collections = {Notation and conventions,History,Easily explained,Integerology},
year = 2010
}
@article{Fractalgeometryofacomplexplumagetraitrevealsbirdsquality,
title = {Fractal geometry of a complex plumage trait reveals bird's quality},
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.},
url = {http://rspb.royalsocietypublishing.org/content/280/1755/20122783},
author = {Lorenzo P{\'{e}}rez-Rodr{\'{i}}guez and Roger Jovani and Fran\ccois Mougeot},
comment = {},
urldate = {2016-09-27},
collections = {Animals,Geometry},
year = 2013
}
@article{AvoidingSquaresandOverlapsOvertheNaturalNumbers,
title = {Avoiding Squares and Overlaps Over the Natural Numbers},
abstract = {We consider avoiding squares and overlaps over the natural numbers, using a
greedy algorithm that chooses the least possible integer at each step; the word
generated is lexicographically least among all such infinite words. In the case
of avoiding squares, the word is 01020103..., the familiar ruler function, and
is generated by iterating a uniform morphism. The case of overlaps is more
challenging. We give an explicitly-defined morphism phi : N* -> N* that
generates the lexicographically least infinite overlap-free word by iteration.
Furthermore, we show that for all h,k in N with h <= k, the word phi^{\{}k-h{\}}(h)
is the lexicographically least overlap-free word starting with the letter h and
ending with the letter k, and give some of its symmetry properties.},
url = {http://arxiv.org/abs/0901.1397v1 http://arxiv.org/pdf/0901.1397v1},
author = {Mathieu Guay-Paquet and Jeffrey Shallit},
comment = {},
urldate = {2016-10-03},
archivePrefix = {arXiv},
eprint = {0901.1397},
primaryClass = {math.CO},
year = 2009,
collections = {Integerology}
}
@article{ADiscreteandBoundedEnvyFreeCakeCuttingProtocolforAnyNumberofAgents,
title = {A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents},
abstract = {We consider the well-studied cake cutting problem in which the goal is to
find an envy-free allocation based on queries from {\$}n{\$} agents. The problem has
received attention in computer science, mathematics, and economics. It has been
a major open problem whether there exists a discrete and bounded envy-free
protocol. We resolve the problem by proposing a discrete and bounded envy-free
protocol for any number of agents. The maximum number of queries required by
the protocol is {\$}n^{\{}n^{\{}n^{\{}n^{\{}n^n{\}}{\}}{\}}{\}}{\$}. We additionally show that even if we do
not run our protocol to completion, it can find in at most {\$}n^{\{}n+1{\}}{\$} queries a
partial allocation of the cake that achieves proportionality (each agent gets
at least {\$}1/n{\$} of the value of the whole cake) and envy-freeness. Finally we
show that an envy-free partial allocation can be computed in {\$}n^{\{}n+1{\}}{\$} queries
such that each agent gets a connected piece that gives the agent at least
{\$}1/(3n){\$} of the value of the whole cake.},
url = {http://arxiv.org/abs/1604.03655v10 http://arxiv.org/pdf/1604.03655v10},
author = {Haris Aziz and Simon Mackenzie},
comment = {},
urldate = {2016-10-13},
archivePrefix = {arXiv},
eprint = {1604.03655},
primaryClass = {cs.DS},
collections = {Attention-grabbing titles,Easily explained,Protocols and strategies,Food,Fun maths facts},
year = 2016
}
@article{GeometricMechanicsofCurvedCreaseOrigami,
title = {Geometric Mechanics of Curved Crease Origami},
abstract = {Folding a sheet of paper along a curve can lead to structures seen in
decorative art and utilitarian packing boxes. Here we present a theory for the
simplest such structure: an annular circular strip that is folded along a
central circular curve to form a three-dimensional buckled structure driven by
geometrical frustration. We quantify this shape in terms of the radius of the
circle, the dihedral angle of the fold and the mechanical properties of the
sheet of paper and the fold itself. When the sheet is isometrically deformed
everywhere except along the fold itself, stiff folds result in creases with
constant curvature and oscillatory torsion. However, relatively softer folds
inherit the broken symmetry of the buckled shape with oscillatory curvature and
torsion. Our asymptotic analysis of the isometrically deformed state is
corroborated by numerical simulations which allow us to generalize our analysis
to study multiply folded structures.},
url = {http://arxiv.org/abs/1206.0461v2 http://arxiv.org/pdf/1206.0461v2},
author = {Marcelo A. Dias and Levi H. Dudte and L. Mahadevan and Christian D. Santangelo},
comment = {},
urldate = {2016-10-14},
archivePrefix = {arXiv},
eprint = {1206.0461},
primaryClass = {cond-mat.soft},
collections = {Art,Basically physics,Things to make and do,Geometry},
year = 2012
}
@article{ASpaceEfficientAlgorithmfortheCalculationoftheDigitDistributionintheKolakoskiSequence,
title = {A Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence},
abstract = {With standard algorithms for generating the classical Kolakoski sequence, the
numerical calculation of the digit distribution requires a linear amount of
space. Here, we present an algorithm for calculating the distribution of the
digits in the classical Kolakoski sequence, that only requires a logarithmic
amount of space and still runs in linear time. The algorithm is easily
adaptable to generalised Kolakoski sequences.},
url = {http://arxiv.org/abs/1110.4228v2 http://arxiv.org/pdf/1110.4228v2},
author = {Johan Nilsson},
comment = {},
urldate = {2016-10-14},
archivePrefix = {arXiv},
eprint = {1110.4228},
primaryClass = {math.CO},
year = 2011
}
@article{Tworemarksonevenandoddtownproblems,
title = {Two remarks on even and oddtown problems},
abstract = {A family {\$}\mathcal A{\$} of subsets of an {\$}n{\$}-element set is called an eventown
(resp. oddtown) if all its sets have even (resp. odd) size and all pairwise
intersections have even size. Using tools from linear algebra, it was shown by
Berlekamp and Graver that the maximum size of an eventown is {\$}2^{\{}\left\lfloor
n/2\right\rfloor{\}}{\$}. On the other hand (somewhat surprisingly), it was proven by
Berlekamp, that oddtowns have size at most {\$}n{\$}. Over the last four decades,
many extensions of this even/oddtown problem have been studied. In this paper
we present new results on two such extensions. First, extending a result of Vu,
we show that a {\$}k{\$}-wise eventown (i.e., intersections of {\$}k{\$} sets are even) has
for {\$}k \geq 3{\$} a unique extremal configuration and obtain a stability result
for this problem. Next we improve some known bounds for the defect version of
an {\$}\ell{\$}-oddtown problem. In this problem we consider sets of size {\$}\not\equiv
0 \pmod \ell{\$} where {\$}\ell{\$} is a prime number {\$}\ell{\$} (not necessarily {\$}2{\$}) and
allow a few pairwise intersections to also have size {\$}\not\equiv 0 \pmod \ell{\$}.},
url = {http://arxiv.org/abs/1610.07907v1 http://arxiv.org/pdf/1610.07907v1},
author = {Benny Sudakov and Pedro Vieira},
comment = {},
urldate = {2016-10-27},
archivePrefix = {arXiv},
eprint = {1610.07907},
primaryClass = {math.CO},
year = 2016,
collections = {Combinatorics}
}
@article{QuasipracticalNumbers,
title = {Quasipractical Numbers},
abstract = {},
url = {http://ieeexplore.ieee.org/document/1090205/},
author = {Harvey J. Hindin},
comment = {},
urldate = {2016-11-28},
collections = {Unusual arithmetic,Integerology},
year = 1980
}
@article{DevelopingaMathematicalModelforBobbinLace,
title = {Developing a Mathematical Model for Bobbin Lace},
abstract = {Bobbin lace is a fibre art form in which intricate and delicate patterns are
created by braiding together many threads. An overview of how bobbin lace is
made is presented and illustrated with a simple, traditional bookmark design.
Research on the topology of textiles and braid theory form a base for the
current work and is briefly summarized. We define a new mathematical model that
supports the enumeration and generation of bobbin lace patterns using an
intelligent combinatorial search. Results of this new approach are presented
and, by comparison to existing bobbin lace patterns, it is demonstrated that
this model reveals new patterns that have never been seen before. Finally, we
apply our new patterns to an original bookmark design and propose future areas
for exploration.},
url = {http://arxiv.org/abs/1406.1532v3 http://arxiv.org/pdf/1406.1532v3},
author = {Veronika Irvine and Frank Ruskey},
comment = {},
urldate = {2016-11-28},
archivePrefix = {arXiv},
eprint = {1406.1532},
primaryClass = {math.CO},
collections = {Art,Geometry,Modelling},
year = 2014
}
@article{SequencesOfConsecutiveNNivenNumbers,
title = {Sequences of consecutive \(n\)-Niven numbers},
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
lengths of sequences of consecutive \(n\)-Niven numbers. The main result is given in the following theorem. },
url = {http://www.fq.math.ca/Scanned/32-2/grundman.pdf},
author = {H.G. Grundman},
comment = {},
urldate = {2016-12-05},
year = 1992,
collections = {Integerology}
}
@article{Everynaturalnumberisthesumoffortyninepalindromes,
title = {Every natural number is the sum of forty-nine palindromes},
abstract = {It is shown that the set of decimal palindromes is an additive basis for the
natural numbers. Specifically, we prove that every natural number can be
expressed as the sum of forty-nine (possibly zero) decimal palindromes.},
url = {http://arxiv.org/abs/1508.04721v1 http://arxiv.org/pdf/1508.04721v1},
author = {William D. Banks},
comment = {},
urldate = {2016-12-17},
archivePrefix = {arXiv},
eprint = {1508.04721},
primaryClass = {math.NT},
year = 2015,
collections = {Fun maths facts,Integerology}
}
@article{TwoshortproofsofthePerfectForestTheorem,
title = {Two short proofs of the Perfect Forest Theorem},
abstract = {A perfect forest is a spanning forest of a connected graph {\$}G{\$}, all of whose
components are induced subgraphs of {\$}G{\$} and such that all vertices have odd
degree in the forest. A perfect forest generalised a perfect matching since, in
a matching, all components are trees on one edge. Scott first proved the
Perfect Forest Theorem, namely, that every connected graph of even order has a
perfect forest. Gutin then gave another proof using linear algebra.
We give here two very short proofs of the Perfect Forest Theorem which use
only elementary notions from graph theory. Both our proofs yield
polynomial-time algorithms for finding a perfect forest in a connected graph of
even order.},
url = {http://arxiv.org/abs/1612.05004v1 http://arxiv.org/pdf/1612.05004v1},
author = {Yair Caro and Josef Lauri and Christina Zarb},
comment = {},
urldate = {2016-12-18},
archivePrefix = {arXiv},
eprint = {1612.05004},
primaryClass = {math.CO},
year = 2016,
collections = {About proof,Fun maths facts}
}
@article{BalloonPolyhedra,
title = {Balloon Polyhedra},
abstract = {},
url = {http://erikdemaine.org/papers/Balloons{\_}ShapingSpace2/paper.pdf},
author = {Erik D. Demaine and Martin L. Demaine and Vi Hart},
comment = {},
urldate = {2016-12-21},
collections = {Art,Easily explained,Things to make and do,Geometry},
year = 2013
}
@article{ASingularMathematicalPromenade,
title = {A Singular Mathematical Promenade},
abstract = {This is neither an elementary introduction to singularity theory nor a
specialized treatise containing many new theorems. The purpose of this little
book is to invite the reader on a mathematical promenade. We will pay a visit
to Hipparchus, Newton and Gauss, but also to many contemporary mathematicians.
We will play with a bit of algebra, topology, geometry, complex analysis and
computer science. Hopefully, some motivated undergraduates and some more
advanced mathematicians will enjoy some of these panoramas.},
url = {http://arxiv.org/abs/1612.06373v1 http://arxiv.org/pdf/1612.06373v1},
author = {Etienne Ghys},
comment = {A delight!},
urldate = {2016-12-22},
archivePrefix = {arXiv},
eprint = {1612.06373},
primaryClass = {math.AG},
collections = {History},
year = 2016
}
@article{HumanInferencesaboutSequencesAMinimalTransitionProbabilityModel,
title = {Human Inferences about Sequences: A Minimal Transition Probability Model},
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.},
url = {http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1005260},
author = {Florent Meyniel and Maxime Maheu and Stanislas Dehaene},
comment = {},
urldate = {2017-01-09},
collections = {The act of doing maths,Probability and statistics,Modelling},
year = 2016
}
@article{RulesforFoldingPolyminoesfromOneLeveltoTwoLevels,
title = {Rules for Folding Polyminoes from One Level to Two Levels},
abstract = {Polyominoes have been the focus of many recreational and research
investigations. In this article, the authors investigate whether a paper cutout
of a polyomino can be folded to produce a second polyomino in the same shape as
the original, but now with two layers of paper. For the folding, only "corner
folds" and "half edge cuts" are allowed, unless the polyomino forms a closed
loop, in which case one is allowed to completely cut two squares in the
polyomino apart. With this set of allowable moves, the authors present
algorithms for folding different types of polyominoes and prove that certain
polyominoes can successfully be folded to two layers. The authors also
establish that other polyominoes cannot be folded to two layers if only these
moves are allowed.},
url = {http://arxiv.org/abs/1701.03461v1 http://arxiv.org/pdf/1701.03461v1},
author = {Julia Martin and Elizabeth Wilcox},
comment = {},
urldate = {2017-01-16},
archivePrefix = {arXiv},
eprint = {1701.03461},
primaryClass = {math.CO},
collections = {Easily explained,Things to make and do,Geometry,Fun maths facts},
year = 2017
}
@article{OntheExistenceofOrdinaryTriangles,
title = {On the Existence of Ordinary Triangles},
abstract = {Let {\$}P{\$} be a finite point set in the plane. A {\$}c{\$}-ordinary triangle in {\$}P{\$} is
a subset of {\$}P{\$} consisting of three non-collinear points such that each of the
three lines determined by the three points contains at most {\$}c{\$} points of {\$}P{\$}.
We prove that there exists a constant {\$}c>0{\$} such that {\$}P{\$} contains a
{\$}c{\$}-ordinary triangle, provided that {\$}P{\$} is not contained in the union of two
lines. Furthermore, the number of {\$}c{\$}-ordinary triangles in {\$}P{\$} is
{\$}\Omega(|P|){\$}.},
url = {http://arxiv.org/abs/1701.08183v1 http://arxiv.org/pdf/1701.08183v1},
author = {Radoslav Fulek and Hossein Nassajian Mojarrad and M{\'{a}}rton Nasz{\'{o}}di and J{\'{o}}zsef Solymosi and Sebastian U. Stich and May Szedl{\'{a}}k},
comment = {},
urldate = {2017-02-06},
archivePrefix = {arXiv},
eprint = {1701.08183},
primaryClass = {math.CO},
year = 2017,
collections = {Geometry,Fun maths facts}
}
@article{TransfiniteVersionofWeltersGame,
title = {Transfinite Version of Welter's Game},
abstract = {We study the transfinite version of Welter's Game, a combinatorial game,
which is played on the belt divided into squares with general ordinal numbers
extended from natural numbers.
In particular, we obtain a straight-forward solution for the transfinite
version based on those of the transfinite version of Nim and the original
version of Welter's Game.},
url = {http://arxiv.org/abs/1701.08928v1 http://arxiv.org/pdf/1701.08928v1},
author = {Tomoaki Abuku},
comment = {},
urldate = {2017-02-06},
archivePrefix = {arXiv},
eprint = {1701.08928},
primaryClass = {math.CO},
collections = {Protocols and strategies,Games to play with friends,Combinatorics},
year = 2017
}
@article{PlanepartitionsintheworkofRichardStanleyandhisschool,
title = {Plane partitions in the work of Richard Stanley and his school},
abstract = {These notes provide a survey of the theory of plane partitions, seen through
the glasses of the work of Richard Stanley and his school.},
url = {http://arxiv.org/abs/1503.05934v2 http://arxiv.org/pdf/1503.05934v2},
author = {C. Krattenthaler},
comment = {},
urldate = {2017-02-06},
archivePrefix = {arXiv},
eprint = {1503.05934},
primaryClass = {math.CO},
collections = {Art,Combinatorics},
year = 2015
}
@article{RandomTrianglesandPolygonsinthePlane,
title = {Random Triangles and Polygons in the Plane},
abstract = {We consider the problem of finding the probability that a random triangle is
obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a
natural correspondence between plane polygons and the Grassmann manifold of
2-planes in real {\$}n{\$}-space proposed by Allen Knutson and Jean-Claude Hausmann.
This correspondence defines a natural probability measure on plane polygons. In
these terms, we answer Caroll's question. We then explore the Grassmannian
geometry of planar quadrilaterals, providing an answer to Sylvester's
four-point problem, and describing explicitly the moduli space of unordered
quadrilaterals. All of this provides a concrete introduction to a family of
metrics used in shape classification and computer vision.},
url = {http://arxiv.org/abs/1702.01027v1 http://arxiv.org/pdf/1702.01027v1},
author = {Jason Cantarella and Tom Needham and Clayton Shonkwiler and Gavin Stewart},
comment = {},
urldate = {2017-02-06},
archivePrefix = {arXiv},
eprint = {1702.01027},
primaryClass = {math.MG},
collections = {Probability and statistics,Geometry},
year = 2017
}
@article{ThreeThoughtsonPrimeSimplicity,
title = {Three Thoughts on “Prime Simplicity”},
abstract = {In 2009, Catherine Woodgold and I published ‘‘Prime Simplicity’’, examining the belief that Euclid’s famous proof of the infinitude of prime numbers was by contradiction. We demonstrated that that belief is widespread among mathematicians and is false: Euclid’s 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.},
url = {https://link.springer.com/article/10.1007/s00283-012-9322-z},
author = {Michael Hardy},
comment = {},
urldate = {2017-02-27},
year = 2012,
collections = {About proof,Integerology}
}
@article{PrimeSimplicity,
title = {Prime Simplicity},
author = {Michael Hardy and Catherine Woodgold},
url = {https://link.springer.com/article/10.1007/s00283-009-9064-8},
abstract = {},
comment = {Euclid's proof of the infinitude of the primes wasn't a proof by contradiction!},
urldate = {2017-02-27},
year = 2009,
collections = {About proof,Integerology}
}
@article{StatisticsDoneWrong,
title = {Statistics Done Wrong},
abstract = {If you’re 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.
This is unfortunate, because statistical errors are rife.
Statistics 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.},
url = {https://www.statisticsdonewrong.com/},
author = {Alex Reinhart},
comment = {},
urldate = {2017-03-01},
collections = {Attention-grabbing titles,Lists and catalogues,Probability and statistics,Education},
year = 2015
}
@article{Themathematicsoflecturehallpartitions,
title = {The mathematics of lecture hall partitions},
abstract = {Over the past twenty years, lecture hall partitions have emerged as
fundamental combinatorial structures, leading to new generalizations and
interpretations of classical theorems and new results. In recent years,
geometric approaches to lecture hall partitions have used polyhedral geometry
to discover further properties of these rich combinatorial objects.
In this paper we give an overview of some of the surprising connections that
have surfaced in the process of trying to understand the lecture hall
partitions.},
url = {http://arxiv.org/abs/1607.01765v1 http://arxiv.org/pdf/1607.01765v1},
author = {Carla D. Savage},
comment = {},
urldate = {2017-03-08},
archivePrefix = {arXiv},
eprint = {1607.01765},
primaryClass = {math.CO},
collections = {Easily explained,Combinatorics},
year = 2016
}
@article{CrazySequentialRepresentationNumbersfrom0to11111intermsofIncreasingandDecreasingOrdersof1to9,
title = {Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9},
abstract = {Natural numbers from 0 to 11111 are written in terms of 1 to 9 in two
different ways. The first one in increasing order of 1 to 9, and the second one
in decreasing order. This is done by using the operations of addition,
multiplication, subtraction, potentiation, and division. In both the situations
there are no missing numbers, except one, i.e., 10958 in the increasing case.},
url = {http://arxiv.org/abs/1302.1479v5 http://arxiv.org/pdf/1302.1479v5},
author = {Inder J. Taneja},
comment = {},
urldate = {2017-03-12},
archivePrefix = {arXiv},
eprint = {1302.1479},
primaryClass = {math.HO},
collections = {Easily explained,Integerology},
year = 2013
}
@article{BeyondFloatingPoint,
title = {Beyond Floating Point: Next-Generation Computer Arithmetic },
abstract = {},
url = {http://web.stanford.edu/class/ee380/Abstracts/170201-slides.pdf},
author = {John L. Gustafson},
comment = {About posits, which beat IEEE floats on pretty much everything. Lots of nice pictures!},
urldate = {2017-03-21},
collections = {Basically computer science,Unusual arithmetic},
year = 2017
}
@article{SENEWSPAPAC00,
title = {PAPAC-00, a Do-It-Yourself Paper Computer},
abstract = {},
url = {http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=5222588},
author = {Rollin P. Mayer},
comment = {},
urldate = {2017-03-21},
collections = {Basically computer science,Easily explained,Things to make and do,Unusual computers},
year = 1958
}
@article{Whitt2013,
title = {The Maximum Throughput Rate for Each Hole on a Golf Course},
author = {Whitt, Ward},
url = {http://www.columbia.edu/{\~{}}ww2040/golf{\_}throughput{\_}POMS{\_}2015.pdf},
urldate = {2013-11-25},
abstract = {},
comment = {},
collections = {Games to play with friends,Basically physics,Probability and statistics},
year = 2015
}
@misc{item61,
title = {Prime numbers in certain arithmetic progressions},
author = {Ram Murty and Nithum Thain},
url = {http://projecteuclid.org/download/pdf{\_}1/euclid.facm/1229442627},
urldate = {2016-05-07},
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 = {},
year = 2001,
collections = {Fun maths facts,Integerology}
}
@article{Plagge1978,
title = {Fractions without Quotients: Arithmetic of Repeating Decimals},
author = {Plagge, Richard},
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},
abstract = {},
comment = {},
collections = {Notation and conventions,Unusual arithmetic,Easily explained},
year = 1978
}
@article{Fink1999,
title = {Designing tie knots by random walks},
author = {Thomas M. Fink and Yong Mao},
url = {http://www.nature.com/nature/journal/v398/n6722/abs/398031a0.html},
urldate = {2011-12-09},
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 = {},
collections = {Probability and statistics},
year = 1999
}
@article{NaguraOntheintervalcontainingatleastoneprimenumber,
title = {On the interval containing at least one prime number},
author = {Jitsuro Nagura},
url = {https://projecteuclid.org/euclid.pja/1195570997 https://projecteuclid.org/download/pdf{\_}1/euclid.pja/1195570997},
urldate = {2016-11-14},
abstract = {},
comment = {},
year = 1952,
collections = {Fun maths facts,Integerology}
}
@misc{item58,
title = {Area and Hausdorff Dimension of Julia Sets of Entire Functions},
author = {Curt McMullen},
url = {http://www.math.harvard.edu/{\~{}}ctm/papers/home/text/papers/entire/entire.pdf},
urldate = {2015-12-17},
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 = {},
year = 1987
}
@article{Kolliopoulos,
title = {The 1-Hyperbolic Projection for User Interfaces},
author = {Kolliopoulos, Alexander},
url = {http://www.dgp.toronto.edu/{\~{}}alexk/hyperproj.pdf},
urldate = {2011-04-12},
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
to those of a standard interface. The results of usability experiments conducted with this prototype are also presented and analyzed.},
comment = {},
collections = {Basically computer science},
year = 2003
}
@article{item36,
title = {Solving Differential Equations by Symmetry Groups},
author = {John Starret},
url = {https://www.jstor.org/stable/27642331},
urldate = {2013-12-06},
abstract = {},
comment = {},
year = 2007
}
@article{Mercer2009,
title = {On Furstenberg's Proof of the Infinitude of Primes},
author = {Mercer, Idris D},
url = {http://www.idmercer.com/monthly355-356-mercer.pdf},
urldate = {2010-09-09},
abstract = {},
comment = {},
year = 2009,
collections = {About proof,Integerology}
}
@online{item17,
title = {Light reflecting off Christmas-tree balls},
author = {Joseph O'Rourke},
url = {http://mathoverflow.net/questions/50150/light-reflecting-off-christmas-tree-balls},
urldate = {2012-04-08},
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 = {},
collections = {Easily explained},
year = 2010
}
@article{CountingCasesInMarchingCubes,
title = {Counting Cases in Marching Cubes: Towards a Generic Algorithm for Producing Substitopes},
author = {David C. Banks and Stephen Linton},
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},
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 = {},
collections = {Basically computer science},
year = 2003
}
@article{Akritas1986,
title = {There is no "Uspensky's method"},
author = {Akritas, AG},
url = {http://portal.acm.org/citation.cfm?id=32457},
urldate = {2011-09-15},
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 = {},
collections = {Attention-grabbing titles,Drama!},
year = 1986
}
@book{Diaconis2006,
title = {Methods for studying coincidences},
author = {Diaconis, P and Mosteller, Frederick},
url = {http://link.springer.com/chapter/10.1007/978-0-387-44956-2{\_}39},
urldate = {2014-10-07},
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’s 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 “close” 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 = {},
collections = {Easily explained,Probability and statistics,Modelling},
year = 2006
}
@article{Shuai,
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},
url = {http://arxiv.org/abs/1107.0681 http://arxiv.org/pdf/1107.0681v1},
urldate = {2011-09-08},
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 = {},
collections = {Basically computer science,Basically physics},
year = 2011
}
@article{Saloffcoste,
title = {Random Walks on Finite Groups},
author = {Saloff-coste, Laurent},
url = {http://statweb.stanford.edu/{\~{}}cgates/PERSI/papers/rwfg.pdf},
urldate = {2012-01-05},
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 “mixing time” 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 “the mixing time”. 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 – Probability, Algebra, Representation Theory, Functional Analysis, Geometry, Combinatorics – have been used to attack special instances of this problem. This article gives a general overview of this area of research.},
comment = {},
collections = {Probability and statistics,The groups group},
year = 2004
}
@article{NonSexistMenage,
title = {Non-sexist solution of the m{\'{e}}nage problem},
author = {Kenneth P. Bogart},
url = {http://www.math.dartmouth.edu/{\~{}}doyle/docs/menage/menage/menage.html},
urldate = {2013-06-16},
abstract = {The m{\'{e}}nage 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 = {},
collections = {Puzzles,Easily explained},
year = 1985
}
@article{Martin1998,
title = {Denser Egyptian Fractions},
author = {Martin, Greg},
url = {http://arxiv.org/abs/math/9811112 http://arxiv.org/pdf/math/9811112v1},
urldate = {2015-07-23},
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{\H{o}}s 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 = {},
collections = {Easily explained,Fun maths facts},
year = 1998
}
@online{item5,
title = {Four questions about fuzzy rankings},
author = {Brian Hayes},
url = {http://bit-player.org/2010/four-questions-about-fuzzy-rankings},
urldate = {2011-01-12},
abstract = {},
comment = {},
year = 2010
}
@article{Onsubsetswithintersectionsofevencardinality,
title = {On subsets with intersections of even cardinality},
author = {E.R. Berlekamp},
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},
abstract = {This paper solves a question by Paul Erd{\H{o}}s},
comment = {},
year = 1969,
collections = {Fun maths facts}
}
@article{item54,
title = {How to Beat Your Wythoff Games' Opponent on Three Fronts},
author = {Aviezri S. Fraenkel},
url = {http://www.jstor.org/stable/2321643},
urldate = {2015-10-23},
abstract = {},
comment = {},
collections = {Protocols and strategies,Games to play with friends},
year = 1982
}
@article{BestLaidPlansofLionsandMen,
title = {Best Laid Plans of Lions and Men},
abstract = {We answer the following question dating back to J.E. Littlewood (1885 -
1977): Can two lions catch a man in a bounded area with rectifiable lakes? The
lions and the man are all assumed to be points moving with at most unit speed.
That the lakes are rectifiable means that their boundaries are finitely long.
This requirement is to avoid pathological examples where the man survives
forever because any path to the lions is infinitely long. We show that the
answer to the question is not always "yes" by giving an example of a region {\$}R{\$}
in the plane where the man has a strategy to survive forever. {\$}R{\$} is a
polygonal region with holes and the exterior and interior boundaries are
pairwise disjoint, simple polygons. Our construction is the first truly
two-dimensional example where the man can survive.
Next, we consider the following game played on the entire plane instead of a
bounded area: There is any finite number of unit speed lions and one fast man
who can run with speed {\$}1+\varepsilon{\$} for some value {\$}\varepsilon>0{\$}. Can the
man always survive? We answer the question in the affirmative for any constant
{\$}\varepsilon>0{\$}.},
url = {http://arxiv.org/abs/1703.03687v1 http://arxiv.org/pdf/1703.03687v1},
author = {Mikkel Abrahamsen and Jacob Holm and Eva Rotenberg and Christian Wulff-Nilsen},
comment = {},
urldate = {2017-03-22},
archivePrefix = {arXiv},
eprint = {1703.03687},
primaryClass = {cs.CG},
collections = {Attention-grabbing titles,Protocols and strategies,Animals,Fun maths facts},
year = 2017
}
@online{Ptak,
title = {A Do-It-Yourself Paper Digital Computer, 1959.},
author = {Ptak, John F.},
url = {http://longstreet.typepad.com/books/2010/11/item-mayer-rollin-p-papac-00-a-do-it-yourself-paper-computer-in-communications-of-the-association-for-computing-machin.html},
urldate = {2013-05-05},
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 “2-register, 1-bit, fixed-instruction binary digital computer” and was submitted to the journal by Rollin P. Mayer (of the MIT Lincoln Lab).},
comment = {},
collections = {Easily explained,Things to make and do},
year = 2010
}
@article{Gale1991,
title = {The Strange and Surprising Saga of the Somos Sequences},
author = {Gale, David},
url = {http://link.springer.com/10.1007/BF03024070},
urldate = {2013-10-09},
abstract = {},
comment = {},
doi = {10.1007/BF03024070},
issn = {0343-6993},
journal = {The Mathematical Intelligencer},
month = {mar},
number = 1,
pages = {40--43},
volume = 13,
year = 1991,
collections = {Fun maths facts,Integerology}
}
@article{Battisti2006,
title = {A Generalized Fibonacci LSB Data Hiding Technique},
author = {Battisti, F and Carli, M and Neri, A and Egiaziarian, K},
url = {http://www.comlab.uniroma3.it/Marco/Articoli{\%}20Battisti/A{\%}20Generalized{\%}20Fibonacci{\%}20LSB{\%}20Data{\%}20Hiding{\%}20Technique.pdf},
urldate = {2012-01-10},
abstract = {},
comment = {},
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},
year = 2006,
collections = {Basically computer science,Fibonaccinalia}
}
@article{Flajolet1998,
title = {Continued fraction algorithms, functional operators, and structure constants},
author = {Flajolet, P. and Vall{\'{e}}e, B.},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0304397597001230},
urldate = {2011-01-20},
abstract = {},
comment = {},
issn = {0304-3975},
journal = {Theoretical Computer Science},
number = {1-2},
pages = {1--34},
publisher = {Elsevier},
volume = 194,
year = 1998,
collections = {}
}
@misc{item2,
title = {Digital halftoning space filling curves},
author = {Luiz C. Velho and Jonas M. Gomes},
url = {http://www.google.co.uk/patents?hl=en{\&}lr={\&}vid=USPAT5602943{\&}id=Mn0jAAAAEBAJ{\&}oi=fnd{\&}dq=hilbert+curve{\&}printsec=abstract{\#}v=onepage{\&}q=hilbert curve{\&}f=false},
urldate = {2010-08-31},
abstract = {},
comment = {},
collections = {Basically computer science},
year = 1997
}
@online{Bauer,
title = {Constructive gem: juggling exponentials},
author = {Bauer, Andrej},
url = {http://math.andrej.com/2009/09/09/constructive-gem-juggling-exponentials/},
urldate = {2010-07-15},
abstract = {},
comment = {},
collections = {Basically computer science},
year = 2009
}
@article{AnalysisofCarriesinSignedDigitExpansions,
title = {Analysis of Carries in Signed Digit Expansions},
author = {Clemens Heuberger and Sara Kropf and Helmut Prodinger},
url = {http://arxiv.org/abs/1503.08816v3 http://arxiv.org/pdf/1503.08816v3},
urldate = {2017-02-06},
abstract = {The number of positive and negative carries in the addition of two
independent random signed digit expansions of given length is analyzed
asymptotically for the {\$}(q, d){\$}-system and the symmetric signed digit
expansion. The results include expectation, variance, covariance between the
positive and negative carries and a central limit theorem.
Dependencies between the digits require determining suitable transition
probabilities to obtain equidistribution on all expansions of given length. A
general procedure is described to obtain such transition probabilities for
arbitrary regular languages.
The number of iterations in von Neumann's parallel addition method for the
symmetric signed digit expansion is also analyzed, again including expectation,
variance and convergence to a double exponential limiting distribution. This
analysis is carried out in a general framework for sequences of generating
functions.},
comment = {},
archivePrefix = {arXiv},
eprint = {1503.08816},
primaryClass = {math.CO},
collections = {Basically computer science},
year = 2015
}
@article{Miquel2007,
title = {The experimental effectiveness of mathematical proof},
author = {Miquel, Alexandre},
url = {https://www.fing.edu.uy/{\~{}}amiquel/publis/effectiveness.pdf},
urldate = {2011-10-30},
abstract = {},
comment = {},
pages = {1--42},
year = 2007,
collections = {History,The act of doing maths,About proof}
}
@book{CajoriNotations,
title = {A history of mathematical notations},
author = {Florian Cajori},
url = {http://www.maths.ed.ac.uk/{\~{}}aar/papers/cajorinot.pdf},
urldate = {2011-01-12},
abstract = {},
comment = {},
year = 1993,
collections = {History,Notation and conventions,Lists and catalogues}
}
@misc{item51,
title = {Representations of Palindromic, Prime and Number Patterns},
author = {Inder J. Taneja},
url = {http://rgmia.org/papers/v18/v18a77.pdf},
urldate = {2015-06-12},
abstract = {},
comment = {},
collections = {Unusual arithmetic,Integerology},
year = 2015
}
@misc{Kahan,
title = {Beastly Numbers},
author = {Kahan, W},
url = {http://www.cs.berkeley.edu/{\~{}}wkahan/tests/numbeast.ps},
urldate = {2012-09-05},
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 = {},
collections = {Basically computer science},
year = 1996
}
@article{Srinivasan1948,
title = {Practical numbers},
author = {Srinivasan, A.K.},
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},
abstract = {},
comment = {},
journal = {Current science},
year = 1948,
month = {jun},
volume = 17,
issue = 6,
collections = {Easily explained,Integerology}
}
@article{Akleman2011,
title = {Cyclic twill-woven objects},
author = {Akleman, Ergun and Chen, Jianer and Chen, YenLin and Xing, Qing and Gross, Jonathan L.},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0097849311000422},
urldate = {2013-05-16},
abstract = {},
comment = {},
doi = {10.1016/j.cag.2011.03.003},
issn = 00978493,
journal = {Computers {\&} Graphics},
month = {jun},
number = 3,
pages = {623--631},
volume = 35,
year = 2011,
collections = {Basically computer science}
}
@article{TheGeneralCounterfeitCoinProblem,
title = {The general counterfeit coin problem},
author = {Lorenz Halbeisen and Norbert Hungerb{\"{u}}hler},
url = {http://user.math.uzh.ch/halbeisen/publications/pdf/coin.pdf},
urldate = {2016-08-24},
abstract = {Given {\$}c{\$} nickels among which there may be a counterfeit coin, which can only be told
apart 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 = {},
collections = {Puzzles,Easily explained},
year = 1995
}
@article{MidpointOfAnInterval,
title = {How do you compute the midpoint of an interval?},
author = {Fr{\'{e}}d{\'{e}}ric Goualard},
url = {https://hal.archives-ouvertes.fr/file/index/docid/576641/filename/computing-midpoint.pdf},
urldate = {2016-07-04},
abstract = {},
comment = {},
collections = {Basically computer science},
year = 2014
}
@article{Dividingbyzerohowbadisitreally,
title = {Dividing by zero - how bad is it, really?},
author = {Takayuki Kihara and Arno Pauly},
url = {http://arxiv.org/abs/1606.04126v1 http://arxiv.org/pdf/1606.04126v1},
urldate = {2016-06-17},
abstract = {In computable analysis testing a real number for being zero is a fundamental
example of a non-computable task. This causes problems for division: We cannot
ensure that the number we want to divide by is not zero. In many cases, any
real number would be an acceptable outcome if the divisor is zero - but even
this cannot be done in a computable way.
In this note we investigate the strength of the computational problem "Robust
division": Given a pair of real numbers, the first not greater than the other,
output their quotient if well-defined and any real number else. The formal
framework is provided by Weihrauch reducibility. One particular result is that
having later calls to the problem depending on the outcomes of earlier ones is
strictly more powerful than performing all calls concurrently. However, having
a nesting depths of two already provides the full power. This solves an open
problem raised at a recent Dagstuhl meeting on Weihrauch reducibility.
As application for "Robust division", we show that it suffices to execute
Gaussian elimination.},
comment = {},
archivePrefix = {arXiv},
eprint = {1606.04126},
primaryClass = {cs.LO},
collections = {Basically computer science},
year = 2016
}
@online{item53,
title = {Dice - Numericana},
author = {G{\'{e}}rard P. Michon},
url = {http://www.numericana.com/answer/dice.htm},
urldate = {2015-10-17},
abstract = {},
comment = {},
keywords = {cast,casting,combinatorics,dice,die,enumeration,roll,rolling,toss,tossing},
collections = {Easily explained,Games to play with friends,Lists and catalogues},
year = 2000
}
@misc{item47,
title = {On Legendre's Prime Number Formula},
author = {Janos Pintz},
url = {http://www.jstor.org/stable/2321863?seq=1{\#}page{\_}scan{\_}tab{\_}contents},
urldate = {2015-03-06},
abstract = {},
comment = {},
collections = {History,Integerology},
year = 1980
}
@article{Fink2013,
title = {LIM is not slim},
author = {Fink, Alex and Fraenkel, Aviezri S. and Santos, Carlos},
url = {http://link.springer.com/10.1007/s00182-013-0380-z},
urldate = {2014-06-11},
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’, 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 = {},
issn = {0020-7276},
journal = {International Journal of Game Theory},
month = {may},
number = 2,
pages = {269--281},
volume = 43,
year = 2013,
collections = {Attention-grabbing titles,Combinatorics}
}
@misc{item40,
title = {A number system with an irrational base},
author = {George Bergman},
url = {http://www.jstor.org/discover/10.2307/3029218?uid=3738032{\&}uid=2{\&}uid=4{\&}sid=21103722830183},
urldate = {2014-03-12},
abstract = {},
comment = {},
collections = {Unusual arithmetic,Fun maths facts,Integerology},
year = 1957
}
@article{Silva2007,
title = {Mathematical Games},
author = {Silva, Jorge Nuno},
url = {http://www.cim.pt/files/cim{\_}games.pdf},
urldate = {2014-01-24},
abstract = {},
comment = {Some slides about mathematical games throughout history.},
year = 2007,
collections = {History,Games to play with friends}
}
@article{item34,
title = {Is POPL Mathematics or Science?},
author = {Andrew W. Appel},
url = {http://www.cs.princeton.edu/{\~{}}appel/papers/science.pdf},
urldate = {2013-11-06},
abstract = {},
comment = {},
collections = {The act of doing maths,Probability and statistics},
year = 1992
}
@article{Althofer2013,
title = {Random Structures from Lego Bricks and Analog Monte Carlo Procedures},
author = {Alth{\"{o}}fer, I},
url = {http://www.althofer.de/random-lego-structures.pdf},
urldate = {2013-11-13},
abstract = {},
comment = {},
journal = {Update},
keywords = {analog monte carlo,artificial life,interactive art,procedure,random lego structures},
year = 2013,
collections = {Attention-grabbing titles,Basically computer science,Easily explained,Basically physics}
}
@misc{item33,
title = {The Ubiquitous Thue-Morse Sequence},
author = {Jeffrey Shallit},
url = {https://cs.uwaterloo.ca/{\~{}}shallit/Talks/green3.pdf},
urldate = {2013-10-21},
abstract = {},
comment = {},
collections = {Easily explained,Fun maths facts},
year = 2006
}
@article{Castellanos2013,
title = {The Ubiquitous Pi},
author = {Castellanos, Dario},
url = {http://www.jstor.org/stable/2690037},
urldate = {2013-06-11},
abstract = {},
comment = {},
number = 2,
pages = {67--98},
volume = 61,
year = 2013,
collections = {History,Easily explained,Fun maths facts}
}
@misc{Kalman,
title = {Six Ways to Sum a Series},
author = {Kalman, Dan},
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},
abstract = {A discussion of the sum of squares of the reciprocals of the positive integers with a review of several proofs.},
comment = {},
collections = {Easily explained,About proof,Fun maths facts},
year = 1994
}
@online{DatingRecommenderSystems,
title = {Online Dating Recommender Systems: The Split-complex Number Approach},
author = {J{\'{e}}rôme Kunegis},
url = {http://www.slideshare.net/kunegis/presentation-2012rsweb},
urldate = {2012-09-13},
abstract = {},
comment = {},
collections = {Basically computer science,Unusual arithmetic,Easily explained,Probability and statistics},
year = 2012
}
@article{Arkin2006,
title = {The Snowblower Problem},
author = {Arkin, Esther M. and Bender, Michael A. and Mitchell, Joseph S. B. and Polishchuk, Valentin},
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},
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 = {},
month = {mar},
pages = 19,
year = 2006,
archivePrefix = {arXiv},
eprint = {cs/0603026},
primaryClass = {cs.DS},
collections = {Basically computer science,Puzzles,Easily explained}
}
@article{Lang2000,
title = {Origami Burrs and Woven Polyhedra},
author = {Lang, Robert J},
url = {http://www.langorigami.com/files/articles/Polypolyhedra{\_}part{\_}1.pdf},
urldate = {2011-12-14},
abstract = {},
comment = {},
pages = {1--26},
volume = 4,
year = 2000,
collections = {Things to make and do}
}
@article{Faure2008,
title = {Hierarchical Position Based Dynamics},
author = {Faure, F. and Teschner, M.},
url = {http://www.matthiasmueller.info/publications/hpbd.pdf},
urldate = {2011-03-11},
abstract = {},
comment = {},
journal = {matthiasmueller.info},
year = 2008,
collections = {Basically computer science}
}
@article{Freeman1967,
title = {Detection of transposition errors in decimal numbers},
author = {Freeman, H},
url = {http://ieeexplore.ieee.org/xpls/abs{\_}all.jsp?arnumber=1447797},
urldate = {2010-08-20},
abstract = {},
comment = {},
journal = {Proceedings of the IEEE},
number = 8,
pages = {1500--1501},
volume = 55,
year = 1967,
collections = {Basically computer science}
}
@article{Berkel2009,
title = {On a curious property of 3435},
author = {Berkel, Daan Van},
url = {http://arxiv.org/abs/0911.3038 http://arxiv.org/pdf/0911.3038v2},
urldate = {2011-02-02},
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 = {},
archivePrefix = {arXiv},
arxivId = {0911.3038v2},
eprint = {0911.3038v2},
journal = {Folklore},
pages = {1--5},
year = 2009,
collections = {Easily explained,Fun maths facts,Integerology}
}
@article{HuntingRabbitsontheHypercube,
title = {Hunting Rabbits on the Hypercube},
author = {Jessalyn Bolkema and Corbin Groothuis},
url = {http://arxiv.org/abs/1701.08726v1 http://arxiv.org/pdf/1701.08726v1},
urldate = {2017-02-06},
abstract = {We explore the Hunters and Rabbits game on the hypercube. In the process, we
find the solution for all classes of graphs with an isoperimetric nesting
property and find the exact hunter number of {\$}Q^n{\$} to be
{\$}1+\sum\limits{\_}{\{}i=0{\}}^{\{}n-2{\}} \binom{\{}i{\}}{\{}\lfloor i/2 \rfloor{\}}{\$}. In addition, we
extend results to the situation where we allow the rabbit to not move between
shots.},
comment = {},
archivePrefix = {arXiv},
eprint = {1701.08726},
primaryClass = {math.CO},
collections = {Attention-grabbing titles,Easily explained,Protocols and strategies,Animals,Food},
year = 2017
}
@article{item32,
title = {Missing Data: Instrument-Level Heffalumps and Item-Level Woozles},
author = {Philip L. Roth and Fred S. Switzer III},
url = {https://archive.is/8Jq0O http://division.aomonline.org/rm/1999{\_}RMD{\_}Forum{\_}Missing{\_}Data.htm},
urldate = {2013-06-26},
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’s treatment of missing data in important situations.},
comment = {},
collections = {Attention-grabbing titles},
year = 1999
}
@article{Oliva2010,
title = {What Sequential Games , the Tychonoff Theorem and the Double-Negation Shift have in Common},
author = {Oliva, Paulo and Escardo, Martin},
url = {http://www.cs.bham.ac.uk/{\~{}}mhe/papers/msfp2010/Escardo-Oliva-MSFP2010.pdf},
urldate = {2010-08-04},
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.
Therefore 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 = {},
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,
collections = {Basically computer science,Protocols and strategies,Games to play with friends}
}
@article{Janson2008,
title = {Plane recursive trees, Stirling permutations and an urn model},
author = {Janson, Svante},
url = {https://arxiv.org/abs/0803.1129v1 http://arxiv.org/pdf/0803.1129},
urldate = {2010-02-25},
abstract = {},
comment = {},
journal = {Preprint},
pages = {1--8},
year = 2008,
collections = {Probability and statistics,Combinatorics}
}
@article{Ito2015,
title = {On the Existence of Generalized Parking Spaces for Complex Reflection Groups},
author = {Ito, Yosuke and Okada, Soichi},
url = {http://arxiv.org/abs/1508.06846 http://arxiv.org/pdf/1508.06846v1},
urldate = {2015-09-06},
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 = {},
month = {aug},
year = 2015,
archivePrefix = {arXiv},
eprint = {1508.06846},
primaryClass = {math.CO},
collections = {Attention-grabbing titles,The groups group}
}
@article{Knauer2011,
title = {How to eat 4/9 of a pizza},
author = {Knauer, Kolja and Micek, Piotr and Ueckerdt, Torsten},
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},
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 = {},
archivePrefix = {arXiv},
arxivId = {0812.2870},
eprint = {0812.2870},
journal = {Discrete Mathematics},
month = {dec},
pages = {1--15},
year = 2011,
primaryClass = {cs.DM},
collections = {Easily explained,Protocols and strategies,Puzzles}
}
@article{Hadley1992,
title = {Problems to sharpen the young},
author = {Hadley, John and Singmaster, David},
url = {http://www.jstor.org/stable/10.2307/3620384},
urldate = {2013-03-18},
abstract = {An annotated translation of Propositiones ad acuendos juvenes, the oldest mathematical problem collection in Latin, attributed to Alcuin of York.},
comment = {},
journal = {The Mathematical Gazette},
number = 475,
pages = {102--126},
volume = 76,
year = 1992,
collections = {Easily explained,Puzzles}
}
@article{Flocchini2010,
title = {Mapping an unfriendly subway system},
author = {Flocchini, Paola and Kellett, Matthew and Mason, P.},
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},
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 = {},
journal = {Fun with Algorithms},
pages = {190--201},
publisher = {Springer},
year = 2010,
collections = {Protocols and strategies}
}
@article{Bazargan2007,
title = {A linear programming approach for aircraft boarding strategy},
author = {Bazargan, M},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0377221706010137},
urldate = {2011-10-04},
abstract = {},
comment = {},
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,
collections = {Basically computer science,Protocols and strategies}
}
@article{item13,
title = {London Calling Philosophy and Engineering: WPE 2008},
author = {Glen Miller},
url = {http://www.ncbi.nlm.nih.gov/pubmed/19543813},
urldate = {2012-01-24},
abstract = {},
comment = {},
booktitle = {The Annals of occupational hygiene},
isbn = 1903496780,
issn = {0003-4878},
number = 2,
pmid = 3752842,
volume = 30,
collections = {},
year = 2009
}
@article{Ge2007,
title = {Interpolating Solid Orientations with a {\$}C^2{\$} -Continuous B-Spline Quaternion Curve},
author = {Ge, Wenbing and Huang, Zhangjin and Wang, Guoping},
url = {http://link.springer.com/chapter/10.1007{\%}2F978-3-540-73011-8{\_}58},
urldate = {2011-01-12},
abstract = {},
comment = {},
keywords = {b-spline curve,c 2 -continuous,computer,interpolation,quaternion},
pages = {606--615},
year = 2007,
collections = {Basically computer science}
}
@article{Stallings1966,
title = {How not to prove the Poincar{\'{e}} conjecture},
author = {Stallings, JR},
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},
abstract = {I have committed the sin of falsely proving Poincar{\'{e}}'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 = {},
year = 1966,
collections = {Attention-grabbing titles,About proof}
}
@misc{item41,
title = {Useful inequalities cheat sheet},
author = {L{\'{a}}szl{\'{o}} Kozma},
url = {http://lkozma.net/inequalities{\_}cheat{\_}sheet/},
urldate = {2014-04-28},
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 = {},
collections = {Lists and catalogues,About proof},
year = 2011
}
@article{Duran2014,
title = {The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?},
author = {Dur{\'{a}}n, Antonio J and P{\'{e}}rez, Mario and Varona, Juan L},
url = {http://www.ams.org/notices/201410/rnoti-p1249.pdf},
urldate = {2014-10-23},
abstract = {},
comment = {},
isbn = 1584704349,
number = {November},
pages = {1249--1252},
year = 2014,
collections = {Drama!}
}
@article{Gould2011,
title = {Table for Fundamentals of Series : Part I : Basic Properties of Series and Products},
author = {Gould, Henry W.},
url = {http://math.wvu.edu/{\~{}}gould/Vol.1.PDF},
urldate = {2014-02-11},
abstract = {},
comment = {},
year = 2011,
collections = {Lists and catalogues}
}
@article{Frieze2013,
title = {The topology of competitively constructed graphs},
author = {Frieze, Alan and Pegden, Wesley},
url = {http://arxiv.org/abs/1312.0964 http://arxiv.org/pdf/1312.0964v2},
urldate = {2013-12-30},
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 = {},
month = {dec},
pages = 9,
year = 2013,
archivePrefix = {arXiv},
eprint = {1312.0964},
primaryClass = {math.CO},
collections = {Games to play with friends,Protocols and strategies}
}
@article{Kikuchi1994,
title = {A Note on Boolos' Proof of the Incompleteness Theorem},
author = {Kikuchi, Makoto},
url = {http://doi.wiley.com/10.1002/malq.19940400409},
urldate = {2010-09-03},
abstract = {We give a proof of G{\"{o}}del's first incompleteness theorem based on Berry's paradox, and from it we also derive the second incompleteness theorem model-theoretically.},
comment = {},
journal = {Mathematical Logic Quarterly},
number = 4,
pages = {528--532},
volume = 40,
year = 1994,
collections = {About proof}
}
@article{Rechnitzer2006,
title = {Haruspicy 3: The anisotropic generating function of directed bond-animals is not D-finite},
author = {Rechnitzer, Andrew},
url = {http://www.sciencedirect.com/science/article/pii/S0097316505001883},
urldate = {2015-11-08},
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) 163–177] 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. 1–234, programs available from http://www.ms.unimelb.edu.au/{\~{}}tonyg].},
comment = {},
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,
collections = {Animals,Attention-grabbing titles}
}
@article{Williams2012,
title = {The paramagnetic and glass transitions in sudoku},
author = {Williams, Alex and Ackland, Graeme . J.},
url = {http://arxiv.org/abs/1212.1649 http://arxiv.org/pdf/1212.1649v1},
urldate = {2012-12-29},
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 = {},
archivePrefix = {arXiv},
arxivId = {1212.1649},
doi = {10.1103/PhysRevE.86.031109},
eprint = {1212.1649},
month = {dec},
year = 2012,
primaryClass = {math-ph},
collections = {Basically physics}
}
@article{Kac,
title = {Can One Hear the Shape of a Drum?},
author = {Kac, Mark},
url = {http://www.jstor.org/stable/2313748},
urldate = {2011-01-12},
abstract = {},
comment = {},
collections = {Basically physics,Easily explained,Music,Geometry,Fun maths facts},
year = 1966
}
@article{MeaninginClassicalMathematicsIsitatOddswithIntuitionism,
title = {Meaning in Classical Mathematics: Is it at Odds with Intuitionism?},
author = {Karin Usadi Katz and Mikhail G. Katz},
url = {http://arxiv.org/abs/1110.5456v1 http://arxiv.org/pdf/1110.5456v1},
urldate = {2017-02-27},
abstract = {We examine the classical/intuitionist divide, and how it reflects on modern
theories of infinitesimals. When leading intuitionist Heyting announced that
"the creation of non-standard analysis is a standard model of important
mathematical research", he was fully aware that he was breaking ranks with
Brouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a
comparative textual analysis of three of Bishop's texts, we analyze the
ideological and/or pedagogical nature of his objections to infinitesimals a la
Robinson. Bishop's famous "debasement" comment at the 1974 Boston workshop,
published as part of his Crisis lecture, in reality was never uttered in front
of an audience. We compare the realist and the anti-realist intuitionist
narratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and
Tennant. Variational principles are important physical applications, currently
lacking a constructive framework. We examine the case of the Hawking-Penrose
singularity theorem, already analyzed by Hellman in the context of the
Quine-Putnam indispensability thesis.},
comment = {},
archivePrefix = {arXiv},
eprint = {1110.5456},
primaryClass = {math.LO},
collections = {History,The act of doing maths},
year = 2011
}
@article{Coman,
title = {The Math Encyclopedia of Smarandache Type Notions},
author = {Coman, Marius},
url = {http://vixra.org/abs/1311.0097},
urldate = {2013-11-21},
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 “a set of Smarandache-Coman divisors of order k of a composite positive integer n with m prime factors”, 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 “navigate” 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 = {},
pages = 135,
collections = {Lists and catalogues},
year = 2013
}
@article{Gauvrit2011,
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},
url = {http://arxiv.org/abs/1101.4470 http://arxiv.org/pdf/1101.4470v2},
urldate = {2013-10-17},
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 = {},
month = {jan},
year = 2011,
archivePrefix = {arXiv},
eprint = {1101.4470},
primaryClass = {math.PR},
collections = {Easily explained,Probability and statistics,The act of doing maths,Integerology}
}
@article{Fathauer,
title = {Statistical Modeling of Gang Violence in Los Angeles},
author = {Fathauer, Chris},
url = {http://paleo.sscnet.ucla.edu/SIURO{\_}revised.pdf},
urldate = {2012-04-13},
abstract = {},
comment = {},
pages = {1--25},
collections = {Basically computer science,Probability and statistics,Modelling},
year = 2010
}
@article{Poonen2012,
title = {Undecidable problems: a sampler},
author = {Poonen, Bjorn},
url = {http://arxiv.org/abs/1204.0299 http://arxiv.org/pdf/1204.0299v2},
urldate = {2012-04-04},
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 = {},
archivePrefix = {arXiv},
arxivId = {1204.0299},
eprint = {1204.0299},
keywords = {and phrases,decision problem,undecidability},
month = {apr},
pages = 28,
year = 2012,
primaryClass = {math.LO},
collections = {Probability and statistics}
}
@article{JewishProblems,
title = {Jewish Problems},
author = {Tanya Khovanova and Alexey Radul},
url = {http://arxiv.org/abs/1110.1556v2 http://arxiv.org/pdf/1110.1556v2},
urldate = {2017-03-23},
abstract = {This is a special collection of problems that were given to select applicants
during oral entrance exams to the math department of Moscow State University.
These problems were designed to prevent Jews and other undesirables from
getting a passing grade. Among problems that were used by the department to
blackball unwanted candidate students, these problems are distinguished by
having a simple solution that is difficult to find. Using problems with a
simple solution protected the administration from extra complaints and appeals.
This collection therefore has mathematical as well as historical value.},
comment = {},
archivePrefix = {arXiv},
eprint = {1110.1556},
primaryClass = {math.HO},
collections = {Attention-grabbing titles,History,Puzzles},
year = 2011
}
@misc{item3,
title = {openttd logic gates},
author = {Heikki Kallasjoki},
url = {http://zem.fi/ttd{\_}logic/},
urldate = {2010-09-30},
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 = {},
collections = {Basically computer science,Unusual computers},
year = 2005
}
@misc{item4,
title = {Push-pull LEGO logic gates},
author = {Randomwraith},
url = {https://www.randomwraith.com/logic.html},
urldate = {2010-09-30},
abstract = {},
comment = {},
collections = {Basically computer science,Unusual computers},
year = 1999
}
@article{item6,
title = {Zaphod Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle},
author = {Andrew Granville},
url = {http://www.dms.umontreal.ca/{\~{}}andrew/PDF/beeb.pdf},
urldate = {2010-10-31},
abstract = {},
comment = {},
collections = {Attention-grabbing titles},
year = 1992
}
@online{item9,
title = {Laying train tracks},
author = {Danny Calegari},
url = {http://lamington.wordpress.com/2011/12/02/laying-train-tracks/},
urldate = {2011-12-03},
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’t tend to get concerned about whether the track closes up to make a loop.},
comment = {Group theory applied to BRIO train tracks},
collections = {Easily explained,Things to make and do,Fun maths facts,The groups group},
year = 2011
}
@online{item10,
title = {Understanding Monads With JavaScript},
author = {Ionuț G. Stan},
url = {http://igstan.ro/posts/2011-05-02-understanding-monads-with-javascript.html},
urldate = {2012-01-18},
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 = {},
collections = {Basically computer science},
year = 2011
}
@book{item11,
title = {Drawings from Angola: living mathematics},
author = {Paulus Gerdes},
url = {http://books.google.com/books?id=rRDbdeWoZEMC{\&}pgis=1},
urldate = {2012-01-19},
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 = {},
pages = 71,
publisher = {Lulu.com},
year = 2007,
collections = {Art,Easily explained,Unusual arithmetic}
}
@online{item14,
title = {Poe, E.: Near A Raven},
author = {Mike Keith},
url = {http://www.cadaeic.net/naraven.htm},
urldate = {2012-02-25},
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 π constraint, in which the number of letters in each successive word "spells out" the digits of π (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 = {},
collections = {Animals,Art},
year = 1995
}
@online{item15,
title = {Gaussian prime spirals},
author = {Joseph O'Rourke},
url = {http://mathoverflow.net/questions/91423/gaussian-prime-spirals},
urldate = {2012-03-24},
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 = {},
collections = {Easily explained},
year = 2012
}
@article{item18,
title = {Nim multiplication},
author = {H. W. Lenstra, Jr.},
url = {https://openaccess.leidenuniv.nl/bitstream/handle/1887/2125/346{\_}027.pdf?sequence=1},
urldate = {2012-04-21},
abstract = {},
comment = {},
collections = {Games to play with friends,Unusual arithmetic},
year = 1978
}
@online{item24,
title = {Magic: the Gathering is Turing Complete},
author = {Alex Churchill},
url = {http://www.toothycat.net/{\~{}}hologram/Turing/},
urldate = {2012-09-11},
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 = {},
collections = {Basically computer science,Games to play with friends,Unusual computers},
year = 2012
}
@online{item22,
title = {A Hamiltonian circuit for Rubik's Cube},
author = {cuBerBruce},
url = {http://bruce.cubing.net/ham333/rubikhamiltonexplanation.html},
urldate = {2012-09-04},
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 = {},
collections = {Easily explained,Puzzles,Fun maths facts},
year = 2012
}
@review{item29,
title = {Review of "Groups" by Georges Papy in New Scientist},
author = {T. H. O'Beirne},
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},
abstract = {},
comment = {},
page = 582,
year = 1964,
month = {sep},
day = 3,
collections = {The groups group}
}
@online{item31,
title = {Pascal's Pyramid Or Pascal's Tetrahedron},
author = {Jim Nugent},
url = {https://web-beta.archive.org/web/20160410142410/http://buckydome.com/math/Article2.htm http://buckydome.com/math/Article2.htm},
urldate = {2013-06-25},
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 = {},
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},
collections = {Easily explained},
year = 1990
}
@online{item37,
title = {Fair but irregular polyhedral dice},
author = {Joseph O'Rourke},
url = {http://mathoverflow.net/questions/46684/fair-but-irregular-polyhedral-dice},
urldate = {2013-12-16},
abstract = {},
comment = {},
collections = {Probability and statistics,Geometry},
year = 2010
}
@online{item39,
title = {Rithmomachia},
author = {Daniel U. Thibault and Michel Boutin},
url = {http://www.gamecabinet.com/rules/Rithmomachia.html},
urldate = {2014-01-24},
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 Boissière, a Frenchman.},
comment = {},
keywords = {Board,GURPS,Games,Medieval,Rithmomachia},
collections = {Games to play with friends,History},
year = 1984
}
@article{item42,
title = {The Number-Pad Game},
author = {Alex Fink and Richard Guy},
url = {http://www.maths.qmul.ac.uk/{\~{}}fink/numbpad2f.pdf},
urldate = {2014-06-11},
abstract = {},
comment = {},
collections = {Easily explained,Games to play with friends,Puzzles},
year = 2007
}
@article{item44,
title = {A Fresh Look at Peg Solitaire},
author = {George I. Bell},
url = {http://recmath.org/pegsolitaire/papers/Bell{\_}AFreshLookatPegSolitaire{\_}MathMag2007.pdf},
urldate = {2014-08-26},
abstract = {},
comment = {},
collections = {Easily explained,Puzzles},
year = 2007
}
@online{item45,
title = {Pondering an Artist's Perplexing Tribute to the Pythagorean Theorem},
author = {Ivars Peterson},
url = {http://mathtourist.blogspot.co.uk/2009/02/pondering-artists-perplexing-tribute-to.html},
urldate = {2014-08-28},
abstract = {},
comment = {},
collections = {Art,Easily explained,Drama!},
year = 2009
}
@misc{item49,
title = {Maximum Matching and a Polyhedron With 0,1-Vertices},
author = {Jack Edmonds},
url = {http://nvlpubs.nist.gov/nistpubs/jres/69B/jresv69Bn1-2p125{\_}A1b.pdf},
urldate = {2015-03-07},
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 = {},
year = 1964,
collections = {}
}
@article{item50,
title = {The Lost Calculus (1637-1670): Tangency and Optimization without Limits},
author = {Jeff Suzuki},
url = {http://www.maa.org/programs/maa-awards/writing-awards/the-lost-calculus-1637-1670-tangency-and-optimization-without-limits},
urldate = {2015-03-12},
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 = {},
collections = {History},
year = 2005
}
@article{item52,
title = {Proposal to Encode the Ganda Currency Mark for Bengali in ISO/IEC 10646},
author = {Anshuman Pandey},
url = {http://std.dkuug.dk/JTC1/SC2/wg2/docs/n3311.pdf},
urldate = {2015-09-30},
abstract = {},
comment = {},
collections = {Easily explained,Notation and conventions},
year = 2007
}
@article{item55,
title = {On the Number of Times an Integer Occurs as a Binomial Coefficient},
author = {H. L. Abbott and P. Erd{\H{o}}s and D. Hanson},
url = {http://www.jstor.org/stable/2319526?seq=4{\#}page{\_}scan{\_}tab{\_}contents},
urldate = {2015-11-30},
abstract = {},
comment = {},
collections = {Integerology},
year = 1974
}
@article{item57,
title = {Approaches to the Enumerative Theory of Meanders},
author = {Michael La Croix},
url = {http://www.math.uwaterloo.ca/{\~{}}malacroi/Latex/Meanders.pdf},
urldate = {2015-12-14},
abstract = {},
comment = {},
collections = {},
year = 2003
}
@article{item60,
title = {Transposable integers in arbitrary bases},
author = {Anne L. Ludington},
url = {http://www.fq.math.ca/Scanned/25-3/ludington.pdf},
urldate = {2016-04-19},
abstract = {},
comment = {},
year = 1985,
collections = {Integerology}
}
@online{Entanglement,
title = {A game for budding knot theorists},
author = {Dave Richeson},
url = {http://divisbyzero.com/2010/08/24/a-game-for-budding-knot-theorists/},
urldate = {2010-08-25},
abstract = {},
comment = {},
year = 2010,
month = {aug},
day = 24,
collections = {Games to play with friends}
}
@misc{item56,
title = {The Theory of Heaps and the Cartier-Foata Monoid},
author = {C. Krattenthaler},
url = {http://www.mat.univie.ac.at/{\~{}}kratt/artikel/heaps.pdf},
urldate = {2015-12-03},
abstract = {We present Viennot’s 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–M{\'{e}}lou and Viennot.},
comment = {},
collections = {Combinatorics},
year = 2006
}
@article{NotesontheFourthDimensionThePublicDomainReview,
title = {Notes on the Fourth Dimension},
author = {Jon Crabb },
url = {http://publicdomainreview.org/2015/10/28/notes-on-the-fourth-dimension/},
urldate = {2016-06-13},
abstract = {Hyperspace, ghosts, and colourful cubes — Jon Crabb on the work of Charles Howard Hinton and the cultural history of higher dimensions.},
comment = {},
collections = {Art,Geometry},
year = 2015
}
@online{item20,
title = {To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction},
author = {David C Keenan},
url = {http://dkeenan.com/Lambda/index.htm},
urldate = {2012-06-03},
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 = {},
year = 1996,
collections = {Animals,Basically computer science}
}
@online{PhotoelectricNumberSieve,
title = {Photoelectric Number Sieve Machine ("Gear Machine")},
author = {D. H. Lehmer and Robert Canepa},
url = {http://www.computerhistory.org/collections/catalog/X85.82},
urldate = {2016-06-13},
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 = {},
collections = {Easily explained,Things to make and do,Integerology},
year = 1932
}
@misc{Feferman,
title = {Penrose's Godelian argument},
author = {Feferman, Solomon},
url = {http://math.stanford.edu/{\~{}}feferman/papers/penrose.pdf},
urldate = {2011-05-16},
abstract = {},
comment = {},
collections = {},
year = 1996
}
@article{Levien2008,
title = {The Euler spiral: a mathematical history},
author = {Levien, Raph},
url = {http://raph.levien.com/phd/euler{\_}hist.pdf},
urldate = {2012-04-07},
abstract = {The beautiful Euler spiral, deﬁned by the linear relationship between curvature and arclength, was ﬁrst proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, ﬁrst 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 conﬂuent 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.},
journal = {Opera},
pages = {1--14},
year = 2008,
collections = {History,Geometry}
}
@online{ScoopingLoopSnooper,
title = {Scooping the Loop Snooper},
author = {Geoffrey K. Pullum},
url = {http://www.lel.ed.ac.uk/{\~{}}gpullum/loopsnoop.html},
urldate = {2012-01-20},
abstract = {},
comment = {A proof that the Halting Problem is undecidable.},
collections = {Attention-grabbing titles,Easily explained,About proof},
year = 2000
}
@article{Good1819,
title = {Pantologia. A new (cabinet) cyclopædia, by J.M. Good, O. Gregory, and N. Bosworth assisted by other gentlemen of eminence},
author = {Good, John Mason and Gregory, Olinthus Gilbert},
url = {http://books.google.com/books?id=72vgv4yVQOAC{\&}pgis=1},
urldate = {2012-02-10},
abstract = {},
comment = {Contains something on the continued surd.},
year = 1819,
collections = {History,Lists and catalogues}
}
@article{Williams2004,
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},
url = {http://books.google.com/books?id=udr3tHHwBl0C{\&}pgis=1},
urldate = {2012-04-12},
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 think 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.},
pages = 392,
publisher = {American Mathematical Soc.},
year = 2004,
collections = {Attention-grabbing titles,Integerology}
}
@article{Guilbaud1963,
title = {Analyse alg{\'{e}}brique d'un scrutin},
author = {Guilbaud, GT and Rosenstiehl, P},
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},
abstract = {},
comment = {Found while looking at the permutohedron.},
journal = {Math. Sci. Hum},
pages = {9--33},
volume = 4,
year = 1963,
collections = {}
}
@online{item48,
title = {Magic squares of seventh powers},
author = {Christian Boyer},
url = {http://www.multimagie.com/English/SquaresOfSeventhPowers.htm},
urldate = {2015-03-06},
abstract = {},
comment = {Magic square whose sum is the first 52 digits of {\$}\pi{\$}.},
collections = {Puzzles},
year = 2004
}
@article{PauliPascalPyramidsPauliFibonacciNumbersandPauliJacobsthalNumbers,
title = {Pauli Pascal Pyramids, Pauli Fibonacci Numbers, and Pauli Jacobsthal Numbers},
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
Jacobsthal numbers, and Pauli Fibonacci numbers of higher order. And the question is: are Pauli rabbits killer rabbits?},
url = {http://arxiv.org/abs/0711.4030v1 http://arxiv.org/pdf/0711.4030v1},
author = {Martin Erik Horn},
comment = {},
urldate = {2017-03-24},
archivePrefix = {arXiv},
eprint = {0711.4030},
primaryClass = {math.GM},
year = 2007,
collections = {Attention-grabbing titles,Fibonaccinalia}
}
@article{ApprovalVotinginProductSocieties,
title = {Approval Voting in Product Societies},
abstract = {In approval voting, individuals vote for all platforms that they find
acceptable. In this situation it is natural to ask: When is agreement possible?
What conditions guarantee that some fraction of the voters agree on even a
single platform? Berg et. al. found such conditions when voters are asked to
make a decision on a single issue that can be represented on a linear spectrum.
In particular, they showed that if two out of every three voters agree on a
platform, there is a platform that is acceptable to a majority of the voters.
Hardin developed an analogous result when the issue can be represented on a
circular spectrum. We examine scenarios in which voters must make two decisions
simultaneously. For example, if voters must decide on the day of the week to
hold a meeting and the length of the meeting, then the space of possible
options forms a cylindrical spectrum. Previous results do not apply to these
multi-dimensional voting societies because a voter's preference on one issue
often impacts their preference on another. We present a general lower bound on
agreement in a two-dimensional voting society, and then examine specific
results for societies whose spectra are cylinders and tori.},
url = {http://arxiv.org/abs/1703.09870v1 http://arxiv.org/pdf/1703.09870v1},
author = {Kristen Mazur and Mutiara Sondjaja and Matthew Wright and Carolyn Yarnall},
comment = {},
urldate = {2017-03-30},
archivePrefix = {arXiv},
eprint = {1703.09870},
primaryClass = {math.CO},
year = 2017,
collections = {Protocols and strategies}
}
@misc{item8,
title = {Packing circles and spheres on surfaces},
author = {Alexander Schiftner and Mathias H{\"{o}}binger and Johannes Wallner and Helmut Pottmann},
url = {http://dl.acm.org/citation.cfm?id=1618485 http://www.geometrie.tugraz.at/wallner/packing.pdf},
urldate = {2011-09-04},
abstract = {Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces’ 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
carry 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
source of geometric structures relevant to architectural geometry.},
comment = {},
year = 2009,
collections = {Basically computer science,Geometry}
}
@article{Tropicaltotallypositivematrices,
title = {Tropical totally positive matrices},
abstract = {We investigate the tropical analogues of totally positive and totally
nonnegative matrices. These arise when considering the images by the
nonarchimedean valuation of the corresponding classes of matrices over a real
nonarchimedean valued field, like the field of real Puiseux series. We show
that the nonarchimedean valuation sends the totally positive matrices precisely
to the Monge matrices. This leads to explicit polyhedral representations of the
tropical analogues of totally positive and totally nonnegative matrices. We
also show that tropical totally nonnegative matrices with a finite permanent
can be factorized in terms of elementary matrices. We finally determine the
eigenvalues of tropical totally nonnegative matrices, and relate them with the
eigenvalues of totally nonnegative matrices over nonarchimedean fields.},
url = {http://arxiv.org/abs/1606.00238v3 http://arxiv.org/pdf/1606.00238v3},
author = {St{\'{e}}phane Gaubert and Adi Niv},
comment = {},
urldate = {2017-05-02},
archivePrefix = {arXiv},
eprint = {1606.00238},
primaryClass = {math.AC},
year = 2016,
collections = {Attention-grabbing titles,Unusual arithmetic}
}
@article{OnFibonacciQuaternions,
title = {On Fibonacci Quaternions},
author = {Serpil Halici},
url = {https://link.springer.com/article/10.1007/s00006-011-0317-1},
urldate = {2017-05-02},
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 = {},
doi = {10.1007/s00006-011-0317-1},
year = 2012,
issue = 2,
volume = 22,
collections = {Fibonaccinalia}
}
@article{NoThisisnotaCircle,
title = {No, This is not a Circle},
abstract = {A curve, also shown in introductory maths textbooks, seems like a circle. But
it is actually a different curve. This paper discusses some easy approaches to
classify the result, including a GeoGebra applet construction.},
url = {http://arxiv.org/abs/1704.08483v2 http://arxiv.org/pdf/1704.08483v2},
author = {Zolt{\'{a}}n Kov{\'{a}}cs},
comment = {},
urldate = {2017-05-02},
archivePrefix = {arXiv},
eprint = {1704.08483},
primaryClass = {math.HO},
year = 2017,
collections = {Attention-grabbing titles,Easily explained,Drama!,Geometry}
}
@article{PVCPolyhedra,
title = {PVC Polyhedra},
author = {David Glickenstein},
url = {http://arxiv.org/abs/1705.00100v1 http://arxiv.org/pdf/1705.00100v1},
urldate = {2017-05-02},
abstract = {We describe how to construct a dodecahedron, tetrahedron, cube, and
octahedron out of pvc pipes using standard fittings.},
comment = {},
archivePrefix = {arXiv},
eprint = {1705.00100},
primaryClass = {math.HO},
year = 2017,
collections = {Easily explained,Things to make and do,Geometry}
}
@article{Homotopytypetheorythelogicofspace,
title = {Homotopy type theory: the logic of space},
abstract = {This is an introduction to type theory, synthetic topology, and homotopy type
theory from a category-theoretic and topological point of view, written as a
chapter for the book "New Spaces for Mathematics and Physics" (ed. Gabriel
Catren and Mathieu Anel).},
url = {http://arxiv.org/abs/1703.03007v1 http://arxiv.org/pdf/1703.03007v1},
author = {Michael Shulman},
comment = {},
urldate = {2017-05-03},
archivePrefix = {arXiv},
eprint = {1703.03007},
primaryClass = {math.CT},
year = 2017,
collections = {}
}
@article{Theopaquesquare,
title = {The opaque square},
abstract = {The problem of finding small sets that block every line passing through a
unit square was first considered by Mazurkiewicz in 1916. We call such a set
{\{}\em opaque{\}} or a {\{}\em barrier{\}} for the square. The shortest known barrier has
length {\$}\sqrt{\{}2{\}}+ \frac{\{}\sqrt{\{}6{\}}{\}}{\{}2{\}}= 2.6389\ldots{\$}. The current best lower
bound for the length of a (not necessarily connected) barrier is {\$}2{\$}, as
established by Jones about 50 years ago. No better lower bound is known even if
the barrier is restricted to lie in the square or in its close vicinity. Under
a suitable locality assumption, we replace this lower bound by {\$}2+10^{\{}-12{\}}{\$},
which represents the first, albeit small, step in a long time toward finding
the length of the shortest barrier. A sharper bound is obtained for interior
barriers: the length of any interior barrier for the unit square is at least {\$}2
+ 10^{\{}-5{\}}{\$}. Two of the key elements in our proofs are: (i) formulas established
by Sylvester for the measure of all lines that meet two disjoint planar convex
bodies, and (ii) a procedure for detecting lines that are witness to the
invalidity of a short bogus barrier for the square.},
url = {http://arxiv.org/abs/1311.3323v1 http://arxiv.org/pdf/1311.3323v1},
author = {Adrian Dumitrescu and Minghui Jiang},
comment = {},
urldate = {2017-05-22},
archivePrefix = {arXiv},
eprint = {1311.3323},
primaryClass = {math.CO},
year = 2013,
collections = {Easily explained,Geometry}
}
@article{CuckooFilterSimplificationandAnalysis,
title = {Cuckoo Filter: Simplification and Analysis},
abstract = {The cuckoo filter data structure of Fan, Andersen, Kaminsky, and Mitzenmacher
(CoNEXT 2014) performs the same approximate set operations as a Bloom filter in
less memory, with better locality of reference, and adds the ability to delete
elements as well as to insert them. However, until now it has lacked
theoretical guarantees on its performance. We describe a simplified version of
the cuckoo filter using fewer hash function calls per query. With this
simplification, we provide the first theoretical performance guarantees on
cuckoo filters, showing that they succeed with high probability whenever their
fingerprint length is large enough.},
url = {http://arxiv.org/abs/1604.06067v1 http://arxiv.org/pdf/1604.06067v1},
author = {David Eppstein},
comment = {},
urldate = {2017-05-23},
archivePrefix = {arXiv},
eprint = {1604.06067},
primaryClass = {cs.DS},
year = 2016,
collections = {Animals,Basically computer science}
}
@article{Gunji2011,
title = {Robust Soldier Crab Ball Gate},
author = {Gunji, YP and Nishiyama, Y},
url = {https://arxiv.org/abs/1204.1749 https://arxiv.org/pdf/1204.1749},
urldate = {2012-04-15},
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 = {},
journal = {AIP Conference Proceedings},
year = 2011,
collections = {Animals,Attention-grabbing titles,Basically computer science,Easily explained,Unusual computers}
}
@article{Arithmeticalstructuresongraphs,
title = {Arithmetical structures on graphs},
abstract = {Arithmetical structures on a graph were introduced by Lorenzini as some
intersection matrices that arise in the study of degenerating curves in
algebraic geometry. In this article we study these arithmetical structures, in
particular we are interested in the arithmetical structures on complete graphs,
paths, and cycles. We begin by looking at the arithmetical structures on a
multidigraph from the general perspective of {\$}M{\$}-matrices. As an application,
we recover the result of Lorenzini about the finiteness of the number of
arithmetical structures on a graph. We give a description on the arithmetical
structures on the graph obtained by merging and splitting a vertex of a graph
in terms of its arithmetical structures. On the other hand, we give a
description of the arithmetical structures on the clique--star transform of a
graph, which generalizes the subdivision of a graph. As an application of this
result we obtain an explicit description of all the arithmetical structures on
the paths and cycles and we show that the number of the arithmetical structures
on a path is a Catalan number.},
url = {http://arxiv.org/abs/1604.02502v3 http://arxiv.org/pdf/1604.02502v3},
author = {Hugo Corrales and Carlos E. Valencia},
comment = {},
urldate = {2017-06-14},
archivePrefix = {arXiv},
eprint = {1604.02502},
primaryClass = {math.CO},
year = 2016,
collections = {Unusual arithmetic}
}
@article{Randomlyjugglingbackwards,
title = {Randomly juggling backwards},
abstract = {We recall the directed graph of {\_}juggling states{\_}, closed walks within which
give juggling patterns, as studied by Ron Graham in [w/Chung, w/Butler].
Various random walks in this graph have been studied before by several authors,
and their equilibrium distributions computed. We motivate a random walk on the
reverse graph (and an enrichment thereof) from a very classical linear algebra
problem, leading to a particularly simple equilibrium: a Boltzmann distribution
closely related to the Poincar\'e series of the b-Grassmannian in
infinite-dimensional space.
We determine the most likely asymptotic state in the limit of many balls,
where in the limit the probability of a 0-throw is kept fixed.},
url = {http://arxiv.org/abs/1601.06391v1 http://arxiv.org/pdf/1601.06391v1},
author = {Allen Knutson},
comment = {},
urldate = {2017-06-14},
archivePrefix = {arXiv},
eprint = {1601.06391},
primaryClass = {math.CO},
year = 2016,
collections = {Attention-grabbing titles,Probability and statistics}
}
@article{AformulagoestocourtPartisangerrymanderingandtheefficiencygap,
title = {A formula goes to court: Partisan gerrymandering and the efficiency gap},
abstract = {Recently, a proposal has been advanced to detect unconstitutional partisan
gerrymandering with a simple formula called the efficiency gap. The efficiency
gap is now working its way towards a possible landmark case in the Supreme
Court. This note explores some of its mathematical properties in light of the
fact that it reduces to a straight proportional comparison of votes to seats.
Though we offer several critiques, we assess that EG can still be a useful
component of a courtroom analysis. But a famous formula can take on a life of
its own and this one will need to be watched closely.},
url = {http://arxiv.org/abs/1705.10812v1 http://arxiv.org/pdf/1705.10812v1},
author = {Mira Bernstein and Moon Duchin},
comment = {},
urldate = {2017-06-16},
archivePrefix = {arXiv},
eprint = {1705.10812},
primaryClass = {physics.soc-ph},
year = 2017,
collections = {Protocols and strategies,Geometry}
}
@article{Everypositiveintegerisasumofthreepalindromes,
title = {Every positive integer is a sum of three palindromes},
abstract = {For integer {\$}g\ge 5{\$}, we prove that any positive integer can be written as a
sum of three palindromes in base {\$}g{\$}.},
url = {http://arxiv.org/abs/1602.06208v2 http://arxiv.org/pdf/1602.06208v2},
author = {Javier Cilleruelo and Florian Luca and Lewis Baxter},
comment = {},
urldate = {2017-06-21},
archivePrefix = {arXiv},
eprint = {1602.06208},
primaryClass = {math.NT},
year = 2016,
collections = {Easily explained,Fun maths facts}
}
@article{item59,
title = {A Dozen Hat Problems},
author = {Ezra Brown and James Tanton},
url = {https://www.math.vt.edu/people/brown/doc/dozen{\_}hats.pdf},
urldate = {2016-04-12},
abstract = {Hat problems are all the rage these days, proliferating on various web sites and generating a great deal of conversation—and research—among mathematicians
and students. But they have been around for quite a while in different forms.},
comment = {Also known as prisoner puzzles},
year = 2009,
collections = {Games to play with friends,Protocols and strategies,Puzzles}
}
@article{OnthedateofCauchyscontributionstothefoundingofthetheoryofgroups,
title = {On the date of Cauchy's contributions to the founding of the theory of groups},
author = {Peter M. Neumann},
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},
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 = {},
doi = {10.1017/S000497270000438X},
year = 1989,
collections = {History,Drama!}
}
@book{TheHandbookofMathematicalDiscourse,
title = {The Handbook of Mathematical Discourse},
author = {Charles Wells},
url = {http://case.edu/artsci/math/wells/pub/html/abouthbk.html http://www.cwru.edu/artsci/math/wells/pub/pdf/hyperhbk.pdf},
urldate = {2017-06-19},
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.
This 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.
Someday, 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.
The 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 = {},
year = 2003,
collections = {Notation and conventions,The act of doing maths,Education}
}
@article{ArrangementsOfStarsOnTheAmericanFlag,
title = {Arrangements of Stars on the American Flag},
author = {Dimitris Koukoulopoulos and Johann Thiel},
url = {http://www.maa.org/sites/default/files/pdf/pubs/monthly{\_}june12-stars.pdf},
urldate = {2017-06-12},
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 = {},
year = 2012,
collections = {Art,Easily explained,Geometry}
}
@article{Polylogarithmicladdershypergeometricseriesandthetenmillionthdigitsof3and5,
title = {Polylogarithmic ladders, hypergeometric series and the ten millionth digits of {\$}ζ(3){\$} and {\$}ζ(5){\$}},
abstract = {We develop ladders that reduce {\$}\zeta(n):=\sum{\_}{\{}k>0{\}}k^{\{}-n{\}}{\$}, for
{\$}n=3,5,7,9,11{\$}, and {\$}\beta(n):=\sum{\_}{\{}k\ge0{\}}(-1)^k(2k+1)^{\{}-n{\}}{\$}, for {\$}n=2,4,6{\$},
to convergent polylogarithms and products of powers of {\$}\pi{\$} and {\$}\log2{\$}. Rapid
computability results because the required arguments of {\$}{\{}\rm
Li{\}}{\_}n(z)=\sum{\_}{\{}k>0{\}}z^k/k^n{\$} satisfy {\$}z^8=1/16^p{\$}, with {\$}p=1,3,5{\$}. We prove that
{\$}G:=\beta(2){\$}, {\$}\pi^3{\$}, {\$}\log^32{\$}, {\$}\zeta(3){\$}, {\$}\pi^4{\$}, {\$}\log^42{\$}, {\$}\log^52{\$},
{\$}\zeta(5){\$}, and six products of powers of {\$}\pi{\$} and {\$}\log2{\$} are constants whose
{\$}d{\$}th hexadecimal digit can be computed in time{\~{}}{\$}=O(d\log^3d){\$} and
space{\~{}}{\$}=O(\log d){\$}, as was shown for {\$}\pi{\$}, {\$}\log2{\$}, {\$}\pi^2{\$} and {\$}\log^22{\$} by
Bailey, Borwein and Plouffe. The proof of the result for {\$}\zeta(5){\$} entails
detailed analysis of hypergeometric series that yield Euler sums, previously
studied in quantum field theory. The other 13 results follow more easily from
Kummer's functional identities. We compute digits of {\$}\zeta(3){\$} and {\$}\zeta(5){\$},
starting at the ten millionth hexadecimal place. These constants result from
calculations of massless Feynman diagrams in quantum chromodynamics. In a
related paper, hep-th/9803091, we show that massive diagrams also entail
constants whose base of super-fast computation is {\$}b=3{\$}.},
url = {http://arxiv.org/abs/math/9803067v1 http://arxiv.org/pdf/math/9803067v1},
author = {D. J. Broadhurst},
comment = {},
urldate = {2017-06-29},
archivePrefix = {arXiv},
eprint = {math/9803067},
primaryClass = {math.CA},
year = 1998,
collections = {Attention-grabbing titles}
}
@online{SteinhausLongimeter,
title = {Steinhaus Longimeter},
author = {Chris Staecker},
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},
abstract = {The longimeter, invented by Hugo Steinhaus, is a device for measuring the length of a curve drawn on paper.
It'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 = {},
year = 2017,
collections = {Easily explained,Things to make and do}
}
@article{UnunfoldablePolyhedrawithConvexFaces,
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},
url = {http://erikdemaine.org/papers/Ununfoldable/},
urldate = {2017-06-26},
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 “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut.},
comment = {},
year = 2003,
collections = {Things to make and do}
}
@online{TheternarycalculatingmachineofThomasFowler,
title = {The ternary calculating machine of Thomas Fowler},
author = {Mark Glusker},
url = {http://www.mortati.com/glusker/fowler/index.htm},
urldate = {2017-06-22},
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 = {},
year = 1997,
collections = {Basically computer science,History,Unusual arithmetic,Unusual computers,Integerology}
}
@software{SetunSimulator,
title = {Сетунь ВС (Setun Web Simulator)},
author = {Trinarygroup},
url = {http://trinary.ru/projects/setunws/},
urldate = {2017-06-22},
abstract = {},
comment = {A simulator of the balanced-ternary Setun computer. },
year = 2017,
collections = {Basically computer science,History,Unusual computers}
}
@online{MathMagic,
title = {Math Magic},
author = {Erich Friedman},
url = {http://www2.stetson.edu/{\~{}}efriedma/mathmagic/},
urldate = {2017-06-21},
abstract = {Math Magic is a web site devoted to original mathematical recreations.},
comment = {A monthly recreational maths problem, stretching back to 1998.},
year = 1998,
collections = {Puzzles}
}
@article{Graphlopedia,
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},
url = {https://sites.math.washington.edu/{\~{}}billey/graphlopedia.html http://www.math.washington.edu/{\~{}}billey/graphlopediaPDF.pdf},
urldate = {2017-06-21},
abstract = {A database of graphs for the use of mathematicians and other graph lovers. The graphs are ordered by degree sequence.},
comment = {},
year = 2017,
collections = {Lists and catalogues}
}
@article{Friedman1998,
title = {Long finite sequences},
author = {Friedman, Harvey M},
url = {http://u.osu.edu/friedman.8/files/2014/01/LongFinSeq98-2f0wmq3.pdf},
urldate = {2012-04-24},
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 = {},
pages = {1--50},
year = 1998,
collections = {}
}
@online{item26,
title = {Earliest Uses of Symbols of Calculus},
author = {Jeff Miller},
url = {http://jeff560.tripod.com/calculus.html},
urldate = {2012-10-09},
abstract = {},
comment = {},
year = 2008,
collections = {Easily explained,History,Lists and catalogues,Notation and conventions}
}
@article{OvalsandEggCurves,
title = {Ovals and Egg Curves},
author = {J{\"{u}}rgen K{\"{o}}ller },
url = {http://www.mathematische-basteleien.de/eggcurves.htm},
urldate = {2017-04-03},
abstract = {},
comment = {},
year = 2000,
collections = {Art,Things to make and do,Food,Geometry}
}
@misc{Devlin,
title = {Good stories, pity they're not true},
author = {Devlin, Keith},
url = {https://www.maa.org/external{\_}archive/devlin/devlin{\_}06{\_}04.html http://www.maa.org/devlin/devlin{\_}06{\_}04.html},
urldate = {2012-02-08},
abstract = {},
comment = {},
year = 2004,
collections = {Attention-grabbing titles,History,Drama!}
}
@article{CONRAD,
title = {Proofs by Descent},
author = {Keith Conrad},
url = {http://www.math.uconn.edu/{\~{}}kconrad/blurbs/ugradnumthy/descent.pdf},
urldate = {2013-11-04},
abstract = {},
comment = {},
year = 2010,
collections = {About proof}
}
@article{item23,
title = {Modiﬁed Pascal Triangle and Pascal Surfaces},
author = {Rely Pellicer and David Alvo},
url = {http://uai.academia.edu/DavidAlvo/Papers/995009/Modi{\_}ed{\_}Pascal{\_}Triangle{\_}and{\_}Pascal{\_}Surfaces},
urldate = {2012-09-05},
abstract = {},
comment = {},
year = 2012,
collections = {}
}
@online{item28,
title = {What are some of the most ridiculous proofs in mathematics?},
author = {Anonymous},
url = {http://www.quora.com/Mathematics/What-are-some-of-the-most-ridiculous-proofs-in-mathematics},
urldate = {2013-02-17},
abstract = {},
comment = {},
year = 2012,
collections = {The act of doing maths,About proof}
}
@online{item27,
title = {Figures for "Impossible fractals"},
author = {Cameron Browne},
url = {http://www.cameronius.com/graphics/impossible-fractals-figures/},
urldate = {2012-12-18},
abstract = {},
comment = {},
year = 2008,
collections = {Art,Easily explained,Geometry}
}
@article{CSCheatSheet,
title = {Theoretical Computer Science Cheat Sheet},
author = {Steve Seiden},
url = {http://www.tug.org/texshowcase/cheat.pdf},
urldate = {2011-03-26},
abstract = {},
comment = {},
journal = {TeX showcase},
year = 2003,
collections = {Basically computer science,Lists and catalogues}
}
@online{item21,
title = {Carcassonne and multivariate calculus},
author = {Douglas Weathers},
url = {http://douglasweathers.com/permalink.php?id=5},
urldate = {2012-08-13},
abstract = {},
comment = {Link is broken and the Internet Archive gives an error :(},
year = 2012,
collections = {Unusual arithmetic}
}
@article{Allis1988,
title = {A knowledge-based approach of connect-four},
author = {Allis, Victor},
url = {http://www.informatik.uni-trier.de/{\~{}}fernau/DSL0607/Masterthesis-Viergewinnt.pdf},
urldate = {2013-12-03},
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 × (2n) board, provided that White does not start at the middle column, as well as on any 6 × (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 × 6 board. Using a database of approximately half a million positions, VICTOR can play real time against opponents on the 7 × 6 board, always winning with White.},
comment = {},
year = 1988,
collections = {Games to play with friends}
}
@article{Computationalcomplexityand3manifoldsandzombies,
title = {Computational complexity and 3-manifolds and zombies},
abstract = {We show the problem of counting homomorphisms from the fundamental group of a
homology {\$}3{\$}-sphere {\$}M{\$} to a finite, non-abelian simple group {\$}G{\$} is
{\#}P-complete, in the case that {\$}G{\$} is fixed and {\$}M{\$} is the computational input.
Similarly, deciding if there is a non-trivial homomorphism is NP-complete. In
both reductions, we can guarantee that every non-trivial homomorphism is a
surjection. As a corollary, for any fixed integer {\$}m \ge 5{\$}, it is NP-complete
to decide whether {\$}M{\$} admits a connected {\$}m{\$}-sheeted covering.
Our construction is inspired by universality results in topological quantum
computation. Given a classical reversible circuit {\$}C{\$}, we construct {\$}M{\$} so that
evaluations of {\$}C{\$} with certain initialization and finalization conditions
correspond to homomorphisms {\$}\pi{\_}1(M) \to G{\$}. An intermediate state of {\$}C{\$}
likewise corresponds to a homomorphism {\$}\pi{\_}1(\Sigma{\_}g) \to G{\$}, where
{\$}\Sigma{\_}g{\$} is a pointed Heegaard surface of {\$}M{\$} of genus {\$}g{\$}. We analyze the
action on these homomorphisms by the pointed mapping class group
{\$}\text{\{}MCG{\}}{\_}*(\Sigma{\_}g){\$} and its Torelli subgroup {\$}\text{\{}Tor{\}}{\_}*(\Sigma{\_}g){\$}. By
results of Dunfield-Thurston, the action of {\$}\text{\{}MCG{\}}{\_}*(\Sigma{\_}g){\$} is as
large as possible when {\$}g{\$} is sufficiently large; we can pass to the Torelli
group using the congruence subgroup property of {\$}\text{\{}Sp{\}}(2g,\mathbb{\{}Z{\}}){\$}. Our
results can be interpreted as a sharp classical universality property of an
associated combinatorial {\$}(2+1){\$}-dimensional TQFT.},
url = {http://arxiv.org/abs/1707.03811v1 http://arxiv.org/pdf/1707.03811v1},
author = {Greg Kuperberg and Eric Samperton},
comment = {},
urldate = {2017-07-13},
archivePrefix = {arXiv},
eprint = {1707.03811},
primaryClass = {math.GT},
year = 2017,
collections = {Attention-grabbing titles,Basically computer science}
}
@article{Overcurvature,
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},
url = {https://www.nature.com/articles/ncomms2311 https://www.nature.com/articles/ncomms2311.pdf},
urldate = {2017-07-20},
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 = {},
year = 2012,
journal = {Nature},
collections = {Art,Basically physics,Easily explained,Things to make and do,Geometry,Modelling}
}
@article{item43,
title = {The Nesting and Roosting Habits of The Laddered Parenthesis},
author = {R. K. Guy and J. L. Selfridge},
url = {http://www.jstor.org/stable/2319392 http://oeis.org/A003018/a003018.pdf},
urldate = {2016-09-23},
abstract = {},
comment = {},
year = 1973,
collections = {Attention-grabbing titles,Easily explained,Combinatorics}
}
@article{AvianeggshapeFormfunctionandevolution,
title = {Avian egg shape: Form, function, and evolution},
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.},
url = {http://science.sciencemag.org/content/356/6344/1249 http://science.sciencemag.org/content/356/6344/1249.full.pdf},
author = {Mary Caswell Stoddard and Ee Hou Yong and Derya Akkaynak and Catherine Sheard and Joseph A. Tobias and L. Mahadevan},
comment = {},
urldate = {2017-07-24},
year = 2017,
doi = {10.1126/science.aaj1945},
collections = {Animals,Food,Geometry,Modelling}
}
@article{Linusson1998,
title = {A Smaller Sleeping Bag For A Baby Snake},
author = {Linusson, Svante and ASTLUND, JW},
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},
abstract = {},
comment = {},
keywords = {blanket,sleeping bag,snake,worm},
pages = {1--5},
year = 1998,
doi = {10.1.1.31.7066},
collections = {Animals,Attention-grabbing titles,Easily explained,Puzzles,Geometry}
}
@article{AnExplicitIsometricReductionoftheUnitSphereintoanArbitrarilySmallBall,
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},
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},
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 60:383–396, 1954) and Kuiper (Indag Math 17:545–555, 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 = {},
year = 2017,
doi = {10.1007/s10208-017-9360-1},
collections = {Geometry}
}
@article{Doyle2006,
title = {Division by three},
author = {Doyle, Peter G. and Conway, John Horton},
url = {https://arxiv.org/abs/math/0605779v1 https://arxiv.org/pdf/math/0605779v1.pdf},
urldate = {2014-11-17},
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 = {},
month = {may},
year = 2006,
collections = {Fun maths facts}
}
@article{ProofofConwaysLostCosmologicalTheorem,
title = {Proof of Conway's Lost Cosmological Theorem},
abstract = {John Horton Conway's Cosmological Theorem, about Audioactive sequences, for
which no extant proof existed, is given a computer-generated proof, hopefully
for good.},
url = {http://arxiv.org/abs/math/9808077v1 http://arxiv.org/pdf/math/9808077v1},
author = {Shalosh B. Ekhad and Doron Zeilberger},
comment = {},
urldate = {2017-07-26},
archivePrefix = {arXiv},
eprint = {math/9808077},
primaryClass = {math.CO},
year = 1998,
collections = {Attention-grabbing titles,Basically computer science,Easily explained,The act of doing maths,About proof}
}
@article{TheCurlingNumberConjecture,
title = {The Curling Number Conjecture},
abstract = {Given a finite nonempty sequence of integers S, by grouping adjacent terms it
is always possible to write it, possibly in many ways, as S = X Y^k, where X
and Y are sequences and Y is nonempty. Choose the version which maximizes the
value of k: this k is the curling number of S. The Curling Number Conjecture is
that if one starts with any initial sequence S, and extends it by repeatedly
appending the curling number of the current sequence, the sequence will
eventually reach 1. The conjecture remains open, but we will report on some
numerical results and conjectures in the case when S consists of only 2's and
3's.},
url = {http://arxiv.org/abs/0912.2382v5 http://arxiv.org/pdf/0912.2382v5},
author = {Benjamin Chaffin and N. J. A. Sloane},
comment = {},
urldate = {2017-07-31},
archivePrefix = {arXiv},
eprint = {0912.2382},
primaryClass = {math.CO},
year = 2009,
collections = {Easily explained,Integerology}
}
@article{MaximumgenusofthegeneralizedJengagame,
title = {Maximum genus of the generalized Jenga game},
abstract = {We treat the boundary of the union of blocks in the Jenga game as a surface
with a polyhedral structure and consider its genus. We generalize the game and
determine the maximum genus of the generalized game.},
url = {http://arxiv.org/abs/1708.01503v1 http://arxiv.org/pdf/1708.01503v1},
author = {Rika Akiyama and Nozomi Abe and Hajime Fujita and Yukie Inaba and Mari Hataoka and Shiori Ito and Satomi Seita},
comment = {},
urldate = {2017-08-07},
archivePrefix = {arXiv},
eprint = {1708.01503},
primaryClass = {math.HO},
year = 2017,
collections = {Attention-grabbing titles,Games to play with friends,Things to make and do}
}
@article{TheBulgariansolitaireandthemathematicsaroundit,
title = {The Bulgarian solitaire and the mathematics around it},
abstract = {The Bulgarian solitaire is a mathematical card game played by one person. A
pack of \(n\) cards is divided into several decks (or "piles"). Each move consists
of the removing of one card from each deck and collecting the removed cards to
form a new deck. The game ends when the same position occurs twice. It has
turned out that when \(n=k(k+1)/2\) is a triangular number, the game reaches the
same stable configuration with size of the piles \(1,2,\ldots,k\). The purpose of the
paper is to tell the (quite amusing) story of the game and to discuss
mathematical problems related with the Bulgarian solitaire.
The paper is dedicated to the memory of Borislav Bojanov (1944-2009), a great
mathematician, person, and friend, and one of the main protagonists in the
story of the Bulgarian solitaire.},
url = {http://arxiv.org/abs/1503.00885v1 http://arxiv.org/pdf/1503.00885v1},
author = {Vesselin Drensky},
comment = {},
urldate = {2017-08-10},
archivePrefix = {arXiv},
eprint = {1503.00885},
primaryClass = {math.CO},
year = 2015,
collections = {Easily explained,Games to play with friends,History,Puzzles}
}
@article{Frustrationsolitaire,
title = {Frustration solitaire},
abstract = {In this expository article, we discuss the rank-derangement problem, which
asks for the number of permutations of a deck of cards such that each card is
replaced by a card of a different rank. This combinatorial problem arises in
computing the probability of winning the game of `frustration solitaire'. The
solution is a prime example of the method of inclusion and exclusion. We also
discuss and announce the solution to Montmort's `Probleme du Treize', a related
problem dating back to circa 1708.},
url = {http://arxiv.org/abs/math/0703900v2 http://arxiv.org/pdf/math/0703900v2},
author = {Peter G. Doyle and Charles M. Grinstead and J. Laurie Snell},
comment = {},
urldate = {2017-08-10},
archivePrefix = {arXiv},
eprint = {math/0703900},
primaryClass = {math.PR},
year = 2007,
collections = {Easily explained,Games to play with friends,Probability and statistics,Puzzles}
}
@online{OfficiallyHomePlateDoesntExist,
title = {Officially, Home Plate doesn’t exist.},
abstract = {The official Major League and Little League rule books require the two “slanty” sides to be 12” 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 8½” (17” divided by 2).
There is no such shape!},
url = {http://gosper.org/homeplate.html},
author = {Bill Gosper},
comment = {},
urldate = {2017-08-16},
year = 2017,
collections = {Easily explained,Geometry}
}
@article{Mosher2013,
title = {Lone Axes in Outer Space},
author = {Mosher, Lee and Pfaff, Catherine},
url = {http://arxiv.org/abs/1311.6855 http://arxiv.org/pdf/1311.6855v3},
urldate = {2013-11-29},
abstract = {Handel and Mosher define the axis bundle for a fully irreducible outer
automorphism in "Axes in Outer Space." In this paper we give a necessary and
sufficient condition for the axis bundle to consist of a unique periodic fold
line. As a consequence, we give a setting, and means for identifying in this
setting, when two elements of an outer automorphism group {\$}Out(F{\_}r){\$} have
conjugate powers.},
comment = {},
month = {nov},
year = 2013,
archivePrefix = {arXiv},
eprint = {1311.6855},
primaryClass = {math.GR},
collections = {Attention-grabbing titles}
}
@article{FactoringintheChickenMcNuggetmonoid,
title = {Factoring in the Chicken McNugget monoid},
abstract = {Every day, 34 million Chicken McNuggets are sold worldwide. At most McDonalds
locations in the United States today, Chicken McNuggets are sold in packs of 4,
6, 10, 20, 40, and 50 pieces. However, shortly after their introduction in 1979
they were sold in packs of 6, 9, and 20. The use of these latter three numbers
spawned the so-called Chicken McNugget problem, which asks: "what numbers of
Chicken McNuggets can be ordered using only packs with 6, 9, or 20 pieces?" In
this paper, we present an accessible introduction to this problem, as well as
several related questions whose motivation comes from the theory of non-unique
factorization.},
url = {http://arxiv.org/abs/1709.01606v1 http://arxiv.org/pdf/1709.01606v1},
author = {Scott Chapman and Christopher O'Neill},
comment = {},
urldate = {2017-09-06},
archivePrefix = {arXiv},
eprint = {1709.01606},
primaryClass = {math.AC},
year = 2017,
collections = {Animals,Attention-grabbing titles,Easily explained,Unusual arithmetic,Food,Integerology}
}
@article{Hsupermagiclabelingsforfirecrackersbananatreesandflowers,
title = {{\$}H{\$}-supermagic labelings for firecrackers, banana trees and flowers},
abstract = {A simple graph {\$}G=(V,E){\$} admits an {\$}H{\$}-covering if every edge in {\$}E{\$} is
contained in a subgraph {\$}H'=(V',E'){\$} of {\$}G{\$} which is isomorphic to {\$}H{\$}. In this
case we say that {\$}G{\$} is {\$}H{\$}-supermagic if there is a bijection {\$}f:V\cup
E\to\{\{}1,\ldots\lvert V\rvert+\lvert E\rvert\{\}}{\$} such that
{\$}f(V)=\{\{}1,\ldots,\lvert V\rvert\{\}}{\$} and {\$}\sum{\_}{\{}v\in V(H'){\}}f(v)+\sum{\_}{\{}e\in
E(H'){\}}f(e){\$} is constant over all subgraphs {\$}H'{\$} of {\$}G{\$} which are isomorphic to
{\$}H{\$}. In this paper, we show that for odd {\$}n{\$} and arbitrary {\$}k{\$}, the firecracker
{\$}F{\_}{\{}k,n{\}}{\$} is {\$}F{\_}{\{}2,n{\}}{\$}-supermagic, the banana tree {\$}B{\_}{\{}k,n{\}}{\$} is
{\$}B{\_}{\{}1,n{\}}{\$}-supermagic and the flower {\$}F{\_}n{\$} is {\$}C{\_}3{\$}-supermagic.},
url = {http://arxiv.org/abs/1607.07911v2 http://arxiv.org/pdf/1607.07911v2},
author = {Rachel Wulan Nirmalasari Wijaya and Andrea Semani{\v{c}}ov{\'{a}}-Fe{\v{n}}ov{\v{c}}{\'{i}}kov{\'{a}} and Joe Ryan and Thomas Kalinowski},
comment = {},
urldate = {2017-09-11},
archivePrefix = {arXiv},
eprint = {1607.07911},
primaryClass = {cs.DM},
year = 2016,
collections = {Attention-grabbing titles,Basically computer science,Food}
}
@online{EarliestUsesofVariousMathematicalSymbols,
title = {Earliest Uses of Various Mathematical Symbols},
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.},
url = {http://jeff560.tripod.com/mathsym.html},
author = {Jeff Miller},
comment = {},
urldate = {2017-09-12},
year = 2008,
collections = {History,Notation and conventions}
}
@article{Becker2003,
title = {Topic-based vector space model},
author = {J{\"{o}}rg Becker and Dominik Kuropka},
url = {http://www.kuropka.net/files/TVSM.pdf},
urldate = {2012-03-24},
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 = {},
year = 2003,
collections = {Basically computer science}
}
@article{Thetaildoesnotdeterminethesizeofthegiant,
title = {The tail does not determine the size of the giant},
abstract = {The size of the giant component in the configuration model is given by a
well-known expression involving the generating function of the degree
distribution. In this note, we argue that the size of the giant is not
determined by the tail behavior of the degree distribution but rather by the
distribution over small degrees. Upper and lower bounds for the component size
are derived for an arbitrary given distribution over small degrees {\$}d\leq L{\$}
and given expected degree, and numerical implementations show that these bounds
are very close already for small values of {\$}L{\$}. On the other hand, examples
illustrate that, for a fixed degree tail, the component size can vary
substantially depending on the distribution over small degrees. Hence the
degree tail does not play the same crucial role for the size of the giant as it
does for many other properties of the graph.},
url = {http://arxiv.org/abs/1710.01208v1 http://arxiv.org/pdf/1710.01208v1},
author = {Maria Deijfen and Sebastian Rosengren and Pieter Trapman},
comment = {},
urldate = {2017-10-04},
archivePrefix = {arXiv},
eprint = {1710.01208},
primaryClass = {math.PR},
year = 2017,
collections = {Attention-grabbing titles}
}
@article{AMidsummerKnotsDream,
title = {A Midsummer Knot's Dream},
abstract = {In this paper, we introduce playing games on shadows of knots. We demonstrate
two novel games, namely, To Knot or Not to Knot and Much Ado about Knotting. We
also discuss winning strategies for these games on certain families of knot
shadows. Finally, we suggest variations of these games for further study.},
url = {http://arxiv.org/abs/1003.4494v1 http://arxiv.org/pdf/1003.4494v1},
author = {Allison Henrich and Noël MacNaughton and Sneha Narayan and Oliver Pechenik and Robert Silversmith and Jennifer Townsend},
comment = {},
urldate = {2017-10-04},
archivePrefix = {arXiv},
eprint = {1003.4494},
primaryClass = {math.GT},
year = 2010,
collections = {Attention-grabbing titles,Games to play with friends,Protocols and strategies}
}
@article{BadgroupsinthesenseofCherlin,
title = {Bad groups in the sense of Cherlin},
author = {Olivier Fr{\'{e}}con},
url = {http://arxiv.org/abs/1607.02994v1 http://arxiv.org/pdf/1607.02994v1},
urldate = {2016-08-02},
abstract = {There exists no bad group (in the sense of Gregory Cherlin), namely any
simple group of Morley rank 3 is isomorphic to {\$}\mathrm{\{}PSL{\_}2{\}}(K){\$} for an algebraically
closed field {\$}K{\$}.},
comment = {},
year = 2016,
collections = {Attention-grabbing titles,The groups group}
}
@article{Curiositiesofarithmeticgases,
title = {Curiosities of arithmetic gases},
author = {Ioannis Bakas and Mark J. Bowick},
url = {http://aip.scitation.org/doi/10.1063/1.529511 http://aip.scitation.org/doi/pdf/10.1063/1.529511},
urldate = {2017-10-16},
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‐parafermion 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 = {},
year = 1991,
collections = {Unusual arithmetic}
}
@article{Mathemagics,
title = {Mathemagics},
abstract = {My thesis is:there is another way of doing mathematics, equally successful, and the two methods should supplement each other and not fight.},
url = {http://ftp.gwdg.de/pub/misc/EMIS/journals/SLC/wpapers/s44cartier1.pdf},
author = {Pierre Cartier},
comment = {},
urldate = {2017-10-24},
year = 2000,
collections = {Notation and conventions,The act of doing maths}
}
@article{TwodimensionalphotonicaperiodiccrystalsbasedonThueMorsesequence,
title = {Two-dimensional photonic aperiodic crystals based on Thue-Morse sequence},
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.},
url = {https://www.osapublishing.org/oe/abstract.cfm?uri=oe-15-23-15314},
author = {Luigi Moretti and Vito Mocella },
comment = {},
urldate = {2017-11-13},
year = 2007,
collections = {Basically physics,Geometry}
}
@article{Thegameofplatesandolives,
title = {The game of plates and olives},
abstract = {The game of plates and olives, introduced by Nicolaescu, begins with an empty
table. At each step either an empty plate is put down, an olive is put down on
a plate, an olive is removed, an empty plate is removed, or the olives on one
plate are moved to another plate and the resulting empty plate is removed.
Plates are indistinguishable from one another, as are olives, and there is an
inexhaustible supply of each.
The game derives from the consideration of Morse functions on the {\$}2{\$}-sphere.
Specifically, the number of topological equivalence classes of excellent Morse
functions on the {\$}2{\$}-sphere that have order {\$}n{\$} (that is, that have {\$}2n+2{\$}
critical points) is the same as the number of ways of returning to an empty
table for the first time after exactly {\$}2n+2{\$} steps. We call this number {\$}M{\_}n{\$}.
Nicolaescu gave the lower bound {\$}M{\_}n \geq (2n-1)!! = (2/e)^{\{}n+o(n){\}}n^n{\$} and
speculated that {\$}\log M{\_}n \sim n\log n{\$}. In this note we confirm this
speculation, showing that {\$}M{\_}n \leq (4/e)^{\{}n+o(n){\}}n^n{\$}.},
url = {http://arxiv.org/abs/1711.10670v1 http://arxiv.org/pdf/1711.10670v1},
author = {Teena Carroll and David Galvin},
comment = {},
urldate = {2017-11-30},
archivePrefix = {arXiv},
eprint = {1711.10670},
primaryClass = {math.CO},
year = 2017,
collections = {Attention-grabbing titles,Easily explained,Food}
}
@article{CryptographicProtocolsWithEverydayObjects,
title = {Cryptographic Protocols with Everyday Objects},
author = {James Heather and Steve Schneider and Vanessa Teague},
url = {http://www.computing.surrey.ac.uk/personal/st/S.Schneider/papers/2011/cryptoforma11a.pdf},
urldate = {2016-11-17},
abstract = {Most security protocols appearing in the literature make use of cryptographic primitives that assume that the participants have access
to some sort of computational device.
However, there are times when there is need for a security mechanism
to evaluate some result without leaking sensitive information, but computational devices are unavailable. We discuss here various protocols for
solving cryptographic problems using everyday objects: coins, dice, cards, and envelopes.},
comment = {},
year = 2011,
collections = {Easily explained,Protocols and strategies,Things to make and do}
}
@online{MechanismsbyTchebyshev,
title = {Mechanisms by Tchebyshev},
abstract = {This project gathers all the mechanisms created by a great Russian mathematician Pafnuty Lvovich Tchebyshev (1821—1894).
Some 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{\'{e}}e des Arts et M{\'{e}}tiers in Paris and in Science Museum (London). There are only photos or descriptions left for some of the mechanisms.
The 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.},
url = {http://en.tcheb.ru/},
author = {},
comment = {},
urldate = {2017-12-04},
year = 2009,
collections = {Basically physics,History,Things to make and do,Geometry}
}
@article{AcompilationofLEGOTechnicpartstosupportlearningexperimentsonlinkages,
title = {A compilation of LEGO Technic parts to support learning experiments on linkages},
abstract = {We present a compilation of LEGO Technic parts to provide easy-to-build
constructions of basic planar linkages. Some technical issues and their
possible solutions are discussed. Fine details -- like deciding whether the
motion is an exactly straight line or not -- are forwarded to the dynamic
mathematics software tool GeoGebra.},
url = {http://arxiv.org/abs/1712.00440v1 http://arxiv.org/pdf/1712.00440v1},
author = {Zolt{\'{a}}n Kov{\'{a}}cs and Benedek Kov{\'{a}}cs},
comment = {},
urldate = {2017-12-04},
archivePrefix = {arXiv},
eprint = {1712.00440},
primaryClass = {math.HO},
year = 2017,
collections = {Basically physics,Easily explained,Things to make and do,Geometry}
}
@article{SpotitSolitaire,
title = {Spot it(R) Solitaire},
abstract = {The game of Spot it(R) is based on an order 7 finite projective plane. This
article presents a solitaire challenge: extract an order 7 affine plane and
arrange those 49 cards into a square such that the symmetries of the affine and
projective planes are obvious. The objective is not to simply create such a
deck already in this solved position. Rather, it is to solve the inverse
problem of arranging the cards of such a deck which has already been created
shuffled.},
url = {http://arxiv.org/abs/1301.7058v1 http://arxiv.org/pdf/1301.7058v1},
author = {Donna A. Dietz},
comment = {},
urldate = {2017-12-18},
archivePrefix = {arXiv},
eprint = {1301.7058},
primaryClass = {math.HO},
year = 2013,
keywords = {dobble},
collections = {Games to play with friends,Puzzles}
}
@article{TheSleepingBeautyControversy,
title = {The Sleeping Beauty Controversy},
author = {Peter Winkler},
url = {http://www.jstor.org/stable/10.4169/amer.math.monthly.124.7.579},
urldate = {2017-08-14},
abstract = {In 2000, Adam Elga posed the following problem:
Some 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?
This 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 = {},
year = 2017,
collections = {Easily explained,Probability and statistics,Drama!}
}
@article{AnUnusualCubicRepresentationProblem,
title = {An unusual cubic representation problem},
author = {Andrew Bremner and Allan Macleod},
url = {http://ami.ektf.hu/uploads/papers/finalpdf/AMI{\_}43{\_}from29to41.pdf},
urldate = {2017-08-07},
abstract = {For a non-zero integer \(N\), we consider the problem of finding \(3\) integers
\( (a, b, c) \) such that
\[ N = \frac{\{}a{\}}{\{}b+c{\}} + \frac{\{}b{\}}{\{}c+a{\}} + \frac{\{}c{\}}{\{}a+b{\}}. \]
We 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 = {},
year = 2014,
collections = {Puzzles}
}
@article{FoldingPolyominoesintoPolyCubes,
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{\'{a}}ndor P. Fekete and Adam Hesterberg and Irina Kostitsyna and Christiane Schmidt},
url = {http://arxiv.org/abs/1712.09317v1 http://arxiv.org/pdf/1712.09317v1},
urldate = {2018-01-03},
year = 2017,
abstract = {We study the problem of folding a polyomino {\$}P{\$} into a polycube {\$}Q{\$}, allowing
faces of {\$}Q{\$} to be covered multiple times. First, we define a variety of
folding models according to whether the folds (a) must be along grid lines of
{\$}P{\$} or can divide squares in half (diagonally and/or orthogonally), (b) must be
mountain or can be both mountain and valley, (c) can remain flat (forming an
angle of {\$}180^\circ{\$}), and (d) must lie on just the polycube surface or can
have interior faces as well. Second, we give all the inclusion relations among
all models that fold on the grid lines of {\$}P{\$}. Third, we characterize all
polyominoes that can fold into a unit cube, in some models. Fourth, we give a
linear-time dynamic programming algorithm to fold a tree-shaped polyomino into
a constant-size polycube, in some models. Finally, we consider the triangular
version of the problem, characterizing which polyiamonds fold into a regular
tetrahedron.},
comment = {},
archivePrefix = {arXiv},
eprint = {1712.09317},
primaryClass = {cs.CG},
collections = {Puzzles,Things to make and do,Geometry}
}
@article{Moler2003,
title = {Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later},
author = {Moler, Cleve and Van Loan, C.},
url = {http://www.jstor.org/stable/10.2307/25054364 http://www.cs.cornell.edu/cv/researchpdf/19ways+.pdf},
urldate = {2012-02-09},
year = 2003,
abstract = {},
comment = {},
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,
collections = {Attention-grabbing titles,Basically computer science}
}
@article{AnyMonotoneBooleanFunctionCanBeRealizedByInterlockedPolygons,
title = {Any Monotone Boolean Function Can Be Realized by Interlocked Polygons},
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 \(S \subseteq S′\) and removing \(S\) frees the polygons, then so does \(S′\). 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.},
url = {http://erikdemaine.org/papers/InterlockedPolygons{\_}CCCG2010/ http://erikdemaine.org/papers/InterlockedPolygons{\_}CCCG2010/paper.pdf},
year = 2010,
author = {Erik D. Demaine and Martin L. Demaine and Ryuhei Uehara},
comment = {},
urldate = {2018-01-08},
collections = {Basically computer science,Easily explained,Geometry,Fun maths facts}
}
@article{Randomrailwaysmodeledasrandom3regulargraphs,
title = {Random railways modeled as random 3-regular graphs},
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.},
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,
author = {Hans Garmo},
comment = {},
urldate = {2018-01-09},
collections = {Attention-grabbing titles,Probability and statistics,Modelling}
}
@article{WhatdidRyserConjecture,
title = {What did Ryser Conjecture?},
abstract = {Two prominent conjectures by Herbert J. Ryser have been falsely attributed to
a somewhat obscure conference proceedings that he wrote in German. Here we
provide a translation of that paper and try to correct the historical record at
least as far as what was conjectured in it. The two conjectures relate to
transversals in Latin squares of odd order and to the relationship between the
covering number and the matching number of multipartite hypergraphs.},
url = {http://arxiv.org/abs/1801.02893v1 http://arxiv.org/pdf/1801.02893v1},
year = 2018,
author = {Darcy Best and Ian M. Wanless},
comment = {},
urldate = {2018-01-10},
archivePrefix = {arXiv},
eprint = {1801.02893},
primaryClass = {math.CO},
collections = {History,Drama!}
}
@article{Richeson2013,
title = {Circular reasoning: who first proved that {\$}C/d{\$} is a constant?},
author = {Richeson, David},
url = {http://arxiv.org/abs/1303.0904 http://arxiv.org/pdf/1303.0904v2},
urldate = {2013-03-06},
year = 2013,
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 = {},
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},
collections = {Geometry,About proof}
}
@article{Markowsky1992,
title = {Misconceptions about the Golden Ratio},
author = {Markowsky, George},
url = {http://www.jstor.org/stable/2686193?origin=crossref},
urldate = {2010-09-29},
year = 1992,
abstract = {},
comment = {},
journal = {The College Mathematics Journal},
month = {jan},
number = 1,
pages = 2,
volume = 23,
collections = {Easily explained,History}
}
@article{PlayingGameswithAlgorithmsAlgorithmicCombinatorialGameTheory,
title = {Playing Games with Algorithms: Algorithmic Combinatorial Game Theory},
abstract = {Combinatorial games lead to several interesting, clean problems in algorithms
and complexity theory, many of which remain open. The purpose of this paper is
to provide an overview of the area to encourage further research. In
particular, we begin with general background in Combinatorial Game Theory,
which analyzes ideal play in perfect-information games, and Constraint Logic,
which provides a framework for showing hardness. Then we survey results about
the complexity of determining ideal play in these games, and the related
problems of solving puzzles, in terms of both polynomial-time algorithms and
computational intractability results. Our review of background and survey of
algorithmic results are by no means complete, but should serve as a useful
primer.},
url = {http://arxiv.org/abs/cs/0106019v2 http://arxiv.org/pdf/cs/0106019v2},
year = 2001,
author = {Erik D. Demaine and Robert A. Hearn},
comment = {},
urldate = {2018-01-23},
archivePrefix = {arXiv},
eprint = {cs/0106019},
primaryClass = {cs.CC},
collections = {Basically computer science,Games to play with friends,Combinatorics,Computational complexity of games}
}
@article{Cormode2004,
title = {The hardness of the Lemmings game, or Oh no, more NP-completeness proofs},
author = {Cormode, Graham},
url = {https://pdfs.semanticscholar.org/9413/3f598974f93b089ca7d2875d59d0dab1e1eb.pdf},
urldate = {2012-01-27},
year = 2004,
abstract = {},
comment = {},
booktitle = {Proceedings of the 3rd International Conference on FUN with Algorithms},
pages = {65--76},
collections = {Animals,Attention-grabbing titles,Basically computer science,Games to play with friends,About proof,Computational complexity of games}
}
@article{Thegrasshopperproblem,
title = {The grasshopper problem},
abstract = {We introduce and physically motivate the following problem in geometric
combinatorics, originally inspired by analysing Bell inequalities. A
grasshopper lands at a random point on a planar lawn of area one. It then jumps
once, a fixed distance {\$}d{\$}, in a random direction. What shape should the lawn
be to maximise the chance that the grasshopper remains on the lawn after
jumping? We show that, perhaps surprisingly, a disc shaped lawn is not optimal
for any {\$}d>0{\$}. We investigate further by introducing a spin model whose ground
state corresponds to the solution of a discrete version of the grasshopper
problem. Simulated annealing and parallel tempering searches are consistent
with the hypothesis that for {\$} d < \pi^{\{}-1/2{\}}{\$} the optimal lawn resembles a
cogwheel with {\$}n{\$} cogs, where the integer {\$}n{\$} is close to {\$} \pi ( \arcsin (
\sqrt{\{}\pi{\}} d /2 ) )^{\{}-1{\}}{\$}. We find transitions to other shapes for {\$}d \gtrsim
\pi^{\{}-1/2{\}}{\$}.},
url = {http://arxiv.org/abs/1705.07621v3 http://arxiv.org/pdf/1705.07621v3},
year = 2017,
author = {Olga Goulko and Adrian Kent},
comment = {},
urldate = {2018-01-24},
archivePrefix = {arXiv},
eprint = {1705.07621},
primaryClass = {cond-mat.stat-mech},
collections = {Animals,Probability and statistics,Puzzles,Geometry}
}
@article{Anemptyexercise,
title = {An empty exercise},
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.},
url = {https://dl.acm.org/citation.cfm?id=122273 http://ftp.cs.wisc.edu/Approx/empty.pdf},
year = 1990,
author = {Carl de Boor},
comment = {},
urldate = {2018-01-24},
collections = {Attention-grabbing titles,Basically computer science,Notation and conventions}
}
@article{MechanicalComputingSystemsUsingOnlyLinksandRotaryJoints,
title = {Mechanical Computing Systems Using Only Links and Rotary Joints},
abstract = {A new paradigm for mechanical computing is demonstrated that requires only
two basic parts, links and rotary joints. These basic parts are combined into
two main higher level structures, locks and balances, and suffice to create all
necessary combinatorial and sequential logic required for a Turing-complete
computational system. While working systems have yet to be implemented using
this new paradigm, the mechanical simplicity of the systems described may lend
themselves better to, e.g., microfabrication, than previous mechanical
computing designs. Additionally, simulations indicate that if molecular-scale
implementations could be realized, they would be far more energy-efficient than
conventional electronic computers.},
url = {http://arxiv.org/abs/1801.03534v1 http://arxiv.org/pdf/1801.03534v1},
year = 2018,
author = {Ralph C. Merkle and Robert A. Freitas Jr. and Tad Hogg and Thomas E. Moore and Matthew S. Moses and James Ryley},
comment = {},
urldate = {2018-01-30},
archivePrefix = {arXiv},
eprint = {1801.03534},
primaryClass = {cs.ET},
collections = {Basically computer science,Things to make and do,Unusual computers}
}
@article{TheMuffinProblem,
title = {The Muffin Problem},
abstract = {You have {\$}m{\$} muffins and {\$}s{\$} students. You want to divide the muffins into
pieces and give the shares to students such that every student has
{\$}\frac{\{}m{\}}{\{}s{\}}{\$} muffins. Find a divide-and-distribute protocol that maximizes the
minimum piece. Let {\$}f(m,s){\$} be the minimum piece in the optimal protocol. We
prove that {\$}f(m,s){\$} exists, is rational, and finding it is computable (though
possibly difficult). We show that {\$}f(m,s){\$} can be derived from {\$}f(s,m){\$}; hence
we need only consider {\$}m\ge s{\$}. For {\$}1\le s\le 6{\$} we find nice formulas for
{\$}f(m,s){\$}. We also find a nice formula for {\$}f(s+1,s){\$}. We give a function
{\$}FC(m,s){\$} such that, for {\$}m\ge s+2{\$}, {\$}f(m,s)\le FC(m,s){\$}. This function
permeates the entire paper since it is often the case that {\$}f(m,s)=FC(m,s){\$}.
More formally, for all {\$}s{\$} there is a nice formula {\$}FORM(m,s){\$} such that, for
all but a finite number of {\$}m{\$}, {\$}f(m,s)=FC(m,s)=FORM(m,s){\$}. For those finite
number of exceptions we have another function {\$}INT(m,s){\$} such that {\$}f(m,s)\le
INT(m,s){\$}. It seems to be the case that when {\$}m\ge s+2{\$},
{\$}f(m,s)=\min\{\{}f(m,s),INT(m,s)\{\}}{\$}. For {\$}s=7{\$} to 60 we have conjectured formulas
for {\$}f(m,s){\$} that include exceptions.},
url = {http://arxiv.org/abs/1709.02452v2 http://arxiv.org/pdf/1709.02452v2},
year = 2017,
author = {Guangiqi Cui and John Dickerson and Naveen Durvasula and William Gasarch and Erik Metz and Naveen Raman and Sung Hyun Yoo},
comment = {},
urldate = {2018-01-30},
archivePrefix = {arXiv},
eprint = {1709.02452},
primaryClass = {math.CO},
collections = {Easily explained,Food,Protocols and strategies}
}
@article{Straightknots,
title = {Straight knots},
abstract = {We introduce a new invariant, the straight number of a knot. We give some
relations to crossing number and petal number. Then we discuss the methods we
used to compute the straight numbers for all the knots in the standard knot
table and present some interesting questions and the full table.},
url = {http://arxiv.org/abs/1801.10428v1 http://arxiv.org/pdf/1801.10428v1},
year = 2018,
author = {Nicholas Owad},
comment = {},
urldate = {2018-02-01},
archivePrefix = {arXiv},
eprint = {1801.10428},
primaryClass = {math.GT},
collections = {Attention-grabbing titles,Easily explained}
}
@article{Fingerprintdatabasesfortheorems,
title = {Fingerprint databases for theorems},
abstract = {We discuss the advantages of searchable, collaborative, language-independent
databases of mathematical results, indexed by "fingerprints" of small and
canonical data. Our motivating example is Neil Sloane's massively influential
On-Line Encyclopedia of Integer Sequences. We hope to encourage the greater
mathematical community to search for the appropriate fingerprints within each
discipline, and to compile fingerprint databases of results wherever possible.
The benefits of these databases are broad - advancing the state of knowledge,
enhancing experimental mathematics, enabling researchers to discover unexpected
connections between areas, and even improving the refereeing process for
journal publication.},
url = {http://arxiv.org/abs/1304.3866v1 http://arxiv.org/pdf/1304.3866v1},
year = 2013,
author = {Sara C. Billey and Bridget E. Tenner},
comment = {},
urldate = {2018-02-12},
archivePrefix = {arXiv},
eprint = {1304.3866},
primaryClass = {math.HO},
collections = {Lists and catalogues,The act of doing maths}
}
@online{almanachoudictionnairedesnombrescuriositsetproprits,
title = {almanach ou dictionnaire des nombres - curiosit{\'{e}}s et propri{\'{e}}t{\'{e}}s},
abstract = {},
url = {http://yoda.guillaume.pagesperso-orange.fr/index.htm},
year = {},
author = {G{\'{e}}rard Villemin},
comment = {},
urldate = {2018-03-05},
collections = {Lists and catalogues,Integerology}
}
@online{NotablePropertiesofSpecificNumbers,
title = {Notable Properties of Specific Numbers},
author = {Robert Munafo},
url = {http://www.mrob.com/pub/math/numbers.html},
urldate = {2018-03-05},
year = {},
abstract = {},
comment = {},
collections = {Lists and catalogues,Integerology}
}
@article{HowdoyoufixanOvalTrackPuzzle,
title = {How do you fix an Oval Track Puzzle?},
abstract = {The oval track group, {\$}OT{\_}{\{}n,k{\}}{\$}, is the subgroup of the symmetric group,
{\$}S{\_}n{\$}, generated by the basic moves available in a generalized oval track
puzzle with {\$}n{\$} tiles and a turntable of size {\$}k{\$}. In this paper we completely
describe the oval track group for all possible {\$}n{\$} and {\$}k{\$} and use this
information to answer the following question: If the tiles are removed from an
oval track puzzle, how must they be returned in order to ensure that the puzzle
is still solvable? As part of this discussion we introduce the parity subgroup
of {\$}S{\_}n{\$} in the case when {\$}n{\$} is even.},
url = {http://arxiv.org/abs/1612.04476v3 http://arxiv.org/pdf/1612.04476v3},
year = 2016,
author = {David A. Nash and Sara Randall},
comment = {},
urldate = {2018-03-13},
archivePrefix = {arXiv},
eprint = {1612.04476},
primaryClass = {math.GR},
collections = {Easily explained,Puzzles}
}
@online{NumeralSystemsoftheWorld,
title = {Numeral Systems of the World},
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.},
url = {https://mpi-lingweb.shh.mpg.de/numeral/},
year = 2006,
author = {Bernard Comrie and Eugene Chan},
comment = {},
urldate = {2018-03-25},
collections = {Lists and catalogues,Notation and conventions}
}
@article{APuzzleForPirates,
title = {A Puzzle for Pirates},
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.},
url = {https://omohundro.files.wordpress.com/2009/03/stewart99{\_}a{\_}puzzle{\_}for{\_}pirates.pdf},
year = 1999,
author = {Ian Stewart},
comment = {},
urldate = {2018-04-03},
collections = {Puzzles}
}
@article{Thematerialityofmathematicspresentingmathematicsattheblackboard,
title = {The materiality of mathematics: presenting mathematics at the blackboard},
author = {Christian Greiffenhagen},
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,
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 ‘mind’ and ‘body’. 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 ‘architecture’ visualizing the overall structure of the proof). The paper argues that doing mathematics really is ‘thinking with eyes and hands’ (Latour 1986). Thinking in mathematics is inextricably interwoven with writing mathematics.},
comment = {},
collections = {The act of doing maths}
}
@article{ChocolategamesthatsatisfytheinequalityforandGrundynumbers,
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},
url = {http://www.mi.sanu.ac.rs/vismath/miyadera2013/index.html},
urldate = {2017-10-18},
year = 2013,
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.
The coordinates \( \{\{}x,y,z\{\}}\) of the chocolates satisfy the inequalities \( y\leq \lfloor \frac{\{}z{\}}{\{}k{\}} \rfloor \) for \( k = 1,2\) .
For \( k = 2\) 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 \(k\).
For \(k = 1\) 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 = {},
collections = {Attention-grabbing titles,Food,Games to play with friends}
}
@article{Knutson2012,
title = {A stratification of the space of all {\$}k{\$}-planes in {\$}\mathbb{\{}C{\}}{\_}n{\$}},
author = {Allen Knutson},
url = {http://www.math.cornell.edu/{\~{}}allenk/joint.pdf},
urldate = {2012-11-19},
year = 2012,
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
on the {\$}k{\$}-plane spanned by the rows, so gives a decomposition of the “Grassmannian” of all {\$}k{\$}-planes in {\$}n{\$}-space.
There 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 “more excited” than another. This same decomposition turns out to naturally arise from totally
positive 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 = {},
collections = {Geometry}
}
@article{NiceNeighbours,
title = {Nice Neighbors: A Brief Adventure in Mathematical Gamification},
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’ll describe the mathematics behind this game and some of the surprises along the way that still have me scratching my head. },
url = {http://cstaecker.fairfield.edu/{\~{}}cstaecker/files/research/nnpublished.pdf},
year = 2016,
author = {Chris Staecker},
comment = {},
urldate = {2018-04-17},
collections = {Easily explained,Puzzles}
}
@article{TheMathematicalColoringBook,
title = {The Mathematical Coloring Book},
abstract = {Due to the author's correspondence with Van der Waerden, Erd{\"{o}}s, 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.},
url = {https://link.springer.com/book/10.1007{\%}2F978-0-387-74642-5},
year = 2009,
author = {Alexander Soifer},
comment = {},
urldate = {2018-04-18},
collections = {Attention-grabbing titles}
}
@article{AnInvitationtoInverseGroupTheory,
title = {An Invitation to Inverse Group Theory},
abstract = {In group theory there are many constructions which produce a new group from a
given one. Often the result is a subgroup: the derived group, centre, socle,
Frattini subgroup, Hall subgroup, Fitting subgroup, and so on. Other
constructions may produce groups in other ways, for example quotients (solvable
residual, derived quotient) or cohomology groups (Schur multiplier). Inverse
group theory refers to problems in which a construction and the resulting group
is given and we want information about the possible original group or groups;
examples are the {\{}\em inverse Schur multiplier problem{\}} (given a finite abelian
group is it the Schur multiplier of some finite group?), or the {\{}\em inverse
derived group{\}} (given a group {\$}G{\$} is there a group {\$}H{\$} such that {\$}H'=G{\$}?). In
1956 B. H. Neumann sent a first invitation to inverse group theory, but
apparently the topic did not receive the attention it deserves, so that we
attempt here at repeating that invitation. Many of the inverse group problems
associated with the constructions referred to above are trivial, but some are
not. Like Neumann we will work mainly on inverse derived groups. We also
explain how the main questions about inverse Frattini subgroups have been
settled.
An integral of a group {\$}G{\$} is a group {\$}H{\$} such that the derived group of {\$}H{\$}
is {\$}G{\$}. Our first goal is to prove a number of general facts about the
integrals of finite groups, and to raise some open questions. Our results
concern orders of non-integrable groups (we give a complete description of the
set of such numbers), the smallest integral of a group (in particular, we show
that if a finite group is integrable it has a finite integral), and groups
which can be integrated infinitely often, a problem already tackled by Neumann.
We also consider integrals of infinite groups. Regarding inverse Frattini, we
explain Neumann's and Eick's results.},
url = {http://arxiv.org/abs/1803.10179v2 http://arxiv.org/pdf/1803.10179v2},
year = 2018,
author = {Jo{\~{a}}o Ara{\'{u}}jo and Peter J. Cameron and Francesco Matucci},
comment = {},
urldate = {2018-04-19},
archivePrefix = {arXiv},
eprint = {1803.10179},
primaryClass = {math.GR},
collections = {Unusual arithmetic,The groups group}
}
@article{Whatisthesmallestprime,
title = {What is the smallest prime?},
author = {Chris K. Caldwell and Yeng Xiong},
url = {http://arxiv.org/abs/1209.2007v2 http://arxiv.org/pdf/1209.2007v2},
urldate = {2016-07-05},
year = 2012,
abstract = {What is the first prime? It seems that the number two should be the obvious
answer, and today it is, but it was not always so. There were times when and
mathematicians for whom the numbers one and three were acceptable answers. To
find the first prime, we must also know what the first positive integer is.
Surprisingly, with the definitions used at various times throughout history,
one was often not the first positive integer (some started with two, and a few
with three). In this article, we survey the history of the primality of one,
from the ancient Greeks to modern times. We will discuss some of the reasons
definitions changed, and provide several examples. We will also discuss the
last significant mathematicians to list the number one as prime.},
comment = {},
archivePrefix = {arXiv},
eprint = {1209.2007},
primaryClass = {math.HO},
collections = {Easily explained,History,Notation and conventions,Integerology}
}
@article{Mostprimitivegroupshavemessyinvariants,
title = {Most primitive groups have messy invariants},
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 \(|G\).},
url = {https://www.sciencedirect.com/science/article/pii/0001870879900380 http://neilsloane.com/doc/Me61.pdf},
year = 1979,
author = {W.C. Huffman and N.J.A. Sloane},
comment = {},
urldate = {2018-05-06},
collections = {Attention-grabbing titles,The groups group}
}
@article{Redefiningtheintegral,
title = {Redefining the integral},
abstract = {In this paper, we discuss a similar functional to that of a standard
integral. The main difference is in its definition: instead of taking a sum, we
are taking a product. It turns out this new "star-integral" may be written in
terms of the standard integral but it has many different (and similar)
interesting properties compared to the regular integral. Further, we define a
"star-derivative" and discuss its relationship to the "star-integral".},
url = {http://arxiv.org/abs/1805.01861v1 http://arxiv.org/pdf/1805.01861v1},
year = 2018,
author = {Derek Orr},
comment = {},
urldate = {2018-05-08},
archivePrefix = {arXiv},
eprint = {1805.01861},
primaryClass = {math.CA},
collections = {Unusual arithmetic}
}
@online{MathCounterexamples,
title = {Math Counterexamples},
abstract = {I initiated this website because for years I have been passionated about Mathematics as a hobby and also by “strange objects”. Mathematical counterexamples combine both topics.
The 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… but that was a positive experience to raise my interest in counterexamples.
According 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.
By 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.
While I’m particularly interested in Topology and Analysis, I will also try to cover Logic and Algebra counterexamples.},
url = {http://www.mathcounterexamples.net/},
year = 2014,
author = {Jean-Pierre Merx},
comment = {},
urldate = {2018-05-09},
collections = {Lists and catalogues}
}
@article{Howlongdoesittaketocatchawildkangaroo,
title = {How long does it take to catch a wild kangaroo?},
abstract = {We develop probabilistic tools for upper and lower bounding the expected time
until two independent random walks on {\$}\ZZ{\$} intersect each other. This leads to
the first sharp analysis of a non-trivial Birthday attack, proving that
Pollard's Kangaroo method solves the discrete logarithm problem {\$}g^x=h{\$} on a
cyclic group in expected time {\$}(2+o(1))\sqrt{\{}b-a{\}}{\$} for an average
{\$}x\in{\_}{\{}uar{\}}[a,b]{\$}. Our methods also resolve a conjecture of Pollard's, by
showing that the same bound holds when step sizes are generalized from powers
of 2 to powers of any fixed {\$}n{\$}.},
url = {http://arxiv.org/abs/0812.0789v2 http://arxiv.org/pdf/0812.0789v2},
year = 2008,
author = {Ravi Montenegro and Prasad Tetali},
comment = {},
urldate = {2018-05-12},
archivePrefix = {arXiv},
eprint = {0812.0789},
primaryClass = {math.PR},
collections = {Animals,Easily explained,Probability and statistics}
}
@article{ExactEnumerationOfGardenOfEdenPartitions,
title = {Exact Enumeration of Garden of Eden Partitions},
abstract = {We give two proofs for a formula that counts the number of partitions of \(n\) that have rank −2 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’s 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.},
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,
author = {Brian Hopkins and James A. Sellers},
comment = {},
urldate = {2018-05-13},
collections = {Attention-grabbing titles,Easily explained,Combinatorics}
}
@article{AdescriptionoftheouterautomorphismofS6andtheinvariantsofsixpointsinprojectivespace,
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},
url = {https://www.sciencedirect.com/science/article/pii/S0097316508000125 http://math.stanford.edu/{\~{}}vakil/files/sixjan2308.pdf},
urldate = {2018-05-14},
year = 2008,
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.},
collections = {Things to make and do,The groups group}
}
@article{TheRearrangementNumber,
title = {The Rearrangement Number},
abstract = {How many permutations of the natural numbers are needed so that every
conditionally convergent series of real numbers can be rearranged to no longer
converge to the same sum? We show that the minimum number of permutations
needed for this purpose, which we call the rearrangement number, is
uncountable, but whether it equals the cardinal of the continuum is independent
of the usual axioms of set theory. We compare the rearrangement number with
several natural variants, for example one obtained by requiring the rearranged
series to still converge but to a new, finite limit. We also compare the
rearrangement number with several well-studied cardinal characteristics of the
continuum. We present some new forcing constructions designed to add
permutations that rearrange series from the ground model in particular ways,
thereby obtaining consistency results going beyond those that follow from
comparisons with familiar cardinal characteristics. Finally we deal briefly
with some variants concerning rearrangements by a special sort of permutations
and with rearranging some divergent series to become (conditionally)
convergent.},
url = {http://arxiv.org/abs/1612.07830v1 http://arxiv.org/pdf/1612.07830v1},
year = 2016,
author = {Andreas Blass and J{\"{o}}rg Brendle and Will Brian and Joel David Hamkins and Michael Hardy and Paul B. Larson},
comment = {},
urldate = {2018-05-16},
archivePrefix = {arXiv},
eprint = {1612.07830},
primaryClass = {math.LO},
collections = {Combinatorics}
}
@article{SovietStreetMathematicsLandausLicensePlateGame,
title = {Soviet Street Mathematics: Landau’s License Plate Game},
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.},
url = {https://link.springer.com/article/10.1007{\%}2Fs00283-017-9743-9},
year = 2018,
author = {Harun {\v{S}}iljak},
comment = {},
urldate = {2018-05-17},
collections = {Easily explained,Games to play with friends,Puzzles}
}
@article{Neumbering,
title = {Neumbering},
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
usual 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’s confusing if we start from zero. This is a good reason to introduce John von Neumann’s convention, which we call ‘‘Neumbering.’’ The authors have been using this name privately, and we apologise for being slangy. This part of the paper starts by using it publicly.},
url = {https://link.springer.com/article/10.1007/s00283-017-9720-3},
year = 2018,
author = {O.G. Cassani and John H. Conway},
comment = {},
urldate = {2018-05-17},
collections = {Easily explained,Notation and conventions}
}
@article{RenyisParkingProblemRevisited,
title = {Renyi's Parking Problem Revisited},
abstract = {R\'enyi's parking problem (or {\$}1D{\$} sequential interval packing problem) dates
back to 1958, when R\'enyi studied the following random process: Consider an
interval {\$}I{\$} of length {\$}x{\$}, and sequentially and randomly pack disjoint unit
intervals in {\$}I{\$} until the remaining space prevents placing any new segment.
The expected value of the measure of the covered part of {\$}I{\$} is {\$}M(x){\$}, so that
the ratio {\$}M(x)/x{\$} is the expected filling density of the random process.
Following recent work by Gargano {\{}\it et al.{\}} \cite{\{}GWML(2005){\}}, we studied the
discretized version of the above process by considering the packing of the {\$}1D{\$}
discrete lattice interval {\$}\{\{}1,2,...,n+2k-1\{\}}{\$} with disjoint blocks of {\$}(k+1){\$}
integers but, as opposed to the mentioned \cite{\{}GWML(2005){\}} result, our
exclusion process is symmetric, hence more natural. Furthermore, we were able
to obtain useful recursion formulas for the expected number of {\$}r{\$}-gaps ({\$}0\le
r\le k{\$}) between neighboring blocks. We also provided very fast converging
series and extensive computer simulations for these expected numbers, so that
the limiting filling density of the long line segment (as {\$}n\to \infty{\$}) is
R\'enyi's famous parking constant, {\$}0.7475979203...{\$}.},
url = {http://arxiv.org/abs/1406.1781v2 http://arxiv.org/pdf/1406.1781v2},
year = 2014,
author = {Matthew P. Clay and Nandor J. Simanyi},
comment = {},
urldate = {2018-05-21},
archivePrefix = {arXiv},
eprint = {1406.1781},
primaryClass = {math.PR},
collections = {Easily explained}
}
@article{OneParameterIsAlwaysEnough,
title = {One parameter is always enough},
abstract = {We construct an elementary equation with a single real valued parameter that is capable of fitting any “scatter plot” 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 “parameter counting” fails as a measure of
model complexity when the class of models under consideration is only slightly broad.},
url = {https://colala.bcs.rochester.edu/papers/piantadosi2018one.pdf},
year = 2018,
author = {Steven T. Piantadosi},
comment = {},
urldate = {2018-06-06},
collections = {Attention-grabbing titles,Probability and statistics,Fun maths facts,Modelling}
}
@article{Programmingquantumcomputersusing3Dpuzzlescoffeecupsanddoughnuts,
title = {Programming quantum computers using 3-D puzzles, coffee cups, and doughnuts},
author = {Simon J. Devitt},
url = {http://arxiv.org/abs/1609.06628v1 http://arxiv.org/pdf/1609.06628v1},
urldate = {2016-09-23},
year = 2016,
abstract = {The task of programming a quantum computer is just as strange as quantum
mechanics itself. But it now looks like a simple 3D puzzle may be the future
tool of quantum software engineers.},
comment = {},
archivePrefix = {arXiv},
eprint = {1609.06628},
primaryClass = {quant-ph},
collections = {Unusual computers}
}
@article{WhenareMultiplesofPolygonalNumbersagainPolygonalNumbers,
title = {When are Multiples of Polygonal Numbers again Polygonal Numbers?},
author = {Jasbir S. Chahal and Nathan Priddis},
url = {http://arxiv.org/abs/1806.07981v1 http://arxiv.org/pdf/1806.07981v1},
urldate = {2018-06-25},
year = 2018,
abstract = {Euler showed that there are infinitely many triangular numbers that are three
times another triangular number. In general, as we prove, it is an easy
consequence of the Pell equation that for a given square-free m > 1, the
relation D = mD' is satisfied by infinitely many pairs of triangular numbers D,
D'. However, due to the erratic behavior of the fundamental solution to the
Pell equation, this problem is more difficult for more general polygonal
numbers. We will show that if one solution exists, then infinitely many exist.
We give an example, however, showing that there are cases where no solution
exists. Finally, we also show in this paper that, given m > n > 1 with obvious
exceptions, the simultaneous relations P = mP', P = nP" has only finitely many
possibilities not just for triangular numbers, but for triplets P, P', P" of
polygonal numbers.},
comment = {},
archivePrefix = {arXiv},
eprint = {1806.07981},
primaryClass = {math.NT},
collections = {Easily explained,Fun maths facts,Integerology}
}
@article{AsurprisinglysimpledeBruijnsequenceconstruction,
title = {A surprisingly simple de Bruijn sequence construction},
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.},
url = {https://www.sciencedirect.com/science/article/pii/S0012365X15002873},
year = 2016,
author = {Joe Sawada and Aaron Williams and DennisWong},
comment = {},
urldate = {2018-06-25},
collections = {Basically computer science,Combinatorics}
}
@article{ProofwithoutWordsFairAllocationofaPizza,
title = {Proof without Words: Fair Allocation of a Pizza},
author = {Larry Carter and Stan Wagon},
url = {https://www.jstor.org/stable/2690845},
urldate = {2018-06-26},
year = 1994,
abstract = {},
comment = {},
collections = {Easily explained,Protocols and strategies,Things to make and do,About proof,Fun maths facts}
}
@article{Howtoheartheshapeofabilliardtable,
title = {How to hear the shape of a billiard table},
abstract = {The bounce spectrum of a polygonal billiard table is the collection of all
bi-infinite sequences of edge labels corresponding to billiard trajectories on
the table. We give methods for reconstructing from the bounce spectrum of a
polygonal billiard table both the cyclic ordering of its edge labels and the
sizes of its angles. We also show that it is impossible to reconstruct the
exact shape of a polygonal billiard table from any finite collection of finite
words from its bounce spectrum.},
url = {http://arxiv.org/abs/1806.09644v1 http://arxiv.org/pdf/1806.09644v1},
year = 2018,
author = {Aaron Calderon and Solly Coles and Diana Davis and Justin Lanier and Andre Oliveira},
comment = {},
urldate = {2018-06-27},
archivePrefix = {arXiv},
eprint = {1806.09644},
primaryClass = {math.DS},
collections = {Basically physics,Easily explained,Geometry,Fun maths facts}
}
@article{NationalCurveBank,
title = {National Curve Bank},
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.},
url = {http://web.calstatela.edu/curvebank/home/home.htm},
year = 2002,
author = {Shirley B. Gray and Stewart Venit and Russ Abbott},
comment = {},
urldate = {2018-07-02},
collections = {Lists and catalogues,Geometry}
}
@article{OnSomeRegularToroids,
title = {On Some Regular Toroids},
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{\'{a}}sz{\'{a}}r-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 {\'{A}}kos Cs{\'{a}}sz{\'{a}}r in 1949, and the latter one was found by the author, in 1977.},
url = {http://symmetry-us.com/Journals/visbook/szilassi/},
year = 2001,
author = {Lajos Szilassi},
comment = {},
urldate = {2018-07-26},
collections = {Easily explained,Things to make and do,Geometry}
}
@article{Euclidstheoremontheinfinitudeofprimesahistoricalsurveyofitsproofs300BC2017andanothernewproof,
title = {Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2017) and another new proof},
abstract = {In this article, we provide a comprehensive historical survey of 183
different proofs of famous Euclid's theorem on the infinitude of prime numbers.
The author is trying to collect almost all the known proofs on infinitude of
primes, including some proofs that can be easily obtained as consequences of
some known problems or divisibility properties. Furthermore, here are listed
numerous elementary proofs of the infinitude of primes in different arithmetic
progressions.
All the references concerning the proofs of Euclid's theorem that use similar
methods and ideas are exposed subsequently. Namely, presented proofs are
divided into 8 subsections of Section 2 in dependence of the methods that are
used in them. {\{}\bf Related new 14 proofs (2012-2017) are given in the last
subsection of Section 2.{\}} In the next section, we survey mainly elementary
proofs of the infinitude of primes in different arithmetic progressions.
Presented proofs are special cases of Dirichlet's theorem. In Section 4, we
give a new simple "Euclidean's proof" of the infinitude of primes.},
url = {http://arxiv.org/abs/1202.3670v3 http://arxiv.org/pdf/1202.3670v3},
year = 2012,
author = {Romeo Me{\v{s}}trovi{\'{c}}},
comment = {},
urldate = {2018-08-21},
archivePrefix = {arXiv},
eprint = {1202.3670},
primaryClass = {math.HO},
collections = {Easily explained,History,Lists and catalogues,The act of doing maths,About proof,Integerology}
}
@article{TheNamerClaimergame,
title = {The Namer-Claimer game},
abstract = {In each round of the Namer-Claimer game, Namer names a distance d, then
Claimer claims a subset of [n] that does not contain two points that differ by
d. Claimer wins once they have claimed sets covering [n]. I show that the
length of this game is of order log log n with optimal play from each side.},
url = {http://arxiv.org/abs/1808.10800v1 http://arxiv.org/pdf/1808.10800v1},
year = 2018,
author = {Ben Barber},
comment = {},
urldate = {2018-09-04},
archivePrefix = {arXiv},
eprint = {1808.10800},
primaryClass = {math.CO},
collections = {Easily explained,Games to play with friends}
}
@article{MathematicalWriting,
title = {Mathematical Writing},
abstract = {This report is based on a course of the same name given at Stanford University during autumn quarter, 1987. Here’s the catalog description:
CS 209. Mathematical Writing — Issues of technical writing and the effective presentation of mathematics and computer science. Preparation of theses, papers, books, and “literate” computer programs. A term paper on a topic of your choice; this paper may be used for credit in another course.},
url = {http://jmlr.csail.mit.edu/reviewing-papers/knuth{\_}mathematical{\_}writing.pdf},
year = 1987,
author = {Donald E. Knuth and Tracy Larrabee and Paul M. Roberts},
comment = {},
urldate = {2018-09-19},
collections = {Notation and conventions,The act of doing maths,Education}
}
@article{TheSplittingAlgorithmforEgyptianFractions,
title = {The Splitting Algorithm for Egyptian Fractions},
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.},
url = {https://www.sciencedirect.com/science/article/pii/S0022314X83710152?via{\%}3Dihub},
year = 1991,
author = {L. Beeckmans},
comment = {},
urldate = {2018-09-20},
collections = {Easily explained}
}
@article{FindingtheBanditinaGraphSequentialSearchandStop,
title = {Finding the Bandit in a Graph: Sequential Search-and-Stop},
abstract = {We consider the problem where an agent wants to find a hidden object that is
randomly located in some vertex of a directed acyclic graph (DAG) according to
a fixed but possibly unknown distribution. The agent can only examine vertices
whose in-neighbors have already been examined. In scheduling theory, this
problem is denoted by {\$}1|prec|\sum w{\_}jC{\_}j{\$}. However, in this paper, we address
learning setting where we allow the agent to stop before having found the
object and restart searching on a new independent instance of the same problem.
The goal is to maximize the total number of hidden objects found under a time
constraint. The agent can thus skip an instance after realizing that it would
spend too much time on it. Our contributions are both to the search theory and
multi-armed bandits. If the distribution is known, we provide a quasi-optimal
greedy strategy with the help of known computationally efficient algorithms for
solving {\$}1|prec|\sum w{\_}jC{\_}j{\$} under some assumption on the DAG. If the
distribution is unknown, we show how to sequentially learn it and, at the same
time, act near-optimally in order to collect as many hidden objects as
possible. We provide an algorithm, prove theoretical guarantees, and
empirically show that it outperforms the na\"ive baseline.},
url = {http://arxiv.org/abs/1806.02282v1 http://arxiv.org/pdf/1806.02282v1},
year = 2018,
author = {Pierre Perrault and Vianney Perchet and Michal Valko},
comment = {},
urldate = {2018-09-22},
archivePrefix = {arXiv},
eprint = {1806.02282},
primaryClass = {stat.ML},
collections = {Attention-grabbing titles,Protocols and strategies}
}
@article{Thelargestsmallhexagon,
title = {The largest small hexagon},
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{\%}.},
url = {https://www.sciencedirect.com/science/article/pii/0097316575900047},
year = 1975,
author = {R. L. Graham},
comment = {},
urldate = {2018-09-22},
collections = {Attention-grabbing titles,Easily explained,Geometry,Fun maths facts}
}
@article{Powerlawdistributionsinempiricaldata,
title = {Power-law distributions in empirical data},
abstract = {Power-law distributions occur in many situations of scientific interest and
have significant consequences for our understanding of natural and man-made
phenomena. Unfortunately, the detection and characterization of power laws is
complicated by the large fluctuations that occur in the tail of the
distribution -- the part of the distribution representing large but rare events
-- and by the difficulty of identifying the range over which power-law behavior
holds. Commonly used methods for analyzing power-law data, such as
least-squares fitting, can produce substantially inaccurate estimates of
parameters for power-law distributions, and even in cases where such methods
return accurate answers they are still unsatisfactory because they give no
indication of whether the data obey a power law at all. Here we present a
principled statistical framework for discerning and quantifying power-law
behavior in empirical data. Our approach combines maximum-likelihood fitting
methods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic
and likelihood ratios. We evaluate the effectiveness of the approach with tests
on synthetic data and give critical comparisons to previous approaches. We also
apply the proposed methods to twenty-four real-world data sets from a range of
different disciplines, each of which has been conjectured to follow a power-law
distribution. In some cases we find these conjectures to be consistent with the
data while in others the power law is ruled out.},
url = {http://arxiv.org/abs/0706.1062v2 http://arxiv.org/pdf/0706.1062v2},
year = 2007,
author = {Aaron Clauset and Cosma Rohilla Shalizi and M. E. J. Newman},
comment = {},
urldate = {2018-09-24},
archivePrefix = {arXiv},
eprint = {0706.1062},
primaryClass = {physics.data-an},
collections = {Drama!,Probability and statistics}
}
@online{Searchingforgeneralizedbinarynumbersystems,
title = {Searching for generalized binary number systems},
author = {Attila Kov{\'{a}}cs},
url = {http://szdg.lpds.sztaki.hu/assets/desc{\_}numsys{\_}es.php},
urldate = {2016-08-22},
year = 2017,
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 = {},
collections = {Unusual arithmetic,Fun maths facts,Integerology}
}
@article{SettingLinearAlgebraProblems,
title = {Setting linear algebra problems},
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.},
url = {http://web.maths.unsw.edu.au/{\~{}}jds/Papers/linalg.pdf},
year = 1997,
author = {John D. Steele},
comment = {},
urldate = {2018-09-27},
collections = {Education}
}
@online{NearMissPolyhedra,
title = {Near Miss Polyhedra},
author = {Jim McNeill},
url = {http://www.orchidpalms.com/polyhedra/acrohedra/nearmiss/nearmiss.htm},
urldate = {2018-01-10},
year = 2002,
abstract = {The polyhedra on this page are not quite regular, but as they are close I present them here as 'near misses'. },
comment = {},
collections = {The act of doing maths,Geometry}
}
@online{Thenumberdictionary,
title = {The number dictionary},
author = {},
url = {http://numdic.com/},
urldate = {2017-04-12},
year = 2008,
abstract = {The purpose is to provide an opportunity to show properties of numbers.},
comment = {},
collections = {Lists and catalogues,Integerology}
}
@article{Jerabek2016,
title = {Division by zero},
author = {Emil Je{\v{r}}{\'{a}}bek},
url = {http://arxiv.org/abs/1604.07309 http://arxiv.org/pdf/1604.07309v1},
urldate = {2016-04-26},
year = 2016,
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 = {},
month = {apr},
pages = 12,
archivePrefix = {arXiv},
eprint = {1604.07309},
primaryClass = {math.LO},
collections = {About proof}
}
@article{VanHarreveld1939,
title = {Doubly-, triply-, quadruply- and quintuply-innervated crustacean muscles},
author = {van Harreveld, A.},
url = {http://doi.wiley.com/10.1002/cne.900700208},
urldate = {2011-08-09},
year = 1939,
abstract = {},
comment = {},
issn = {0021-9967},
journal = {The Journal of Comparative Neurology},
month = {apr},
number = 2,
pages = {285--296},
volume = 70,
collections = {Animals}
}
@article{Szekely2005,
title = {Half of a coin: negative probabilities},
author = {Sz{\'{e}}kely, GJ},
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,
abstract = {},
comment = {},
journal = {Wilmott Magazine},
pages = {66--68},
collections = {Probability and statistics,Unusual arithmetic,Fun maths facts}
}
@article{Pierce2009,
title = {The Circle-Squaring Problem Decomposed},
author = {Pierce, Pamela and Ramsay, John},
url = {https://www.maa.org/sites/default/files/pdf/upload{\_}library/22/Evans/Pierce.pdf},
urldate = {2013-03-17},
year = 2009,
abstract = {},
comment = {},
journal = {Math \ldots},
number = {November},
pages = {19--22},
collections = {Geometry}
}
@article{Hutter2002,
title = {The Fastest and Shortest Algorithm for All Well-Defined Problems},
author = {Hutter, Marcus},
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,
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.
},
comment = {},
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,
collections = {Attention-grabbing titles,Basically computer science}
}
@article{Shelah1992,
title = {Cardinal arithmetic for skeptics},
author = {Shelah, Saharon},
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,
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 = {},
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,
collections = {Attention-grabbing titles,The act of doing maths,Unusual arithmetic}
}
@article{Shoemake1985,
title = {Animating rotation with quaternion curves},
author = {Shoemake, Ken},
url = {http://portal.acm.org/citation.cfm?doid=325334.325242},
urldate = {2011-01-12},
year = 1985,
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 = {},
journal = {International Conference on Computer Graphics and Interactive Techniques},
keywords = {B-spline,B{\'{e}}zier curve,animation,approximation,in-betweening,interpolation,quaternion,rotation,spherical geometry,spline},
number = 3,
pages = 245,
volume = 19,
collections = {Basically computer science}
}
@misc{item38,
title = {The Stick Problem},
author = {Augustine Bertagnolli},
url = {http://ajbertagnolli.com/wp-content/uploads/2013/10/sticks2.pdf},
urldate = {2014-01-13},
year = 2013,
abstract = {Given sticks of possible sizes one through six, what is the smallest number of sticks you can have
to ensure that you are able to form a perfect square? The Pigeonhole Principle tells us that if we have
nineteen sticks we would have at least four of one of the sizes, but can we do better if we take partitions
into account? This is one case of the stick problem which, though simple in statement, proves to be
not so simple in solution. In this paper, we define the stick problem clearly, discuss our methods for
approaching and simplifying the problem, provide an algorithm for generating solutions, and present
some computer generated solutions for specific cases.},
comment = {},
collections = {Easily explained,Geometry,Puzzles}
}
@article{Romano,
title = {Deriving Uniform Polyhedra with Wythoff's Construction},
author = {Romano, Don},
url = {http://math.ucdenver.edu/{\~{}}mferrara/seminar/F2010/Don{\_}Romano{\_}Seminar.pdf},
urldate = {2011-05-10},
year = 2010,
abstract = {},
comment = {},
collections = {Geometry}
}
@online{CalculatorForensics,
title = {Calculator Forensics},
abstract = {Results from the evaluation of this equation in degrees mode: arcsin (arccos (arctan (tan (cos (sin (9) ) ) ) ) ) },
url = {http://www.rskey.org/{\~{}}mwsebastian/miscprj/results.htm},
year = 2000,
author = {Mike Sebastian},
comment = {},
urldate = {2018-10-16},
collections = {Basically computer science,Easily explained,Lists and catalogues}
}
@article{Shima2012,
title = {How far can Tarzan jump?},
author = {Shima, Hiroyuki},
url = {http://arxiv.org/abs/1208.4355 http://arxiv.org/pdf/1208.4355v1},
urldate = {2012-09-02},
year = 2012,
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 = {},
archivePrefix = {arXiv},
arxivId = {1208.4355},
eprint = {1208.4355},
month = {aug},
pages = 8,
primaryClass = {physics.pop-ph},
collections = {Basically physics,Easily explained}
}
@misc{Henderson,
title = {The Collatz Fractal},
author = {Henderson, Xander},
url = {http://yozh.org/2012/01/12/the{\_}collatz{\_}fractal/},
urldate = {2012-01-15},
year = 2012,
abstract = {},
comment = {},
collections = {Easily explained,Fun maths facts}
}
@article{item35,
title = {WHAT IS Lehmer's number?},
author = {Eriko Hironaka},
url = {http://www.math.fsu.edu/{\~{}}aluffi/archive/paper355.pdf},
urldate = {2013-12-03},
year = 2009,
abstract = {Lehmer's number \(\lambda \approx 1.17628\) is the largest real root of the polynomial \(f{\_}\lambda(x) = x^10 + x^9 - x^7 - x^6 -x^5 -x^4 - x^3 + x + 1\).
This number appears in various contexts in number theory and topology as the (sometimes conjectural) answer to natural questions involving ``minimality'' and ``small complexity''.},
comment = {},
collections = {Attention-grabbing titles,Integerology}
}
@article{Enumerationofmarycacti,
title = {Enumeration of m-ary cacti},
abstract = {The purpose of this paper is to enumerate various classes of cyclically
colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is
motivated by the topological classification of complex polynomials having at
most m critical values, studied by Zvonkin and others. We obtain explicit
formulae for both labelled and unlabelled m-ary cacti, according to i) the
number of polygons, ii) the vertex-color distribution, iii) the vertex-degree
distribution of each color. We also enumerate m-ary cacti according to the
order of their automorphism group. Using a generalization of Otter's formula,
we express the species of m-ary cacti in terms of rooted and of pointed cacti.
A variant of the m-dimensional Lagrange inversion is then used to enumerate
these structures. The method of Liskovets for the enumeration of unrooted
planar maps can also be adapted to m-ary cacti.},
url = {http://arxiv.org/abs/math/9804119v2 http://arxiv.org/pdf/math/9804119v2},
year = 1998,
author = {Miklos Bona and Michel Bousquet and Gilbert Labelle and Pierre Leroux},
comment = {The number of binary cacti is the number of Truchet tiles.},
urldate = {2018-10-23},
archivePrefix = {arXiv},
eprint = {math/9804119},
primaryClass = {math.CO},
collections = {Attention-grabbing titles,Combinatorics,Things to make and do}
}
@article{SomeFundamentalTheoremsInMathematics,
title = {Some Fundamental Theorems in Mathematics},
abstract = {An expository hitchhiker's guide to some theorems in mathematics.},
url = {http://www.math.harvard.edu/{\~{}}knill/graphgeometry/papers/fundamental.pdf},
year = 2018,
author = {Oliver Knill},
comment = {},
urldate = {2018-10-24},
collections = {Lists and catalogues}
}
@online{ATilingDatabase,
title = {A Tiling Database},
abstract = {This database has three aims:
1. Provide a comprehensive collection of high quality images of geometric tiling patterns;
2. Provide a means of locating images by means of their geometric properties;
3. Provide an authoritative source for such patterns. },
url = {http://www.tilingsearch.org/index1.htm},
year = 2011,
author = {Brian Wichmann and Tony Lee},
comment = {},
urldate = {2018-10-24},
collections = {}
}
@article{Noncrossingpartitionsunderrotationandreflection,
title = {Noncrossing partitions under rotation and reflection},
abstract = {We consider noncrossing partitions of [n] under the action of (i) the
reflection group (of order 2), (ii) the rotation group (cyclic of order n) and
(iii) the rotation/reflection group (dihedral of order 2n). First, we exhibit a
bijection from rotation classes to bicolored plane trees on n edges, and
consider its implications. Then we count noncrossing partitions of [n]
invariant under reflection and show that, somewhat surprisingly, they are
equinumerous with rotation classes invariant under reflection. The proof uses a
pretty involution originating in work of Germain Kreweras. We conjecture that
the "equinumerous" result also holds for arbitrary partitions of [n].},
url = {http://arxiv.org/abs/math/0510447v3 http://arxiv.org/pdf/math/0510447v3},
year = 2005,
author = {David Callan and Len Smiley},
comment = {},
urldate = {2018-10-27},
archivePrefix = {arXiv},
eprint = {math/0510447},
primaryClass = {math.CO},
collections = {Combinatorics,Easily explained,Geometry,Things to make and do}
}
@article{Conwaysdoughnuts,
title = {Conway's doughnuts},
abstract = {Morley's Theorem about angle trisectors can be viewed as the statement that a
certain diagram `exists', meaning that triangles of prescribed shapes meet in a
prescribed pattern. This diagram is the case n=3 of a class of diagrams we call
`Conway's doughnuts'. These diagrams can be proven to exist using John
Smillie's holonomy method, recently championed by Eric Braude: `Guess the
shapes; check the holonomy.' For n = 2, 3, 4 the existence of the doughnut
happens to be easy to prove because the hole is absent or triangular.},
url = {http://arxiv.org/abs/1804.04024v1 http://arxiv.org/pdf/1804.04024v1},
year = 2018,
author = {Peter Doyle and Shikhin Sethi},
comment = {},
urldate = {2018-11-04},
archivePrefix = {arXiv},
eprint = {1804.04024},
primaryClass = {math.HO},
collections = {Food,Fun maths facts,Geometry}
}
@misc{Elsholtz2002,
title = {A combinatorial approach to sums of two squares and related problems},
author = {Elsholtz, Christian},
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,
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 = {},
collections = {About proof,Fun maths facts}
}
@article{PrimeNumberRaces,
title = {Prime Number Races},
abstract = {This is a survey article on prime number races. Chebyshev noticed in the
first half of the nineteenth century that for any given value of x, there
always seem to be more primes of the form 4n+3 less than x then there are of
the form 4n+1. Similar observations have been made with primes of the form 3n+2
and 3n+1, with primes of the form 10n+3/10n+7 and 10n+1/10n+9, and many others
besides. More generally, one can consider primes of the form qn+a, qn+b, qn+c,
>... for our favorite constants q, a, b, c, ... and try to figure out which
forms are "preferred" over the others. In this paper, we describe these
phenomena in greater detail and explain the efforts that have been made at
understanding them.},
url = {http://arxiv.org/abs/math/0408319v1 http://arxiv.org/pdf/math/0408319v1},
year = 2004,
author = {Andrew Granville and Greg Martin},
comment = {},
urldate = {2018-11-12},
archivePrefix = {arXiv},
eprint = {math/0408319},
primaryClass = {math.NT},
collections = {Attention-grabbing titles,Easily explained,Fun maths facts,Integerology}
}
@article{DuotoneTruchetliketilings,
title = {Duotone Truchet-like tilings},
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.},
url = {https://www.tandfonline.com/doi/full/10.1080/17513470902718252?scroll=top{\&}needAccess=true},
year = 2009,
author = {Cameron Browne},
comment = {},
urldate = {2018-11-13},
collections = {Art,Easily explained,Geometry,Things to make and do}
}
@article{SevenTreesinOne,
title = {Seven Trees in One},
abstract = {Following a remark of Lawvere, we explicitly exhibit a particularly
elementary bijection between the set T of finite binary trees and the set T^7
of seven-tuples of such trees. "Particularly elementary" means that the
application of the bijection to a seven-tuple of trees involves case
distinctions only down to a fixed depth (namely four) in the given seven-tuple.
We clarify how this and similar bijections are related to the free commutative
semiring on one generator X subject to X=1+X^2. Finally, our main theorem is
that the existence of particularly elementary bijections can be deduced from
the provable existence, in intuitionistic type theory, of any bijections at
all.},
url = {http://arxiv.org/abs/math/9405205v1 http://arxiv.org/pdf/math/9405205v1},
year = 1994,
author = {Andreas Blass},
comment = {},
urldate = {2018-11-26},
archivePrefix = {arXiv},
eprint = {math/9405205},
primaryClass = {math.LO},
collections = {Fun maths facts,Unusual arithmetic}
}
@article{RectangleArithmetic,
title = {Rectangle Arithmetic},
author = {Bill Gosper},
url = {http://gosper.org/rectarith12.pdf},
urldate = {2018-11-27},
year = 2006,
abstract = {Another slant on fractions},
comment = {},
collections = {Easily explained,Geometry,Unusual arithmetic}
}
@article{Mathematicsappliedtodressmaking,
title = {Mathematics applied to dressmaking},
author = {Christopher Zeeman},
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,
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 = {},
collections = {Easily explained,Fun maths facts,Things to make and do}
}
@article{AmusingPermutationRepresentationsofGroupExtensions,
title = {Amusing Permutation Representations of Group Extensions},
abstract = {Wreath products of finite groups have permutation representations that are
constructed from the permutation representations of their constituents. One can
envision these in a metaphoric sense in which a rope is made from a bundle of
threads. In this way, subgroups and quotients are easily visualized. The
general idea is applied to the finite subgroups of the special unitary group of
{\$}(2\times 2){\$}-matrices. Amusing diagrams are developed that describe the unit
quaternions, the binary tetrahedral, octahedral, and icosahedral group as well
as the dicyclic groups. In all cases, the quotients as subgroups of the
permutation group are readily apparent. These permutation representations lead
to injective homomorphisms into wreath products.},
url = {http://arxiv.org/abs/1812.08475v1 http://arxiv.org/pdf/1812.08475v1},
year = 2018,
author = {Yongju Bae and J. Scott Carter and Byeorhi Kim},
comment = {},
urldate = {2019-01-01},
archivePrefix = {arXiv},
eprint = {1812.08475},
primaryClass = {math.GT},
collections = {Attention-grabbing titles,Fun maths facts,The groups group}
}
@article{DoingMathinJestReflectionsonUselessMaththeUnreasonableEffectivenessofMathematicsandtheEthicalObligationsofMathematicians,
title = {Doing Math in Jest: Reflections on Useless Math, the Unreasonable Effectiveness of Mathematics, and the Ethical Obligations of Mathematicians},
abstract = {Mathematicians occasionally discover interesting truths even when they are
playing with mathematical ideas with no thoughts about possible consequences of
their actions. This paper describes two specific instances of this phenomenon.
The discussion touches upon the theme of the unreasonable effectiveness of
mathematics as well as the ethical obligations of mathematicians.},
url = {http://arxiv.org/abs/1812.09601v1 http://arxiv.org/pdf/1812.09601v1},
year = 2018,
author = {Gizem Karaali},
comment = {},
urldate = {2019-01-01},
archivePrefix = {arXiv},
eprint = {1812.09601},
primaryClass = {math.HO},
collections = {The act of doing maths}
}
@article{Whatisaclosedformnumber,
title = {What is a closed-form number?},
abstract = {If a student asks for an antiderivative of exp(x^2), there is a standard
reply: the answer is not an elementary function. But if a student asks for a
closed-form expression for the real root of x = cos(x), there is no standard
reply. We propose a definition of a closed-form expression for a number (as
opposed to a *function*) that we hope will become standard. With our
definition, the question of whether the root of x = cos(x) has a closed form
is, perhaps surprisingly, still open. We show that Schanuel's conjecture in
transcendental number theory resolves questions like this, and we also sketch
some connections with Tarski's problem of the decidability of the first-order
theory of the reals with exponentiation. Many (hopefully accessible) open
problems are described.},
url = {http://arxiv.org/abs/math/9805045v1 http://arxiv.org/pdf/math/9805045v1},
year = 1998,
author = {Timothy Y. Chow},
comment = {},
urldate = {2019-01-09},
archivePrefix = {arXiv},
eprint = {math/9805045},
primaryClass = {math.NT},
collections = {Notation and conventions,The act of doing maths}
}
@article{Smarandache1991,
title = {Only problems, not solutions!},
author = {Smarandache, Florentin},
url = {http://vixra.org/abs/1005.0049 http://vixra.org/pdf/1005.0049v1.pdf},
urldate = {2013-07-29},
year = 1991,
abstract = {},
comment = {},
isbn = 1879585006,
volume = 1993,
collections = {Attention-grabbing titles,Puzzles}
}
@article{WhatIsanEnvelope,
title = {What Is an Envelope?},
abstract = {},
url = {https://www.jstor.org/stable/3617131?seq=1{\#}metadata{\_}info{\_}tab{\_}contents},
year = 1981,
author = {J.W. Bruce and P.J. Giblin},
comment = {},
urldate = {2019-02-04},
collections = {Easily explained,Geometry}
}
@online{item16,
title = {Navigating Hyperbolic Space with Fibonacci Trees},
author = {moniker},
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,
abstract = {},
comment = {},
collections = {Basically computer science,Fibonaccinalia,Geometry}
}
@article{CodesLowerBoundsandPhaseTransitionsintheSymmetricRendezvousProblem,
title = {Codes, Lower Bounds, and Phase Transitions in the Symmetric Rendezvous Problem},
abstract = {In the rendezvous problem, two parties with different labelings of the
vertices of a complete graph are trying to meet at some vertex at the same
time. It is well-known that if the parties have predetermined roles, then the
strategy where one of them waits at one vertex, while the other visits all {\$}n{\$}
vertices in random order is optimal, taking at most {\$}n{\$} steps and averaging
about {\$}n/2{\$}. Anderson and Weber considered the symmetric rendezvous problem,
where both parties must use the same randomized strategy. They analyzed
strategies where the parties repeatedly play the optimal asymmetric strategy,
determining their role independently each time by a biased coin-flip. By tuning
the bias, Anderson and Weber achieved an expected meeting time of about {\$}0.829
n{\$}, which they conjectured to be asymptotically optimal.
We change perspective slightly: instead of minimizing the expected meeting
time, we seek to maximize the probability of meeting within a specified time
{\$}T{\$}. The Anderson-Weber strategy, which fails with constant probability when
{\$}T= \Theta(n){\$}, is not asymptotically optimal for large {\$}T{\$} in this setting.
Specifically, we exhibit a symmetric strategy that succeeds with probability
{\$}1-o(1){\$} in {\$}T=4n{\$} steps. This is tight: for any {\$}\alpha < 4{\$}, any symmetric
strategy with {\$}T = \alpha n{\$} fails with constant probability. Our strategy uses
a new combinatorial object that we dub a "rendezvous code," which may be of
independent interest.
When {\$}T \le n{\$}, we show that the probability of meeting within {\$}T{\$} steps is
indeed asymptotically maximized by the Anderson-Weber strategy. Our results
imply new lower bounds, showing that the best symmetric strategy takes at least
{\$}0.638 n{\$} steps in expectation. We also present some partial results for the
symmetric rendezvous problem on other vertex-transitive graphs.},
url = {http://arxiv.org/abs/1609.01582v1 http://arxiv.org/pdf/1609.01582v1},
year = 2016,
author = {Varsha Dani and Thomas P. Hayes and Cristopher Moore and Alexander Russell},
comment = {},
urldate = {2019-03-02},
archivePrefix = {arXiv},
eprint = {1609.01582},
primaryClass = {math.CO},
collections = {Easily explained,Protocols and strategies}
}
@article{YetAnotherSingleLawforLattices,
title = {Yet Another Single Law for Lattices},
abstract = {In this note we show that the equational theory of all lattices is defined by
a single absorption law. The identity of length 29 with 8 variables is shorter
than previously known such equations defining lattices.},
url = {http://arxiv.org/abs/math/0307284v1 http://arxiv.org/pdf/math/0307284v1},
year = 2003,
author = {William McCune and Ranganathan Padmanabhan and Robert Veroff},
comment = {},
urldate = {2019-03-02},
archivePrefix = {arXiv},
eprint = {math/0307284},
primaryClass = {math.LO},
collections = {Attention-grabbing titles,Fun maths facts}
}
@article{TheInstructorsGuidetoRealInduction,
title = {The Instructor's Guide to Real Induction},
abstract = {We introduce real induction, a proof technique analogous to mathematical
induction but applicable to statements indexed by an interval on the real line.
More generally we give an inductive principle applicable in any Dedekind
complete linearly ordered set. Real and ordered induction is then applied to
give streamlined, conceptual proofs of basic results in honors calculus,
elementary real analysis and topology.},
url = {http://arxiv.org/abs/1208.0973v1 http://arxiv.org/pdf/1208.0973v1},
year = 2012,
author = {Pete L. Clark},
comment = {},
urldate = {2019-03-02},
archivePrefix = {arXiv},
eprint = {1208.0973},
primaryClass = {math.HO},
collections = {About proof,Easily explained,Fun maths facts,The act of doing maths}
}
@online{Leystatistics,
title = {Ley statistics},
abstract = {},
url = {https://www.cantab.net/users/michael.behrend/ley{\_}stats/index.html},
year = 2014,
author = {Michael Behrend},
comment = {},
urldate = {2019-03-02},
collections = {Geometry,Probability and statistics}
}
@article{TheTakagiFunctionandItsProperties,
title = {The Takagi Function and Its Properties},
abstract = {The Takagi function is a continuous non-differentiable function on [0,1]
introduced by Teiji Takagi in 1903. It has since appeared in a surprising
number of different mathematical contexts, including mathematical analysis,
probability theory and number theory. This paper surveys the known properties
of this function as it relates to these fields.},
url = {http://arxiv.org/abs/1112.4205v2 http://arxiv.org/pdf/1112.4205v2},
year = 2011,
author = {Jeffrey C. Lagarias},
comment = {},
urldate = {2019-03-04},
archivePrefix = {arXiv},
eprint = {1112.4205},
primaryClass = {math.CA},
collections = {Easily explained,Fun maths facts}
}
@article{BrazilianPrimesWhichAreAlsoSophieGermainPrimes,
title = {Brazilian Primes Which Are Also Sophie Germain Primes},
abstract = {We disprove a conjecture of Schott that no Brazilian primes are Sophie
Germain primes. We enumerate all counterexamples up to {\$}10^{\{}44{\}}{\$}.},
url = {http://arxiv.org/abs/1903.04577v1 http://arxiv.org/pdf/1903.04577v1},
year = 2019,
author = {Jon Grantham and Hester Graves},
comment = {},
urldate = {2019-03-13},
archivePrefix = {arXiv},
eprint = {1903.04577},
primaryClass = {math.NT},
collections = {Attention-grabbing titles,Easily explained,Fun maths facts,Integerology}
}
@article{PerformingMathematicalOperationswithMetamaterials,
title = {Performing Mathematical Operations with Metamaterials},
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’s 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.},
url = {http://science.sciencemag.org/content/343/6167/160},
year = 2014,
author = {Alexandre Silva and Francesco Monticone and Giuseppe Castaldi and Vincenzo Galdi and Andrea Al{\`{u}} and Nader Engheta},
comment = {},
urldate = {2019-03-26},
collections = {Basically physics,Unusual computers}
}
@book{CharlesBabbageNotation,
title = {Charles Babbage's thoughts on notation},
author = {Charles Babbage},
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,
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.},
collections = {Notation and conventions}
}
@article{Kamel1994,
title = {Hilbert R-tree: An improved R-tree using fractals},
author = {Kamel, Ibrahim and Faloutsos, Christos},
url = {http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.9180{\&}rep=rep1{\&}type=pdf},
urldate = {2010-08-31},
year = 1994,
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).
Following [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.},
booktitle = {PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON VERY LARGE DATA BASES},
pages = {500--500},
publisher = {Citeseer},
volume = 8958546,
collections = {Basically computer science}
}
@article{Glenis2008,
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},
url = {https://arxiv.org/abs/math/0611062 https://arxiv.org/pdf/math/0611062},
urldate = {2011-02-03},
year = 2008,
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.},
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},
collections = {Geometry}
}
@book{BrainfillingCurvesaFractalBestiary,
title = {Brainfilling Curves - a Fractal Bestiary},
author = {Jeffrey Ventrella},
url = {https://archive.org/details/BrainfillingCurves-AFractalBestiary},
urldate = {2019-05-09},
year = 2012,
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.
This 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.
This book would be of interest to educated people of all backgrounds, especially geometers, computer scientists, and artists of the Escher ilk. },
comment = {},
collections = {Art,Easily explained,Fun maths facts,Things to make and do}
}
@article{AContributiontotheMathematicalTheoryofBigGameHunting,
title = {A Contribution to the Mathematical Theory of Big Game Hunting},
abstract = {Problem: To Catch a Lion in the Sahara Desert.},
url = {https://stuff.mit.edu/people/dpolicar/writing/netsam/lionhunt.html},
year = 1938,
author = {Ralph Boas},
comment = {},
urldate = {2019-05-09},
collections = {Animals,Attention-grabbing titles}
}
@article{PlanarHypohamiltonianGraphson40Vertices,
title = {Planar Hypohamiltonian Graphs on 40 Vertices},
abstract = {A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any
single vertex gives a Hamiltonian graph. Until now, the smallest known planar
hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That
result is here improved upon by 25 planar hypohamiltonian graphs of order 40,
which are found through computer-aided generation of certain families of planar
graphs with girth 4 and a fixed number of 4-faces. It is further shown that
planar hypohamiltonian graphs exist for all orders greater than or equal to 42.
If Hamiltonian cycles are replaced by Hamiltonian paths throughout the
definition of hypohamiltonian graphs, we get the definition of hypotraceable
graphs. It is shown that there is a planar hypotraceable graph of order 154 and
of all orders greater than or equal to 156. We also show that the smallest
hypohamiltonian planar graph of girth 5 has 45 vertices.},
url = {http://arxiv.org/abs/1302.2698v4 http://arxiv.org/pdf/1302.2698v4},
year = 2013,
author = {Mohammadreza Jooyandeh and Brendan D. McKay and Patric R. J. {\"{O}}sterg{\aa}rd and Ville H. Pettersson and Carol T. Zamfirescu},
comment = {},
urldate = {2019-05-09},
archivePrefix = {arXiv},
eprint = {1302.2698},
primaryClass = {math.CO},
collections = {Fun maths facts}
}
@article{TheSwissCheeseOperad,
title = {The Swiss-Cheese Operad},
abstract = {We introduce a new operad, which we call the Swiss-cheese operad. It mixes
naturally the little disks and the little intervals operads. The Swiss-cheese
operad is related to the configuration spaces of points on the upper half-plane
and points on the real line, considered by Kontsevich for the sake of
deformation quantization. This relation is similar to the relation between the
little disks operad and the configuration spaces of points on the plane. The
Swiss-cheese operad may also be regarded as a finite-dimensional model of the
moduli space of genus-zero Riemann surfaces appearing in the open-closed string
theory studied recently by Zwiebach. We describe algebras over the homology of
the Swiss-cheese operad.},
url = {http://arxiv.org/abs/math/9807037v1 http://arxiv.org/pdf/math/9807037v1},
year = 1998,
author = {Alexander A. Voronov},
comment = {},
urldate = {2019-05-09},
archivePrefix = {arXiv},
eprint = {math/9807037},
primaryClass = {math.QA},
collections = {Attention-grabbing titles,Food}
}
@article{ThegraphsbehindReuleauxpolyhedra,
title = {The graphs behind Reuleaux polyhedra},
abstract = {This work is about graphs arising from Reuleaux polyhedra. Such graphs must
necessarily be planar, {\$}3{\$}-connected and strongly self-dual. We study the
question of when these conditions are sufficient.
If {\$}G{\$} is any such a graph with isomorphism {\$}\tau : G \to G^*{\$} (where {\$}G^*{\$}
is the unique dual graph), a metric mapping is a map {\$}\eta : V(G) \to \mathbb
R^3{\$} such that the diameter of {\$}\eta(G){\$} is {\$}1{\$} and for every pair of vertices
{\$}(u,v){\$} such that {\$}u\in \tau(v){\$} we have dist{\$}(\eta(u),\eta(v)) = 1{\$}. If {\$}\eta{\$}
is injective, it is called a metric embedding. Note that a metric embedding
gives rise to a Reuleaux Polyhedra.
Our contributions are twofold: Firstly, we prove that any planar,
{\$}3{\$}-connected, strongly self-dual graph has a metric mapping by proving that
the chromatic number of the diameter graph (whose vertices are {\$}V(G){\$} and whose
edges are pairs {\$}(u,v){\$} such that {\$}u\in \tau(v){\$}) is at most {\$}4{\$}, which means
there exists a metric mapping to the tetrahedron. Furthermore, we use the
Lov\'asz neighborhood-complex theorem in algebraic topology to prove that the
chromatic number of the diameter graph is exactly {\$}4{\$}.
Secondly, we develop algorithms that allow us to obtain every such graph with
up to {\$}14{\$} vertices. Furthermore, we numerically construct metric embeddings
for every such graph. From the theorem and this computational evidence we
conjecture that every such graph is realizable as a Reuleaux polyhedron in
{\$}\mathbb R^3{\$}.
In previous work the first and last authors described a method to construct a
constant-width body from a Reuleaux polyhedron. So in essence, we also
construct hundreds of new examples of constant-width bodies.
This is related to a problem of V\'azsonyi, and also to a problem of
Blaschke-Lebesgue.},
url = {http://arxiv.org/abs/1904.12761v1 http://arxiv.org/pdf/1904.12761v1},
year = 2019,
author = {Luis Montejano and Eric Pauli and Miguel Raggi and Edgardo Rold{\'{a}}n-Pensado},
comment = {},
urldate = {2019-05-09},
archivePrefix = {arXiv},
eprint = {1904.12761},
primaryClass = {cs.CG},
collections = {Geometry}
}
@article{Auniversaldifferentialequation,
title = {A universal differential equation},
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)\]},
url = {https://projecteuclid.org/euclid.bams/1183548125},
year = 1981,
author = {Lee A. Rubel},
comment = {},
urldate = {2019-05-09},
collections = {Fun maths facts,Unusual computers}
}
@book{TheSensualquadraticForm,
title = {The Sensual (quadratic) Form},
author = {John Horton Conway},
url = {https://bookstore.ams.org/car-26 https://www.maths.ed.ac.uk/{\~{}}v1ranick/papers/conwaysens.pdf},
urldate = {2019-05-09},
year = 1997,
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.
The 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 = {},
collections = {Attention-grabbing titles}
}
@book{HexAStrategyGuide,
title = {Hex: A Strategy Guide},
abstract = {},
url = {http://www.mseymour.ca/hex{\_}book/hexstrat.html},
year = 2019,
author = {Matthew Seymour},
comment = {},
urldate = {2019-05-31},
collections = {Games to play with friends}
}
@article{ChordsofanellipseLucaspolynomialsandcubicequations,
title = {Chords of an ellipse, Lucas polynomials, and cubic equations},
abstract = {A beautiful result of Thomas Price links the Fibonacci numbers and the Lucas
polynomials to the plane geometry of an ellipse. We give a conceptually
transparent development of this result that provides a tour of several gems of
classical mathematics: It is inspired by Girolamo Cardano's solution of the
cubic equation, uses Newton's theorem connecting power sums and elementary
symmetric polynomials, and yields for free an alternative proof of the Binet
formula for the generalized Lucas polynomials.},
url = {http://arxiv.org/abs/1810.00492v3 http://arxiv.org/pdf/1810.00492v3},
year = 2018,
author = {Ben Blum-Smith and Japheth Wood},
comment = {},
urldate = {2019-06-30},
archivePrefix = {arXiv},
eprint = {1810.00492},
primaryClass = {math.HO},
collections = {Easily explained,Fun maths facts,Puzzles}
}
@article{FiboquadraticSequencesandExtensionsoftheCassiniIdentityRaisedFromtheStudyofRithmomachia,
title = {Fiboquadratic Sequences and Extensions of the Cassini Identity Raised From the Study of Rithmomachia},
abstract = {In this paper, we introduce fiboquadratic sequences as an extension to
infinity of the board of Rithmomachia and we prove that this extension gives
raise to fiboquadratic sequences which we define here. Also, fiboquadratic
sequences provide extensions of Cassini's Identity.},
url = {http://arxiv.org/abs/1509.03177v3 http://arxiv.org/pdf/1509.03177v3},
year = 2015,
author = {Tom{\'{a}}s Guardia and Douglas Jim{\'{e}}nez},
comment = {},
urldate = {2019-07-18},
archivePrefix = {arXiv},
eprint = {1509.03177},
primaryClass = {math.HO},
collections = {Fibonaccinalia,Games to play with friends,History}
}
@article{NumbersforMasochistsAGuidetoMentalFactoring,
title = {Numbers for Masochists: A Guide to Mental Factoring},
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.},
url = {http://www.purplestreak.com/g4g13/mentalfactoring.pdf},
year = 2018,
author = {Hilarie Orman and Richard Schroeppel},
comment = {},
urldate = {2019-07-31},
collections = {Easily explained,Integerology}
}
@article{CreationofHyperbolicOrnaments,
title = {Creation of Hyperbolic Ornaments},
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.},
url = {http://martin.von-gagern.net/publications/2014-phd/ http://mediatum.ub.tum.de/doc/1210572/document.pdf},
year = 2014,
author = {Martin von Gagern},
comment = {},
urldate = {2019-08-08},
collections = {Art,Basically computer science,Geometry}
}
@online{FractalSequences,
title = {Fractal Sequences},
abstract = { Fractal sequences have in common with the more familiar geometric fractals the property of self-containment. An example of a fractal sequence is
1, 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, . . .
If 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.)},
url = {https://faculty.evansville.edu/ck6/integer/fractals.html},
year = 1999,
author = {Clark Kimberling},
comment = {},
urldate = {2019-08-21},
collections = {Easily explained,Fun maths facts,Integerology,Puzzles}
}
@article{PalindromesinDifferentBasesAConjectureofJErnestWilkins,
title = {Palindromes in Different Bases: A Conjecture of J. Ernest Wilkins},
abstract = {We show that there exist exactly 203 positive integers {\$}N{\$} such that for some
integer {\$}d \geq 2{\$} this number is a {\$}d{\$}-digit palindrome base 10 as well as a
{\$}d{\$}-digit palindrome for some base {\$}b{\$} different from 10. To be more precise,
such {\$}N{\$} range from 22 to 9986831781362631871386899.},
url = {http://arxiv.org/abs/0909.5452v1 http://arxiv.org/pdf/0909.5452v1},
year = 2009,
author = {Edray Herber Goins},
comment = {},
urldate = {2019-09-14},
archivePrefix = {arXiv},
eprint = {0909.5452},
primaryClass = {math.NT},
collections = {Easily explained,Fun maths facts,Integerology}
}
@article{PercolationisOdd,
title = {Percolation is Odd},
abstract = {We discuss the number of spanning configurations in site percolation. We show
that for a large class of lattices, the number of spanning configrations is odd
for all lattice sizes. This class includes site percolation on the square
lattice and on the hypercubic lattice in any dimension.},
url = {http://arxiv.org/abs/1909.01484v1 http://arxiv.org/pdf/1909.01484v1},
year = 2019,
author = {Stephan Mertens and Cristopher Moore},
comment = {},
urldate = {2019-09-14},
archivePrefix = {arXiv},
eprint = {1909.01484},
primaryClass = {cond-mat.stat-mech},
collections = {Attention-grabbing titles,Combinatorics}
}
@article{FindingaprincessinapalaceApursuitevasionproblem,
title = {Finding a princess in a palace: A pursuit-evasion problem},
abstract = {This paper solves a pursuit-evasion problem in which a prince must find a
princess who is constrained to move on each day from one vertex of a finite
graph to another. Unlike the related and much studied `Cops and Robbers Game',
the prince has no knowledge of the position of the princess; he may, however,
visit any single room he wishes on each day. We characterize the graphs for
which the prince has a winning strategy, and determine, for each such graph,
the minimum number of days the prince requires to guarantee to find the
princess.},
url = {http://arxiv.org/abs/1204.5490v1 http://arxiv.org/pdf/1204.5490v1},
year = 2012,
author = {John R. Britnell and Mark Wildon},
comment = {},
urldate = {2019-10-08},
archivePrefix = {arXiv},
eprint = {1204.5490},
primaryClass = {math.CO},
collections = {Combinatorics,Easily explained,Protocols and strategies,Puzzles}
}
@article{Catchingamouseonatree,
title = {Catching a mouse on a tree},
abstract = {In this paper we consider a pursuit-evasion game on a graph. A team of cats,
which may choose any vertex of the graph at any turn, tries to catch an
invisible mouse, which is constrained to moving along the vertices of the
graph. Our main focus shall be on trees. We prove that {\$}\lceil
(1/2)\log{\_}2(n)\rceil{\$} cats can always catch a mouse on a tree of order {\$}n{\$} and
give a collection of trees where the mouse can avoid being caught by {\$} (1/4 -
o(1))\log{\_}2(n){\$} cats.},
url = {http://arxiv.org/abs/1502.06591v1 http://arxiv.org/pdf/1502.06591v1},
year = 2015,
author = {Vytautas Gruslys and Arès M{\'{e}}roueh},
comment = {},
urldate = {2019-10-08},
archivePrefix = {arXiv},
eprint = {1502.06591},
primaryClass = {math.CO},
collections = {Animals,Attention-grabbing titles,Combinatorics}
}
@article{HowtoHuntanInvisibleRabbitonaGraph,
title = {How to Hunt an Invisible Rabbit on a Graph},
abstract = {We investigate Hunters {\&} Rabbit game, where a set of hunters tries to catch
an invisible rabbit that slides along the edges of a graph. We show that the
minimum number of hunters required to win on an (n\times m)-grid is \lfloor
min{\{}n,m{\}}/2\rfloor+1. We also show that the extremal value of this number on
n-vertex trees is between \Omega(log n/log log n) and O(log n).},
url = {http://arxiv.org/abs/1502.05614v2 http://arxiv.org/pdf/1502.05614v2},
year = 2015,
author = {Tatjana V. Abramovskaya and Fedor V. Fomin and Petr A. Golovach and Micha{\l} Pilipczuk},
comment = {},
urldate = {2019-10-08},
archivePrefix = {arXiv},
eprint = {1502.05614},
primaryClass = {math.CO},
collections = {Animals,Attention-grabbing titles,Combinatorics,Easily explained,Protocols and strategies,Puzzles}
}
@article{portandsweepsolitaire,
title = {Port-and-Sweep Solitaire},
abstract = {How does this happen? I just wanted a nice game where I didn’t have to count higher than two, and I ended up dealing with imaginary numbers. But let me back up: I’ve been a little obsessed with a puzzle lately, and I would like to explain what’s puzzling me and how the square root of –1 can sneak in where you least expect it. },
url = {http://homepages.gac.edu/{\~{}}jsiehler/Articles/port-and-sweep-article.pdf},
year = 2010,
author = {Jacob Siehler},
comment = {},
urldate = {2019-10-12},
collections = {Easily explained,Games to play with friends,Puzzles}
}
@article{TheGraphMenagerieAbstractAlgebraAndTheMadVeterinarian,
title = {The Graph Menagerie: Abstract Algebra and the Mad Veterinarian},
author = {Gene Abrams and Jessica K. Sklar},
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,
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 = {},
collections = {Animals,Attention-grabbing titles,Easily explained,Puzzles}
}
@article{LightsOutandVariants,
title = {“Lights Out” and Variants},
author = {Martin Kreh},
url = {https://www.tandfonline.com/doi/abs/10.4169/amer.math.monthly.124.10.937},
urldate = {2019-12-09},
year = 2017,
abstract = {In this article, we investigate the puzzle “Lights Out” 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 = {},
journal = {The American Mathematical Monthly: Vol 124},
volume = 124,
number = 10,
collections = {Puzzles}
}
@article{Thenothreeinlineproblemonatorus,
title = {The no-three-in-line problem on a torus},
author = {Jim Fowler and Andrew Groot and Deven Pandya and Bart Snapp},
url = {http://arxiv.org/abs/1203.6604v1 http://arxiv.org/pdf/1203.6604v1},
urldate = {2019-12-10},
year = 2012,
abstract = {Let {\$}T(\mathbb{\{}Z{\}}{\_}m \times \mathbb{\{}Z{\}}{\_}n){\$} denote the maximal number of points that can be
placed on an {\$}m \times n{\$} discrete torus with "no three in a line," meaning no
three in a coset of a cyclic subgroup of {\$}\mathbb{\{}Z{\}}{\_}m \times \mathbb{\{}Z{\}}{\_}n{\$}. By proving upper
bounds and providing explicit constructions, for distinct primes {\$}p{\$} and {\$}q{\$},
we show that {\$}T(\mathbb{\{}Z{\}}{\_}p \times \mathbb{\{}Z{\}}{\_}{\{}p^2{\}}) = 2p{\$} and {\$}T(\mathbb{\{}Z{\}}{\_}p \times \mathbb{\{}Z{\}}{\_}{\{}pq{\}}) = p+1{\$}.
Via Grobner bases, we compute {\$}T(\mathbb{\{}Z{\}}{\_}m \times \mathbb{\{}Z{\}}{\_}n){\$} for {\$}2 \leq m \leq 7{\$} and
{\$}2 \leq n \leq 19{\$}.},
comment = {},
archivePrefix = {arXiv},
eprint = {1203.6604},
primaryClass = {math.CO},
collections = {Fun maths facts,Geometry,Puzzles}
}
@article{Auniquepairoftriangles,
title = {A unique pair of triangles},
abstract = {A rational triangle is a triangle with sides of rational lengths. In this
short note, we prove that there exists a unique pair of a rational right
triangle and a rational isosceles triangle which have the same perimeter and
the same area. In the proof, we determine the set of rational points on a
certain hyperelliptic curve by a standard but sophisticated argument which is
based on the 2-descent on its Jacobian variety and Coleman's theory of {\$}p{\$}-adic
abelian integrals.},
url = {http://arxiv.org/abs/1809.09936v1 http://arxiv.org/pdf/1809.09936v1},
year = 2018,
author = {Yoshinosuke Hirakawa and Hideki Matsumura},
comment = {},
urldate = {2019-12-10},
archivePrefix = {arXiv},
eprint = {1809.09936},
primaryClass = {math.NT},
collections = {Easily explained,Fun maths facts,Geometry}
}
@article{Smalldatacomputingcorrectcalculatorarithmetic,
title = {Small-data computing: correct calculator arithmetic},
abstract = {Rounding errors are usually avoidable, and sometimes we can afford to avoid them.},
url = {https://dl.acm.org/doi/10.1145/2911981},
year = 2017,
author = {Hans-J. Boehm},
comment = {Description of how the built-in Android calculator uses constructive real numbers to avoid rounding errors.},
urldate = {2020-01-13},
collections = {Basically computer science,Unusual arithmetic}
}
@article{Theoryandapplicationsofthedoublebasenumbersystem,
title = {Theory and applications of the double-base number system},
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.},
url = {https://ieeexplore.ieee.org/document/805158},
year = 1999,
author = { V.S. Dimitrov and G.A. Jullien and W.C. Miller },
comment = {},
urldate = {2020-02-03},
collections = {Basically computer science,Easily explained,Integerology,Unusual arithmetic}
}
@article{Sandwichsemigroupsindiagramcategories,
title = {Sandwich semigroups in diagram categories},
abstract = {This paper concerns a number of diagram categories, namely the partition,
planar partition, Brauer, partial Brauer, Motzkin and Temperley-Lieb
categories. If {\$}\mathcal K{\$} denotes any of these categories, and if
{\$}\sigma\in\mathcal K{\_}{\{}nm{\}}{\$} is a fixed morphism, then an associative operation
{\$}\star{\_}\sigma{\$} may be defined on {\$}\mathcal K{\_}{\{}mn{\}}{\$} by
{\$}\alpha\star{\_}\sigma\beta=\alpha\sigma\beta{\$}. The resulting semigroup {\$}\mathcal
K{\_}{\{}mn{\}}^\sigma=(\mathcal K{\_}{\{}mn{\}},\star{\_}\sigma){\$} is called a sandwich semigroup.
We conduct a thorough investigation of these sandwich semigroups, with an
emphasis on structural and combinatorial properties such as Green's relations
and preorders, regularity, stability, mid-identities, ideal structure,
(products of) idempotents, and minimal generation. It turns out that the Brauer
category has many remarkable properties not shared by any of the other diagram
categories we study. Because of these unique properties, we may completely
classify isomorphism classes of sandwich semigroups in the Brauer category,
calculate the rank (smallest size of a generating set) of an arbitrary sandwich
semigroup, enumerate Green's classes and idempotents, and calculate ranks (and
idempotent ranks, where appropriate) of the regular subsemigroup and its
ideals, as well as the idempotent-generated subsemigroup. Several illustrative
examples are considered throughout, partly to demonstrate the sometimes-subtle
differences between the various diagram categories.},
url = {http://arxiv.org/abs/1910.10286v1 http://arxiv.org/pdf/1910.10286v1},
year = 2019,
author = {Ivana Đurđev and Igor Dolinka and James East},
comment = {},
urldate = {2020-02-03},
archivePrefix = {arXiv},
eprint = {1910.10286},
primaryClass = {math.GR},
collections = {Food,The groups group}
}
@article{Henwood1980,
title = {Eponymy in Mathematical Nomenclature: What's in a Name, and What Should Be?},
author = {Henwood, Mervyn R. and Rival, Ivan},
url = {http://ivanrival.com/docs/eponymy.pdf http://link.springer.com/10.1007/BF03028604},
urldate = {2014-02-11},
year = 1980,
abstract = {},
comment = {},
doi = {10.1007/BF03028604},
issn = {0343-6993},
journal = {The Mathematical Intelligencer},
month = {dec},
number = 4,
pages = {204--205},
volume = 2,
collections = {The act of doing maths}
}
@article{OnSometwowayClassificationsofIntegers,
title = {On Some two way Classifications of Integers},
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.},
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,
author = {J. Lambek and L. Moser},
comment = {},
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},
collections = {Fun maths facts,Integerology}
}
@article{TheRoleofNumberNotationSignValueNotationNumberProcessingisEasierthanPlaceValue,
title = {The Role of Number Notation: Sign-Value Notation Number Processing is Easier than Place-Value},
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.},
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,
author = {Krajcsi, Attila and Szab{\'{o}}, Eszter},
comment = {},
urldate = {2020-03-19},
volume = 3,
journal = {Frontiers in Psychology},
publisher = {Frontiers},
issn = {1664-1078},
doi = {10.3389/fpsyg.2012.00463},
pages = 463,
collections = {Education,Notation and conventions}
}
@book{item46,
title = {Music: a Mathematical Offering},
author = {Dave Benson},
url = {https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-music.html},
urldate = {2014-12-02},
year = 2006,
abstract = {},
comment = {},
collections = {Music}
}
@article{ACatalogueOfMathematicalFormulasInvolvingPiWithAnalysis,
title = {A catalogue of mathematical formulas involving π, with analysis},
abstract = {This paper presents a catalogue of mathematical formulas and iterative algorithms for evaluating the mathematical constant π, ranging from Archimedes’ 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.},
url = {https://www.davidhbailey.com/dhbpapers/pi-formulas.pdf},
year = 2020,
author = {David H. Bailey},
comment = {},
urldate = {2020-03-27},
collections = {Basically computer science,History,Lists and catalogues}
}
@article{FusiblenumbersandPeanoArithmetic,
title = {Fusible numbers and Peano Arithmetic},
author = {Jeff Erickson and Gabriel Nivasch and Junyan Xu},
url = {https://arxiv.org/abs/2003.14342v1 https://arxiv.org/pdf/2003.14342v1},
urldate = {2020-04-01},
year = 2020,
abstract = {Inspired by a mathematical riddle involving fuses, we define the "fusible
numbers" as follows: {\$}0{\$} is fusible, and whenever {\$}x,y{\$} are fusible with
{\$}|y-x|<1{\$}, the number {\$}(x+y+1)/2{\$} is also fusible. We prove that the set of
fusible numbers, ordered by the usual order on {\$}\mathbb R{\$}, is well-ordered,
with order type {\$}\varepsilon{\_}0{\$}. Furthermore, we prove that the density of the
fusible numbers along the real line grows at an incredibly fast rate: Letting
{\$}g(n){\$} be the largest gap between consecutive fusible numbers in the interval
{\$}[n,\infty){\$}, we have {\$}g(n)^{\{}-1{\}} \ge F{\_}{\{}\varepsilon{\_}0{\}}(n-c){\$} for some constant
{\$}c{\$}, where {\$}F{\_}\alpha{\$} denotes the fast-growing hierarchy. Finally, we derive
some true statements that can be formulated but not proven in Peano Arithmetic,
of a different flavor than previously known such statements. For example, PA
cannot prove the true statement "For every natural number {\$}n{\$} there exists a
smallest fusible number larger than {\$}n{\$}."},
comment = {},
archivePrefix = {arXiv},
eprint = {2003.14342},
primaryClass = {cs.LO},
collections = {Unusual arithmetic}
}
@article{Ufnarovski2003,
title = {How to differentiate a number},
author = {Ufnarovski, Victor and {\AA}hlander, B},
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,
abstract = {},
comment = {},
collections = {Easily explained,Integerology,Unusual arithmetic}
}
@article{HowLongIsMyToiletRoll,
title = {How long is my toilet roll? – a simple exercise in mathematical modelling},
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.},
url = {https://www.tandfonline.com/doi/abs/10.1080/0020739X.2013.790502?journalCode=tmes20},
year = 2013,
author = {Peter R. Johnston},
comment = {},
urldate = {2020-04-09},
collections = {Attention-grabbing titles,Easily explained,The act of doing maths,Modelling}
}
@article{KnotsAndLinksInSpatialGraphs,
title = {Knots and links in spatial graphs},
author = {J. H. Conway and C. McA. Gordon},
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,
abstract = {The main purpose of this paper is to show that any embedding of \(K{\_}7\) in three‐dimensional 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 = {},
collections = {Easily explained,Fun maths facts,Geometry}
}
@article{ExtremeProofsITheIrrationalityof2,
title = {Extreme Proofs I: The Irrationality of √2},
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’s book. Aigner and Ziegler (1998) have attempted to reconstruct some of this Book. Here we take a different, and more tolerant approach.},
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,
author = {John H. Conway and Joseph Shipman and John H. Conway and Joseph Shipman},
comment = {},
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},
collections = {About proof}
}
@article{ComputingLinkages,
title = {Computing Linkages},
author = {Andries de Man},
url = {https://sites.google.com/site/calculatinghistory/home/computing-linkages},
urldate = {2020-04-06},
year = 2010,
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 “computing linkages” and the work of Antonin Svoboda on their systematic development.},
comment = {},
collections = {Easily explained,History,Things to make and do,Unusual computers}
}
@article{ConwaysTilingGroupsonJSTOR,
title = {Conway's Tiling Groups on JSTOR},
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.},
url = {https://www.jstor.org/stable/2324578},
year = 1990,
author = {William P. Thurston},
comment = {},
urldate = {2020-04-15},
collections = {Fun maths facts,Geometry,The groups group}
}
@article{ProjectivizingSet,
title = {Projectivizing Set},
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.},
url = {https://www.tandfonline.com/doi/abs/10.1080/10724117.2020.1714377?journalCode=umho20},
year = 2020,
author = {Cathy Hsu and Jonah Ostroff and Lucas Van Meter},
comment = {},
urldate = {2020-04-19},
collections = {Easily explained,Games to play with friends}
}
@article{FourPagesAreIndeedNecessaryforPlanarGraphs,
title = {Four Pages Are Indeed Necessary for Planar Graphs},
abstract = {An embedding of a graph in a book consists of a linear order of its vertices
along the spine of the book and of an assignment of its edges to the pages of
the book, so that no two edges on the same page cross. The book thickness of a
graph is the minimum number of pages over all its book embeddings. Accordingly,
the book thickness of a class of graphs is the maximum book thickness over all
its members. In this paper, we address a long-standing open problem regarding
the exact book thickness of the class of planar graphs, which previously was
known to be either three or four. We settle this problem by demonstrating
planar graphs that require four pages in any of their book embeddings, thus
establishing that the book thickness of the class of planar graphs is four.},
url = {http://arxiv.org/abs/2004.07630v1 http://arxiv.org/pdf/2004.07630v1},
year = 2020,
author = {Michael A. Bekos and Michael Kaufmann and Fabian Klute and Sergey Pupyrev and Chrysanthi Raftopoulou and Torsten Ueckerdt},
comment = {},
urldate = {2020-04-19},
archivePrefix = {arXiv},
eprint = {2004.07630},
primaryClass = {cs.DS},
collections = {Attention-grabbing titles,Fun maths facts}
}
@article{Gradwohl2007,
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.},
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,
abstract = {},
comment = {},
booktitle = {Fun with Algorithms},
number = 860,
pages = {166--182},
publisher = {Springer},
collections = {About proof,Easily explained,Protocols and strategies}
}
@article{Shadowmoviesnotarisingfromknots,
title = {Shadow movies not arising from knots},
abstract = {A shadow diagram is a knot diagram with under-over information omitted; a
shadow movie is a sequence of shadow diagrams related by shadow Reidemeister
moves. We show that not every shadow movie arises as the shadow of a
Reidemeister movie, meaning a sequence of classical knot diagrams related by
classical Reidemeister moves. This means that in Kaufman's theory of virtual
knots, virtual crossings cannot simply be viewed as classical crossings where
which strand is over has been left `to be determined'.},
url = {http://arxiv.org/abs/1106.3545v1 http://arxiv.org/pdf/1106.3545v1},
year = 2011,
author = {Daniel Denton and Peter Doyle},
comment = {},
urldate = {2020-04-21},
archivePrefix = {arXiv},
eprint = {1106.3545},
primaryClass = {math.GT},
collections = {Attention-grabbing titles}
}
@article{ThedistanceofapermutationfromasubgroupofSn,
title = {The distance of a permutation from a subgroup of \(S{\_}n\)},
author = {Richard G. E. Pinch},
url = {http://arxiv.org/abs/math/0511501v1 http://arxiv.org/pdf/math/0511501v1},
urldate = {2020-04-30},
year = 2005,
abstract = {We show that the problem of computing the distance of a given permutation
from a subgroup {\$}H{\$} of {\$}S{\_}n{\$} is in general NP-complete, even under the
restriction that {\$}H{\$} is elementary Abelian of exponent 2. The problem is shown
to be polynomial-time equivalent to a problem related to finding a maximal
partition of the edges of an Eulerian directed graph into cycles and this
problem is in turn equivalent to the standard NP-complete problem of Boolean
satisfiability.},
comment = {},
archivePrefix = {arXiv},
eprint = {math/0511501},
primaryClass = {math.CO},
collections = {basically-computer-science,computational-complexity-of-games,the-groups-group}
}
@article{Demaine2012,
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},
url = {http://arxiv.org/abs/1203.3602 http://arxiv.org/pdf/1203.3602v2},
urldate = {2012-11-03},
year = 2012,
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 = {},
archivePrefix = {arXiv},
arxivId = {1203.3602},
eprint = {1203.3602},
isbn = 9783642303470,
month = {mar},
pages = 17,
primaryClass = {cs.DS},
collections = {easily-explained,puzzles,the-groups-group}
}
@article{WordcalculusinthefundamentalgroupoftheMengercurve,
title = {Word calculus in the fundamental group of the Menger curve},
abstract = {The fundamental group of the Menger universal curve is uncountable and not
free, although all of its finitely generated subgroups are free. It contains an
isomorphic copy of the fundamental group of every one-dimensional separable
metric space and an isomorphic copy of the fundamental group of every planar
Peano continuum. We give an explicit and systematic combinatorial description
of the fundamental group of the Menger universal curve and its generalized
Cayley graph in terms of word sequences. The word calculus, which requires only
two letters and their inverses, is based on Pasynkov's partial topological
product representation and can be expressed in terms of a variation on the
classical puzzle known as the Towers of Hanoi.},
url = {http://arxiv.org/abs/1310.7968v1 http://arxiv.org/pdf/1310.7968v1},
year = 2013,
author = {Hanspeter Fischer and Andreas Zastrow},
comment = {},
urldate = {2020-05-02},
archivePrefix = {arXiv},
eprint = {1310.7968},
primaryClass = {math.GT},
collections = {combinatorics,fun-maths-facts,games-to-play-with-friends,the-groups-group}
}
@article{38406501359372282063949allthatMonodromyofFanoProblems,
title = {38406501359372282063949 {\&} all that: Monodromy of Fano Problems},
abstract = {A Fano problem is an enumerative problem of counting {\$}r{\$}-dimensional linear
subspaces on a complete intersection in {\$}\mathbb{\{}P{\}}^n{\$} over a field of
arbitrary characteristic, whenever the corresponding Fano scheme is finite. A
classical example is enumerating lines on a cubic surface. We study the
monodromy of finite Fano schemes {\$}F{\_}{\{}r{\}}(X){\$} as the complete intersection {\$}X{\$}
varies. We prove that the monodromy group is either symmetric or alternating in
most cases. In the exceptional cases, the monodromy group is one of the Weyl
groups {\$}W(E{\_}6){\$} or {\$}W(D{\_}k){\$}.},
url = {http://arxiv.org/abs/2002.04580v1 http://arxiv.org/pdf/2002.04580v1},
year = 2020,
author = {Sachi Hashimoto and Borys Kadets},
comment = {},
urldate = {2020-05-06},
archivePrefix = {arXiv},
eprint = {2002.04580},
primaryClass = {math.AG},
collections = {attention-grabbing-titles,combinatorics,integerology}
}
@article{Crepeau1987,
title = {A zero-knowledge Poker protocol that achieves confidentiality of the players' strategy or How to achieve an electronic Poker face},
author = {Cr{\'{e}}peau, C.},
url = {https://link.springer.com/chapter/10.1007/3-540-47721-7{\_}18},
urldate = {2012-01-10},
year = 1987,
abstract = {},
comment = {},
pages = {239--247},
publisher = {Springer},
collections = {attention-grabbing-titles,games-to-play-with-friends,protocols-and-strategies}
}
@article{InfinitudeofPrimesUsingFormalLanguageTheory,
title = {Infinitude of Primes Using Formal Language Theory},
abstract = {Formal languages are sets of strings of symbols described by a set of rules
specific to them. In this note, we discuss a certain class of formal languages,
called regular languages, and put forward some elementary results. The
properties of these languages are then employed to prove that there are
infinitely many prime numbers.},
url = {http://arxiv.org/abs/2005.10372v1 http://arxiv.org/pdf/2005.10372v1},
year = 2020,
author = {Aalok Thakkar},
comment = {},
urldate = {2020-05-25},
archivePrefix = {arXiv},
eprint = {2005.10372},
primaryClass = {cs.FL},
collections = {about-proof,basically-computer-science,easily-explained,fun-maths-facts,integerology}
}
@article{GergonnesCardTrickPositionalNotationandRadixSort,
title = {Gergonne's Card Trick, Positional Notation, and Radix Sort},
author = {Ethan D. Bolker},
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,
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},
collections = {easily-explained,fun-maths-facts}
}
@article{IndigenousperspectivesinmathsUnderstandingGurruu,
title = {Indigenous perspectives in maths: Understanding Gurruṯu},
abstract = {Discusses Yolŋu mathematics and the interconnected relationships of Gurruṯu, and shares an activity for teachers and students to explore the connections and patterns in family trees.},
url = {https://www.teachermagazine.com.au/articles/indigenous-perspectives-in-maths-understanding-gurruu},
year = 2020,
author = {Chris Matthews},
comment = {},
urldate = {2020-06-08},
collections = {easily-explained,fun-maths-facts}
}
@article{GoldbugVariations,
title = {Goldbug Variations},
abstract = {This "Mathematical Entertainments" column from the Intelligencer is an
exposition of current investigations, rooted in recent work of Jim Propp, into
"quasirandom" analogues of random walk and random aggregation processes.
Featured are the "Goldbugs" and the "Rotor-router". These are deterministic
processes which simulate the random ones, for example having the same limiting
states, but with faster convergence.
The paper includes three large illustrations, which appear twice in the
submission, as both raster image (.png) and postscript (.eps) files. The latter
are much larger but needed for latex inclusion; the former are smaller, used by
pdflatex, and better for pixel-level viewing.},
url = {http://arxiv.org/abs/math/0501497v1 http://arxiv.org/pdf/math/0501497v1},
year = 2005,
author = {Michael Kleber},
comment = {},
urldate = {2020-06-12},
archivePrefix = {arXiv},
eprint = {math/0501497},
primaryClass = {math.CO},
collections = {animals,attention-grabbing-titles,easily-explained,fun-maths-facts,puzzles}
}
@article{SomeDoublyExponentialSequences,
title = {Some Doubly Exponential Sequences},
author = {A. V. Aho and N. J. A. Sloane},
url = {http://neilsloane.com/doc/doubly.html},
urldate = {2020-07-13},
year = 1970,
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 = {},
journal = {Fibonacci Quarterly},
collections = {fun-maths-facts,integerology}
}