Interesting Esoterica
https://read.somethingorotherwhatever.com/
en-GBFri, 19 Apr 2024 22:23:50 +0000Squigonometry: The Study of Imperfect Circles
https://read.somethingorotherwhatever.com/entry/SquigonometryTheStudyofImperfectCircles
This textbook introduces generalized trigonometric functions through the exploration of imperfect circles: curves defined by \(|x|^p + |y|^p = 1\) where \(p \geq 1\). Grounded in visualization and computations, this accessible, modern perspective encompasses new and old results, casting a fresh light on duality, special functions, geometric curves, and differential equations. Projects and opportunities for research abound, as we explore how similar (or different) the trigonometric and squigonometric worlds might be.
Comprised of many short chapters, the book begins with core definitions and techniques. Successive chapters cover inverse squigonometric functions, the many possible re-interpretations of π, two deeper dives into parameterizing the squigonometric functions, and integration. Applications include a celebration of Piet Hein’s work in design. From here, more technical pathways offer further exploration. Topics include infinite series; hyperbolic, exponential, and logarithmic functions; metrics and norms; and lemniscatic and elliptic functions. Illuminating illustrations accompany the text throughout, along with historical anecdotes, engaging exercises, and wry humor.
Squigonometry: The Study of Imperfect Circles invites readers to extend familiar notions from trigonometry into a new setting. Ideal for an undergraduate reading course in mathematics or a senior capstone, this book offers scaffolding for active discovery. Knowledge of the trigonometric functions, single-variable calculus, and initial-value problems is assumed, while familiarity with multivariable calculus and linear algebra will allow additional insights into certain later material.SquigonometryTheStudyofImperfectCirclesWed, 10 Apr 2024 00:00:00 +0000 Robert D. Poodiack and William E. WoodThe Cardboard Computer
https://read.somethingorotherwhatever.com/entry/TheCardboardComputer
The Cardboard Computer is a free (Public Domain) do-it-yourself circular slide rule that you can print out and put together at home or in a classroom. It's available in two versions: a basic version that can do basic ratios, multiplication, and division, and an advanced version that adds more features, including square and cube roots. TheCardboardComputerWed, 10 Apr 2024 00:00:00 +0000David MegginsonPolyamorous Scheduling
https://read.somethingorotherwhatever.com/entry/PolyamorousScheduling
Finding schedules for pairwise meetings between the members of a complex
social group without creating interpersonal conflict is challenging, especially
when different relationships have different needs. We formally define and study
the underlying optimisation problem: Polyamorous Scheduling.
In Polyamorous Scheduling, we are given an edge-weighted graph and try to
find a periodic schedule of matchings in this graph such that the maximal
weighted waiting time between consecutive occurrences of the same edge is
minimised. We show that the problem is NP-hard and that there is no efficient
approximation algorithm with a better ratio than 4/3 unless P = NP. On the
positive side, we obtain an $O(\log n)$-approximation algorithm; indeed, a
$O(\log \Delta)$-approximation for $\Delta$ the maximum degree, i.e., the
largest number of relationships of any individual. We also define a
generalisation of density from the Pinwheel Scheduling Problem, "poly density",
and ask whether there exists a poly-density threshold similar to the
5/6-density threshold for Pinwheel Scheduling [Kawamura, STOC 2024].
Polyamorous Scheduling is a natural generalisation of Pinwheel Scheduling with
respect to its optimisation variant, Bamboo Garden Trimming.
Our work contributes the first nontrivial hardness-of-approximation reduction
for any periodic scheduling problem, and opens up numerous avenues for further
study of Polyamorous Scheduling.PolyamorousSchedulingWed, 10 Apr 2024 00:00:00 +0000Leszek Gąsieniec and Benjamin Smith and Sebastian WildA simple group of order 44,352,000
https://read.somethingorotherwhatever.com/entry/Asimplegroupoforder44352000
The group \(G\) of the title is obtained as a primitive permutation group of degree 100 in which the stabilizer of a point has orbits of lengths 1, 22 and 77 and is isomorphic to the Mathieu group \(M_{22}\). Thus \(G\) has rank 3 in the sense
of [1]. \(G\) is an automorphism group of a graph constructed from the Steiner system \(\mathfrak{S}(3, 6, 22)\).Asimplegroupoforder44352000Tue, 02 Apr 2024 00:00:00 +0000Higman, Donald G. and Sims, Charles C.The Flapping Birds in the Pentagram Zoo
https://read.somethingorotherwhatever.com/entry/TheFlappingBirdsinthePentagramZoo
We study the $(k+1,k)$ diagonal map for $k=2,3,4,...$. We call this map
$\Delta_k$. The map $\Delta_1$ is the pentagram map and $\Delta_k$ is a
generalization. $\Delta_k$ does not preserve convexity, but we prove that
$\Delta_k$ preserves a subset $B_k$ of certain star-shaped polygons which we
call $k$-birds. The action of $\Delta_k$ on $B_k$ seems similar to the action
of $\Delta_1$ on the space of convex polygons. We show that some classic
geometric results about $\Delta_1$ generalize to this setting.TheFlappingBirdsinthePentagramZooThu, 14 Mar 2024 00:00:00 +0000Richard Evan SchwartzThe Flower Calculus
https://read.somethingorotherwhatever.com/entry/TheFlowerCalculus
We introduce the flower calculus, a deep inference proof system for intuitionistic first-order logic inspired by Peirce's existential graphs. It works as a rewriting system over inductive objects called "flowers", that enjoy both a graphical interpretation as topological diagrams, and a textual presentation as nested sequents akin to coherent formulas. Importantly, the calculus dispenses completely with the traditional notion of symbolic connective, operating solely on nested flowers containing atomic predicates. We prove both the soundness of the full calculus and the completeness of an analytic fragment with respect to Kripke semantics. This provides to our knowledge the first analyticity result for a proof system based on existential graphs, adapting semantic cut-elimination techniques to a deep inference setting. Furthermore, the kernel of rules targetted by completeness is fully invertible, a desirable property for both automated and interactive proof search.TheFlowerCalculusMon, 11 Mar 2024 00:00:00 +0000Pablo DonatoRandom Formula Generators
https://read.somethingorotherwhatever.com/entry/RandomFormulaGenerators
In this article, we provide three generators of propositional formulae for
arbitrary languages, which uniformly sample three different formulae spaces.
They take the same three parameters as input, namely, a desired depth, a set of
atomics and a set of logical constants (with specified arities). The first
generator returns formulae of exactly the given depth, using all or some of the
propositional letters. The second does the same but samples up-to the given
depth. The third generator outputs formulae with exactly the desired depth and
all the atomics in the set. To make the generators uniform (i.e. to make them
return every formula in their space with the same probability), we will prove
various cardinality results about those spaces.RandomFormulaGeneratorsMon, 11 Mar 2024 00:00:00 +0000Ariel J. Roffe and Joaquin S. Toranzo CalderonMy Favorite Math Jokes
https://read.somethingorotherwhatever.com/entry/MyFavoriteMathJokes
For many years, I have been collecting math jokes and posting them on my
website. I have more than 400 jokes there. In this paper, which is an extended
version of my talk at the G4G15, I would like to present 66 of them.MyFavoriteMathJokesWed, 06 Mar 2024 00:00:00 +0000Tanya KhovanovaThe Sine of a Single Degree
https://read.somethingorotherwhatever.com/entry/TheSineOfASingleDegree
Ostensibly a derivation of an algebraically exact formula for the value of the sine of 1 degree, we present this calculation as a “historical romp” looking at the problem through the tools of geometry, then algebra, and finally complex analysis. Each one of these approaches gets the reader nearer to the correct value, but also serves to frame a vignette of surprising or beautiful mathematics.TheSineOfASingleDegreeMon, 15 Jan 2024 00:00:00 +0000Travis KowalskiHow big a table do you need for your jigsaw puzzle?
https://read.somethingorotherwhatever.com/entry/Howbigatabledoyouneedforyourjigsawpuzzle
Jigsaw puzzles are typically labeled with their finished area and number of
pieces. With this information, is it possible to estimate the area required to
lay each piece flat before assembly? We derive a simple formula based on
two-dimensional circular packing and show that the unassembled puzzle area is
$\sqrt{3}$ times the assembled puzzle area, independent of the number of
pieces. We perform measurements on 9 puzzles ranging from 333 cm$^2$ (9 pieces)
to 6798 cm$^2$ (2000 pieces) and show that the formula accurately predicts
realistic assembly scenarios.HowbigatabledoyouneedforyourjigsawpuzzleMon, 15 Jan 2024 00:00:00 +0000Madeleine Bonsma-Fisher and Kent Bonsma-FisherMathBases
https://read.somethingorotherwhatever.com/entry/MathBases
MathBases.org brings together a searchable index of existing mathematical databases and resources for mathematicians interested in creating new mathematical databases.MathBasesSat, 09 Dec 2023 00:00:00 +0000Katja BerčičSorry, the nilpotents are in the center
https://read.somethingorotherwhatever.com/entry/Sorrythenilpotentsareinthecenter
The behavior of nilpotents can reveal valuable information about the algebra.
We give a simple proof of a classic result that a finite ring is commutative if
all its nilpotents lie in the center.SorrythenilpotentsareinthecenterFri, 17 Nov 2023 00:00:00 +0000Vineeth ChintalaThe Euclidean Algorithm Generates Traditional Musical Rhythms
https://read.somethingorotherwhatever.com/entry/TheEuclideanAlgorithmGeneratesTraditionalMusicalRhythms
The Euclidean algorithm (which comes down to us from Euclid’s Elements) computes the greatest common divisor of two given integers. It is shown here that the structure of the Euclidean algorithm may be used to automatically generate, very efficiently, a large family of rhythms used as timelines (rhythmic ostinatos), in traditional world music. These rhythms, here dubbed Euclidean rhythms, have the property that their onset patterns are distributed as evenly as possible in a mathematically precise sense, and optimal manner. Euclidean rhythms are closely related to the family of Aksak rhythms studied by ethnomusicologists, and occur in a wide variety of other disciplines as well. For example they characterize algorithms for drawing digital straight lines in computer graphics, as well as algorithms for calculating
leap years in calendar design. Euclidean rhythms also find application in nuclear physics accelerators and in computer science, and are closely related to several families of words and sequences of interest
in the study of the combinatorics of words, such as mechanical words, Sturmian words, two-distance
sequences, and Euclidean strings, to which the Euclidean rhythms are compared.TheEuclideanAlgorithmGeneratesTraditionalMusicalRhythmsWed, 25 Oct 2023 00:00:00 +0000Godfried ToussaintWhen Can You Tile an Integer Rectangle with Integer Squares?
https://read.somethingorotherwhatever.com/entry/WhenCanYouTileanIntegerRectanglewithIntegerSquares
This paper characterizes when an $m \times n$ rectangle, where $m$ and $n$
are integers, can be tiled (exactly packed) by squares where each has an
integer side length of at least 2. In particular, we prove that tiling is
always possible when both $m$ and $n$ are sufficiently large (at least 10).
When one dimension $m$ is small, the behavior is eventually periodic in $n$
with period 1, 2, or 3. When both dimensions $m,n$ are small, the behavior is
determined computationally by an exhaustive search.WhenCanYouTileanIntegerRectanglewithIntegerSquaresMon, 09 Oct 2023 00:00:00 +0000MIT CompGeom Group and Zachary Abel and Hugo A. Akitaya and Erik D. Demaine and Adam C. Hesterberg and Jayson LynchDubious Identities: A Visit to the Borwein Zoo
https://read.somethingorotherwhatever.com/entry/DubiousIdentitiesAVisittotheBorweinZoo
We contribute to the zoo of dubious identities established by J.M. and P.B.
Borwein in their 1992 paper, "Strange Series and High Precision Fraud" with
five new entries, each of a different variety than the last. Some of these
identities are again a high precision fraud and picking out the true from the
bogus can be a challenging task with many unexpected twists along the way.DubiousIdentitiesAVisittotheBorweinZooMon, 09 Oct 2023 00:00:00 +0000Zachary P. Bradshaw and Christophe VignatConway and Doyle Can Divide by Three, But I Can't
https://read.somethingorotherwhatever.com/entry/ConwayandDoyleCanDividebyThreeButICant
Conway and Doyle have claimed to be able to divide by three. We attempt to
replicate their achievement and fail. In the process, we get tangled up in some
shoes and socks and forget how to multiply.ConwayandDoyleCanDividebyThreeButICantFri, 29 Sep 2023 00:00:00 +0000Patrick LutzRopes, fractions, and moduli spaces
https://read.somethingorotherwhatever.com/entry/Ropesfractionsandmodulispaces
This is an exposition of John H. Conway's tangle trick. We discuss what the
trick is, how to perform it, why it works mathematically, and finally offer a
conceptual explanation for why a trick like this should exist in the first
place. The mathematical centerpiece is the relationship between braids on three
strands and elliptic curves, and we a draw a line from the tangle trick back to
work of Weierstrass, Abel, and Jacobi in the 19th century. For the most part we
assume only a familiarity with the language of group actions, but some prior
exposure to the fundamental group is beneficial in places.RopesfractionsandmodulispacesFri, 29 Sep 2023 00:00:00 +0000Nick SalterFlat origami is Turing Complete
https://read.somethingorotherwhatever.com/entry/FlatorigamiisTuringComplete
Flat origami refers to the folding of flat, zero-curvature paper such that
the finished object lies in a plane. Mathematically, flat origami consists of a
continuous, piecewise isometric map $f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$
along with a layer ordering $\lambda_f:P\times P\to \{-1,1\}$ that tracks which
points of $P$ are above/below others when folded. The set of crease lines that
a flat origami makes (i.e., the set on which the mapping $f$ is
non-differentiable) is called its \textit{crease pattern}. Flat origami
mappings and their layer orderings can possess surprisingly intricate
structure. For instance, determining whether or not a given straight-line
planar graph drawn on $P$ is the crease pattern for some flat origami has been
shown to be an NP-complete problem, and this result from 1996 led to numerous
explorations in computational aspects of flat origami. In this paper we prove
that flat origami, when viewed as a computational device, is Turing complete.
We do this by showing that flat origami crease patterns with \textit{optional
creases} (creases that might be folded or remain unfolded depending on
constraints imposed by other creases or inputs) can be constructed to simulate
Rule 110, a one-dimensional cellular automaton that was proven to be Turing
complete by Matthew Cook in 2004.FlatorigamiisTuringCompleteFri, 29 Sep 2023 00:00:00 +0000Thomas C. Hull and Inna ZakharevichMathematical proofs with cardboard and paper
https://read.somethingorotherwhatever.com/entry/asgunziMathproofscardboardpaperMathematicalartwithpaperandcardboard
In this tutorial, we'll show visual mathematical proofs using cardboard and paper. This ludic technique is especially good for children. These math proofs are like puzzles, and, best of all, you can do it yourself, at home.asgunziMathproofscardboardpaperMathematicalartwithpaperandcardboardThu, 24 Aug 2023 00:00:00 +0000Arnaldo Gunzi and Ernée Kozyroff FilhoDistant decimals of $π$
https://read.somethingorotherwhatever.com/entry/DistantdecimalsofPi
We describe how to compute very far decimals of \(\pi\) and how to provide
formal guarantees that the decimals we compute are correct. In particular, we
report on an experiment where 1 million decimals of \(\pi\) and the billionth
hexadecimal (without the preceding ones) have been computed in a formally
verified way. Three methods have been studied, the first one relying on a
spigot formula to obtain at a reasonable cost only one distant digit (more
precisely a hexadecimal digit, because the numeration basis is 16) and the
other two relying on arithmetic-geometric means. All proofs and computations
can be made inside the Coq system. We detail the new formalized material that
was necessary for this achievement and the techniques employed to guarantee the
accuracy of the computed digits, in spite of the necessity to work with fixed
precision numerical computation.DistantdecimalsofPiMon, 14 Aug 2023 00:00:00 +0000Yves Bertot and Laurence Rideau and Laurent ThéryHyperbolic Minesweeper is in P
https://read.somethingorotherwhatever.com/entry/HyperbolicMinesweeperisinP
We show that, while Minesweeper is NP-complete, its hyperbolic variant is in
P. Our proof does not rely on the rules of Minesweeper, but is valid for any
puzzle based on satisfying local constraints on a graph embedded in the
hyperbolic plane.HyperbolicMinesweeperisinPMon, 14 Aug 2023 00:00:00 +0000Eryk KopczyńskiA sign that used to annoy me, and still does
https://read.somethingorotherwhatever.com/entry/Asignthatusedtoannoymeandstilldoes
We provide a proof of the following fact: if a complex scheme $Y$ has Behrend
function constantly equal to a sign $\sigma \in \{\pm 1\}$, then all of its
components $Z \subset Y$ are generically reduced and satisfy
$(-1)^{\mathrm{dim}_{\mathbb C} T_pY} = \sigma = (-1)^{\mathrm{dim}Z}$ for $p
\in Z$ a general point. Given the recent counterexamples to the parity
conjecture for the Hilbert scheme of points $\mathrm{Hilb}^n(\mathbb A^3)$, our
argument suggests a possible path to disprove the constancy of the Behrend
function of $\mathrm{Hilb}^n(\mathbb A^3)$.AsignthatusedtoannoymeandstilldoesMon, 14 Aug 2023 00:00:00 +0000Andrea T. RicolfiLyons Taming
https://read.somethingorotherwhatever.com/entry/LyonsTaming
Based on Kantor's geometry, we give a new Highly symmetric construction of
Lyons' sporadic simple group $Ly$ via its minimal representation over $\mathbb
F_5^{111}$, thus obtaining elementary existence proofs for both the group and
the representation at one stroke.LyonsTamingMon, 14 Aug 2023 00:00:00 +0000Wolfram NeutschMaximum mutational robustness in genotype–phenotype maps follows a self-similar blancmange-like curve
https://read.somethingorotherwhatever.com/entry/Maximummutationalrobustnessingenotypephenotypemapsfollowsaselfsimilarblancmangelikecurve
Phenotype robustness, defined as the average mutational robustness of all the genotypes that map to a given phenotype, plays a key role in facilitating neutral exploration of novel phenotypic variation by an evolving population. By applying results from coding theory, we prove that the maximum phenotype robustness occurs when genotypes are organized as bricklayer’s graphs, so-called because they resemble the way in which a bricklayer would fill in a Hamming graph. The value of the maximal robustness is given by a fractal continuous everywhere but differentiable nowhere sums-of-digits function from number theory. Interestingly, genotype–phenotype maps for RNA secondary structure and the hydrophobic-polar (HP) model for protein folding can exhibit phenotype robustness that exactly attains this upper bound. By exploiting properties of the sums-of-digits function, we prove a lower bound on the deviation of the maximum robustness of phenotypes with multiple neutral components from the bricklayer’s graph bound, and show that RNA secondary structure phenotypes obey this bound. Finally, we show how robustness changes when phenotypes are coarse-grained and derive a formula and associated bounds for the transition probabilities between such phenotypes.MaximummutationalrobustnessingenotypephenotypemapsfollowsaselfsimilarblancmangelikecurveWed, 02 Aug 2023 00:00:00 +0000Vaibhav Mohanty , Sam F. Greenbury , Tasmin Sarkany , Shyam Narayanan , Kamaludin Dingle , Sebastian E. Ahnert, Ard A. LouisWallpaper Functions
https://read.somethingorotherwhatever.com/entry/WallpaperFunctions
Instead of making wallpaper by repeating copies of a motif, we construct wallpaper functions. These are functions on \(\mathbb{R}^2\) that are invariant under the action of one of the 17 planar crystallographic groups. We also construct functions with antisymmetries, and offer a complete analysis of types. Techniques include exhibiting bases for various spaces of wallpaper functions, and an algebraic definition of equivalence of pattern type.WallpaperFunctionsTue, 20 Jun 2023 00:00:00 +0000Frank A. Farris and Rima LanningSolving Rush Hour, the Puzzle
https://read.somethingorotherwhatever.com/entry/SolvingRushHour
Rush Hour is a 6x6 sliding block puzzle invented by Nob Yoshigahara in the 1970s. It was first sold in the United States in 1996.
I played a clone of this game on my first iPhone several years ago. Recently, I stumbled on the physical incarnation of it and instantly bought it on Amazon for my kids to play. We've been having fun with it, but naturally I was most interested in writing some code to solve the puzzles.
After writing a solver, I wrote a puzzle generator that would create more starting positions for us to try (the game comes with 40 levels printed on playing cards). The generator used simulated annealing to try to maximize the number of moves required to solve the puzzle.
Unsatisfied with the results, I then decided to try generating all possible puzzles. Ultimately I ended up with a complete database of every "interesting" starting position. It was quite challenging (and exciting!) and that's what I want to talk about in this article. My code is open source with a permissive license and the resulting database is available for download.SolvingRushHourWed, 22 Mar 2023 00:00:00 +0000Michael FoglemanFoldings and Meanders
https://read.somethingorotherwhatever.com/entry/FoldingsandMeanders
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.FoldingsandMeandersWed, 22 Mar 2023 00:00:00 +0000Stéphane LegendreMetaNumbers - Number encyclopedia
https://read.somethingorotherwhatever.com/entry/MetaNumbers
MetaNumbers is a free math tool providing information about any positive integer (up to 9223372036854775807), such as its factorized form, its divisors, its classification, or its arithmetic properties (widely used in the field of number theory). MetaNumbersWed, 22 Mar 2023 00:00:00 +0000MetaNumbersErich's Packing Center
https://read.somethingorotherwhatever.com/entry/ErichsPackingCenter
ErichsPackingCenterMon, 20 Mar 2023 00:00:00 +0000Erich FriedmanA curious result related to Kempner's series
https://read.somethingorotherwhatever.com/entry/AcuriousresultrelatedtoKempnersseries
It is well known since A. J. Kempner's work that the series of the
reciprocals of the positive integers whose the decimal representation does not
contain any digit 9, is convergent. This result was extended by F. Irwin and
others to deal with the series of the reciprocals of the positive integers
whose the decimal representation contains only a limited quantity of each digit
of a given nonempty set of digits. Actually, such series are known to be all
convergent. Here, letting $S^{(r)}$ $(r \in \mathbb{N})$ denote the series of
the reciprocal of the positive integers whose the decimal representation
contains the digit 9 exactly $r$ times, the impressive obtained result is that
$S^{(r)}$ tends to $10 \log{10}$ as $r$ tends to infinity!AcuriousresultrelatedtoKempnersseriesMon, 27 Feb 2023 00:00:00 +0000Bakir FarhiHedra Zoo
https://read.somethingorotherwhatever.com/entry/HedraZoo
Encyclopedia of Combinatorial Polytope SequencesHedraZooThu, 02 Feb 2023 00:00:00 +0000Stefan ForceyThe Packing Chromatic Number of the Infinite Square Grid is 15
https://read.somethingorotherwhatever.com/entry/ThePackingChromaticNumberoftheInfiniteSquareGridis15
A packing $k$-coloring is a natural variation on the standard notion of graph
$k$-coloring, where vertices are assigned numbers from $\{1, \ldots, k\}$, and
any two vertices assigned a common color $c \in \{1, \ldots, k\}$ need to be at
a distance greater than $c$ (as opposed to $1$, in standard graph colorings).
Despite a sequence of incremental work, determining the packing chromatic
number of the infinite square grid has remained an open problem since its
introduction in 2002. We culminate the search by proving this number to be 15.
We achieve this result by improving the best-known method for this problem by
roughly two orders of magnitude. The most important technique to boost
performance is a novel, surprisingly effective propositional encoding for
packing colorings. Additionally, we developed an alternative symmetry-breaking
method. Since both new techniques are more complex than existing techniques for
this problem, a verified approach is required to trust them. We include both
techniques in a proof of unsatisfiability, reducing the trusted core to the
correctness of the direct encoding.ThePackingChromaticNumberoftheInfiniteSquareGridis15Thu, 26 Jan 2023 00:00:00 +0000Bernardo Subercaseaux and Marijn J. H. HeuleSub shoot!
https://read.somethingorotherwhatever.com/entry/Subshoot
By day, Leonidas Kontothanassis works as my colleague at Google. By night, he runs a gaudy carnival booth on the boardwalk outside of
town. "Step right up and try your luck! Shoot the sub and win a prize," Leonidas was calling out one fine Fall evening. He was standing in the middle of the booth, surrounded by a moat packed with plastic toy submarines. The subs were of every shape and size, and they circled the moat at all different speeds, propelled by a complex system of currents.SubshootSun, 08 Jan 2023 00:00:00 +0000Kleber, MichaelLucy and Lily: A Game of Geometry and Number Theory: The American Mathematical Monthly: Vol 109, No 1
https://read.somethingorotherwhatever.com/entry/LucyandLilyAGameofGeometryandNumberTheoryTheAmericanMathematicalMonthlyVol109No1
The purpose of this article is to describe a computer game I created. I named the game “Lucy and Lily,” after my two daughters. Many people have expressed enthusiasm for the game, and their enthusiasm has encouraged me to write this article. At some point, Daniel Allcock and Brian Conrad worked out an informal but careful
analysis of “Lucy and Lily.” The several challenges I issue to the reader, during the
course of the article, derive from facts that one or both of them establishedLucyandLilyAGameofGeometryandNumberTheoryTheAmericanMathematicalMonthlyVol109No1Mon, 14 Nov 2022 00:00:00 +0000Richard Evan SchwartzPIGTIKAL (puzzles in geometry that I know and love)
https://read.somethingorotherwhatever.com/entry/PIGTIKALpuzzlesingeometrythatIknowandlove
Problems for the graduate students who want to improve problem-solving skills
in geometry. Every problem has a short elegant solution -- this gives a hint
which was not available when the problem was discovered.PIGTIKALpuzzlesingeometrythatIknowandloveFri, 14 Oct 2022 00:00:00 +0000Anton PetruninMaclaurin Integration: A Weapon Against Infamous Integrals
https://read.somethingorotherwhatever.com/entry/MaclaurinIntegrationAWeaponAgainstInfamousIntegrals
Maclaurin Integration is a new series-based technique for solving infamously
difficult integrals in terms of elementary functions. It has fairly liberal
conditions for sound use, making it one of the most versatile integration
techniques. Additionally, there is essentially zero human labor involved in
calculating integrals using this technique, making it one of the easiest
integration techniques to use. Its scope is mainly in pure mathematics.MaclaurinIntegrationAWeaponAgainstInfamousIntegralsSat, 08 Oct 2022 00:00:00 +0000Glenn BrudaArbitrarily Close
https://read.somethingorotherwhatever.com/entry/ArbitrarilyClose
Mathematicians tend to use the phrase "arbitrarily close" to mean something
along the lines of "every neighborhood of a point intersects a set". Taking the
latter statement as a technical definition for arbitrarily close leads to an
alternative development of classic concepts in real analysis such as supremum,
closure, convergence and limits of sequences, closure, connectedness,
compactness, and continuity. The goal of this text is to provide readers with
an introduction to real analysis by taking deliberate steps to parse these
difficult concepts using arbitrarily close as the kernel.ArbitrarilyCloseSat, 08 Oct 2022 00:00:00 +0000John A. RockGroupNames
https://read.somethingorotherwhatever.com/entry/GroupNames
GroupNames.org is a database, under construction, of names, extensions, properties and character tables of finite groups of small order.GroupNamesFri, 09 Sep 2022 00:00:00 +0000Tim DokchitserEuclidean traveller in hyperbolic worlds
https://read.somethingorotherwhatever.com/entry/Euclideantravellerinhyperbolicworlds
We will discuss all possible closures of a Euclidean line in various
geometric spaces. Imagine the Euclidean traveller, who travels only along a
Euclidean line. She will be travelling to many different geometric worlds, and
our question will be "what places does she get to see in each world?". Here is
the itinerary of our Euclidean traveller: In 1884, she travels to the torus of
any dimension, guided by Kronecker. In 1936, she travels to the world, called a
closed hyperbolic surface, guided by Hedlund. In 1991, she then travels to a
closed hyperbolic manifold of higher dimension $n\ge 3$ guided by Ratner.
Finally, she adventures into hyperbolic manifolds of infinite volume guided by
Dal'bo in dimension $2$ in 2000, by McMullen-Mohammadi-Oh in dimension $3$ in
2016 and by Lee-Oh in all higher dimensions in 2019.EuclideantravellerinhyperbolicworldsFri, 09 Sep 2022 00:00:00 +0000Hee OhMartin Gardner column to book mapping project
https://read.somethingorotherwhatever.com/entry/MartinGardnercolumntobookmappingproject
A table mapping each column written by Martin Gardner in Scientific American to the book it's reproduced in.MartinGardnercolumntobookmappingprojectThu, 08 Sep 2022 00:00:00 +0000Peter RowlettStrange Expectations and the Winnie-the-Pooh Problem
https://read.somethingorotherwhatever.com/entry/StrangeExpectationsandtheWinniethePoohProblem
Motivated by the study of simultaneous cores, we give three proofs (in
varying levels of generality) for the expected norm of a weight in a highest
weight representation of a complex simple Lie algebra. First, we argue directly
using the polynomial method and the Weyl character formula. Second, we use the
combinatorics of semistandard tableaux to obtain the result in type A. Third,
and most interestingly, we relate this problem to the "Winnie-the-Pooh problem"
regarding orthogonal decompositions of Lie algebras; although this approach
offers the most explanatory power, it applies only to Cartan types other than A
and C. We conclude with computations of many combinatorial cumulants.StrangeExpectationsandtheWinniethePoohProblemMon, 22 Aug 2022 00:00:00 +0000Marko Thiel and Nathan WilliamsDistanceRegular.org
https://read.somethingorotherwhatever.com/entry/DistanceRegular
An online repository of distance-regular graphs. Here you will find distance-regular graphs available for download in a variety of formats.DistanceRegularMon, 22 Aug 2022 00:00:00 +0000Robert BaileyList of numbers - Googology Wiki
https://read.somethingorotherwhatever.com/entry/ListofnumbersGoogologyWiki
This is a list of googolisms (names for numbers) in ascending order.
This list contains ill-defined large numbers, e.g. BEAF numbers beyond tetrational arrays, BIG FOOT, Little Bigeddon, Sasquatch, and large numbers whose well-definedness is not known, e.g. large numbers defined by Taranovsky's ordinal notation and Bashicu matrix number with respect to Bashicu matrix system version 2.3. ListofnumbersGoogologyWikiTue, 09 Aug 2022 00:00:00 +0000Mathematical Marbling
https://read.somethingorotherwhatever.com/entry/MathematicalMarbling
Marbling refers to painting techniques for creating a stone-like appearance or intricate flowing designs.
Marbling originated in Asia more than 800 years ago and spread to Europe in the 1500s, where it was used for endpapers and book covers.
My web-pages are about generating marbling designs mathematically. MathematicalMarblingTue, 12 Jul 2022 00:00:00 +0000Aubrey JafferStuttering look and say sequences and a challenger to Conway's most complicated algebraic number from the silliest source
https://read.somethingorotherwhatever.com/entry/StutteringlookandsaysequencesandachallengertoConwaysmostcomplicatedalgebraicnumberfromthesilliestsource
We introduce stuttering look and say sequences and describe their chemical
structure in the spirit of Conway's work on audioactive decay. We show the
growth rate of a stuttering look and say sequence is an algebraic integer of
degree 415.StutteringlookandsaysequencesandachallengertoConwaysmostcomplicatedalgebraicnumberfromthesilliestsourceFri, 08 Jul 2022 00:00:00 +0000Jonathan ComesOn The Euclidean Algorithm: Rhythm Without Recursion
https://read.somethingorotherwhatever.com/entry/OnTheEuclideanAlgorithmRhythmWithoutRecursion
A modified form of Euclid's algorithm has gained popularity among musical
composers following Toussaint's 2005 survey of so-called Euclidean rhythms in
world music. We offer a method to easily calculate Euclid's algorithm by hand
as a modification of Bresenham's line-drawing algorithm. Notably, this modified
algorithm is a non-recursive matrix construction, using only modular arithmetic
and combinatorics. This construction does not outperform the traditional
divide-with-remainder method; it is presented for combinatorial interest and
ease of hand computation.OnTheEuclideanAlgorithmRhythmWithoutRecursionFri, 08 Jul 2022 00:00:00 +0000Thomas MorrillThe mathematics of burger flipping
https://read.somethingorotherwhatever.com/entry/Themathematicsofburgerflipping
What is the most effective way to grill food? Timing is everything, since
only one surface is exposed to heat at a given time. Should we flip only once,
or many times? We present a simple model of cooking by flipping, and some
interesting observations emerge. The rate of cooking depends on the spectrum of
a linear operator, and on the fixed point of a map. If the system has symmetric
thermal properties, the rate of cooking becomes independent of the sequence of
flips, as long as the last point to be cooked is the midpoint. After numerical
optimization, the flipping intervals become roughly equal in duration as their
number is increased, though the final interval is significantly longer. We find
that the optimal improvement in cooking time, given an arbitrary number of
flips, is about 29% over a single flip. This toy problem has some
characteristics reminiscent of turbulent thermal convection, such as a uniform
average interior temperature with boundary layers.ThemathematicsofburgerflippingFri, 08 Jul 2022 00:00:00 +0000Jean-Luc ThiffeaultThe Strong Law of Small Numbers
https://read.somethingorotherwhatever.com/entry/TheStrongLawOfSmallNumbers
This article is in two parts, the first of which is a do-it-yourself operation, in which I'll show you 35 examples of patterns that seem to appear when we look at several small values of n, in various problems whose answers depend on n. The question will be, in each case: do you think that the pattern persists for all n, or do you believe that it is a figment of the smallness of the values of n that are worked out in the examples?
Caution: examples of both kinds appear; they are not all figments!
In the second part I'll give you the answers, insofar as I know them, together with references.TheStrongLawOfSmallNumbersTue, 05 Jul 2022 00:00:00 +0000Richard K. GuyFilling space with hypercubes of two sizes -- The pythagorean tiling in higher dimensions
https://read.somethingorotherwhatever.com/entry/FillingspacewithhypercubesoftwosizesThepythagoreantilinginhigherdimensions
We construct a unilateral lattice tiling of $\mathbb{R}^n$ into hypercubes of
two differnet side lengths $p$ or $q$. This generalizes the Pythagorean tiling
in $\mathbb{R}^2$. We also show that this tiling is unique up to symmetries,
which proves a variation of a conjecture by B\"olcskei from 2001. For positive
integers $p$ and $q$ this tiling also provides a tiling of
$(\mathbb{Z}/(p^n+q^n)\mathbb{Z})^n$.FillingspacewithhypercubesoftwosizesThepythagoreantilinginhigherdimensionsFri, 13 May 2022 00:00:00 +0000Jakob FührerWho's Afraid of Mathematical Diagrams?
https://read.somethingorotherwhatever.com/entry/WhosAfraidofMathematicalDiagrams
Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability. I then consider whether diagrams can be essential to the proofs in which they appear. WhosAfraidofMathematicalDiagramsFri, 13 May 2022 00:00:00 +0000Silvia De ToffoliCharacteristic Polynomial Database
https://read.somethingorotherwhatever.com/entry/CharacteristicPolynomialDatabase
The characteristic polynomial database contains characteristic polynomials, minimal polynomials and properties for a variety of families of Bohemian matrices.CharacteristicPolynomialDatabaseSun, 24 Apr 2022 00:00:00 +0000Steven E. ThorntonA simple mnemonic to compute sums of powers
https://read.somethingorotherwhatever.com/entry/Asimplemnemonictocomputesumsofpowers
We give a simple recursive formula to obtain the general sum of the first $N$
natural numbers to the $r$th power. Our method allows one to obtain the general
formula for the $(r+1)$th power once one knows the general formula for the
$r$th power. The method is very simple to remember owing to an analogy with
differentiation and integration. Unlike previously known methods, no knowledge
of additional specific constants (such as the Bernoulli numbers) is needed.
This makes it particularly suitable for applications in cases when one cannot
consult external references, for example mathematics competitions.AsimplemnemonictocomputesumsofpowersSun, 24 Apr 2022 00:00:00 +0000Alessandro MarianiA note on the Screaming Toes game
https://read.somethingorotherwhatever.com/entry/AnoteontheScreamingToesgame
We investigate properties of random mappings whose core is composed of
derangements as opposed to permutations. Such mappings arise as the natural
framework to study the Screaming Toes game described, for example, by Peter
Cameron. This mapping differs from the classical case primarily in the
behaviour of the small components, and a number of explicit results are
provided to illustrate these differences.AnoteontheScreamingToesgameSun, 24 Apr 2022 00:00:00 +0000Simon TavaréConvex Equipartitions: The Spicy Chicken Theorem
https://read.somethingorotherwhatever.com/entry/ConvexEquipartitionsTheSpicyChickenTheorem
We show that, for any prime power n and any convex body K (i.e., a compact
convex set with interior) in Rd, there exists a partition of K into n convex
sets with equal volumes and equal surface areas. Similar results regarding
equipartitions with respect to continuous functionals and absolutely continuous
measures on convex bodies are also proven. These include a generalization of
the ham-sandwich theorem to arbitrary number of convex pieces confirming a
conjecture of Kaneko and Kano, a similar generalization of perfect partitions
of a cake and its icing, and a generalization of the Gromov-Borsuk-Ulam theorem
for convex sets in the model spaces of constant curvature.
Most of the results in this paper appear in arxiv:1011.4762 and in
arxiv:1010.4611. Since the main results and techniques there are essentially
the same, we have merged the papers for journal publication. In this version we
also provide a technical alternative to a part of the proof of the main
topological result that avoids the use of compactly supported homology.ConvexEquipartitionsTheSpicyChickenTheoremSun, 24 Apr 2022 00:00:00 +0000Roman Karasev and Alfredo Hubard and Boris AronovDescending Dungeons and Iterated Base-Changing
https://read.somethingorotherwhatever.com/entry/DescendingDungeonsandIteratedBaseChanging
For real numbers a, b> 1, let as a_b denote the result of interpreting a in
base b instead of base 10. We define ``dungeons'' (as opposed to ``towers'') to
be numbers of the form a_b_c_d_..._e, parenthesized either from the bottom
upwards (preferred) or from the top downwards. Among other things, we show that
the sequences of dungeons with n-th terms 10_11_12_..._(n-1)_n or
n_(n-1)_..._12_11_10 grow roughly like 10^{10^{n log log n}}, where the
logarithms are to the base 10. We also investigate the behavior as n increases
of the sequence a_a_a_..._a, with n a's, parenthesized from the bottom upwards.
This converges either to a single number (e.g. to the golden ratio if a = 1.1),
to a two-term limit cycle (e.g. if a = 1.05) or else diverges (e.g. if a =
frac{100{99).DescendingDungeonsandIteratedBaseChangingMon, 28 Feb 2022 00:00:00 +0000David Applegate and Marc LeBrun and N. J. A. SloaneEasy Proof of Three Recursive $π$-Algorithms -- Einfacher Beweis dreier rekursiver $π$-Algorithmen
https://read.somethingorotherwhatever.com/entry/EasyProofofThreeRecursiveAlgorithmsEinfacherBeweisdreierrekursiverAlgorithmen
This paper consists of three independent parts: First we use only elementary
algebra to prove that the quartic algorithm of the Borwein brothers has exactly
the same output as the Brent-Salamin algorithm, but that the latter needs twice
as many iterations. Second we use integral calculus to prove that the
Brent-Salamin algorithm approximates $\pi$. Combining these results proves that
the Borwein brothers' quartic algorithm also approximates $\pi$. Third, we
prove the quadratic convergence of the Brent-Salamin algorithm, which also
proves the quartic convergence of Borwein's algorithm.
-----
Dieses Paper besteht aus drei unabh\"angigen Teilen: Erstens beweisen wir mit
elementarer Algebra, dass der Borwein-Algorithmus vierter Ordnung die gleichen
Ergebnisse liefert wie der Brent-Salamin-Algorithmus, wobei letzterer doppelt
so viele Iterationen ben\"otigt. Zweitens beweisen wir mit Integralrechnung,
dass der Brent-Salamin-Algorithmus gegen $\pi$ konvergiert. Hieraus folgt, dass
der Borwein-Algorithmus vierter Ordnung ebenfalls gegen $\pi$ konvergiert.
Drittens beweisen wir die quadratische Konvergenz des Brent-Salamin-Algorithmus
und somit auch die quartische Konvergenz des Borwein-Algorithmus.EasyProofofThreeRecursiveAlgorithmsEinfacherBeweisdreierrekursiverAlgorithmenFri, 25 Feb 2022 00:00:00 +0000Lorenz MillaTiling with arbitrary tiles
https://read.somethingorotherwhatever.com/entry/Tilingwitharbitrarytiles
Let $T$ be a tile in $\mathbb{Z}^n$, meaning a finite subset of
$\mathbb{Z}^n$. It may or may not tile $\mathbb{Z}^n$, in the sense of
$\mathbb{Z}^n$ having a partition into copies of $T$. However, we prove that
$T$ does tile $\mathbb{Z}^d$ for some $d$. This resolves a conjecture of
Chalcraft.TilingwitharbitrarytilesWed, 23 Feb 2022 00:00:00 +0000Vytautas Gruslys and Imre Leader and Ta Sheng TanLess Mundane Applications of the Most Mundane Functions
https://read.somethingorotherwhatever.com/entry/LessMundaneApplicationsoftheMostMundaneFunctions
Linear functions are arguably the most mundane among all functions. However,
the basic fact that a multi-variable linear function has a constant gradient
field can provide simple geometric insights into several familiar results such
as the Cauchy-Schwarz inequality, the GM-AM inequality, and some distance
formulae, as we shall show.LessMundaneApplicationsoftheMostMundaneFunctionsWed, 16 Feb 2022 00:00:00 +0000Pisheng DingMATHREPO - Mathematical Data and Software
https://read.somethingorotherwhatever.com/entry/MATHREPO
This is the repository website of the Max Planck Institute for Mathematics in the Sciences in Leipzig, dedicated to software, computations, and research data in mathematics.
The purpose of this webpage is to collect and explain mathematical software developed in various projects among members and collaborators of this institute and its respective applications. The website also contains supplementary material to publications and materials created at events held at MPI MiS.MATHREPOWed, 16 Feb 2022 00:00:00 +0000Claudia Fevola and Christiane GörgenHow to Solve "The Hardest Logic Puzzle Ever" and Its Generalization
https://read.somethingorotherwhatever.com/entry/HowtoSolveTheHardestLogicPuzzleEverandItsGeneralization
Raymond Smullyan came up with a puzzle that George Boolos called "The Hardest
Logic Puzzle Ever".[1] The puzzle has truthful, lying, and random gods who
answer yes or no questions with words that we don't know the meaning of. The
challenge is to figure out which type each god is. Various "top-down" solutions
to the puzzle have been developed.[1,2] Here a systematic bottom-up approach to
the puzzle and its generalization is presented. We prove that an n gods puzzle
is solvable if and only if the random gods are less than the non-random gods.
We develop a solution using 4.13 questions to the 5 gods variant with 2 random
and 3 lying gods. There is also an aside on mathematical vs. computational
thinking.HowtoSolveTheHardestLogicPuzzleEverandItsGeneralizationWed, 16 Feb 2022 00:00:00 +0000Daniel VallstromGenerating graphs randomly
https://read.somethingorotherwhatever.com/entry/Generatinggraphsrandomly
Graphs are used in many disciplines to model the relationships that exist
between objects in a complex discrete system. Researchers may wish to compare a
network of interest to a "typical" graph from a family (or ensemble) of graphs
which are similar in some way. One way to do this is to take a sample of
several random graphs from the family, to gather information about what is
"typical". Hence there is a need for algorithms which can generate graphs
uniformly (or approximately uniformly) at random from the given family. Since a
large sample may be required, the algorithm should also be computationally
efficient.
Rigorous analysis of such algorithms is often challenging, involving both
combinatorial and probabilistic arguments. We will focus mainly on the set of
all simple graphs with a particular degree sequence, and describe several
different algorithms for sampling graphs from this family uniformly, or almost
uniformly.GeneratinggraphsrandomlyWed, 16 Feb 2022 00:00:00 +0000Catherine GreenhillMax/Min Puzzles in Geometry
https://read.somethingorotherwhatever.com/entry/MaxMinPuzzlesinGeometry
The objective here is to find the maximum polygon, in area, which can be
enclosed in a given triangle, for the polygons: parallelograms, rectangles and
squares. It will initially be assumed that the choices are inscribed polygons,
that is all vertices of the polygon are on the sides of the triangle. This
concept will be generalized later to include wedged polygons.MaxMinPuzzlesinGeometryWed, 16 Feb 2022 00:00:00 +0000James M ParksPop-Up Geometry
https://read.somethingorotherwhatever.com/entry/PopUpGeometry
Anyone browsing at the stationery store will see an incredible array of pop-up cards available for any occasion. The workings of pop-up cards and pop-up books can be remarkably intricate. Behind such designs lies beautiful geometry involving the intersection of circles, cones, and spheres, the movements of linkages, and other constructions. The geometry can be modelled by algebraic equations, whose solutions explain the dynamics. For example, several pop-up motions rely on the intersection of three spheres, a computation made every second for GPS location. Connecting the motions of the card structures with the algebra and geometry reveals abstract mathematics performing tangible calculations. Beginning with the nephroid in the 19th-century, the mathematics of pop-up design is now at the frontiers of rigid origami and algorithmic computational complexity. All topics are accessible to those familiar with high-school mathematics; no calculus required. Explanations are supplemented by 140+ figures and 20 animations.PopUpGeometryWed, 16 Feb 2022 00:00:00 +0000Joseph O'RourkeClark Kimberling's Encyclopedia of Triangle Centers
https://read.somethingorotherwhatever.com/entry/EncyclopediaOfTriangleCenters
Long ago, someone drew a triangle and three segments across it. Each segment started at a vertex and stopped at the midpoint of the opposite side. The segments met in a point. The person was impressed and repeated the experiment on a different shape of triangle. Again the segments met in a point. The person drew yet a third triangle, very carefully, with the same result. He told his friends. To their surprise and delight, the coincidence worked for them, too. Word spread, and the magic of the three segments was regarded as the work of a higher power.
Centuries passed, and someone proved that the three medians do indeed concur in a point, now called the centroid. The ancients found other points, too, now called the incenter, circumcenter, and orthocenter. More centuries passed, more special points were discovered, and a definition of triangle center emerged. Like the definition of continuous function, this definition is satisfied by infinitely many objects, of which only finitely many will ever be published. The Encyclopedia of Triangle Centers (ETC) extends a list of 400 triangle centers published in the 1998 book Triangle Centers and Central Triangles.EncyclopediaOfTriangleCentersWed, 16 Feb 2022 00:00:00 +0000Clark KimberlingDatabase of Permutation Pattern Avoidance
https://read.somethingorotherwhatever.com/entry/DatabaseofPermutationPatternAvoidance
The aim of this database is to provide a resource of phenomena characterized by avoiding a finite number of permutation patterns.DatabaseofPermutationPatternAvoidanceWed, 16 Feb 2022 00:00:00 +0000Bridget TennerFindStat
https://read.somethingorotherwhatever.com/entry/FindStat
This collaborative project is a database of combinatorial statistics and maps on combinatorial collections and a search engine, identifying your data as the composition of known maps and statistics.FindStatWed, 16 Feb 2022 00:00:00 +0000Martin Rubey and Christian StumpHouse of Graphs
https://read.somethingorotherwhatever.com/entry/HouseofGraphs
Most graph theorists will agree that among the vast number of graphs that exist there are only a few that can be considered really interesting.
It is the aim of this House of Graphs project to find a workable definition of 'interesting' and provide a searchable database of graphs that conform to this definition. And to allow users to add additional graphs which they find interesting. In order to avoid abuse, only registered users can add new graphs.
We would also like to serve as a repository for lists of graphs (which can be downloaded in several formats) and graph generators. Currently we only provide a small selection. HouseofGraphsWed, 16 Feb 2022 00:00:00 +0000Gunnar Brinkmann and Kris Coolsaet and Jan Goedgebeur and Hadrien MélotA catalog of matchstick graphs
https://read.somethingorotherwhatever.com/entry/Acatalogofmatchstickgraphs
Classification of planar unit-distance graphs with up to 9 edges, by
homeomorphism and isomorphism classes. With exactly nine edges, there are 633
nonisomorphic connected matchstick graphs, of which 196 are topologically
distinct from each other. Increasing edges' number, their quantities rise more
than exponentially, in a still unclear way.AcatalogofmatchstickgraphsWed, 16 Feb 2022 00:00:00 +0000Raffaele SalviaMathematics ClipArt
https://read.somethingorotherwhatever.com/entry/MathematicsClipArt
The Mathematics ClipArt collection includes 9,820 images for algebra, geometry, trigonometry, probability, money, number sense, and more., conveniently arranged in 222 galleries. This comprehensive set of illustrations for teachers and students consists of ClipArt for all levels of K-12 math classes. Included are coordinate and polar grids, graphs, number lines, clocks, patterns, flashcards, protractors, thermometers, fractions, geometric shapes & solids, angles & lines, bags of marbles, spinners, constructions, theorems & proofs, and dice. From the common place to the hard-to-find ClipArt, everything an educator needs for math activities, assessments, and presentations can be found here.MathematicsClipArtTue, 18 Jan 2022 00:00:00 +0000Florida Center for Instructional TechnologyCheck Digits
https://read.somethingorotherwhatever.com/entry/CheckDigits
A decimal (or alphanumeric) digit added to a number for the purpose of detecting the sorts of errors humans typically make on data entry.CheckDigitsTue, 18 Jan 2022 00:00:00 +0000Jonathan MohrOn Kaprekar's Junction Numbers
https://read.somethingorotherwhatever.com/entry/OnKaprekarsJunctionNumbers
A base b junction number u has the property that there are at least two ways
to write it as u = v + s(v), where s(v) is the sum of the digits in the
expansion of the number v in base b. For the base 10 case, Kaprekar in the
1950's and 1960's studied the problem of finding K(n), the smallest u such that
the equation u=v+s(v) has exactly n solutions. He gave the values K(2)=101,
K(3)=10^13+1, and conjectured that K(4)=10^24+102. In 1966 Narasinga Rao gave
the upper bound 10^1111111111124+102 for K(5), as well as upper bounds for
K(6), K(7), K(8), and K(16). In the present work, we derive a set of
recurrences, which determine K(n) for any base b and in particular imply that
these conjectured values of K(n) are correct. The key to our approach is an
apparently new recurrence for F(u), the number of solutions to u=v+s(v). We
have applied our method to compute K(n) for n <= 16 and bases b <= 10. These
sequences grow extremely rapidly. Rather surprisingly, the solution to the base
5 problem is determined by the classical Thue-Morse sequence. For a fixed b, it
appears that K(n) grows as a tower of height about log_2(n).OnKaprekarsJunctionNumbersMon, 10 Jan 2022 00:00:00 +0000Max A. Alekseyev and Donovan Johnson and N. J. A. SloaneFibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-Style Square-Filling Curves
https://read.somethingorotherwhatever.com/entry/FibbinaryZippers
Within the recursive subdivision of the \(n \times n\) square, what characterizes a Hilbert-style space-filling curve motif of length \(n^2\) when—under iterated, self-similar, pure edge-replacement—a sequence of always self-avoiding lattice paths results? How many motifs are there and what do they look like? Such motifs are composable elements of a monoid, where all such motifs map to a particular subset of Hamiltonian cycles on the \(n \times n\) toroidal grid-graph. We prove that for any odd \(n \geq 1\) each motif has a shape that falls into exactly one of \(F_{(n−3)/2}\) boundary “zipping” modes, where \(F_i\) is the \(i\)th Fibonacci number; for even \(n\) no solution motifs exist. Each mode is governed by a special palindromic Fibbinary bit sequence (i.e., having no adjacent 1 bits). To varying degrees, each zipping mode emanates further combinatorial constraint inward from the square’s
boundary, especially at the corners. The zipping mode whose Fibbinary bits have the most consecutive 0s freezes over half of the \(n^2\) edges of an order-\(n\) motif into only one distinct (either left- or right-handed) configuration. Manual and machine enumeration for small \(n\) is significantly enhanced by these results. For \(n = 1, 3, 5, 7, 9, 11\) there are 1, 0, 1, 7, 10101, 20305328 distinct, globally self-avoiding motifs, falling into \(F_{(n−3)/2} = 1, 0, 1, 1, 2, 3\) zipping modes, respectively. For \(n \geq 5\), each such motif, when infinitely exponentiated within its monoid, converges to an open-ended, square-filling, continuous curve.FibbinaryZippersWed, 08 Dec 2021 00:00:00 +0000Douglas M. McKennaRustan Leino's Puzzles
https://read.somethingorotherwhatever.com/entry/RustanLeinoPuzzles
Here are some mathematical puzzles that I have enjoyed. Most of them are of the kind that you can discuss and solve at a dinner table, usually without pen and paper. So as not to spoil your fun, no solutions are given on this page, but for some problems I have provided some hints.RustanLeinoPuzzlesTue, 23 Nov 2021 00:00:00 +0000Rustan LeinoHow Not to Compute a Fourier Transform
https://read.somethingorotherwhatever.com/entry/HowNottoComputeaFourierTransform
We revisit the Fourier transform of a Hankel function, of considerable
importance in the theory of knife edge diffraction. Our approach is based
directly upon the underlying Bessel equation, which admits manipulation into an
alternate second order differential equation, one of whose solutions is
precisely the desired transform, apart from an {\em{a priori}} unknown
constant, and a second, undesired solution of logarithmic type. A modest amount
of analysis is then required to exhibit that constant as having its proper
value, and to purge the logarithmic accompaniment. The intervention of this
analysis, which relies upon an interplay of asymptotic and close-in functional
behaviors, prompts our somewhat ironic, mildly puckish caveat, our negation
{\em{"not"}} in the title. In a concluding section we show that this same
transform is still more readily exhibited as an easy by-product of the
inhomogeneous wave equation in two dimensions satisfied by the Green's function
$G,$ itself proportional to a Hankel function. This latter discussion lapses of
course into the argot of physicists.HowNottoComputeaFourierTransformSat, 20 Nov 2021 00:00:00 +0000J. A. GrzesikBell-ringing methods as polyhedra
https://read.somethingorotherwhatever.com/entry/Bellringingmethodsaspolyhedra
In his popular book "The Bob Caller's companion" Mr. Steve Coleman notes how any bell-ringing method can be represented as a directed graph, with a node for each lead head and an edge for each plain lead or bob lead. He further notes that if the graph can be drawn without any of the edges crossing, then it can be made into a polyhedron, with the edges and nodes of the graph being the edges and nodes of the polyhedron. He gives Grandsire Doubles and Plain Bob Doubles as examples of this.
This left me asking myself the question: how many different polyhedra are there whose nodes and edges map onto the graph of the lead-heads of a ringing method, popular or otherwise? I enumerated the possible cases for lead heads of plain doubles and minor methods and found the following cases, some of which have polyhedral graphs and some of which do not.BellringingmethodsaspolyhedraSat, 20 Nov 2021 00:00:00 +0000Hugh C. PumphreyYet another Proof of an old Hat
https://read.somethingorotherwhatever.com/entry/YetanotherProofofanoldHat
Every odd prime number \(p\) can be written in exactly \((p + 1)/2\) ways as a sum \(ab + cd\) with \(\min(a, b) > \max(c, d)\) of two ordered products. This gives a new proof of Fermat's Theorem expressing primes of the form \(1 + 4\mathbb{N}\) as sums of two squares.YetanotherProofofanoldHatSat, 20 Nov 2021 00:00:00 +0000Roland BacherDatabase of Ring Theory
https://read.somethingorotherwhatever.com/entry/DatabaseOfRingTheory
A repository of rings, their properties, and more ring theory stuff.DatabaseOfRingTheorySat, 20 Nov 2021 00:00:00 +0000Ryan C. SchwiebertIconicity in Mathematical Notation: Commutativity and Symmetry
https://read.somethingorotherwhatever.com/entry/IconicityinMathematicalNotationCommutativityandSymmetry
Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects – those which visually resemble in some way the concepts they represent – offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd’s hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance. IconicityinMathematicalNotationCommutativityandSymmetrySat, 20 Nov 2021 00:00:00 +0000Theresa Elise Wege and Sophie Batchelor and Matthew Inglis and Honali Mistry and Dirk SchlimmThe design of mathematical language
https://read.somethingorotherwhatever.com/entry/Thedesignofmathematicallanguage
As idealized descriptions of mathematical language, there is a sense in which formal systems specify too little, and there is a sense in which they specify too much. They are silent with respect to a number of features of mathematical language that are essential to the communicative and inferential goals of the subject, while many of these features are independent of a specific choice of foundation. This chapter begins to map out the design features of mathematical language without descending to the level of formal implementation, drawing on examples from the mathematical literature and insights from the design of computational proof assistants.ThedesignofmathematicallanguageSat, 20 Nov 2021 00:00:00 +0000Jeremy AvigadThe Mathematical Movie Database
https://read.somethingorotherwhatever.com/entry/TheMathematicalMovieDatabase
This page, along with our TV Database, complements our book Mathematics goes to the movies. It contains a list of titles and short descriptions of about 800 movies that contain mathematics. Also included are links to the respective entries in the International Movie Database (IMDB) and links to clips from some of the movies.TheMathematicalMovieDatabaseSat, 20 Nov 2021 00:00:00 +0000Burkard Polster and Marty RossOuroboros Functionals, Families of Ouroboros Functions, and Their Relationship to Partial Differential Equations and Probability Theory
https://read.somethingorotherwhatever.com/entry/OuroborosFunctionalsFamiliesofOuroborosFunctionsandTheirRelationshiptoPartialDifferentialEquationsandProbabilityTheory
Previously, we have introduced a very small number of examples of what we
call Ouroboros functions. Using our already established theory of Ouroboros
spaces and their functions, we will provide a set of families of Ouroboros
functions that bolster our overall understanding of the Ouroboros spaces. From
here, we extend the theory of Ouroboros functions by introducing Ouroboros
functionals and Ouroboros functional spaces. Furthermore, we re-frame the
expected value of a random variable as an Ouroboros functional, which proves to
be more intuitive in view of probabilistic measure theory. We then show that
these Ouroboros functions have additional applications, as they are general
solutions to certain elementary linear first order partial differential
equations (PDEs). We conclude by elaborating upon this connection and
discussing future endeavors, which will be centered on answering a given
hypothesis.OuroborosFunctionalsFamiliesofOuroborosFunctionsandTheirRelationshiptoPartialDifferentialEquationsandProbabilityTheorySun, 29 Aug 2021 00:00:00 +0000Nathan Thomas ProvostSome instructive mathematical errors
https://read.somethingorotherwhatever.com/entry/Someinstructivemathematicalerrors
We describe various errors in the mathematical literature, and consider how
some of them might have been avoided, or at least detected at an earlier stage,
using tools such as Maple or Sage. Our examples are drawn from three broad
categories of errors. First, we consider some significant errors made by
highly-regarded mathematicians. In some cases these errors were not detected
until many years after their publication. Second, we consider in some detail an
error that was recently detected by the author. This error in a refereed
journal led to further errors by at least one author who relied on the
(incorrect) result. Finally, we mention some instructive errors that have been
detected in the author's own published papers.SomeinstructivemathematicalerrorsSun, 29 Aug 2021 00:00:00 +0000Richard P. BrentSET with a Twist
https://read.somethingorotherwhatever.com/entry/SETwithaTwist
If you can’t get enough of the card game SET and enjoyed our version of Projective SET (see “Projectivizing Set” in the April 2020 issue of this magazine), then get excited, because we are back with another round of variations on the game. This time, we’ll explore the game from a purely algebraic perspective. This article is self-contained—so feel free to just keep reading—but we encourage you to check out our last article for a more geometric take on game modifications.SETwithaTwistSun, 29 Aug 2021 00:00:00 +0000Cathy Hsu, Jonah Ostroff and Lucas Van MeterFruit Diophantine Equation
https://read.somethingorotherwhatever.com/entry/FruitDiophantineEquation
We show that the Diophantine equation given by X^3+ XYZ = Y^2+Z^2+5 has no
integral solution. As a consequence, we show that the family of elliptic curve
given by the Weierstrass equations Y^2-kXY = X^3 - (k^2+5) has no integral
point.FruitDiophantineEquationSun, 29 Aug 2021 00:00:00 +0000Dipramit Majumdar and B. SuryYule's "Nonsense Correlation" Solved!
https://read.somethingorotherwhatever.com/entry/YulesNonsenseCorrelationSolved
In this paper, we resolve a longstanding open statistical problem. The
problem is to mathematically confirm Yule's 1926 empirical finding of "nonsense
correlation" (\cite{Yule}). We do so by analytically determining the second
moment of the empirical correlation coefficient
\beqn \theta := \frac{\int_0^1W_1(t)W_2(t) dt - \int_0^1W_1(t) dt \int_0^1
W_2(t) dt}{\sqrt{\int_0^1 W^2_1(t) dt - \parens{\int_0^1W_1(t) dt}^2}
\sqrt{\int_0^1 W^2_2(t) dt - \parens{\int_0^1W_2(t) dt}^2}}, \eeqn of two {\em
independent} Wiener processes, $W_1,W_2$. Using tools from Fred- holm integral
equation theory, we successfully calculate the second moment of $\theta$ to
obtain a value for the standard deviation of $\theta$ of nearly .5. The
"nonsense" correlation, which we call "volatile" correlation, is volatile in
the sense that its distribution is heavily dispersed and is frequently large in
absolute value. It is induced because each Wiener process is "self-correlated"
in time. This is because a Wiener process is an integral of pure noise and thus
its values at different time points are correlated. In addition to providing an
explicit formula for the second moment of $\theta$, we offer implicit formulas
for higher moments of $\theta$.YulesNonsenseCorrelationSolvedSun, 29 Aug 2021 00:00:00 +0000Philip Ernst and Larry Shepp and Abraham WynerSkateboard Tricks and Topological Flips
https://read.somethingorotherwhatever.com/entry/SkateboardTricksandTopologicalFlips
We study the motion of skateboard flip tricks by modeling them as continuous
curves in the group \(SO(3)\) of special orthogonal matrices. We show that up to
continuous deformation there are only four flip tricks. The proof relies on an
analysis of the lift of such curves to the unit 3-sphere. We also derive
explicit formulas for a number of tricks and continuous deformations between
them.SkateboardTricksandTopologicalFlipsSun, 29 Aug 2021 00:00:00 +0000Justus Carlisle and Kyle Hammer and Robert Hingtgen and Gabriel MartinsA Gambler that Bets Forever and the Strong Law of Large Numbers
https://read.somethingorotherwhatever.com/entry/AGamblerthatBetsForeverandtheStrongLawofLargeNumbers
In this expository note, we give a simple proof that a gambler repeating a
game with positive expected value never goes broke with a positive probability.
This does not immediately follow from the strong law of large numbers or other
basic facts on random walks. Using this result, we provide an elementary proof
of the strong law of large numbers. The ideas of the proofs come from the
maximal ergodic theorem and Birkhoff's ergodic theorem.AGamblerthatBetsForeverandtheStrongLawofLargeNumbersMon, 07 Jun 2021 00:00:00 +0000Calvin Wooyoung ChinGamilaraay kinship revisited: incidence of recessive disease is dynamically traded-off against benefits of cooperative behaviours
https://read.somethingorotherwhatever.com/entry/Gamilaraaykinshiprevisitedincidenceofrecessivediseaseisdynamicallytradedoffagainstbenefitsofcooperativebehaviours
Traditional Indigenous marriage rules have been studied extensively since the
mid 1800s. Despite this, they have historically been cast aside as having very
little utility. This is, in large part, due to a focus on trying to understand
broad-stroke marriage restrictions or how they may evolve. Here, taking the
Gamilaraay system as a case study, we instead ask how relatedness may be
distributed under such a system. We show, remarkably, that this system
dynamically trades off kin avoidance to minimise incidence of recessive
diseases against expected levels of cooperation, as understood formally through
Hamilton's rule.GamilaraaykinshiprevisitedincidenceofrecessivediseaseisdynamicallytradedoffagainstbenefitsofcooperativebehavioursMon, 07 Jun 2021 00:00:00 +0000Jared M. FieldReal Analysis in Reverse
https://read.somethingorotherwhatever.com/entry/RealAnalysisinReverse
Many of the theorems of real analysis, against the background of the ordered
field axioms, are equivalent to Dedekind completeness, and hence can serve as
completeness axioms for the reals. In the course of demonstrating this, the
article offers a tour of some less-familiar ordered fields, provides some of
the relevant history, and considers pedagogical implications.RealAnalysisinReverseMon, 07 Jun 2021 00:00:00 +0000James ProppEvery Salami has two ends
https://read.somethingorotherwhatever.com/entry/EverySalamihastwoends
A salami is a connected, locally finite, weighted graph with non-negative
Ollivier Ricci curvature and at least two ends of infinite volume. We show that
every salami has exactly two ends and no vertices with positive curvature. We
moreover show that every salami is recurrent and admits harmonic functions with
constant gradient. The proofs are based on extremal Lipschitz extensions, a
variational principle and the study of harmonic functions. Assuming a lower
bound on the edge weight, we prove that salamis are quasi-isometric to the
line, that the space of all harmonic functions has finite dimension, and that
the space of subexponentially growing harmonic functions is two-dimensional.
Moreover, we give a Cheng-Yau gradient estimate for harmonic functions on
balls.EverySalamihastwoendsWed, 26 May 2021 00:00:00 +0000Bobo Hua and Florentin MünchTilings Encyclopedia
https://read.somethingorotherwhatever.com/entry/TilingsEncyclopedia
The tilings encyclopedia shows a wealth of examples of nonperiodic substitution tilings. TilingsEncyclopediaTue, 18 May 2021 00:00:00 +0000Dirk Frettlöh and Edmund Harriss and Franz GählerPerimeter-minimizing pentagonal tilings
https://read.somethingorotherwhatever.com/entry/Perimeterminimizingpentagonaltilings
We provide examples of perimeter-minimizing tilings of the plane by convex pentagons and examples of perimeter-minimizing tilings of certain small flat tori. PerimeterminimizingpentagonaltilingsTue, 04 May 2021 00:00:00 +0000Chung, Ping Ngai and Fernandez, Miguel and Shah, Niralee and Sordo Vieira, Luis and Wikner, ElenaThe fluid mechanics of poohsticks
https://read.somethingorotherwhatever.com/entry/Thefluidmechanicsofpoohsticks
2019 is the bicentenary of George Gabriel Stokes, who in 1851 described the
drag - Stokes drag - on a body moving immersed in a fluid, and 2020 is the
centenary of Christopher Robin Milne, for whom the game of poohsticks was
invented; his father A. A. Milne's "The House at Pooh Corner", in which it was
first described in print, appeared in 1928. So this is an apt moment to review
the state of the art of the fluid mechanics of a solid body in a complex fluid
flow, and one floating at the interface between two fluids in motion.
Poohsticks pertains to the latter category, when the two fluids are water and
air.ThefluidmechanicsofpoohsticksTue, 20 Apr 2021 00:00:00 +0000Julyan H. E. Cartwright and Oreste PiroOn Mathematical Symbols in China
https://read.somethingorotherwhatever.com/entry/OnMathematicalSymbolsinChina
When studying the history of mathematical symbols, one finds that the
development of mathematical symbols in China is a significant piece of Chinese
history; however, between the beginning of mathematics and modern day
mathematics in China, there exists a long blank period. Let us focus on the
development of Chinese mathematical symbols, and find out the significance of
their origin, evolution, rise and fall within Chinese mathematics.OnMathematicalSymbolsinChinaTue, 20 Apr 2021 00:00:00 +0000Fang Li and Yong ZhangAn integral's journey over the real line
https://read.somethingorotherwhatever.com/entry/Anintegralsjourneyovertherealline
In 1826 Cauchy presented an Integral over the real line. Al and I thought a
derivation would be mighty fine. So we packed our contour integral bags that
day, and we now present an analytic continuation this time.AnintegralsjourneyoverthereallineTue, 20 Apr 2021 00:00:00 +0000Robert Reynolds and Allan StaufferOrdner: index of real numbers
https://read.somethingorotherwhatever.com/entry/Ordner
Ordner is indexed by 30-digit floating-point decimal keys such as 0.707106781186547524400844362105. For each key, Ordner lists constant symbolic expressions (for example Div(1, Sqrt(2))) with numerical value within ±1 ulp of the key. For each expression x, Ordner also links to the Fungrim entries where x appears.
Ordner is generated automatically by searching all Fungrim formulas for constant subexpressions that Arb can evaluate numerically. Only expressions that appear explicitly in Fungrim are covered, with the following exceptions. For tables with numerical data, all the listed instances of the ground expression are included. All decimal keys in Ordner are normalized to be nonnegative, so expressions x representing negative values are indexed as Neg(x) in Ordner. Complex numbers are indexed by the real and imaginary parts (Re(x), Im(x)), as well as the absolute value and complex argument (Abs(x), Arg(x)) when both the real and imaginary parts are nonzero. The number 0 is a special case: a vanishing expression is only included when the numerical evaluation code can prove that the expression exactly represents 0. Some trivially zero-valued expressions are excluded to prevent bloat. Finally, since the Fungrim formula language normally uses Exp(x) instead of Pow(ConstE, x) to represent the exponential function, formulas containing Exp(...) are listed under 2.71828182845904523536028747135 as a special case, so as to represent this fundamental constant fairly!
ORDNER stands for Online Real Decimal Number Encyclopedia Reference.OrdnerTue, 20 Apr 2021 00:00:00 +0000Fredrik JohanssonThe best known packings of equal circles in a square
https://read.somethingorotherwhatever.com/entry/Thebestknownpackingsofequalcirclesinasquare
ThebestknownpackingsofequalcirclesinasquareTue, 20 Apr 2021 00:00:00 +0000Eckard SpechtFungrim: The Mathematical Functions Grimoire
https://read.somethingorotherwhatever.com/entry/FungrimTheMathematicalFunctionsGrimoire
The Mathematical Functions Grimoire (Fungrim) is an open source library of formulas and data for special functions.FungrimTheMathematicalFunctionsGrimoireTue, 20 Apr 2021 00:00:00 +0000Fredrik JohanssonEqWorld - The World of Mathematical Equations
https://read.somethingorotherwhatever.com/entry/EqWorld
Equations play a crucial role in modern mathematics and form the basis for mathematical modelling of numerous phenomena and processes in science and engineering.
The international scientific-educational website EqWorld presents extensive information on solutions to various classes of ordinary differential, partial differential, integral, functional, and other mathematical equations. It also outlines some methods for solving equations, includes interesting articles, gives links to mathematical websites and software packages, lists useful handbooks and monographs, and refers to scientific publishers, journals, etc. The website includes a dynamic section Equation Archive which allows authors to quickly publish their equations (differential, integral, and other) and also exact solutions, first integrals, and transformations.
The EqWorld website is intended for researchers, university teachers, engineers, and students all over the world. It contains about 2000 webpages and is visited by over 3000 users a day (coming from 200 countries worldwide). All resources presented on this site are free to its users.EqWorldThu, 15 Apr 2021 00:00:00 +0000Alexei I. Zhurov and Alexander L. Levitin and Dmitry A. PolyaninDependency Graph of Propositions in Euclid’s Elements
https://read.somethingorotherwhatever.com/entry/DependencyGraphOfPropositionsInEuclidsElements
This is a dependency graph of propositions from the first book of Euclid’s Elements. We say that a proposition \(A\) depends on \(B\) iff proposition \(B\) is necessary in the proof of proposition \(A\). In the dependency graph below,this will be denoted by an arrow starting at \(A\) and pointing at \(B\).
Figure 1 is a dependency graph of all propositions in the first book. Figure 2 is a dependency graph of all propositions that state a relation between two objects, while Figure 3 is a dependency graph of all propositions that state the existence of an unmarked straightedge and compass construction of something.The dependencies were gratefully extracted from Richard Fitzpatrick’s edition of Euclid’s Elements. The graph itself was written in DOT and converted to pslatex with dot2tex. The motivation for this graph was from Mariusz Wodzicki’s Spring 2007 History of Mathematics course at the University of California, Berkeley. Corrections and comments are always appreciated at thomson@ocf.berkeley.edu.DependencyGraphOfPropositionsInEuclidsElementsWed, 14 Apr 2021 00:00:00 +0000Thomson NguyenGeneration of real algebraic loci via complex detours
https://read.somethingorotherwhatever.com/entry/Generationofrealalgebraiclociviacomplexdetours
We discuss the locus generation algorithm used by the dynamic geometry
software Cinderella, and how it uses complex detours to resolve singularities.
We show that the algorithm is independent of the orientation of its complex
detours. We conjecture that the algorithm terminates if it takes small enough
complex detours and small enough steps on every complex detour. Moreover, we
introduce a variant of the algorithm that possibly generates entire real
connected components of real algebraic loci. Several examples illustrate its
use for organic generation of real algebraic loci. Another example shows how we
can apply the algorithm to simulate mechanical linkages. Apparently, the use of
complex detours produces physically reasonable motion of such linkages.GenerationofrealalgebraiclociviacomplexdetoursSat, 10 Apr 2021 00:00:00 +0000Stefan KranichGödel for Goldilocks: A Rigorous, Streamlined Proof of (a variant of) Gödel's First Incompleteness Theorem
https://read.somethingorotherwhatever.com/entry/GdelforGoldilocksARigorousStreamlinedProofofavariantofGdelsFirstIncompletenessTheorem
Most discussions of G\"odel's theorems fall into one of two types: either
they emphasize perceived philosophical, cultural "meanings" of the theorems,
and perhaps sketch some of the ideas of the proofs, usually relating G\"odel's
proofs to riddles and paradoxes, but do not attempt to present rigorous,
complete proofs; or they do present rigorous proofs, but in the traditional
style of mathematical logic, with all of its heavy notation and difficult
definitions, and technical issues which reflect G\"odel's original approach and
broader logical issues. Many non-specialists are frustrated by these two
extreme types of expositions and want a complete, rigorous proof that they can
understand. Such an exposition is possible, because many people have realized
that variants of G\"odel's first incompleteness theorem can be rigorously
proved by a simpler middle approach, avoiding philosophical discussions and
hand-waiving at one extreme; and also avoiding the heavy machinery of
traditional mathematical logic, and many of the harder detail's of G\"odel's
original proof, at the other extreme. This is the just-right Goldilocks
approach. In this exposition we give a short, self-contained Goldilocks
exposition of G\"odel's first theorem, aimed at a broad, undergraduate
audience.GdelforGoldilocksARigorousStreamlinedProofofavariantofGdelsFirstIncompletenessTheoremSat, 10 Apr 2021 00:00:00 +0000Dan GusfieldA Self-Referential Property of Zimin Words
https://read.somethingorotherwhatever.com/entry/ASelfReferentialPropertyofZiminWords
This paper gives a short overview of Zimin words, and proves an interesting
property of their distribution. Let $L_q^m$ to be the lexically ordered
sequence of $q$-ary words of length $m$, and let $T_n(L_q^m)$ to be the binary
sequence where the $i$-th term is $1$ if and only if the $i$-th word of $L_q^m$
encounters the $n$-th Zimin word, $Z_n$. We show that the sequence $T_n(L_q^m)$
is an instance of $Z_{n+1}$ when $1 < n$ and $m=2^n-1$.ASelfReferentialPropertyofZiminWordsWed, 31 Mar 2021 00:00:00 +0000John ConnorThe cocked hat
https://read.somethingorotherwhatever.com/entry/Thecockedhat
We revisit the cocked hat -- an old problem from navigation -- and examine
under what conditions its old solution is valid.ThecockedhatTue, 23 Mar 2021 00:00:00 +0000Imre Bárány and William Steiger and Sivan ToledoThree friendly walkers
https://read.somethingorotherwhatever.com/entry/Threefriendlywalkers
More than 15 years ago Guttmann and Vöge (2002 J. Stat. Plan. Inference 101 107), introduced a model of friendly walkers. Since then it has remained unsolved. In this paper we provide the exact solution to a closely allied model which essentially only differs in the boundary conditions. The exact solution is expressed in terms of the reciprocal of the generating function for vicious walkers which is a D-finite function. However, ratios of D-finite functions are inherently not D-finite and in this case we prove that the friendly walkers generating function is the solution to a non-linear differential equation with polynomial coefficients, it is in other words D-algebraic. We find using numerically exact calculations a conjectured expression for the generating function of the original model as a ratio of a D-finite function and the generating function for vicious walkers. We obtain an expression for this D-finite function in terms of a \({{}_{2}}{{F}_{1}}\) hypergeometric function with a rational pullback and its first and second derivatives.ThreefriendlywalkersMon, 22 Mar 2021 00:00:00 +0000Iwan JensenMaximum overhang
https://read.somethingorotherwhatever.com/entry/Maximumoverhang
How far can a stack of $n$ identical blocks be made to hang over the edge of
a table? The question dates back to at least the middle of the 19th century and
the answer to it was widely believed to be of order $\log n$. Recently,
Paterson and Zwick constructed $n$-block stacks with overhangs of order
$n^{1/3}$, exponentially better than previously thought possible. We show here
that order $n^{1/3}$ is indeed best possible, resolving the long-standing
overhang problem up to a constant factor.MaximumoverhangMon, 15 Mar 2021 00:00:00 +0000Mike Paterson and Yuval Peres and Mikkel Thorup and Peter Winkler and Uri ZwickThe Amazing $3^n$ Theorem and its even more Amazing Proof [Discovered by Xavier G. Viennot and his École Bordelaise gang]
https://read.somethingorotherwhatever.com/entry/TheAmazing3nTheoremanditsevenmoreAmazingProofDiscoveredbyXavierGViennotandhiscoleBordelaisegang
The most amazing (at least to me) result in Enumerative Combinatorics is
Dominique Gouyou-Beauchamps and Xavier Viennot's theorem that states that the
number of so-called directed animals with compact source (that are equivalent,
via Viennot's beautiful concept of heaps, to towers of dominoes, that I take
the liberty of renaming xaviers) with n+1 points equals 3^n. This amazing
result received an even more amazing proof by Jean B\'etrema and Jean-Guy
Penaud. Both theorem and proof deserve to be better known! Hence this article,
that is also accompanied by a comprehensive Maple package
http://www.math.rutgers.edu/~zeilberg/tokhniot/BORDELAISE that implements
everything (and much more)TheAmazing3nTheoremanditsevenmoreAmazingProofDiscoveredbyXavierGViennotandhiscoleBordelaisegangMon, 15 Mar 2021 00:00:00 +0000Doron ZeilbergerThe term `angle' in the international system of units
https://read.somethingorotherwhatever.com/entry/Thetermangleintheinternationalsystemofunits
The concept of an angle is one that often causes difficulties in metrology.
These are partly caused by a confusing mixture of several mathematical terms,
partly by real mathematical difficulties and finally by imprecise terminology.
The purpose of this publication is to clarify misunderstandings and to explain
why strict terminology is important. It will also be shown that most
misunderstandings regarding the `radian' can be avoided if some simple rules
are obeyed.ThetermangleintheinternationalsystemofunitsMon, 15 Mar 2021 00:00:00 +0000Michael P. KrystekProgramming the Hilbert curve
https://read.somethingorotherwhatever.com/entry/ProgrammingtheHilbertcurve
The Hilbert curve has previously been constructed recursively, using \(p\) levels of recursion of \(n\)‐bit Gray codes to attain a precision of \(p\) bits in \(n\) dimensions. Implementations have reflected the awkwardness of aligning the recursive steps to preserve geometrical adjacency. We point out that a single global Gray code can instead be applied to all \(np\) bits of a Hilbert length. Although this “over‐transforms” the length, the excess work can be undone in a single pass over the bits, leading to compact and efficient computer code.ProgrammingtheHilbertcurveFri, 12 Feb 2021 00:00:00 +0000John SkillingPythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers
https://read.somethingorotherwhatever.com/entry/PythagoreanTriplesComplexNumbersAbelianGroupsandPrimeNumbers
It is well-known that pythagorean triples can be represented by points of the
unit circle with rational coordinates. These points form an abelian group, and
we describe its structure. This structural description yields, almost
immediately, an enumeration of the normalized pythagorean triples with a given
hypotenuse, and also to an effective method for producing all such triples.
This effective method seems to be new.
This paper is intended for the general mathematical audience, including
undergraduate mathematics students, and therefore it contains plenty of
background material, some history and several examples and exercises.PythagoreanTriplesComplexNumbersAbelianGroupsandPrimeNumbersFri, 12 Feb 2021 00:00:00 +0000Amnon Yekutieli"It is like egg": Paul Lorenzen and the collapse of proofs of consistency
https://read.somethingorotherwhatever.com/entry/ItislikeeggPaulLorenzenandthecollapseofproofsofconsistency
Paul Lorenzen, mathematician and philosopher of the 20th century, mentions
October 1947 as the date of a crisis in his mathematical and philosophical
investigations. An autograph dated 15 October 1947 documents this crisis. This
article proposes a traduction and a commentary of it and sketches the
circumstances of its writing on the base of his correspondence with Paul
Bernays. A lettter from Lorenzen to Carl Friedrich Gethmann dated 14 January
1988 carves out the story of this crisis by showing how he soaks up the
indications of his correspondents and transmutes them into an absolutely
original research.ItislikeeggPaulLorenzenandthecollapseofproofsofconsistencyFri, 12 Feb 2021 00:00:00 +0000Stefan NeuwirthBraids which can be plaited with their threads tied together at each end
https://read.somethingorotherwhatever.com/entry/Braidswhichcanbeplaitedwiththeirthreadstiedtogetherateachend
The group of braids, which can be plaited from n untwisted threads tied together at each end, is examined and its structure is determined. An algorithm is derived for deciding whether or not a given braid can be so plaited and a calculation procedure is described. The problem arises from a process of manufacturing braids by a machine which plaits by passing a shuttle, on which the constructed braid is wound, between the threads, which are supplied from bobbins effectively fixed and inaccessible. Every plait on three threads can be constructed in this way. For more than three threads, examples are given both of plaits which can be so constructed and of plaits which cannot.BraidswhichcanbeplaitedwiththeirthreadstiedtogetherateachendFri, 12 Feb 2021 00:00:00 +0000J.A.H. ShepperdHilbert 13: Are there any genuine continuous multivariate real-valued functions?
https://read.somethingorotherwhatever.com/entry/Hilbert13Arethereanygenuinecontinuousmultivariaterealvaluedfunctions
This article begins with a provocative question: Are there any genuine continuous multivariate real-valued functions? This may seem to be a silly question, but it is in essence what David Hilbert asked as one of the 23 problems he posed at the second International Congress of Mathematicians, held in Paris in 1900. These problems guided a large portion of the research in mathematics of the 20th century. Hilbert's 13th problem conjectured that there exists a continuous function $ f:\mathbb{I}^3\to \mathbb{R}$, where $ {\mathbb{I}=[0,1]}$, which cannot be expressed in terms of composition and addition of continuous functions from $ \mathbb{R}^2 \to \mathbb{R}$, that is, as composition and addition of continuous real-valued functions of two variables. It took over 50 years to prove that Hilbert's conjecture is false. This article discusses the solution. Hilbert13ArethereanygenuinecontinuousmultivariaterealvaluedfunctionsTue, 02 Feb 2021 00:00:00 +0000Morris, SidneyAre There More Finite Rings than Finite Groups?
https://read.somethingorotherwhatever.com/entry/AreThereMoreFiniteRingsthanFiniteGroups
We compare the number of finite groups of order \(n\) with the number of finite rings of order \(n\), with some surprising results.AreThereMoreFiniteRingsthanFiniteGroupsWed, 06 Jan 2021 00:00:00 +0000Desmond MacHaleCounting Candy Crush Configurations
https://read.somethingorotherwhatever.com/entry/CountingCandyCrushConfigurations
A k-stable c-coloured Candy Crush grid is a weak proper c-colouring of a
particular type of k-uniform hypergraph. In this paper we introduce a fully
polynomial randomised approximation scheme (FPRAS) which counts the number of
k-stable c-coloured Candy Crush grids of a given size (m, n) for certain values
of c and k. We implemented this algorithm on Matlab, and found that in a Candy
Crush grid with7 available colours there are approximately 4.3*10^61 3-stable
colourings. (Note that, typical Candy Crush games are played with 6 colours and
our FPRAS is not guaranteed to work in expected polynomial time with k= 3 and
c= 6.) We also discuss the applicability of this FPRAS to the problem of
counting the number of weak c-colourings of other, more general hypergraphs.CountingCandyCrushConfigurationsWed, 06 Jan 2021 00:00:00 +0000Adam Hamilton and Giang T. Nguyen and Matthew RoughanEnvelopes are solving machines for quadratics and cubics and certain polynomials of arbitrary degree
https://read.somethingorotherwhatever.com/entry/Envelopesaresolvingmachinesforquadraticsandcubicsandcertainpolynomialsofarbitrarydegree
Everybody knows from school how to solve a quadratic equation of the form
$x^2-px+q=0$ graphically. But this method can become tedious if several
equations ought to be solved, as for each pair $(p,q)$ a new parabola has to be
drawn. Stunningly, there is one single curve that can be used to solve every
quadratic equation via drawing tangent lines through a given point $(p,q)$ to
this curve.
In this article we derive this method in an elementary way and generalize it
to equations of the form $x^n-px+q=0$ for arbitrary $n \ge 2$. Moreover, the
number of solutions of a specific equation of this form can be seen immediately
with this technique. Concluding the article we point out connections to the
duality of points and lines in the plane and to the the concept of Legendre
transformation.EnvelopesaresolvingmachinesforquadraticsandcubicsandcertainpolynomialsofarbitrarydegreeWed, 06 Jan 2021 00:00:00 +0000Michael Schmitz and André StreicherThe real numbers - a survey of constructions
https://read.somethingorotherwhatever.com/entry/Therealnumbersasurveyofconstructions
We present a comprehensive survey of constructions of the real numbers (from
either the rationals or the integers) in a unified fashion, thus providing an
overview of most (if not all) known constructions ranging from the earliest
attempts to recent results, and allowing for a simple comparison-at-a-glance
between different constructions.TherealnumbersasurveyofconstructionsWed, 06 Jan 2021 00:00:00 +0000Ittay Weiss"Match-Stick" Geometry
https://read.somethingorotherwhatever.com/entry/MatchStickGeometry
The following paper arose out of an attempt to improve upon the familiar "match puzzles" one sees in the popular press and which are generally too puerile to be interesting. Following some desultory experiments, it developed that the methods of constructions postulated are capable of determining all points obtainable with ruler and compass, but no others.MatchStickGeometrySun, 20 Dec 2020 00:00:00 +0000T.R. DawsonGraphs, friends and acquaintances
https://read.somethingorotherwhatever.com/entry/Graphsfriendsandacquaintances
As is well known, a graph is a mathematical object modeling the existence of
a certain relation between pairs of elements of a given set. Therefore, it is
not surprising that many of the first results concerning graphs made reference
to relationships between people or groups of people. In this article, we
comment on four results of this kind, which are related to various general
theories on graphs and their applications: the Handshake lemma (related to
graph colorings and Boolean algebra), a lemma on known and unknown people at a
cocktail party (to Ramsey theory), a theorem on friends in common (to
distance-regularity and coding theory), and Hall's Marriage theorem (to the
theory of networks). These four areas of graph theory, often with problems
which are easy to state but difficult to solve, are extensively developed and
currently give rise to much research work. As examples of representative
problems and results of these areas, which are discussed in this paper, we may
cite the following: the Four Colors Theorem (4CTC), the Ramsey numbers,
problems of the existence of distance-regular graphs and completely regular
codes, and finally the study of topological proprieties of interconnection
networks.GraphsfriendsandacquaintancesWed, 25 Nov 2020 00:00:00 +0000C. Dalfó and M. A. FiolBig fields that are not large
https://read.somethingorotherwhatever.com/entry/Bigfieldsthatarenotlarge
A subfield \(K\) of \(\bar {\mathbb{Q}}\) is large if every smooth curve \(C\) over \(K\) with a \(K\)-rational point has infinitely many \(K\)-rational points. A subfield \(K\) of \(\bar {\mathbb{Q}}\) is big if for every positive integer \(n\), \(K\) contains a number field \(F\) with \([F:\mathbb{Q}]\) divisible by \(n\). The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large. BigfieldsthatarenotlargeTue, 17 Nov 2020 00:00:00 +0000Mazur, Barry and Rubin, KarlA Prime-Representing Constant
https://read.somethingorotherwhatever.com/entry/APrimeRepresentingConstant
We present a constant and a recursive relation to define a sequence $f_n$
such that the floor of $f_n$ is the $n$th prime. Therefore, this constant
generates the complete sequence of primes. We also show this constant is
irrational and consider other sequences that can be generated using the same
method.APrimeRepresentingConstantFri, 06 Nov 2020 00:00:00 +0000Dylan Fridman and Juli Garbulsky and Bruno Glecer and James Grime and Massi Tron FlorentinConfiguration spaces of hard squares in a rectangle
https://read.somethingorotherwhatever.com/entry/Configurationspacesofhardsquaresinarectangle
We study the configuration spaces C(n;p,q) of n labeled hard squares in a p
by q rectangle, a generalization of the well-known "15 Puzzle". Our main
interest is in the topology of these spaces. Our first result is to describe a
cubical cell complex and prove that is homotopy equivalent to the configuration
space. We then focus on determining for which n, j, p, and q the homology group
$H_j [ C(n;p,q) ]$ is nontrivial. We prove three homology-vanishing theorems,
based on discrete Morse theory on the cell complex. Then we describe several
explicit families of nontrivial cycles, and a method for interpolating between
parameters to fill in most of the picture for "large-scale" nontrivial
homology.ConfigurationspacesofhardsquaresinarectangleFri, 06 Nov 2020 00:00:00 +0000Hannah Alpert and Ulrich Bauer and Matthew Kahle and Robert MacPherson and Kelly SpendloveGenerating functions for generating trees
https://read.somethingorotherwhatever.com/entry/Generatingfunctionsforgeneratingtrees
Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.GeneratingfunctionsforgeneratingtreesSat, 24 Oct 2020 00:00:00 +0000Cyril Banderier and Mireille Bousquet-Mélou and Alain Denise and Philippe Flajolet and Danièle Gardy and Dominique Gouyou-BeauchampsLeast Significant Non-Zero Digit of n!
https://read.somethingorotherwhatever.com/entry/LeastSignificantNonZeroDigitofn
Let \(p(k)\) be the least significant non-zero decimal digit of \(k!\) Can we directly determine the \(k\)th term for any given \(k\)? LeastSignificantNonZeroDigitofnThu, 22 Oct 2020 00:00:00 +0000Kevin S. BrownThe Penney's Game with Group Action
https://read.somethingorotherwhatever.com/entry/ThePenneysGamewithGroupAction
We generalize word avoidance theory by equipping the alphabet $\mathcal{A}$
with a group action. We call equivalence classes of words patterns. We extend
the notion of word correlation to patterns using group stabilizers. We extend
known word avoidance results to patterns. We use these results to answer
standard questions for the Penney's game on patterns and show non-transitivity
for the game on patterns as the length of the pattern tends to infinity. We
also analyze bounds on the pattern-based Conway leading number and expected
wait time, and further explore the game under the cyclic and symmetric group
actions.ThePenneysGamewithGroupActionFri, 16 Oct 2020 00:00:00 +0000Tanya Khovanova and Sean LiOn the Dreaded Right Bousfield Localization
https://read.somethingorotherwhatever.com/entry/OntheDreadedRightBousfieldLocalization
I verify the existence of right Bousfield localizations of right semimodel
categories, and I apply this to construct a model of the homotopy limit of a
left Quillen presheaf as a right semimodel category.OntheDreadedRightBousfieldLocalizationFri, 16 Oct 2020 00:00:00 +0000Clark BarwickWorld's shortest explanation of Gödel's theorem
https://read.somethingorotherwhatever.com/entry/WorldsshortestexplanationofGdelstheorem
A while back I started writing up an article titled "World's shortest explanation of Gödel's theorem". But I didn't finish it, and later I encountered Raymond Smullyan's version, which is much shorter anyway. So here, shamelessly stolen from Smullyan, is the World's shortest explanation of Gödel's theorem.WorldsshortestexplanationofGdelstheoremFri, 16 Oct 2020 00:00:00 +0000Mark DominusIce cream and orbifold Riemann-Roch
https://read.somethingorotherwhatever.com/entry/IcecreamandorbifoldRiemannRoch
We give an orbifold Riemann-Roch formula in closed form for the Hilbert
series of a quasismooth polarized n-fold X,D, under the assumption that X is
projectively Gorenstein with only isolated orbifold points. Our formula is a
sum of parts each of which is integral and Gorenstein symmetric of the same
canonical weight; the orbifold parts are called "ice cream functions". This
form of the Hilbert series is particularly useful for computer algebra, and we
illustrate it on examples of K3 surfaces and Calabi-Yau 3-folds.
These results apply also with higher dimensional orbifold strata (see [A.
Buckley and B. Szendroi, Orbifold Riemann-Roch for 3-folds with an application
to Calabi-Yau geometry, J. Algebraic Geometry 14 (2005) 601--622] and
[Shengtian Zhou, Orbifold Riemann-Roch and Hilbert series, University of
Warwick PhD thesis, March 2011, 91+vii pp.], although the correct statements
are considerably trickier. We expect to return to this in future publications.IcecreamandorbifoldRiemannRochFri, 16 Oct 2020 00:00:00 +0000Anita Buckley and Miles Reid and Shengtian ZhouFertility Numbers
https://read.somethingorotherwhatever.com/entry/FertilityNumbers
A nonnegative integer is called a fertility number if it is equal to the
number of preimages of a permutation under West's stack-sorting map. We prove
structural results concerning permutations, allowing us to deduce information
about the set of fertility numbers. In particular, the set of fertility numbers
is closed under multiplication and contains every nonnegative integer that is
not congruent to $3$ modulo $4$. We show that the lower asymptotic density of
the set of fertility numbers is at least $1954/2565\approx 0.7618$. We also
exhibit some positive integers that are not fertility numbers and conjecture
that there are infinitely many such numbers.FertilityNumbersFri, 16 Oct 2020 00:00:00 +0000Colin DefantThe Phillip Island penguin parade (a mathematical treatment)
https://read.somethingorotherwhatever.com/entry/ThePhillipIslandpenguinparadeamathematicaltreatment
Penguins are flightless, so they are forced to walk while on land. In
particular, they show rather specific behaviors in their homecoming, which are
interesting to observe and to describe analytically. In this paper, we present
a simple mathematical formulation to describe the little penguins parade in
Phillip Island. We observed that penguins have the tendency to waddle back and
forth on the shore to create a sufficiently large group and then walk home
compactly together. The mathematical framework that we introduce describes this
phenomenon, by taking into account "natural parameters" such as the eye-sight
of the penguins, their cruising speed and the possible "fear" of animals. On
the one hand, this favors the formation of conglomerates of penguins that
gather together, but, on the other hand, this may lead to the "panic" of
isolated and exposed individuals. The model that we propose is based on a set
of ordinary differential equations. Due to the discontinuous behavior of the
speed of the penguins, the mathematical treatment (to get existence and
uniqueness of the solution) is based on a "stop-and-go" procedure. We use this
setting to provide rigorous examples in which at least some penguins manage to
safely return home (there are also cases in which some penguins freeze due to
panic). To facilitate the intuition of the model, we also present some simple
numerical simulations that can be compared with the actual movement of the
penguins parade.ThePhillipIslandpenguinparadeamathematicaltreatmentFri, 16 Oct 2020 00:00:00 +0000Serena Dipierro and Luca Lombardini and Pietro Miraglio and Enrico ValdinociMathematics with a metamathematical flavour.
https://read.somethingorotherwhatever.com/entry/MathematicsWithAMetamathematicalFlavour
Among the most fascinating results of mathematics are unprovability theorems, that is, rigorous proofs that certain statements cannot be deduced from certain axioms. A very famous example is Paul Cohen's demonstration that the continuum hypothesis cannot be deduced from the ZFC axioms . For this, Cohen invented a technique known as forcing, which is far too advanced for a page like this. (Indeed, I am incapable of presenting it anyway - if you are curious you could try visiting this site for some notes on forcing. They seem all right, but I don't know enough to be able to judge with any confidence.) Instead, I shall present here a few examples of low-level unprovability theorems, by which I mean purely mathematical results that, in one way or another, tell us that proofs of certain theorems must necessarily have certain properties. Such conclusions I shall loosely refer to as metamathematics. MathematicsWithAMetamathematicalFlavourFri, 16 Oct 2020 00:00:00 +0000Timothy GowersMathematics, morally
https://read.somethingorotherwhatever.com/entry/MathematicsMorally
A source of tension between Philosophers of Mathematics and Mathematicians is the fact that each group feels ignored by the other; daily mathematical practie seems barely affected by the questions the Philosophers are considering. In this talk I will describe an issue that does have an impact on mathematical practice, and a philosophical stance on mathematics that is detectable in the work of practising mathematicians. No doubt controversially, I will call this issue `morality', but the term is not of my coining: there are mathematicians across the world who use the word `morally' to greate effect in private, and I propose that there should be a public theory of what they mean by this.MathematicsMorallyFri, 16 Oct 2020 00:00:00 +0000Eugenia ChengA Midsummer Knot's Dream
https://read.somethingorotherwhatever.com/entry/AMidsummerKnotsDream
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.AMidsummerKnotsDreamMon, 21 Sep 2020 00:00:00 +0000Allison Henrich and Noël MacNaughton and Sneha Narayan and Oliver Pechenik and Robert Silversmith and Jennifer TownsendAn Interesting Serendipitous Real Number
https://read.somethingorotherwhatever.com/entry/AnInterestingSerendipitousRealNumber
This is the story of a remarkable real number, the discovery of which was due to a misprint. Namely, in the mid-seventies, while Ciprian was at the University of Bucharest, one of his former students approached him with the following question:
If \(x_1 \gt 0\) and \(x_{n+1} = \left(1 + \frac{1}{x_n}\right)^n\), can \(x_n \to \infty\)?AnInterestingSerendipitousRealNumberMon, 21 Sep 2020 00:00:00 +0000John Ewing and Ciprian FoiasThe No-Flippancy Game
https://read.somethingorotherwhatever.com/entry/TheNoFlippancyGame
We analyze a coin-based game with two players where, before starting the
game, each player selects a string of length $n$ comprised of coin tosses. They
alternate turns, choosing the outcome of a coin toss according to specific
rules. As a result, the game is deterministic. The player whose string appears
first wins. If neither player's string occurs, then the game must be infinite.
We study several aspects of this game. We show that if, after $4n-4$ turns,
the game fails to cease, it must be infinite. Furthermore, we examine how a
player may select their string to force a desired outcome. Finally, we describe
the result of the game for particular cases.TheNoFlippancyGameMon, 21 Sep 2020 00:00:00 +0000Isha Agarwal and Matvey Borodin and Aidan Duncan and Kaylee Ji and Tanya Khovanova and Shane Lee and Boyan Litchev and Anshul Rastogi and Garima Rastogi and Andrew ZhaoA minus sign that used to annoy me but now I know why it is there
https://read.somethingorotherwhatever.com/entry/AminussignthatusedtoannoymebutnowIknowwhyitisthere
We consider two well known constructions of link invariants. One uses skein
theory: you resolve each crossing of the link as a linear combination of things
that don't cross, until you eventually get a linear combination of links with
no crossings, which you turn into a polynomial. The other uses quantum groups:
you construct a functor from a topological category to some category of
representations in such a way that (directed framed) links get sent to
endomorphisms of the trivial representation, which are just rational functions.
Certain instances of these two constructions give rise to essentially the same
invariants, but when one carefully matches them there is a minus sign that
seems out of place. We discuss exactly how the constructions match up in the
case of the Jones polynomial, and where the minus sign comes from. On the
quantum group side, one is led to use a non-standard ribbon element, which then
allows one to consider a larger topological category.AminussignthatusedtoannoymebutnowIknowwhyitisthereMon, 21 Sep 2020 00:00:00 +0000Peter TingleyYou Could Have Invented Spectral Sequences
https://read.somethingorotherwhatever.com/entry/YouCouldHaveInventedSpectralSequences
The subject of spectral sequences has a reputation for being difficult for the beginner. Even G. W. Whitehead (quoted in John McCleary) once remarked, “The machinery of spectral sequences, stemming from the algebraic work of Lyndon and Koszul, seemed complicated and obscure to many topologists.”YouCouldHaveInventedSpectralSequencesMon, 21 Sep 2020 00:00:00 +0000Timothy Y. ChowMangoes and Blueberries
https://read.somethingorotherwhatever.com/entry/MangoesandBlueberries
We prove the following conjecture of Erdős and Hajnal:
For every integer \(k\) there is an \(f(k)\) such that if for a graph \(G\), every subgraph \(H\) of \(G\) has a stable set containing vertices, then \(G\) contains a set \(X\) of at most \(f(k)\) vertices such that \(G−X\) is bipartite.
This conjecture was related to me by Paul Erdős at a conference held in Annecy during July of 1996. I regret not being able to share the answer with him.MangoesandBlueberriesMon, 21 Sep 2020 00:00:00 +0000Bruce ReedPaperfolding morphisms, planefilling curves, and fractal tiles
https://read.somethingorotherwhatever.com/entry/Paperfoldingmorphismsplanefillingcurvesandfractaltiles
An interesting class of automatic sequences emerges from iterated
paperfolding. The sequences generate curves in the plane with an almost
periodic structure. We generalize the results obtained by Davis and Knuth on
the self-avoiding and planefilling properties of these curves, giving simple
geometric criteria for a complete classification. Finally, we show how the
automatic structure of the sequences leads to self-similarity of the curves,
which turns the planefilling curves in a scaling limit into fractal tiles. For
some of these tiles we give a particularly simple formula for the Hausdorff
dimension of their boundary.PaperfoldingmorphismsplanefillingcurvesandfractaltilesMon, 21 Sep 2020 00:00:00 +0000Michel DekkingPlane-filling curves on all uniform grids
https://read.somethingorotherwhatever.com/entry/Planefillingcurvesonalluniformgrids
We describe a search for plane-filling curves traversing all edges of a grid
once. The curves are given by Lindenmayer systems with only one non-constant
letter. All such curves for small orders on three grids have been found. For
all uniform grids we show how curves traversing all points once can be obtained
from the curves found. Curves traversing all edges once are described for the
four uniform grids where they exist.PlanefillingcurvesonalluniformgridsMon, 21 Sep 2020 00:00:00 +0000Jörg ArndtParking Functions: Choose Your Own Adventure
https://read.somethingorotherwhatever.com/entry/ParkingFunctionsChooseYourOwnAdventure
Warning. The reading of this paper will send you down many winding roads
toward new and exciting research topics enumerating generalized parking
functions. Buckle up!ParkingFunctionsChooseYourOwnAdventureMon, 21 Sep 2020 00:00:00 +0000Joshua Carlson and Alex Christensen and Pamela E. Harris and Zakiya Jones and Andrés Ramos RodríguezThere are not Exactly Five Objects
https://read.somethingorotherwhatever.com/entry/TherearenotExactlyFiveObjects
The purpose of this note is to present a solution to a problem posed informally by A. Wilkie at the 1977 ASL meeting in Wrocław: Formulate "the cardinality of the universe is not exactly five" as a Horn sentence. Although this solution is not new -- I found it and M. Morley found another solution during the same meeting -- continuing sporadic inquiries about it suggest that its publication may be appropriate, even at this late date.TherearenotExactlyFiveObjectsMon, 21 Sep 2020 00:00:00 +0000Andreas BlassAstonishing Numbers
https://read.somethingorotherwhatever.com/entry/AstonishingNumbers
We say that an ordered pair of positive integers \(a,b\) with \(a \lt b\) is astonishing if the sum of the integers from \(a\) to \(b\), inclusive, is equal to the digits of \(a\) followed by the digits of \(b\). Determine all astonishing ordered pairs.AstonishingNumbersFri, 28 Aug 2020 00:00:00 +0000Richard HoshinoAn Optimal Solution for the Muffin Problem
https://read.somethingorotherwhatever.com/entry/AnOptimalSolutionfortheMuffinProblem
The muffin problem asks us to divide $m$ muffins into pieces and assign each
of those pieces to one of $s$ students so that the sizes of the pieces assigned
to each student total $m/s$, with the objective being to maximize the size of
the smallest piece in the solution. We present a recursive algorithm for
solving any muffin problem and demonstrate that it always produces an optimal
solution.AnOptimalSolutionfortheMuffinProblemMon, 17 Aug 2020 00:00:00 +0000Richard E. ChatwinSome Doubly Exponential Sequences
https://read.somethingorotherwhatever.com/entry/SomeDoublyExponentialSequences
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 \(x_0,x_1,x_2,\cdots\)!SomeDoublyExponentialSequencesMon, 13 Jul 2020 00:00:00 +0000A. V. Aho and N. J. A. SloaneGoldbug Variations
https://read.somethingorotherwhatever.com/entry/GoldbugVariations
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.GoldbugVariationsFri, 12 Jun 2020 00:00:00 +0000Michael KleberIndigenous perspectives in maths: Understanding Gurruṯu
https://read.somethingorotherwhatever.com/entry/IndigenousperspectivesinmathsUnderstandingGurruu
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.IndigenousperspectivesinmathsUnderstandingGurruuMon, 08 Jun 2020 00:00:00 +0000Chris MatthewsGergonne's Card Trick, Positional Notation, and Radix Sort
https://read.somethingorotherwhatever.com/entry/GergonnesCardTrickPositionalNotationandRadixSort
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.GergonnesCardTrickPositionalNotationandRadixSortSat, 30 May 2020 00:00:00 +0000Ethan D. BolkerInfinitude of Primes Using Formal Language Theory
https://read.somethingorotherwhatever.com/entry/InfinitudeofPrimesUsingFormalLanguageTheory
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.InfinitudeofPrimesUsingFormalLanguageTheoryMon, 25 May 2020 00:00:00 +0000Aalok Thakkar38406501359372282063949 & all that: Monodromy of Fano Problems
https://read.somethingorotherwhatever.com/entry/38406501359372282063949allthatMonodromyofFanoProblems
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 $W(D_k)$.38406501359372282063949allthatMonodromyofFanoProblemsWed, 06 May 2020 00:00:00 +0000Sachi Hashimoto and Borys KadetsWord calculus in the fundamental group of the Menger curve
https://read.somethingorotherwhatever.com/entry/WordcalculusinthefundamentalgroupoftheMengercurve
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.WordcalculusinthefundamentalgroupoftheMengercurveSat, 02 May 2020 00:00:00 +0000Hanspeter Fischer and Andreas ZastrowThe distance of a permutation from a subgroup of \(S_n\)
https://read.somethingorotherwhatever.com/entry/ThedistanceofapermutationfromasubgroupofSn
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.ThedistanceofapermutationfromasubgroupofSnThu, 30 Apr 2020 00:00:00 +0000Richard G. E. PinchShadow movies not arising from knots
https://read.somethingorotherwhatever.com/entry/Shadowmoviesnotarisingfromknots
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'.ShadowmoviesnotarisingfromknotsTue, 21 Apr 2020 00:00:00 +0000Daniel Denton and Peter DoyleProjectivizing Set
https://read.somethingorotherwhatever.com/entry/ProjectivizingSet
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.ProjectivizingSetSun, 19 Apr 2020 00:00:00 +0000Cathy Hsu and Jonah Ostroff and Lucas Van MeterFour Pages Are Indeed Necessary for Planar Graphs
https://read.somethingorotherwhatever.com/entry/FourPagesAreIndeedNecessaryforPlanarGraphs
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.FourPagesAreIndeedNecessaryforPlanarGraphsSun, 19 Apr 2020 00:00:00 +0000Michael A. Bekos and Michael Kaufmann and Fabian Klute and Sergey Pupyrev and Chrysanthi Raftopoulou and Torsten UeckerdtConway's Tiling Groups on JSTOR
https://read.somethingorotherwhatever.com/entry/ConwaysTilingGroupsonJSTOR
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.ConwaysTilingGroupsonJSTORWed, 15 Apr 2020 00:00:00 +0000William P. ThurstonExtreme Proofs I: The Irrationality of √2
https://read.somethingorotherwhatever.com/entry/ExtremeProofsITheIrrationalityof2
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.ExtremeProofsITheIrrationalityof2Mon, 13 Apr 2020 00:00:00 +0000John H. Conway and Joseph Shipman and John H. Conway and Joseph ShipmanKnots and links in spatial graphs
https://read.somethingorotherwhatever.com/entry/KnotsAndLinksInSpatialGraphs
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.KnotsAndLinksInSpatialGraphsSun, 12 Apr 2020 00:00:00 +0000J. H. Conway and C. McA. GordonHow long is my toilet roll? – a simple exercise in mathematical modelling
https://read.somethingorotherwhatever.com/entry/HowLongIsMyToiletRoll
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.HowLongIsMyToiletRollThu, 09 Apr 2020 00:00:00 +0000Peter R. JohnstonComputing Linkages
https://read.somethingorotherwhatever.com/entry/ComputingLinkages
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.ComputingLinkagesMon, 06 Apr 2020 00:00:00 +0000Andries de ManFusible numbers and Peano Arithmetic
https://read.somethingorotherwhatever.com/entry/FusiblenumbersandPeanoArithmetic
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 $n$."FusiblenumbersandPeanoArithmeticWed, 01 Apr 2020 00:00:00 +0000Jeff Erickson and Gabriel Nivasch and Junyan XuA catalogue of mathematical formulas involving π, with analysis
https://read.somethingorotherwhatever.com/entry/ACatalogueOfMathematicalFormulasInvolvingPiWithAnalysis
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.ACatalogueOfMathematicalFormulasInvolvingPiWithAnalysisFri, 27 Mar 2020 00:00:00 +0000David H. BaileyThe Role of Number Notation: Sign-Value Notation Number Processing is Easier than Place-Value
https://read.somethingorotherwhatever.com/entry/TheRoleofNumberNotationSignValueNotationNumberProcessingisEasierthanPlaceValue
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.TheRoleofNumberNotationSignValueNotationNumberProcessingisEasierthanPlaceValueThu, 19 Mar 2020 00:00:00 +0000Krajcsi, Attila and Szabó, EszterOn Some two way Classifications of Integers
https://read.somethingorotherwhatever.com/entry/OnSometwowayClassificationsofIntegers
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.OnSometwowayClassificationsofIntegersFri, 13 Mar 2020 00:00:00 +0000J. Lambek and L. MoserTheory and applications of the double-base number system
https://read.somethingorotherwhatever.com/entry/Theoryandapplicationsofthedoublebasenumbersystem
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.TheoryandapplicationsofthedoublebasenumbersystemMon, 03 Feb 2020 00:00:00 +0000 V.S. Dimitrov and G.A. Jullien and W.C. Miller Sandwich semigroups in diagram categories
https://read.somethingorotherwhatever.com/entry/Sandwichsemigroupsindiagramcategories
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.SandwichsemigroupsindiagramcategoriesMon, 03 Feb 2020 00:00:00 +0000Ivana Đurđev and Igor Dolinka and James EastSmall-data computing: correct calculator arithmetic
https://read.somethingorotherwhatever.com/entry/Smalldatacomputingcorrectcalculatorarithmetic
Rounding errors are usually avoidable, and sometimes we can afford to avoid them.SmalldatacomputingcorrectcalculatorarithmeticMon, 13 Jan 2020 00:00:00 +0000Hans-J. BoehmThe no-three-in-line problem on a torus
https://read.somethingorotherwhatever.com/entry/Thenothreeinlineproblemonatorus
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
$2 \leq n \leq 19$.ThenothreeinlineproblemonatorusTue, 10 Dec 2019 00:00:00 +0000Jim Fowler and Andrew Groot and Deven Pandya and Bart SnappA unique pair of triangles
https://read.somethingorotherwhatever.com/entry/Auniquepairoftriangles
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.AuniquepairoftrianglesTue, 10 Dec 2019 00:00:00 +0000Yoshinosuke Hirakawa and Hideki Matsumura“Lights Out” and Variants
https://read.somethingorotherwhatever.com/entry/LightsOutandVariants
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.LightsOutandVariantsMon, 09 Dec 2019 00:00:00 +0000Martin KrehThe Graph Menagerie: Abstract Algebra and the Mad Veterinarian
https://read.somethingorotherwhatever.com/entry/TheGraphMenagerieAbstractAlgebraAndTheMadVeterinarian
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.TheGraphMenagerieAbstractAlgebraAndTheMadVeterinarianWed, 06 Nov 2019 00:00:00 +0000Gene Abrams and Jessica K. SklarPort-and-Sweep Solitaire
https://read.somethingorotherwhatever.com/entry/portandsweepsolitaire
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. portandsweepsolitaireSat, 12 Oct 2019 00:00:00 +0000Jacob SiehlerFinding a princess in a palace: A pursuit-evasion problem
https://read.somethingorotherwhatever.com/entry/FindingaprincessinapalaceApursuitevasionproblem
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.FindingaprincessinapalaceApursuitevasionproblemTue, 08 Oct 2019 00:00:00 +0000John R. Britnell and Mark WildonCatching a mouse on a tree
https://read.somethingorotherwhatever.com/entry/Catchingamouseonatree
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.CatchingamouseonatreeTue, 08 Oct 2019 00:00:00 +0000Vytautas Gruslys and Arès MérouehHow to Hunt an Invisible Rabbit on a Graph
https://read.somethingorotherwhatever.com/entry/HowtoHuntanInvisibleRabbitonaGraph
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).HowtoHuntanInvisibleRabbitonaGraphTue, 08 Oct 2019 00:00:00 +0000Tatjana V. Abramovskaya and Fedor V. Fomin and Petr A. Golovach and Michał PilipczukPalindromes in Different Bases: A Conjecture of J. Ernest Wilkins
https://read.somethingorotherwhatever.com/entry/PalindromesinDifferentBasesAConjectureofJErnestWilkins
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.PalindromesinDifferentBasesAConjectureofJErnestWilkinsSat, 14 Sep 2019 00:00:00 +0000Edray Herber GoinsPercolation is Odd
https://read.somethingorotherwhatever.com/entry/PercolationisOdd
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.PercolationisOddSat, 14 Sep 2019 00:00:00 +0000Stephan Mertens and Cristopher MooreFractal Sequences
https://read.somethingorotherwhatever.com/entry/FractalSequences
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.)FractalSequencesWed, 21 Aug 2019 00:00:00 +0000Clark KimberlingCreation of Hyperbolic Ornaments
https://read.somethingorotherwhatever.com/entry/CreationofHyperbolicOrnaments
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.CreationofHyperbolicOrnamentsThu, 08 Aug 2019 00:00:00 +0000Martin von GagernNumbers for Masochists: A Guide to Mental Factoring
https://read.somethingorotherwhatever.com/entry/NumbersforMasochistsAGuidetoMentalFactoring
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.NumbersforMasochistsAGuidetoMentalFactoringWed, 31 Jul 2019 00:00:00 +0000Hilarie Orman and Richard SchroeppelFiboquadratic Sequences and Extensions of the Cassini Identity Raised From the Study of Rithmomachia
https://read.somethingorotherwhatever.com/entry/FiboquadraticSequencesandExtensionsoftheCassiniIdentityRaisedFromtheStudyofRithmomachia
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.FiboquadraticSequencesandExtensionsoftheCassiniIdentityRaisedFromtheStudyofRithmomachiaThu, 18 Jul 2019 00:00:00 +0000Tomás Guardia and Douglas JiménezChords of an ellipse, Lucas polynomials, and cubic equations
https://read.somethingorotherwhatever.com/entry/ChordsofanellipseLucaspolynomialsandcubicequations
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.ChordsofanellipseLucaspolynomialsandcubicequationsSun, 30 Jun 2019 00:00:00 +0000Ben Blum-Smith and Japheth WoodHex: A Strategy Guide
https://read.somethingorotherwhatever.com/entry/HexAStrategyGuide
HexAStrategyGuideFri, 31 May 2019 00:00:00 +0000Matthew SeymourBrainfilling Curves - a Fractal Bestiary
https://read.somethingorotherwhatever.com/entry/BrainfillingCurvesaFractalBestiary
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. BrainfillingCurvesaFractalBestiaryThu, 09 May 2019 00:00:00 +0000Jeffrey VentrellaA Contribution to the Mathematical Theory of Big Game Hunting
https://read.somethingorotherwhatever.com/entry/AContributiontotheMathematicalTheoryofBigGameHunting
Problem: To Catch a Lion in the Sahara Desert.AContributiontotheMathematicalTheoryofBigGameHuntingThu, 09 May 2019 00:00:00 +0000Ralph BoasPlanar Hypohamiltonian Graphs on 40 Vertices
https://read.somethingorotherwhatever.com/entry/PlanarHypohamiltonianGraphson40Vertices
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.PlanarHypohamiltonianGraphson40VerticesThu, 09 May 2019 00:00:00 +0000Mohammadreza Jooyandeh and Brendan D. McKay and Patric R. J. Östergård and Ville H. Pettersson and Carol T. ZamfirescuThe Swiss-Cheese Operad
https://read.somethingorotherwhatever.com/entry/TheSwissCheeseOperad
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.TheSwissCheeseOperadThu, 09 May 2019 00:00:00 +0000Alexander A. VoronovThe graphs behind Reuleaux polyhedra
https://read.somethingorotherwhatever.com/entry/ThegraphsbehindReuleauxpolyhedra
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.ThegraphsbehindReuleauxpolyhedraThu, 09 May 2019 00:00:00 +0000Luis Montejano and Eric Pauli and Miguel Raggi and Edgardo Roldán-PensadoA universal differential equation
https://read.somethingorotherwhatever.com/entry/Auniversaldifferentialequation
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)\]AuniversaldifferentialequationThu, 09 May 2019 00:00:00 +0000Lee A. RubelThe Sensual (quadratic) Form
https://read.somethingorotherwhatever.com/entry/TheSensualquadraticForm
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.TheSensualquadraticFormThu, 09 May 2019 00:00:00 +0000John Horton ConwayPerforming Mathematical Operations with Metamaterials
https://read.somethingorotherwhatever.com/entry/PerformingMathematicalOperationswithMetamaterials
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.PerformingMathematicalOperationswithMetamaterialsTue, 26 Mar 2019 00:00:00 +0000Alexandre Silva and Francesco Monticone and Giuseppe Castaldi and Vincenzo Galdi and Andrea Alù and Nader EnghetaBrazilian Primes Which Are Also Sophie Germain Primes
https://read.somethingorotherwhatever.com/entry/BrazilianPrimesWhichAreAlsoSophieGermainPrimes
We disprove a conjecture of Schott that no Brazilian primes are Sophie
Germain primes. We enumerate all counterexamples up to $10^{44}$.BrazilianPrimesWhichAreAlsoSophieGermainPrimesWed, 13 Mar 2019 00:00:00 +0000Jon Grantham and Hester GravesThe Takagi Function and Its Properties
https://read.somethingorotherwhatever.com/entry/TheTakagiFunctionandItsProperties
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.TheTakagiFunctionandItsPropertiesMon, 04 Mar 2019 00:00:00 +0000Jeffrey C. LagariasCodes, Lower Bounds, and Phase Transitions in the Symmetric Rendezvous Problem
https://read.somethingorotherwhatever.com/entry/CodesLowerBoundsandPhaseTransitionsintheSymmetricRendezvousProblem
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.CodesLowerBoundsandPhaseTransitionsintheSymmetricRendezvousProblemSat, 02 Mar 2019 00:00:00 +0000Varsha Dani and Thomas P. Hayes and Cristopher Moore and Alexander RussellYet Another Single Law for Lattices
https://read.somethingorotherwhatever.com/entry/YetAnotherSingleLawforLattices
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.YetAnotherSingleLawforLatticesSat, 02 Mar 2019 00:00:00 +0000William McCune and Ranganathan Padmanabhan and Robert VeroffThe Instructor's Guide to Real Induction
https://read.somethingorotherwhatever.com/entry/TheInstructorsGuidetoRealInduction
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.TheInstructorsGuidetoRealInductionSat, 02 Mar 2019 00:00:00 +0000Pete L. ClarkLey statistics
https://read.somethingorotherwhatever.com/entry/Leystatistics
LeystatisticsSat, 02 Mar 2019 00:00:00 +0000Michael BehrendWhat Is an Envelope?
https://read.somethingorotherwhatever.com/entry/WhatIsanEnvelope
WhatIsanEnvelopeMon, 04 Feb 2019 00:00:00 +0000J.W. Bruce and P.J. GiblinWhat is a closed-form number?
https://read.somethingorotherwhatever.com/entry/Whatisaclosedformnumber
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.WhatisaclosedformnumberWed, 09 Jan 2019 00:00:00 +0000Timothy Y. ChowAmusing Permutation Representations of Group Extensions
https://read.somethingorotherwhatever.com/entry/AmusingPermutationRepresentationsofGroupExtensions
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.AmusingPermutationRepresentationsofGroupExtensionsTue, 01 Jan 2019 00:00:00 +0000Yongju Bae and J. Scott Carter and Byeorhi KimDoing Math in Jest: Reflections on Useless Math, the Unreasonable Effectiveness of Mathematics, and the Ethical Obligations of Mathematicians
https://read.somethingorotherwhatever.com/entry/DoingMathinJestReflectionsonUselessMaththeUnreasonableEffectivenessofMathematicsandtheEthicalObligationsofMathematicians
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.DoingMathinJestReflectionsonUselessMaththeUnreasonableEffectivenessofMathematicsandtheEthicalObligationsofMathematiciansTue, 01 Jan 2019 00:00:00 +0000Gizem KaraaliMathematics applied to dressmaking
https://read.somethingorotherwhatever.com/entry/Mathematicsappliedtodressmaking
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.MathematicsappliedtodressmakingThu, 29 Nov 2018 00:00:00 +0000Christopher ZeemanRectangle Arithmetic
https://read.somethingorotherwhatever.com/entry/RectangleArithmetic
Another slant on fractionsRectangleArithmeticTue, 27 Nov 2018 00:00:00 +0000Bill GosperSeven Trees in One
https://read.somethingorotherwhatever.com/entry/SevenTreesinOne
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.SevenTreesinOneMon, 26 Nov 2018 00:00:00 +0000Andreas BlassDuotone Truchet-like tilings
https://read.somethingorotherwhatever.com/entry/DuotoneTruchetliketilings
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.DuotoneTruchetliketilingsTue, 13 Nov 2018 00:00:00 +0000Cameron BrownePrime Number Races
https://read.somethingorotherwhatever.com/entry/PrimeNumberRaces
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.PrimeNumberRacesMon, 12 Nov 2018 00:00:00 +0000Andrew Granville and Greg MartinConway's doughnuts
https://read.somethingorotherwhatever.com/entry/Conwaysdoughnuts
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.ConwaysdoughnutsSun, 04 Nov 2018 00:00:00 +0000Peter Doyle and Shikhin SethiNoncrossing partitions under rotation and reflection
https://read.somethingorotherwhatever.com/entry/Noncrossingpartitionsunderrotationandreflection
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].NoncrossingpartitionsunderrotationandreflectionSat, 27 Oct 2018 00:00:00 +0000David Callan and Len SmileySome Fundamental Theorems in Mathematics
https://read.somethingorotherwhatever.com/entry/SomeFundamentalTheoremsInMathematics
An expository hitchhiker's guide to some theorems in mathematics.SomeFundamentalTheoremsInMathematicsWed, 24 Oct 2018 00:00:00 +0000Oliver KnillA Tiling Database
https://read.somethingorotherwhatever.com/entry/ATilingDatabase
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. ATilingDatabaseWed, 24 Oct 2018 00:00:00 +0000Brian Wichmann and Tony LeeEnumeration of m-ary cacti
https://read.somethingorotherwhatever.com/entry/Enumerationofmarycacti
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.EnumerationofmarycactiTue, 23 Oct 2018 00:00:00 +0000Miklos Bona and Michel Bousquet and Gilbert Labelle and Pierre LerouxCalculator Forensics
https://read.somethingorotherwhatever.com/entry/CalculatorForensics
Results from the evaluation of this equation in degrees mode: arcsin (arccos (arctan (tan (cos (sin (9) ) ) ) ) ) CalculatorForensicsTue, 16 Oct 2018 00:00:00 +0000Mike SebastianSetting linear algebra problems
https://read.somethingorotherwhatever.com/entry/SettingLinearAlgebraProblems
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.SettingLinearAlgebraProblemsThu, 27 Sep 2018 00:00:00 +0000John D. SteelePower-law distributions in empirical data
https://read.somethingorotherwhatever.com/entry/Powerlawdistributionsinempiricaldata
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.PowerlawdistributionsinempiricaldataMon, 24 Sep 2018 00:00:00 +0000Aaron Clauset and Cosma Rohilla Shalizi and M. E. J. NewmanFinding the Bandit in a Graph: Sequential Search-and-Stop
https://read.somethingorotherwhatever.com/entry/FindingtheBanditinaGraphSequentialSearchandStop
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.FindingtheBanditinaGraphSequentialSearchandStopSat, 22 Sep 2018 00:00:00 +0000Pierre Perrault and Vianney Perchet and Michal ValkoThe largest small hexagon
https://read.somethingorotherwhatever.com/entry/Thelargestsmallhexagon
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%.ThelargestsmallhexagonSat, 22 Sep 2018 00:00:00 +0000R. L. GrahamThe Splitting Algorithm for Egyptian Fractions
https://read.somethingorotherwhatever.com/entry/TheSplittingAlgorithmforEgyptianFractions
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.TheSplittingAlgorithmforEgyptianFractionsThu, 20 Sep 2018 00:00:00 +0000L. BeeckmansMathematical Writing
https://read.somethingorotherwhatever.com/entry/MathematicalWriting
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.MathematicalWritingWed, 19 Sep 2018 00:00:00 +0000Donald E. Knuth and Tracy Larrabee and Paul M. RobertsThe Namer-Claimer game
https://read.somethingorotherwhatever.com/entry/TheNamerClaimergame
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.TheNamerClaimergameTue, 04 Sep 2018 00:00:00 +0000Ben BarberCharles Babbage's thoughts on notation
https://read.somethingorotherwhatever.com/entry/CharlesBabbageNotation
CharlesBabbageNotationTue, 04 Sep 2018 00:00:00 +0000Charles BabbageEuclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2017) and another new proof
https://read.somethingorotherwhatever.com/entry/Euclidstheoremontheinfinitudeofprimesahistoricalsurveyofitsproofs300BC2017andanothernewproof
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.Euclidstheoremontheinfinitudeofprimesahistoricalsurveyofitsproofs300BC2017andanothernewproofTue, 21 Aug 2018 00:00:00 +0000Romeo MeštrovićOn Some Regular Toroids
https://read.somethingorotherwhatever.com/entry/OnSomeRegularToroids
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 "Császár-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 Ákos Császár in 1949, and the latter one was found by the author, in 1977.OnSomeRegularToroidsThu, 26 Jul 2018 00:00:00 +0000Lajos SzilassiNational Curve Bank
https://read.somethingorotherwhatever.com/entry/NationalCurveBank
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.NationalCurveBankMon, 02 Jul 2018 00:00:00 +0000Shirley B. Gray and Stewart Venit and Russ AbbottHow to hear the shape of a billiard table
https://read.somethingorotherwhatever.com/entry/Howtoheartheshapeofabilliardtable
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.HowtoheartheshapeofabilliardtableWed, 27 Jun 2018 00:00:00 +0000Aaron Calderon and Solly Coles and Diana Davis and Justin Lanier and Andre OliveiraProof without Words: Fair Allocation of a Pizza
https://read.somethingorotherwhatever.com/entry/ProofwithoutWordsFairAllocationofaPizza
ProofwithoutWordsFairAllocationofaPizzaTue, 26 Jun 2018 00:00:00 +0000Larry Carter and Stan WagonWhen are Multiples of Polygonal Numbers again Polygonal Numbers?
https://read.somethingorotherwhatever.com/entry/WhenareMultiplesofPolygonalNumbersagainPolygonalNumbers
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.WhenareMultiplesofPolygonalNumbersagainPolygonalNumbersMon, 25 Jun 2018 00:00:00 +0000Jasbir S. Chahal and Nathan PriddisA surprisingly simple de Bruijn sequence construction
https://read.somethingorotherwhatever.com/entry/AsurprisinglysimpledeBruijnsequenceconstruction
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.AsurprisinglysimpledeBruijnsequenceconstructionMon, 25 Jun 2018 00:00:00 +0000Joe Sawada and Aaron Williams and DennisWongOne parameter is always enough
https://read.somethingorotherwhatever.com/entry/OneParameterIsAlwaysEnough
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.OneParameterIsAlwaysEnoughWed, 06 Jun 2018 00:00:00 +0000Steven T. PiantadosiRenyi's Parking Problem Revisited
https://read.somethingorotherwhatever.com/entry/RenyisParkingProblemRevisited
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, $0.7475979203...$.RenyisParkingProblemRevisitedMon, 21 May 2018 00:00:00 +0000Matthew P. Clay and Nandor J. SimanyiSoviet Street Mathematics: Landau’s License Plate Game
https://read.somethingorotherwhatever.com/entry/SovietStreetMathematicsLandausLicensePlateGame
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.SovietStreetMathematicsLandausLicensePlateGameThu, 17 May 2018 00:00:00 +0000Harun ŠiljakNeumbering
https://read.somethingorotherwhatever.com/entry/Neumbering
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.NeumberingThu, 17 May 2018 00:00:00 +0000O.G. Cassani and John H. ConwayThe Rearrangement Number
https://read.somethingorotherwhatever.com/entry/TheRearrangementNumber
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.TheRearrangementNumberWed, 16 May 2018 00:00:00 +0000Andreas Blass and Jörg Brendle and Will Brian and Joel David Hamkins and Michael Hardy and Paul B. LarsonA description of the outer automorphism of \(S_6\), and the invariants of six points in projective space
https://read.somethingorotherwhatever.com/entry/AdescriptionoftheouterautomorphismofS6andtheinvariantsofsixpointsinprojectivespace
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 \(\mathbb{P}^3\).AdescriptionoftheouterautomorphismofS6andtheinvariantsofsixpointsinprojectivespaceMon, 14 May 2018 00:00:00 +0000Ben Howard and John Millson and Andrew Snowden and Ravi VakilExact Enumeration of Garden of Eden Partitions
https://read.somethingorotherwhatever.com/entry/ExactEnumerationOfGardenOfEdenPartitions
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.ExactEnumerationOfGardenOfEdenPartitionsSun, 13 May 2018 00:00:00 +0000Brian Hopkins and James A. SellersHow long does it take to catch a wild kangaroo?
https://read.somethingorotherwhatever.com/entry/Howlongdoesittaketocatchawildkangaroo
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 $n$.HowlongdoesittaketocatchawildkangarooSat, 12 May 2018 00:00:00 +0000Ravi Montenegro and Prasad TetaliMath Counterexamples
https://read.somethingorotherwhatever.com/entry/MathCounterexamples
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.MathCounterexamplesWed, 09 May 2018 00:00:00 +0000Jean-Pierre MerxRedefining the integral
https://read.somethingorotherwhatever.com/entry/Redefiningtheintegral
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".RedefiningtheintegralTue, 08 May 2018 00:00:00 +0000Derek OrrMost primitive groups have messy invariants
https://read.somethingorotherwhatever.com/entry/Mostprimitivegroupshavemessyinvariants
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 \(|G\).MostprimitivegroupshavemessyinvariantsSun, 06 May 2018 00:00:00 +0000W.C. Huffman and N.J.A. SloaneAn Invitation to Inverse Group Theory
https://read.somethingorotherwhatever.com/entry/AnInvitationtoInverseGroupTheory
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.AnInvitationtoInverseGroupTheoryThu, 19 Apr 2018 00:00:00 +0000João Araújo and Peter J. Cameron and Francesco MatucciThe Mathematical Coloring Book
https://read.somethingorotherwhatever.com/entry/TheMathematicalColoringBook
Due to the author's correspondence with Van der Waerden, Erdös, 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.TheMathematicalColoringBookWed, 18 Apr 2018 00:00:00 +0000Alexander SoiferNice Neighbors: A Brief Adventure in Mathematical Gamification
https://read.somethingorotherwhatever.com/entry/NiceNeighbours
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. NiceNeighboursTue, 17 Apr 2018 00:00:00 +0000Chris StaeckerThe materiality of mathematics: presenting mathematics at the blackboard
https://read.somethingorotherwhatever.com/entry/Thematerialityofmathematicspresentingmathematicsattheblackboard
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.ThematerialityofmathematicspresentingmathematicsattheblackboardFri, 06 Apr 2018 00:00:00 +0000Christian GreiffenhagenA Puzzle for Pirates
https://read.somethingorotherwhatever.com/entry/APuzzleForPirates
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.APuzzleForPiratesTue, 03 Apr 2018 00:00:00 +0000Ian StewartNumeral Systems of the World
https://read.somethingorotherwhatever.com/entry/NumeralSystemsoftheWorld
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.NumeralSystemsoftheWorldSun, 25 Mar 2018 00:00:00 +0000Bernard Comrie and Eugene ChanHow do you fix an Oval Track Puzzle?
https://read.somethingorotherwhatever.com/entry/HowdoyoufixanOvalTrackPuzzle
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.HowdoyoufixanOvalTrackPuzzleTue, 13 Mar 2018 00:00:00 +0000David A. Nash and Sara RandallNotable Properties of Specific Numbers
https://read.somethingorotherwhatever.com/entry/NotablePropertiesofSpecificNumbers
NotablePropertiesofSpecificNumbersMon, 05 Mar 2018 00:00:00 +0000Robert Munafoalmanach ou dictionnaire des nombres - curiosités et propriétés
https://read.somethingorotherwhatever.com/entry/almanachoudictionnairedesnombrescuriositsetproprits
almanachoudictionnairedesnombrescuriositsetpropritsMon, 05 Mar 2018 00:00:00 +0000Gérard VilleminFingerprint databases for theorems
https://read.somethingorotherwhatever.com/entry/Fingerprintdatabasesfortheorems
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.FingerprintdatabasesfortheoremsMon, 12 Feb 2018 00:00:00 +0000Sara C. Billey and Bridget E. TennerStraight knots
https://read.somethingorotherwhatever.com/entry/Straightknots
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.StraightknotsThu, 01 Feb 2018 00:00:00 +0000Nicholas OwadMechanical Computing Systems Using Only Links and Rotary Joints
https://read.somethingorotherwhatever.com/entry/MechanicalComputingSystemsUsingOnlyLinksandRotaryJoints
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.MechanicalComputingSystemsUsingOnlyLinksandRotaryJointsTue, 30 Jan 2018 00:00:00 +0000Ralph C. Merkle and Robert A. Freitas Jr. and Tad Hogg and Thomas E. Moore and Matthew S. Moses and James RyleyThe Muffin Problem
https://read.somethingorotherwhatever.com/entry/TheMuffinProblem
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.TheMuffinProblemTue, 30 Jan 2018 00:00:00 +0000Guangiqi Cui and John Dickerson and Naveen Durvasula and William Gasarch and Erik Metz and Naveen Raman and Sung Hyun YooThe grasshopper problem
https://read.somethingorotherwhatever.com/entry/Thegrasshopperproblem
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 $d \gtrsim
\pi^{-1/2}$.ThegrasshopperproblemWed, 24 Jan 2018 00:00:00 +0000Olga Goulko and Adrian KentAn empty exercise
https://read.somethingorotherwhatever.com/entry/Anemptyexercise
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.AnemptyexerciseWed, 24 Jan 2018 00:00:00 +0000Carl de BoorPlaying Games with Algorithms: Algorithmic Combinatorial Game Theory
https://read.somethingorotherwhatever.com/entry/PlayingGameswithAlgorithmsAlgorithmicCombinatorialGameTheory
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.PlayingGameswithAlgorithmsAlgorithmicCombinatorialGameTheoryTue, 23 Jan 2018 00:00:00 +0000Erik D. Demaine and Robert A. HearnWhat did Ryser Conjecture?
https://read.somethingorotherwhatever.com/entry/WhatdidRyserConjecture
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.WhatdidRyserConjectureWed, 10 Jan 2018 00:00:00 +0000Darcy Best and Ian M. WanlessNear Miss Polyhedra
https://read.somethingorotherwhatever.com/entry/NearMissPolyhedra
The polyhedra on this page are not quite regular, but as they are close I present them here as 'near misses'. NearMissPolyhedraWed, 10 Jan 2018 00:00:00 +0000Jim McNeillRandom railways modeled as random 3-regular graphs
https://read.somethingorotherwhatever.com/entry/Randomrailwaysmodeledasrandom3regulargraphs
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.Randomrailwaysmodeledasrandom3regulargraphsTue, 09 Jan 2018 00:00:00 +0000Hans GarmoAny Monotone Boolean Function Can Be Realized by Interlocked Polygons
https://read.somethingorotherwhatever.com/entry/AnyMonotoneBooleanFunctionCanBeRealizedByInterlockedPolygons
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.AnyMonotoneBooleanFunctionCanBeRealizedByInterlockedPolygonsMon, 08 Jan 2018 00:00:00 +0000Erik D. Demaine and Martin L. Demaine and Ryuhei UeharaFolding Polyominoes into (Poly)Cubes
https://read.somethingorotherwhatever.com/entry/FoldingPolyominoesintoPolyCubes
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.FoldingPolyominoesintoPolyCubesWed, 03 Jan 2018 00:00:00 +0000Oswin Aichholzer and Michael Biro and Erik D. Demaine and Martin L. Demaine and David Eppstein and Sándor P. Fekete and Adam Hesterberg and Irina Kostitsyna and Christiane SchmidtSpot it(R) Solitaire
https://read.somethingorotherwhatever.com/entry/SpotitSolitaire
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.SpotitSolitaireMon, 18 Dec 2017 00:00:00 +0000Donna A. DietzMechanisms by Tchebyshev
https://read.somethingorotherwhatever.com/entry/MechanismsbyTchebyshev
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 Musée des Arts et Métiers 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.MechanismsbyTchebyshevMon, 04 Dec 2017 00:00:00 +0000A compilation of LEGO Technic parts to support learning experiments on linkages
https://read.somethingorotherwhatever.com/entry/AcompilationofLEGOTechnicpartstosupportlearningexperimentsonlinkages
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.AcompilationofLEGOTechnicpartstosupportlearningexperimentsonlinkagesMon, 04 Dec 2017 00:00:00 +0000Zoltán Kovács and Benedek KovácsThe game of plates and olives
https://read.somethingorotherwhatever.com/entry/Thegameofplatesandolives
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 $M_n \leq (4/e)^{n+o(n)}n^n$.ThegameofplatesandolivesThu, 30 Nov 2017 00:00:00 +0000Teena Carroll and David GalvinTwo-dimensional photonic aperiodic crystals based on Thue-Morse sequence
https://read.somethingorotherwhatever.com/entry/TwodimensionalphotonicaperiodiccrystalsbasedonThueMorsesequence
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.TwodimensionalphotonicaperiodiccrystalsbasedonThueMorsesequenceMon, 13 Nov 2017 00:00:00 +0000Luigi Moretti and Vito Mocella Mathemagics
https://read.somethingorotherwhatever.com/entry/Mathemagics
My thesis is:there is another way of doing mathematics, equally successful, and the two methods should supplement each other and not fight.MathemagicsTue, 24 Oct 2017 00:00:00 +0000Pierre CartierChocolate games that satisfy the inequality \(y \leq \left \lfloor \frac{z}{k} \right\rfloor\) for \(k=1,2\) and Grundy numbers
https://read.somethingorotherwhatever.com/entry/ChocolategamesthatsatisfytheinequalityforandGrundynumbers
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.ChocolategamesthatsatisfytheinequalityforandGrundynumbersWed, 18 Oct 2017 00:00:00 +0000Shunsuke Nakamura and Ryo Hanafusa and Wataru Ogasa and Takeru Kitagawa and Ryohei MiyaderaCuriosities of arithmetic gases
https://read.somethingorotherwhatever.com/entry/Curiositiesofarithmeticgases
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.CuriositiesofarithmeticgasesMon, 16 Oct 2017 00:00:00 +0000Ioannis Bakas and Mark J. BowickThe tail does not determine the size of the giant
https://read.somethingorotherwhatever.com/entry/Thetaildoesnotdeterminethesizeofthegiant
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.ThetaildoesnotdeterminethesizeofthegiantWed, 04 Oct 2017 00:00:00 +0000Maria Deijfen and Sebastian Rosengren and Pieter TrapmanEarliest Uses of Various Mathematical Symbols
https://read.somethingorotherwhatever.com/entry/EarliestUsesofVariousMathematicalSymbols
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.EarliestUsesofVariousMathematicalSymbolsTue, 12 Sep 2017 00:00:00 +0000Jeff Miller$H$-supermagic labelings for firecrackers, banana trees and flowers
https://read.somethingorotherwhatever.com/entry/Hsupermagiclabelingsforfirecrackersbananatreesandflowers
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.HsupermagiclabelingsforfirecrackersbananatreesandflowersMon, 11 Sep 2017 00:00:00 +0000Rachel Wulan Nirmalasari Wijaya and Andrea Semaničová-Feňovčíková and Joe Ryan and Thomas KalinowskiFactoring in the Chicken McNugget monoid
https://read.somethingorotherwhatever.com/entry/FactoringintheChickenMcNuggetmonoid
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.FactoringintheChickenMcNuggetmonoidWed, 06 Sep 2017 00:00:00 +0000Scott Chapman and Christopher O'NeillOfficially, Home Plate doesn’t exist.
https://read.somethingorotherwhatever.com/entry/OfficiallyHomePlateDoesntExist
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!OfficiallyHomePlateDoesntExistWed, 16 Aug 2017 00:00:00 +0000Bill GosperThe Sleeping Beauty Controversy
https://read.somethingorotherwhatever.com/entry/TheSleepingBeautyControversy
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.TheSleepingBeautyControversyMon, 14 Aug 2017 00:00:00 +0000Peter WinklerThe Bulgarian solitaire and the mathematics around it
https://read.somethingorotherwhatever.com/entry/TheBulgariansolitaireandthemathematicsaroundit
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.TheBulgariansolitaireandthemathematicsarounditThu, 10 Aug 2017 00:00:00 +0000Vesselin DrenskyFrustration solitaire
https://read.somethingorotherwhatever.com/entry/Frustrationsolitaire
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.FrustrationsolitaireThu, 10 Aug 2017 00:00:00 +0000Peter G. Doyle and Charles M. Grinstead and J. Laurie SnellMaximum genus of the generalized Jenga game
https://read.somethingorotherwhatever.com/entry/MaximumgenusofthegeneralizedJengagame
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.MaximumgenusofthegeneralizedJengagameMon, 07 Aug 2017 00:00:00 +0000Rika Akiyama and Nozomi Abe and Hajime Fujita and Yukie Inaba and Mari Hataoka and Shiori Ito and Satomi SeitaAn unusual cubic representation problem
https://read.somethingorotherwhatever.com/entry/AnUnusualCubicRepresentationProblem
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).AnUnusualCubicRepresentationProblemMon, 07 Aug 2017 00:00:00 +0000Andrew Bremner and Allan MacleodThe Curling Number Conjecture
https://read.somethingorotherwhatever.com/entry/TheCurlingNumberConjecture
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.TheCurlingNumberConjectureMon, 31 Jul 2017 00:00:00 +0000Benjamin Chaffin and N. J. A. SloaneProof of Conway's Lost Cosmological Theorem
https://read.somethingorotherwhatever.com/entry/ProofofConwaysLostCosmologicalTheorem
John Horton Conway's Cosmological Theorem, about Audioactive sequences, for
which no extant proof existed, is given a computer-generated proof, hopefully
for good.ProofofConwaysLostCosmologicalTheoremWed, 26 Jul 2017 00:00:00 +0000Shalosh B. Ekhad and Doron ZeilbergerAn Explicit Isometric Reduction of the Unit Sphere into an Arbitrarily Small Ball
https://read.somethingorotherwhatever.com/entry/AnExplicitIsometricReductionoftheUnitSphereintoanArbitrarilySmallBall
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.AnExplicitIsometricReductionoftheUnitSphereintoanArbitrarilySmallBallTue, 25 Jul 2017 00:00:00 +0000Evangelis Bartzos and Vincent Borrelli and Roland Denis and Francis Lazarus and Damien Rohmer and Boris ThibertAvian egg shape: Form, function, and evolution
https://read.somethingorotherwhatever.com/entry/AvianeggshapeFormfunctionandevolution
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.AvianeggshapeFormfunctionandevolutionMon, 24 Jul 2017 00:00:00 +0000Mary Caswell Stoddard and Ee Hou Yong and Derya Akkaynak and Catherine Sheard and Joseph A. Tobias and L. MahadevanOvercurvature describes the buckling and folding of rings from curved origami to foldable tents
https://read.somethingorotherwhatever.com/entry/Overcurvature
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.OvercurvatureThu, 20 Jul 2017 00:00:00 +0000Pierre-Olivier Mouthuy and Michael Coulombier and Thomas Pardoen and Jean-Pierre Raskin and Alain M. JonasComputational complexity and 3-manifolds and zombies
https://read.somethingorotherwhatever.com/entry/Computationalcomplexityand3manifoldsandzombies
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.Computationalcomplexityand3manifoldsandzombiesThu, 13 Jul 2017 00:00:00 +0000Greg Kuperberg and Eric SampertonSteinhaus Longimeter
https://read.somethingorotherwhatever.com/entry/SteinhausLongimeter
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.SteinhausLongimeterSun, 02 Jul 2017 00:00:00 +0000Chris StaeckerPolylogarithmic ladders, hypergeometric series and the ten millionth digits of $ζ(3)$ and $ζ(5)$
https://read.somethingorotherwhatever.com/entry/Polylogarithmicladdershypergeometricseriesandthetenmillionthdigitsof3and5
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 $b=3$.Polylogarithmicladdershypergeometricseriesandthetenmillionthdigitsof3and5Thu, 29 Jun 2017 00:00:00 +0000D. J. BroadhurstUnunfoldable Polyhedra with Convex Faces
https://read.somethingorotherwhatever.com/entry/UnunfoldablePolyhedrawithConvexFaces
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.UnunfoldablePolyhedrawithConvexFacesMon, 26 Jun 2017 00:00:00 +0000Marshall Bern and Erik D. Demaine and David Eppstein and Eric Kuo and Andrea Mantler and Jack SnoeyinkThe ternary calculating machine of Thomas Fowler
https://read.somethingorotherwhatever.com/entry/TheternarycalculatingmachineofThomasFowler
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.TheternarycalculatingmachineofThomasFowlerThu, 22 Jun 2017 00:00:00 +0000Mark GluskerСетунь ВС (Setun Web Simulator)
https://read.somethingorotherwhatever.com/entry/SetunSimulator
SetunSimulatorThu, 22 Jun 2017 00:00:00 +0000TrinarygroupEvery positive integer is a sum of three palindromes
https://read.somethingorotherwhatever.com/entry/Everypositiveintegerisasumofthreepalindromes
For integer $g\ge 5$, we prove that any positive integer can be written as a
sum of three palindromes in base $g$.EverypositiveintegerisasumofthreepalindromesWed, 21 Jun 2017 00:00:00 +0000Javier Cilleruelo and Florian Luca and Lewis BaxterOn the date of Cauchy's contributions to the founding of the theory of groups
https://read.somethingorotherwhatever.com/entry/OnthedateofCauchyscontributionstothefoundingofthetheoryofgroups
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.OnthedateofCauchyscontributionstothefoundingofthetheoryofgroupsWed, 21 Jun 2017 00:00:00 +0000Peter M. NeumannGraphlopedia
https://read.somethingorotherwhatever.com/entry/Graphlopedia
A database of graphs for the use of mathematicians and other graph lovers. The graphs are ordered by degree sequence.GraphlopediaWed, 21 Jun 2017 00:00:00 +0000Sara 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 WarnerMath Magic
https://read.somethingorotherwhatever.com/entry/MathMagic
Math Magic is a web site devoted to original mathematical recreations.MathMagicWed, 21 Jun 2017 00:00:00 +0000Erich FriedmanA Handbook of Mathematical Discourse
https://read.somethingorotherwhatever.com/entry/TheHandbookofMathematicalDiscourse
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.TheHandbookofMathematicalDiscourseMon, 19 Jun 2017 00:00:00 +0000Charles WellsA formula goes to court: Partisan gerrymandering and the efficiency gap
https://read.somethingorotherwhatever.com/entry/AformulagoestocourtPartisangerrymanderingandtheefficiencygap
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.AformulagoestocourtPartisangerrymanderingandtheefficiencygapFri, 16 Jun 2017 00:00:00 +0000Mira Bernstein and Moon DuchinArithmetical structures on graphs
https://read.somethingorotherwhatever.com/entry/Arithmeticalstructuresongraphs
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.ArithmeticalstructuresongraphsWed, 14 Jun 2017 00:00:00 +0000Hugo Corrales and Carlos E. ValenciaRandomly juggling backwards
https://read.somethingorotherwhatever.com/entry/Randomlyjugglingbackwards
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.RandomlyjugglingbackwardsWed, 14 Jun 2017 00:00:00 +0000Allen KnutsonArrangements of Stars on the American Flag
https://read.somethingorotherwhatever.com/entry/ArrangementsOfStarsOnTheAmericanFlag
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.ArrangementsOfStarsOnTheAmericanFlagMon, 12 Jun 2017 00:00:00 +0000Dimitris Koukoulopoulos and Johann ThielCuckoo Filter: Simplification and Analysis
https://read.somethingorotherwhatever.com/entry/CuckooFilterSimplificationandAnalysis
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.CuckooFilterSimplificationandAnalysisTue, 23 May 2017 00:00:00 +0000David EppsteinThe opaque square
https://read.somethingorotherwhatever.com/entry/Theopaquesquare
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.TheopaquesquareMon, 22 May 2017 00:00:00 +0000Adrian Dumitrescu and Minghui JiangHomotopy type theory: the logic of space
https://read.somethingorotherwhatever.com/entry/Homotopytypetheorythelogicofspace
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).HomotopytypetheorythelogicofspaceWed, 03 May 2017 00:00:00 +0000Michael ShulmanTropical totally positive matrices
https://read.somethingorotherwhatever.com/entry/Tropicaltotallypositivematrices
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.TropicaltotallypositivematricesTue, 02 May 2017 00:00:00 +0000Stéphane Gaubert and Adi NivOn Fibonacci Quaternions
https://read.somethingorotherwhatever.com/entry/OnFibonacciQuaternions
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.OnFibonacciQuaternionsTue, 02 May 2017 00:00:00 +0000Serpil HaliciPVC Polyhedra
https://read.somethingorotherwhatever.com/entry/PVCPolyhedra
We describe how to construct a dodecahedron, tetrahedron, cube, and
octahedron out of pvc pipes using standard fittings.PVCPolyhedraTue, 02 May 2017 00:00:00 +0000David GlickensteinNo, This is not a Circle
https://read.somethingorotherwhatever.com/entry/NoThisisnotaCircle
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.NoThisisnotaCircleTue, 02 May 2017 00:00:00 +0000Zoltán KovácsThe number dictionary
https://read.somethingorotherwhatever.com/entry/Thenumberdictionary
The purpose is to provide an opportunity to show properties of numbers.ThenumberdictionaryWed, 12 Apr 2017 00:00:00 +0000Ovals and Egg Curves
https://read.somethingorotherwhatever.com/entry/OvalsandEggCurves
OvalsandEggCurvesMon, 03 Apr 2017 00:00:00 +0000Jürgen Köller Approval Voting in Product Societies
https://read.somethingorotherwhatever.com/entry/ApprovalVotinginProductSocieties
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.ApprovalVotinginProductSocietiesThu, 30 Mar 2017 00:00:00 +0000Kristen Mazur and Mutiara Sondjaja and Matthew Wright and Carolyn YarnallPauli Pascal Pyramids, Pauli Fibonacci Numbers, and Pauli Jacobsthal Numbers
https://read.somethingorotherwhatever.com/entry/PauliPascalPyramidsPauliFibonacciNumbersandPauliJacobsthalNumbers
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?PauliPascalPyramidsPauliFibonacciNumbersandPauliJacobsthalNumbersFri, 24 Mar 2017 00:00:00 +0000Martin Erik HornJewish Problems
https://read.somethingorotherwhatever.com/entry/JewishProblems
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.JewishProblemsThu, 23 Mar 2017 00:00:00 +0000Tanya Khovanova and Alexey RadulBest Laid Plans of Lions and Men
https://read.somethingorotherwhatever.com/entry/BestLaidPlansofLionsandMen
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
$\varepsilon>0$.BestLaidPlansofLionsandMenWed, 22 Mar 2017 00:00:00 +0000Mikkel Abrahamsen and Jacob Holm and Eva Rotenberg and Christian Wulff-NilsenBeyond Floating Point: Next-Generation Computer Arithmetic
https://read.somethingorotherwhatever.com/entry/BeyondFloatingPoint
BeyondFloatingPointTue, 21 Mar 2017 00:00:00 +0000John L. GustafsonPAPAC-00, a Do-It-Yourself Paper Computer
https://read.somethingorotherwhatever.com/entry/SENEWSPAPAC00
SENEWSPAPAC00Tue, 21 Mar 2017 00:00:00 +0000Rollin P. MayerCrazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9
https://read.somethingorotherwhatever.com/entry/CrazySequentialRepresentationNumbersfrom0to11111intermsofIncreasingandDecreasingOrdersof1to9
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.CrazySequentialRepresentationNumbersfrom0to11111intermsofIncreasingandDecreasingOrdersof1to9Sun, 12 Mar 2017 00:00:00 +0000Inder J. TanejaThe mathematics of lecture hall partitions
https://read.somethingorotherwhatever.com/entry/Themathematicsoflecturehallpartitions
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.ThemathematicsoflecturehallpartitionsWed, 08 Mar 2017 00:00:00 +0000Carla D. SavageStatistics Done Wrong
https://read.somethingorotherwhatever.com/entry/StatisticsDoneWrong
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.StatisticsDoneWrongWed, 01 Mar 2017 00:00:00 +0000Alex ReinhartThree Thoughts on “Prime Simplicity”
https://read.somethingorotherwhatever.com/entry/ThreeThoughtsonPrimeSimplicity
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.ThreeThoughtsonPrimeSimplicityMon, 27 Feb 2017 00:00:00 +0000Michael HardyPrime Simplicity
https://read.somethingorotherwhatever.com/entry/PrimeSimplicity
PrimeSimplicityMon, 27 Feb 2017 00:00:00 +0000Michael Hardy and Catherine WoodgoldMeaning in Classical Mathematics: Is it at Odds with Intuitionism?
https://read.somethingorotherwhatever.com/entry/MeaninginClassicalMathematicsIsitatOddswithIntuitionism
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.MeaninginClassicalMathematicsIsitatOddswithIntuitionismMon, 27 Feb 2017 00:00:00 +0000Karin Usadi Katz and Mikhail G. KatzOn the Existence of Ordinary Triangles
https://read.somethingorotherwhatever.com/entry/OntheExistenceofOrdinaryTriangles
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
$\Omega(|P|)$.OntheExistenceofOrdinaryTrianglesMon, 06 Feb 2017 00:00:00 +0000Radoslav Fulek and Hossein Nassajian Mojarrad and Márton Naszódi and József Solymosi and Sebastian U. Stich and May SzedlákTransfinite Version of Welter's Game
https://read.somethingorotherwhatever.com/entry/TransfiniteVersionofWeltersGame
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.TransfiniteVersionofWeltersGameMon, 06 Feb 2017 00:00:00 +0000Tomoaki AbukuPlane partitions in the work of Richard Stanley and his school
https://read.somethingorotherwhatever.com/entry/PlanepartitionsintheworkofRichardStanleyandhisschool
These notes provide a survey of the theory of plane partitions, seen through
the glasses of the work of Richard Stanley and his school.PlanepartitionsintheworkofRichardStanleyandhisschoolMon, 06 Feb 2017 00:00:00 +0000C. KrattenthalerRandom Triangles and Polygons in the Plane
https://read.somethingorotherwhatever.com/entry/RandomTrianglesandPolygonsinthePlane
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.RandomTrianglesandPolygonsinthePlaneMon, 06 Feb 2017 00:00:00 +0000Jason Cantarella and Tom Needham and Clayton Shonkwiler and Gavin StewartAnalysis of Carries in Signed Digit Expansions
https://read.somethingorotherwhatever.com/entry/AnalysisofCarriesinSignedDigitExpansions
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.AnalysisofCarriesinSignedDigitExpansionsMon, 06 Feb 2017 00:00:00 +0000Clemens Heuberger and Sara Kropf and Helmut ProdingerHunting Rabbits on the Hypercube
https://read.somethingorotherwhatever.com/entry/HuntingRabbitsontheHypercube
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.HuntingRabbitsontheHypercubeMon, 06 Feb 2017 00:00:00 +0000Jessalyn Bolkema and Corbin GroothuisRules for Folding Polyminoes from One Level to Two Levels
https://read.somethingorotherwhatever.com/entry/RulesforFoldingPolyminoesfromOneLeveltoTwoLevels
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.RulesforFoldingPolyminoesfromOneLeveltoTwoLevelsMon, 16 Jan 2017 00:00:00 +0000Julia Martin and Elizabeth WilcoxHuman Inferences about Sequences: A Minimal Transition Probability Model
https://read.somethingorotherwhatever.com/entry/HumanInferencesaboutSequencesAMinimalTransitionProbabilityModel
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.HumanInferencesaboutSequencesAMinimalTransitionProbabilityModelMon, 09 Jan 2017 00:00:00 +0000Florent Meyniel and Maxime Maheu and Stanislas DehaeneA Singular Mathematical Promenade
https://read.somethingorotherwhatever.com/entry/ASingularMathematicalPromenade
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.ASingularMathematicalPromenadeThu, 22 Dec 2016 00:00:00 +0000Etienne GhysBalloon Polyhedra
https://read.somethingorotherwhatever.com/entry/BalloonPolyhedra
BalloonPolyhedraWed, 21 Dec 2016 00:00:00 +0000Erik D. Demaine and Martin L. Demaine and Vi HartTwo short proofs of the Perfect Forest Theorem
https://read.somethingorotherwhatever.com/entry/TwoshortproofsofthePerfectForestTheorem
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.TwoshortproofsofthePerfectForestTheoremSun, 18 Dec 2016 00:00:00 +0000Yair Caro and Josef Lauri and Christina ZarbEvery natural number is the sum of forty-nine palindromes
https://read.somethingorotherwhatever.com/entry/Everynaturalnumberisthesumoffortyninepalindromes
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.EverynaturalnumberisthesumoffortyninepalindromesSat, 17 Dec 2016 00:00:00 +0000William D. BanksSequences of consecutive \(n\)-Niven numbers
https://read.somethingorotherwhatever.com/entry/SequencesOfConsecutiveNNivenNumbers
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. SequencesOfConsecutiveNNivenNumbersMon, 05 Dec 2016 00:00:00 +0000H.G. GrundmanQuasipractical Numbers
https://read.somethingorotherwhatever.com/entry/QuasipracticalNumbers
QuasipracticalNumbersMon, 28 Nov 2016 00:00:00 +0000Harvey J. HindinDeveloping a Mathematical Model for Bobbin Lace
https://read.somethingorotherwhatever.com/entry/DevelopingaMathematicalModelforBobbinLace
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.DevelopingaMathematicalModelforBobbinLaceMon, 28 Nov 2016 00:00:00 +0000Veronika Irvine and Frank RuskeyCryptographic Protocols with Everyday Objects
https://read.somethingorotherwhatever.com/entry/CryptographicProtocolsWithEverydayObjects
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.CryptographicProtocolsWithEverydayObjectsThu, 17 Nov 2016 00:00:00 +0000James Heather and Steve Schneider and Vanessa TeagueOn the interval containing at least one prime number
https://read.somethingorotherwhatever.com/entry/NaguraOntheintervalcontainingatleastoneprimenumber
NaguraOntheintervalcontainingatleastoneprimenumberMon, 14 Nov 2016 00:00:00 +0000Jitsuro NaguraOn subsets with intersections of even cardinality
https://read.somethingorotherwhatever.com/entry/Onsubsetswithintersectionsofevencardinality
This paper solves a question by Paul ErdősOnsubsetswithintersectionsofevencardinalityTue, 01 Nov 2016 00:00:00 +0000E.R. BerlekampTwo remarks on even and oddtown problems
https://read.somethingorotherwhatever.com/entry/Tworemarksonevenandoddtownproblems
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 $\not\equiv 0 \pmod \ell$.TworemarksonevenandoddtownproblemsThu, 27 Oct 2016 00:00:00 +0000Benny Sudakov and Pedro VieiraGeometric Mechanics of Curved Crease Origami
https://read.somethingorotherwhatever.com/entry/GeometricMechanicsofCurvedCreaseOrigami
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.GeometricMechanicsofCurvedCreaseOrigamiFri, 14 Oct 2016 00:00:00 +0000Marcelo A. Dias and Levi H. Dudte and L. Mahadevan and Christian D. SantangeloA Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence
https://read.somethingorotherwhatever.com/entry/ASpaceEfficientAlgorithmfortheCalculationoftheDigitDistributionintheKolakoskiSequence
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.ASpaceEfficientAlgorithmfortheCalculationoftheDigitDistributionintheKolakoskiSequenceFri, 14 Oct 2016 00:00:00 +0000Johan NilssonA Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents
https://read.somethingorotherwhatever.com/entry/ADiscreteandBoundedEnvyFreeCakeCuttingProtocolforAnyNumberofAgents
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.ADiscreteandBoundedEnvyFreeCakeCuttingProtocolforAnyNumberofAgentsThu, 13 Oct 2016 00:00:00 +0000Haris Aziz and Simon MackenzieAvoiding Squares and Overlaps Over the Natural Numbers
https://read.somethingorotherwhatever.com/entry/AvoidingSquaresandOverlapsOvertheNaturalNumbers
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.AvoidingSquaresandOverlapsOvertheNaturalNumbersMon, 03 Oct 2016 00:00:00 +0000Mathieu Guay-Paquet and Jeffrey ShallitCounting Cases in Marching Cubes: Towards a Generic Algorithm for Producing Substitopes
https://read.somethingorotherwhatever.com/entry/CountingCasesInMarchingCubes
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.CountingCasesInMarchingCubesWed, 28 Sep 2016 00:00:00 +0000David C. Banks and Stephen LintonFractal geometry of a complex plumage trait reveals bird's quality
https://read.somethingorotherwhatever.com/entry/Fractalgeometryofacomplexplumagetraitrevealsbirdsquality
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.FractalgeometryofacomplexplumagetraitrevealsbirdsqualityTue, 27 Sep 2016 00:00:00 +0000Lorenzo Pérez-Rodríguez and Roger Jovani and Fran\ccois MougeotThe Nesting and Roosting Habits of The Laddered Parenthesis
https://read.somethingorotherwhatever.com/entry/item43
item43Fri, 23 Sep 2016 00:00:00 +0000R. K. Guy and J. L. SelfridgeProgramming quantum computers using 3-D puzzles, coffee cups, and doughnuts
https://read.somethingorotherwhatever.com/entry/Programmingquantumcomputersusing3Dpuzzlescoffeecupsanddoughnuts
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.Programmingquantumcomputersusing3DpuzzlescoffeecupsanddoughnutsFri, 23 Sep 2016 00:00:00 +0000Simon J. DevittHistorical methods for multiplication
https://read.somethingorotherwhatever.com/entry/HistoricalMethodsForMultiplication
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.HistoricalMethodsForMultiplicationThu, 22 Sep 2016 00:00:00 +0000Bjørn Smestad and Konstantinos NikolantonakisPonytail Motion
https://read.somethingorotherwhatever.com/entry/PonytailMotion
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.PonytailMotionMon, 19 Sep 2016 00:00:00 +0000Joseph B. KellerSeven Puzzles You Think You Must Not Have Heard Correctly
https://read.somethingorotherwhatever.com/entry/WinklerSevenPuzzles
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.WinklerSevenPuzzlesTue, 30 Aug 2016 00:00:00 +0000Peter WinklerTopologically Distinct Sets of Non-intersecting Circles in the Plane
https://read.somethingorotherwhatever.com/entry/TopologicallyDistinctSetsofNonintersectingCirclesinthePlane
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.TopologicallyDistinctSetsofNonintersectingCirclesinthePlaneThu, 25 Aug 2016 00:00:00 +0000Richard J. MatharBeckett-Gray Codes
https://read.somethingorotherwhatever.com/entry/BeckettGrayCodes
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.BeckettGrayCodesWed, 24 Aug 2016 00:00:00 +0000Mark Cooke and Chris North and Megan Dewar and Brett StevensThe general counterfeit coin problem
https://read.somethingorotherwhatever.com/entry/TheGeneralCounterfeitCoinProblem
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.TheGeneralCounterfeitCoinProblemWed, 24 Aug 2016 00:00:00 +0000Lorenz Halbeisen and Norbert HungerbühlerSearching for generalized binary number systems
https://read.somethingorotherwhatever.com/entry/Searchingforgeneralizedbinarynumbersystems
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.SearchingforgeneralizedbinarynumbersystemsMon, 22 Aug 2016 00:00:00 +0000Attila KovácsThe denominators of convergents for continued fractions
https://read.somethingorotherwhatever.com/entry/Thedenominatorsofconvergentsforcontinuedfractions
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.ThedenominatorsofconvergentsforcontinuedfractionsSat, 06 Aug 2016 00:00:00 +0000Lulu Fang and Min Wu and Bing LiTen Lessons I Wish I Had Learned Before I Started Teaching Differential Equations
https://read.somethingorotherwhatever.com/entry/TenLessonsRota
TenLessonsRotaWed, 03 Aug 2016 00:00:00 +0000Giancarlo RotaMatters Computational - Ideas, Algorithms, Source Code
https://read.somethingorotherwhatever.com/entry/MattersComputational
This is the book "Matters Computational" (formerly titled "Algorithms for Programmers"), published with Springer.MattersComputationalWed, 03 Aug 2016 00:00:00 +0000Jörg ArndtRational Polynomials That Take Integer Values at the Fibonacci Numbers
https://read.somethingorotherwhatever.com/entry/RationalPolynomialsThatTakeIntegerValuesattheFibonacciNumbers
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.RationalPolynomialsThatTakeIntegerValuesattheFibonacciNumbersTue, 02 Aug 2016 00:00:00 +0000Keith Johnson and Kira ScheibelhutBad groups in the sense of Cherlin
https://read.somethingorotherwhatever.com/entry/BadgroupsinthesenseofCherlin
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 $K$.BadgroupsinthesenseofCherlinTue, 02 Aug 2016 00:00:00 +0000Olivier FréconAn Irrationality Measure for Regular Paperfolding Numbers
https://read.somethingorotherwhatever.com/entry/AnIrrationalityMeasureforRegularPaperfoldingNumbers
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.AnIrrationalityMeasureforRegularPaperfoldingNumbersMon, 11 Jul 2016 00:00:00 +0000Michael Coons and Paul VrbikWhat is the smallest prime?
https://read.somethingorotherwhatever.com/entry/Whatisthesmallestprime
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.WhatisthesmallestprimeTue, 05 Jul 2016 00:00:00 +0000Chris K. Caldwell and Yeng XiongHow do you compute the midpoint of an interval?
https://read.somethingorotherwhatever.com/entry/MidpointOfAnInterval
MidpointOfAnIntervalMon, 04 Jul 2016 00:00:00 +0000Frédéric GoualardComplexity and Completeness of Finding Another solution and its Application to Puzzles
https://read.somethingorotherwhatever.com/entry/Yato2003
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.Yato2003Sat, 18 Jun 2016 00:00:00 +0000Takayushi YatoDividing by zero - how bad is it, really?
https://read.somethingorotherwhatever.com/entry/Dividingbyzerohowbadisitreally
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.DividingbyzerohowbadisitreallyFri, 17 Jun 2016 00:00:00 +0000Takayuki Kihara and Arno PaulyFuzzy plane geometry I: Points and lines
https://read.somethingorotherwhatever.com/entry/FuzzyGeometry
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. FuzzyGeometryFri, 17 Jun 2016 00:00:00 +0000J.J. Buckley and E. AslamiContinued Logarithms And Associated Continued Fractions
https://read.somethingorotherwhatever.com/entry/ContinuedLogarithms
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.ContinuedLogarithmsWed, 15 Jun 2016 00:00:00 +0000Jonathan M. Borwein and Neil J. Calkin and Scott B. Lindstrom and Andrew MattinglyDr Mitchill and the Mathematical Tetrodon
https://read.somethingorotherwhatever.com/entry/DrMitchillandtheMathematicalTetrodonThePublicDomainReview
DrMitchillandtheMathematicalTetrodonThePublicDomainReviewMon, 13 Jun 2016 00:00:00 +0000Kevin DannThe snail lemma
https://read.somethingorotherwhatever.com/entry/Thesnaillemma
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.ThesnaillemmaMon, 13 Jun 2016 00:00:00 +0000Enrico M. VitaleNotes on the Fourth Dimension
https://read.somethingorotherwhatever.com/entry/NotesontheFourthDimensionThePublicDomainReview
Hyperspace, ghosts, and colourful cubes — Jon Crabb on the work of Charles Howard Hinton and the cultural history of higher dimensions.NotesontheFourthDimensionThePublicDomainReviewMon, 13 Jun 2016 00:00:00 +0000Jon Crabb Photoelectric Number Sieve Machine ("Gear Machine")
https://read.somethingorotherwhatever.com/entry/PhotoelectricNumberSieve
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.PhotoelectricNumberSieveMon, 13 Jun 2016 00:00:00 +0000D. H. Lehmer and Robert CanepaChallenging mathematical problems with elementary solutions
https://read.somethingorotherwhatever.com/entry/ChallengingProblems
ChallengingProblemsWed, 08 Jun 2016 00:00:00 +0000A.M. Yaglom and I.M. YaglomOn Pellegrino's 20-Caps in $S_{4,3}$
https://read.somethingorotherwhatever.com/entry/OnPellegrinos20CapsinS43
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 $S_{5,3}$.OnPellegrinos20CapsinS43Wed, 01 Jun 2016 00:00:00 +0000R. HillCounting groups: gnus, moas and other exotica
https://read.somethingorotherwhatever.com/entry/CountingGroups
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.CountingGroupsFri, 20 May 2016 00:00:00 +0000John H. Conway and Heiko Dietrich and E.A. O’BrienFibonacci Jigsaw Puzzle
https://read.somethingorotherwhatever.com/entry/FIBONACCIJIGSAWPUZZLE
FIBONACCIJIGSAWPUZZLEThu, 19 May 2016 00:00:00 +0000Akio HizumeDismal Arithmetic
https://read.somethingorotherwhatever.com/entry/DismalArithmetic
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.DismalArithmeticThu, 19 May 2016 00:00:00 +0000David Applegate and Marc LeBrun and N. J. A. SloaneTwo notes on notation
https://read.somethingorotherwhatever.com/entry/TwoNotesOnNotation
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.TwoNotesOnNotationThu, 19 May 2016 00:00:00 +0000Donald E. KnuthOn the Cookie Monster Problem
https://read.somethingorotherwhatever.com/entry/OntheCookieMonsterProblem
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.OntheCookieMonsterProblemThu, 19 May 2016 00:00:00 +0000Leigh Marie Braswell and Tanya KhovanovaPrime numbers in certain arithmetic progressions
https://read.somethingorotherwhatever.com/entry/item61
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.item61Sat, 07 May 2016 00:00:00 +0000Ram Murty and Nithum ThainDivision by zero
https://read.somethingorotherwhatever.com/entry/Jerabek2016
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.Jerabek2016Tue, 26 Apr 2016 00:00:00 +0000Emil JeřábekTransposable integers in arbitrary bases
https://read.somethingorotherwhatever.com/entry/item60
item60Tue, 19 Apr 2016 00:00:00 +0000Anne L. LudingtonA Dozen Hat Problems
https://read.somethingorotherwhatever.com/entry/item59
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.item59Tue, 12 Apr 2016 00:00:00 +0000Ezra Brown and James TantonDe Bruijn's Combinatorics
https://read.somethingorotherwhatever.com/entry/Hung
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.HungFri, 05 Feb 2016 00:00:00 +0000Hung, J.W.Nienhuys (Ling-Ju and Eds.), Ton KloksComparative kinetics of the snowball respect to other dynamical objects
https://read.somethingorotherwhatever.com/entry/Diaz2003
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.Diaz2003Mon, 18 Jan 2016 00:00:00 +0000Diaz, Rodolfo A. and Gonzalez, Diego L. and Marin, Francisco and Martinez, R.On gardeners, dukes and mathematical instruments
https://read.somethingorotherwhatever.com/entry/BlancoAbellan2015
Postprint (author's final draft)BlancoAbellan2015Mon, 18 Jan 2016 00:00:00 +0000Blanco Abellán, MónicaArea and Hausdorff Dimension of Julia Sets of Entire Functions
https://read.somethingorotherwhatever.com/entry/item58
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.item58Thu, 17 Dec 2015 00:00:00 +0000Curt McMullenApproaches to the Enumerative Theory of Meanders
https://read.somethingorotherwhatever.com/entry/item57
item57Mon, 14 Dec 2015 00:00:00 +0000Michael La CroixThe Theory of Heaps and the Cartier-Foata Monoid
https://read.somethingorotherwhatever.com/entry/item56
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–Mélou and Viennot.item56Thu, 03 Dec 2015 00:00:00 +0000C. KrattenthalerPlanar graph is on fire
https://read.somethingorotherwhatever.com/entry/Gordinowicz2015
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.Gordinowicz2015Wed, 02 Dec 2015 00:00:00 +0000Gordinowicz, PrzemysławWhat to do when the trisector comes
https://read.somethingorotherwhatever.com/entry/Dudley
DudleyMon, 30 Nov 2015 00:00:00 +0000Dudley, UnderwoodOn the Number of Times an Integer Occurs as a Binomial Coefficient
https://read.somethingorotherwhatever.com/entry/item55
item55Mon, 30 Nov 2015 00:00:00 +0000H. L. Abbott and P. Erdős and D. HansonWhen is .999... less than 1?
https://read.somethingorotherwhatever.com/entry/Katz2010
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.Katz2010Thu, 19 Nov 2015 00:00:00 +0000Katz, Karin Usadi and Katz, Mikhail G.The effective content of Reverse Nonstandard Mathematics and the nonstandard content of effective Reverse Mathematics
https://read.somethingorotherwhatever.com/entry/Sanders2015
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.Sanders2015Thu, 19 Nov 2015 00:00:00 +0000Sanders, SamHaruspicy and anisotropic generating functions
https://read.somethingorotherwhatever.com/entry/Rechnitzer2003
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.Rechnitzer2003Sun, 08 Nov 2015 00:00:00 +0000Rechnitzer, AndrewSpiralling self-avoiding walks: an exact solution
https://read.somethingorotherwhatever.com/entry/Blote1984
Blote1984Sun, 08 Nov 2015 00:00:00 +0000Blote, H W J and Hilhorst, H JHaruspicy 3: The anisotropic generating function of directed bond-animals is not D-finite
https://read.somethingorotherwhatever.com/entry/Rechnitzer2006
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].Rechnitzer2006Sun, 08 Nov 2015 00:00:00 +0000Rechnitzer, AndrewHow to Beat Your Wythoff Games' Opponent on Three Fronts
https://read.somethingorotherwhatever.com/entry/item54
item54Fri, 23 Oct 2015 00:00:00 +0000Aviezri S. FraenkelDice - Numericana
https://read.somethingorotherwhatever.com/entry/item53
item53Sat, 17 Oct 2015 00:00:00 +0000Gérard P. MichonProposal to Encode the Ganda Currency Mark for Bengali in ISO/IEC 10646
https://read.somethingorotherwhatever.com/entry/item52
item52Wed, 30 Sep 2015 00:00:00 +0000Anshuman PandeyAnother Proof of Segre's Theorem about Ovals
https://read.somethingorotherwhatever.com/entry/Muller2013
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.Muller2013Tue, 29 Sep 2015 00:00:00 +0000Müller, PeterFair Dice
https://read.somethingorotherwhatever.com/entry/Diaconis1989a
Diaconis1989aTue, 15 Sep 2015 00:00:00 +0000Diaconis, Persi and Keller, Joseph BOn the Existence of Generalized Parking Spaces for Complex Reflection Groups
https://read.somethingorotherwhatever.com/entry/Ito2015
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.Ito2015Sun, 06 Sep 2015 00:00:00 +0000Ito, Yosuke and Okada, SoichiMind the Croc! Rationality Gaps vis-à-vis the Crocodile Paradox
https://read.somethingorotherwhatever.com/entry/Gerogiorgakis2015
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.Gerogiorgakis2015Thu, 03 Sep 2015 00:00:00 +0000Gerogiorgakis, StamatiosDenser Egyptian Fractions
https://read.somethingorotherwhatever.com/entry/Martin1998
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 Erdős 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.Martin1998Thu, 23 Jul 2015 00:00:00 +0000Martin, GregReversible quantum cellular automata
https://read.somethingorotherwhatever.com/entry/Schumacher2004
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.Schumacher2004Sun, 28 Jun 2015 00:00:00 +0000Schumacher, B. and Werner, R. F.Representations of Palindromic, Prime and Number Patterns
https://read.somethingorotherwhatever.com/entry/item51
item51Fri, 12 Jun 2015 00:00:00 +0000Inder J. TanejaHave you been using the wrong estimator? These guys bound average fidelity using this one weird trick von Neumann didn't want you to know
https://read.somethingorotherwhatever.com/entry/Ferrie2015
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.Ferrie2015Thu, 26 Mar 2015 00:00:00 +0000Ferrie, Christopher and Kueng, RichardThe Lost Calculus (1637-1670): Tangency and Optimization without Limits
https://read.somethingorotherwhatever.com/entry/item50
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.item50Thu, 12 Mar 2015 00:00:00 +0000Jeff SuzukiThe accuracy of Buffon's needle: a rule of thumb used by ants to estimate area
https://read.somethingorotherwhatever.com/entry/Mugford2001
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.Mugford2001Mon, 09 Mar 2015 00:00:00 +0000Mugford, S. T.Finding long chains in kidney exchange using the traveling salesman problem
https://read.somethingorotherwhatever.com/entry/Anderson2015
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.Anderson2015Sat, 07 Mar 2015 00:00:00 +0000Anderson, Ross and Ashlagi, Itai and Gamarnik, David and Roth, Alvin E.Maximum Matching and a Polyhedron With 0,1-Vertices
https://read.somethingorotherwhatever.com/entry/item49
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.item49Sat, 07 Mar 2015 00:00:00 +0000Jack EdmondsOn Legendre's Prime Number Formula
https://read.somethingorotherwhatever.com/entry/item47
item47Fri, 06 Mar 2015 00:00:00 +0000Janos PintzMagic squares of seventh powers
https://read.somethingorotherwhatever.com/entry/item48
item48Fri, 06 Mar 2015 00:00:00 +0000Christian BoyerA Brief Critique of Pure Hypercomputation
https://read.somethingorotherwhatever.com/entry/Cotogno2009
Cotogno2009Thu, 05 Mar 2015 00:00:00 +0000Cotogno, PaoloComplexity and Algorithms for Graph and Hypergraph Sandwich Problems
https://read.somethingorotherwhatever.com/entry/Golumbic1998
Golumbic1998Tue, 24 Feb 2015 00:00:00 +0000Golumbic, Martin Charles and Wassermann, AmirA combinatorial theorem in plane geometry
https://read.somethingorotherwhatever.com/entry/Chvatal1975
Chvatal1975Mon, 23 Feb 2015 00:00:00 +0000Chvátal, VThe dying rabbit problem revisited
https://read.somethingorotherwhatever.com/entry/Oller2007
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.Oller2007Wed, 18 Feb 2015 00:00:00 +0000Oller, Antonio M.Efficient Algorithms for Zeckendorf Arithmetic
https://read.somethingorotherwhatever.com/entry/Ahlbach2012
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.Ahlbach2012Mon, 16 Feb 2015 00:00:00 +0000Ahlbach, Connor and Usatine, Jeremy and Pippenger, NicholasRational approximations to $\pi$ and some other numbers
https://read.somethingorotherwhatever.com/entry/Hata1993
Hata1993Mon, 16 Feb 2015 00:00:00 +0000Hata, Masayoshi and Mignotte, M and Chudnovsky, G V and Beukers, FExact Approximations of Omega Numbers
https://read.somethingorotherwhatever.com/entry/Calude2006
Calude2006Tue, 03 Feb 2015 00:00:00 +0000Calude, C.S and Dinneen, MichaelEnumeration of symmetry classes of convex polyominoes on the honeycomb lattice
https://read.somethingorotherwhatever.com/entry/Gouyou-Beauchamps2005
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.Gouyou-Beauchamps2005Mon, 02 Feb 2015 00:00:00 +0000Gouyou-Beauchamps, Dominique and Leroux, PierreOn dice and coins: Models of computation for random generation
https://read.somethingorotherwhatever.com/entry/Feldman1993
Feldman1993Mon, 26 Jan 2015 00:00:00 +0000Feldman, D and Impagliazzo, R and Naor, MThis is the (co)end, my only (co)friend
https://read.somethingorotherwhatever.com/entry/Loregian2015
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.Loregian2015Tue, 13 Jan 2015 00:00:00 +0000Loregian, FoscoThe Eudoxus Real Numbers
https://read.somethingorotherwhatever.com/entry/Arthan2004
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.Arthan2004Wed, 07 Jan 2015 00:00:00 +0000Arthan, R. D.Bells, Motels and Permutation Groups
https://read.somethingorotherwhatever.com/entry/McGuire2012
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.McGuire2012Wed, 17 Dec 2014 00:00:00 +0000McGuire, GaryMusic: a Mathematical Offering
https://read.somethingorotherwhatever.com/entry/item46
item46Tue, 02 Dec 2014 00:00:00 +0000Dave BensonIrrationality From The Book
https://read.somethingorotherwhatever.com/entry/Miller2009a
We generalize Tennenbaum's geometric proof of the irrationality of sqrt(2) to
sqrt(n) for n = 3, 5, 6 and 10.Miller2009aMon, 01 Dec 2014 00:00:00 +0000Miller, Steven J. and Montague, DavidHow not to prove the Poincaré conjecture
https://read.somethingorotherwhatever.com/entry/Stallings1966
I have committed the sin of falsely proving Poincaré'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!Stallings1966Mon, 17 Nov 2014 00:00:00 +0000Stallings, JRDivision by three
https://read.somethingorotherwhatever.com/entry/Doyle2006
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.Doyle2006Mon, 17 Nov 2014 00:00:00 +0000Doyle, Peter G. and Conway, John HortonSolving Triangular Peg Solitaire
https://read.somethingorotherwhatever.com/entry/Bell2007
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).Bell2007Mon, 10 Nov 2014 00:00:00 +0000Bell, George I.An Application of Elementary Group Theory to Central Solitaire
https://read.somethingorotherwhatever.com/entry/Bialostocki1998
Bialostocki1998Mon, 10 Nov 2014 00:00:00 +0000Bialostocki, ArieThe Super Patalan Numbers
https://read.somethingorotherwhatever.com/entry/Richardson2014
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.Richardson2014Thu, 23 Oct 2014 00:00:00 +0000Richardson, Thomas M.The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?
https://read.somethingorotherwhatever.com/entry/Duran2014
Duran2014Thu, 23 Oct 2014 00:00:00 +0000Durán, Antonio J and Pérez, Mario and Varona, Juan LProofs without syntax
https://read.somethingorotherwhatever.com/entry/Hughes2006
Hughes2006Thu, 09 Oct 2014 00:00:00 +0000Hughes, DJDMethods for studying coincidences
https://read.somethingorotherwhatever.com/entry/Diaconis2006
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.Diaconis2006Tue, 07 Oct 2014 00:00:00 +0000Diaconis, P and Mosteller, FrederickSudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes
https://read.somethingorotherwhatever.com/entry/Bailey2008
Bailey2008Thu, 02 Oct 2014 00:00:00 +0000Bailey, RAHow often should you clean your room?
https://read.somethingorotherwhatever.com/entry/Martin2013
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.Martin2013Tue, 02 Sep 2014 00:00:00 +0000Martin, Kimball and Shankar, KrishnanPondering an Artist's Perplexing Tribute to the Pythagorean Theorem
https://read.somethingorotherwhatever.com/entry/item45
item45Thu, 28 Aug 2014 00:00:00 +0000Ivars PetersonA Fresh Look at Peg Solitaire
https://read.somethingorotherwhatever.com/entry/item44
item44Tue, 26 Aug 2014 00:00:00 +0000George I. BellThe shape of a Mobius band
https://read.somethingorotherwhatever.com/entry/Mahadevan1993
Mahadevan1993Wed, 20 Aug 2014 00:00:00 +0000Mahadevan, L and Keller, JBFoldings and Meanders
https://read.somethingorotherwhatever.com/entry/Legendre2013
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.Legendre2013Tue, 19 Aug 2014 00:00:00 +0000Legendre, StéphaneMathematics and group theory in music
https://read.somethingorotherwhatever.com/entry/Papadopoulos2014
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.Papadopoulos2014Thu, 24 Jul 2014 00:00:00 +0000Papadopoulos, AthanaseAn arctic circle theorem for groves
https://read.somethingorotherwhatever.com/entry/Petersen2004
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.Petersen2004Tue, 01 Jul 2014 00:00:00 +0000Petersen, T. K. and Speyer, D.History-dependent random processes
https://read.somethingorotherwhatever.com/entry/Clifford2008
Ulam has defined a history-dependent random sequence by the recursion Xn+1=Xn+XU(n), where (U(n); n[≥]1) is a sequence of independent random variables with U(n) uniformly distributed on {1, ..., n} and X1=1. We introduce a new class of continuous-time history-dependent random processes regulated by Poisson processes. The simplest of these, a univariate process regulated by a homogeneous Poisson process, replicates in continuous time the essential properties of Ulam's sequence, and greatly facilitates its analysis. We consider several generalizations and extensions of this, including bivariate and multivariate coupled history-dependent processes, and cases when the dependence on the past is not uniform. The analysis of the discrete-time formulations of these models would be at the very least an extremely formidable project, but we determine the asymptotic growth rates of their means and higher moments with relative ease.Clifford2008Mon, 30 Jun 2014 00:00:00 +0000Clifford, P. and Stirzaker, D.LIM is not slim
https://read.somethingorotherwhatever.com/entry/Fink2013
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.Fink2013Wed, 11 Jun 2014 00:00:00 +0000Fink, Alex and Fraenkel, Aviezri S. and Santos, CarlosThe Number-Pad Game
https://read.somethingorotherwhatever.com/entry/item42
item42Wed, 11 Jun 2014 00:00:00 +0000Alex Fink and Richard GuyNim Fractals
https://read.somethingorotherwhatever.com/entry/Khovanova2014
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.Khovanova2014Thu, 05 Jun 2014 00:00:00 +0000Khovanova, Tanya and Xiong, JoshuaGeneralizing Zeckendorf's Theorem to f-decompositions
https://read.somethingorotherwhatever.com/entry/Demontigny2013
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.Demontigny2013Mon, 28 Apr 2014 00:00:00 +0000Demontigny, Philippe and Do, Thao and Kulkarni, Archit and Miller, Steven J. and Moon, David and Varma, UmangUseful inequalities cheat sheet
https://read.somethingorotherwhatever.com/entry/item41
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.item41Mon, 28 Apr 2014 00:00:00 +0000László KozmaA Mathematical Coloring Book
https://read.somethingorotherwhatever.com/entry/Hampton2009
Hampton2009Thu, 24 Apr 2014 00:00:00 +0000Hampton, MarshallOn the diagram of 132-avoiding permutations
https://read.somethingorotherwhatever.com/entry/Reifegerste2003
Reifegerste2003Mon, 31 Mar 2014 00:00:00 +0000Reifegerste, AstridA number system with an irrational base
https://read.somethingorotherwhatever.com/entry/item40
item40Wed, 12 Mar 2014 00:00:00 +0000George BergmanEponymy in Mathematical Nomenclature: What's in a Name, and What Should Be?
https://read.somethingorotherwhatever.com/entry/Henwood1980
Henwood1980Tue, 11 Feb 2014 00:00:00 +0000Henwood, Mervyn R. and Rival, IvanTable for Fundamentals of Series : Part I : Basic Properties of Series and Products
https://read.somethingorotherwhatever.com/entry/Gould2011
Gould2011Tue, 11 Feb 2014 00:00:00 +0000Gould, Henry W.More ties than we thought
https://read.somethingorotherwhatever.com/entry/Hirsch2014
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.Hirsch2014Thu, 06 Feb 2014 00:00:00 +0000Hirsch, Dan and Patterson, Meredith L and Sandberg, Anders and Vejdemo-Johansson, MikaelMathematical Games
https://read.somethingorotherwhatever.com/entry/Silva2007
Silva2007Fri, 24 Jan 2014 00:00:00 +0000Silva, Jorge NunoRithmomachia
https://read.somethingorotherwhatever.com/entry/item39
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.item39Fri, 24 Jan 2014 00:00:00 +0000Daniel U. Thibault and Michel BoutinLinear recurrences through tilings and Markov chains
https://read.somethingorotherwhatever.com/entry/Benjamin2003
Benjamin2003Tue, 21 Jan 2014 00:00:00 +0000Benjamin, AT and Hanusa, CRH and Su, FEThe Stick Problem
https://read.somethingorotherwhatever.com/entry/item38
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.item38Mon, 13 Jan 2014 00:00:00 +0000Augustine BertagnolliCircular orbits on a warped spandex fabric
https://read.somethingorotherwhatever.com/entry/Middleton2013
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.Middleton2013Tue, 07 Jan 2014 00:00:00 +0000Middleton, Chad A. and Langston, MichaelThe topology of competitively constructed graphs
https://read.somethingorotherwhatever.com/entry/Frieze2013
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.Frieze2013Mon, 30 Dec 2013 00:00:00 +0000Frieze, Alan and Pegden, WesleyThe mathematics of Septoku
https://read.somethingorotherwhatever.com/entry/Bell2008
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.Bell2008Sun, 22 Dec 2013 00:00:00 +0000Bell, George I.Fair but irregular polyhedral dice
https://read.somethingorotherwhatever.com/entry/item37
item37Mon, 16 Dec 2013 00:00:00 +0000Joseph O'RourkeSolving Differential Equations by Symmetry Groups
https://read.somethingorotherwhatever.com/entry/item36
item36Fri, 06 Dec 2013 00:00:00 +0000John StarretA knowledge-based approach of connect-four
https://read.somethingorotherwhatever.com/entry/Allis1988
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.Allis1988Tue, 03 Dec 2013 00:00:00 +0000Allis, VictorWHAT IS Lehmer's number?
https://read.somethingorotherwhatever.com/entry/item35
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''.item35Tue, 03 Dec 2013 00:00:00 +0000Eriko HironakaAnalyse algébrique d'un scrutin
https://read.somethingorotherwhatever.com/entry/Guilbaud1963
Guilbaud1963Sun, 01 Dec 2013 00:00:00 +0000Guilbaud, GT and Rosenstiehl, PLone Axes in Outer Space
https://read.somethingorotherwhatever.com/entry/Mosher2013
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.Mosher2013Fri, 29 Nov 2013 00:00:00 +0000Mosher, Lee and Pfaff, CatherineThe Maximum Throughput Rate for Each Hole on a Golf Course
https://read.somethingorotherwhatever.com/entry/Whitt2013
Whitt2013Mon, 25 Nov 2013 00:00:00 +0000Whitt, WardWolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
https://read.somethingorotherwhatever.com/entry/Mestrovic2011
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.Mestrovic2011Thu, 21 Nov 2013 00:00:00 +0000Mestrovic, RomeoThe Math Encyclopedia of Smarandache Type Notions
https://read.somethingorotherwhatever.com/entry/Coman
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.ComanThu, 21 Nov 2013 00:00:00 +0000Coman, Marius2178 And All That
https://read.somethingorotherwhatever.com/entry/Sloane2013
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.Sloane2013Wed, 20 Nov 2013 00:00:00 +0000Sloane, NJAFibonacci numbers and Leonardo numbers
https://read.somethingorotherwhatever.com/entry/Dijkstra1981
Dijkstra1981Tue, 19 Nov 2013 00:00:00 +0000Dijkstra, E.W.Pancake Flipping is Hard
https://read.somethingorotherwhatever.com/entry/Bulteau2011
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.Bulteau2011Fri, 15 Nov 2013 00:00:00 +0000Bulteau, Laurent and Fertin, Guillaume and Rusu, IrenaGiuga Numbers and the arithmetic derivative
https://read.somethingorotherwhatever.com/entry/Grau2011
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.Grau2011Fri, 15 Nov 2013 00:00:00 +0000Grau, José María and Oller-Marcén, Antonio M.Swiss cheeses, rational approximation and universal plane curves
https://read.somethingorotherwhatever.com/entry/Feinstein2010
Feinstein2010Thu, 14 Nov 2013 00:00:00 +0000Feinstein, JF and Heath, MJPlaying pool with $\pi$ (the number $\pi$ from a billiard point of view)
https://read.somethingorotherwhatever.com/entry/Galperin2003
Galperin2003Thu, 14 Nov 2013 00:00:00 +0000Galperin, GRandom Structures from Lego Bricks and Analog Monte Carlo Procedures
https://read.somethingorotherwhatever.com/entry/Althofer2013
Althofer2013Wed, 13 Nov 2013 00:00:00 +0000Althöfer, IIs POPL Mathematics or Science?
https://read.somethingorotherwhatever.com/entry/item34
item34Wed, 06 Nov 2013 00:00:00 +0000Andrew W. AppelProofs by Descent
https://read.somethingorotherwhatever.com/entry/CONRAD
CONRADMon, 04 Nov 2013 00:00:00 +0000Keith ConradPractical numbers
https://read.somethingorotherwhatever.com/entry/Srinivasan1948
Srinivasan1948Thu, 31 Oct 2013 00:00:00 +0000Srinivasan, A.K.How to differentiate a number
https://read.somethingorotherwhatever.com/entry/Ufnarovski2003
Ufnarovski2003Thu, 31 Oct 2013 00:00:00 +0000Ufnarovski, Victor and Åhlander, BThe Ubiquitous Thue-Morse Sequence
https://read.somethingorotherwhatever.com/entry/item33
item33Mon, 21 Oct 2013 00:00:00 +0000Jeffrey ShallitSloane's Gap: Do Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS?
https://read.somethingorotherwhatever.com/entry/Gauvrit2011
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.Gauvrit2011Thu, 17 Oct 2013 00:00:00 +0000Gauvrit, Nicolas and Delahaye, Jean-Paul and Zenil, HectorCookie Monster Devours Naccis
https://read.somethingorotherwhatever.com/entry/Braswell2013a
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.Braswell2013aWed, 09 Oct 2013 00:00:00 +0000Braswell, Leigh Marie and Khovanova, TanyaPerfect Matchings and the Octahedron Recurrence
https://read.somethingorotherwhatever.com/entry/Speyer2004a
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.Speyer2004aWed, 09 Oct 2013 00:00:00 +0000Speyer, David EAn Infinite Set of Heron Triangles with Two Rational Medians
https://read.somethingorotherwhatever.com/entry/Buchholz2013
Buchholz2013Wed, 09 Oct 2013 00:00:00 +0000Buchholz, Ralph H and Rathbun, Randall LThe Laurent phenomenon
https://read.somethingorotherwhatever.com/entry/Fomin2001
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.Fomin2001Wed, 09 Oct 2013 00:00:00 +0000Fomin, Sergey and Zelevinsky, AndreiThe Strange and Surprising Saga of the Somos Sequences
https://read.somethingorotherwhatever.com/entry/Gale1991
Gale1991Wed, 09 Oct 2013 00:00:00 +0000Gale, DavidOn n-Dimensional Polytope Schemes
https://read.somethingorotherwhatever.com/entry/Fouhey2013
Fouhey2013Fri, 02 Aug 2013 00:00:00 +0000Fouhey, David F and Maturana, DanielOnly problems, not solutions!
https://read.somethingorotherwhatever.com/entry/Smarandache1991
Smarandache1991Mon, 29 Jul 2013 00:00:00 +0000Smarandache, FlorentinFrom Unicode to Typography, a Case Study the Greek Script
https://read.somethingorotherwhatever.com/entry/Haralambous1999
Haralambous1999Mon, 22 Jul 2013 00:00:00 +0000Haralambous, YannisHalf of a coin: negative probabilities
https://read.somethingorotherwhatever.com/entry/Szekely2005
Szekely2005Wed, 17 Jul 2013 00:00:00 +0000Székely, GJSmooth neighbors
https://read.somethingorotherwhatever.com/entry/Conrey2012
We give a new algorithm that quickly finds smooth neighbors.Conrey2012Thu, 11 Jul 2013 00:00:00 +0000Conrey, Brian and Holmstrom, Mark and McLaughlin, TaraOn a problem of Störmer
https://read.somethingorotherwhatever.com/entry/Lehmer1964
Lehmer1964Thu, 11 Jul 2013 00:00:00 +0000Lehmer, DHMissing Data: Instrument-Level Heffalumps and Item-Level Woozles
https://read.somethingorotherwhatever.com/entry/item32
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.item32Wed, 26 Jun 2013 00:00:00 +0000Philip L. Roth and Fred S. Switzer IIIPascal's Pyramid Or Pascal's Tetrahedron
https://read.somethingorotherwhatever.com/entry/item31
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.item31Tue, 25 Jun 2013 00:00:00 +0000Jim NugentA Line of Sages
https://read.somethingorotherwhatever.com/entry/Khovanova2013
Khovanova2013Thu, 20 Jun 2013 00:00:00 +0000Khovanova, TanyaNon-sexist solution of the ménage problem
https://read.somethingorotherwhatever.com/entry/NonSexistMenage
The ménage 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.NonSexistMenageSun, 16 Jun 2013 00:00:00 +0000Kenneth P. BogartThe Ubiquitous Pi
https://read.somethingorotherwhatever.com/entry/Castellanos2013
Castellanos2013Tue, 11 Jun 2013 00:00:00 +0000Castellanos, DarioSix Ways to Sum a Series
https://read.somethingorotherwhatever.com/entry/Kalman
A discussion of the sum of squares of the reciprocals of the positive integers with a review of several proofs.KalmanMon, 10 Jun 2013 00:00:00 +0000Kalman, DanUsing Monoidal Categories in the Transformational Study of Musical Time-Spans and Rhythms
https://read.somethingorotherwhatever.com/entry/Popoff2013
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.Popoff2013Sun, 02 Jun 2013 00:00:00 +0000Popoff, AlexandreKindergarten Quantum Mechanics
https://read.somethingorotherwhatever.com/entry/Coecke2005
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.Coecke2005Tue, 28 May 2013 00:00:00 +0000Coecke, BobCyclic twill-woven objects
https://read.somethingorotherwhatever.com/entry/Akleman2011
Akleman2011Thu, 16 May 2013 00:00:00 +0000Akleman, Ergun and Chen, Jianer and Chen, YenLin and Xing, Qing and Gross, Jonathan L.Division of labor in child care: A game-theoretic approach
https://read.somethingorotherwhatever.com/entry/Vierling-Claassen2013
Vierling-Claassen2013Tue, 07 May 2013 00:00:00 +0000Vierling-Claassen, a.A Do-It-Yourself Paper Digital Computer, 1959.
https://read.somethingorotherwhatever.com/entry/Ptak
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).PtakSun, 05 May 2013 00:00:00 +0000Ptak, John F.Familial sinistrals avoid exact numbers.
https://read.somethingorotherwhatever.com/entry/Sauerland2013
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.Sauerland2013Mon, 15 Apr 2013 00:00:00 +0000Sauerland, Uli and Gotzner, NicoleCircuitry in 3D chess
https://read.somethingorotherwhatever.com/entry/Goucher
GoucherWed, 03 Apr 2013 00:00:00 +0000Goucher, AdamThe urinal problem
https://read.somethingorotherwhatever.com/entry/Kranakis2010
Kranakis2010Thu, 28 Mar 2013 00:00:00 +0000Kranakis, Evangelos and Krizanc, DannyA Smaller Sleeping Bag For A Baby Snake
https://read.somethingorotherwhatever.com/entry/Linusson1998
Linusson1998Wed, 20 Mar 2013 00:00:00 +0000Linusson, Svante and ASTLUND, JWConstructing the Tits ovoid from an elliptic quadric
https://read.somethingorotherwhatever.com/entry/Cherowitzo2006
Cherowitzo2006Tue, 19 Mar 2013 00:00:00 +0000Cherowitzo, WEDas 2: 3-Ei-ein praktikables Eimodell
https://read.somethingorotherwhatever.com/entry/Moller2009
Moller2009Tue, 19 Mar 2013 00:00:00 +0000Möller, HProblems to sharpen the young
https://read.somethingorotherwhatever.com/entry/Hadley1992
An annotated translation of Propositiones ad acuendos juvenes, the oldest mathematical problem collection in Latin, attributed to Alcuin of York.Hadley1992Mon, 18 Mar 2013 00:00:00 +0000Hadley, John and Singmaster, DavidReview of "Groups" by Georges Papy in New Scientist
https://read.somethingorotherwhatever.com/entry/item29
item29Sun, 17 Mar 2013 00:00:00 +0000T. H. O'BeirneThe Circle-Squaring Problem Decomposed
https://read.somethingorotherwhatever.com/entry/Pierce2009
Pierce2009Sun, 17 Mar 2013 00:00:00 +0000Pierce, Pamela and Ramsay, JohnZeroless Arithmetic: Representing Integers ONLY using ONE
https://read.somethingorotherwhatever.com/entry/Ghang2013
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.Ghang2013Thu, 07 Mar 2013 00:00:00 +0000Ghang, EK and Zeilberger, DoronCircular reasoning: who first proved that $C/d$ is a constant?
https://read.somethingorotherwhatever.com/entry/Richeson2013
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.Richeson2013Wed, 06 Mar 2013 00:00:00 +0000Richeson, DavidEmbedding countable groups in 2-generator groups
https://read.somethingorotherwhatever.com/entry/Galvin1993
Galvin1993Tue, 19 Feb 2013 00:00:00 +0000Galvin, FredThe Muddy Children : A logic for public announcement
https://read.somethingorotherwhatever.com/entry/Hughes2007
Hughes2007Tue, 19 Feb 2013 00:00:00 +0000Hughes, JesseWhat are some of the most ridiculous proofs in mathematics?
https://read.somethingorotherwhatever.com/entry/item28
item28Sun, 17 Feb 2013 00:00:00 +0000AnonymousMarkets are efficient if and only if P = NP
https://read.somethingorotherwhatever.com/entry/Maymin2011
I prove that if markets are weak-form efficient, meaning current prices fully reflect all information available in past prices, then P = NP, meaning every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. I also prove the converse by showing how we can "program" the market to solve NP-complete problems. Since P probably does not equal NP, markets are probably not efficient. Specifically, markets become increasingly inefficient as the time series lengthens or becomes more frequent. An illustration by way of partitioning the excess returns to momentum strategies based on data availability confirms this prediction. Maymin2011Mon, 11 Feb 2013 00:00:00 +0000Maymin, PZConway's Rational Tangles
https://read.somethingorotherwhatever.com/entry/Davis2012
Davis2012Wed, 06 Feb 2013 00:00:00 +0000Davis, TomA note on paradoxical metric spaces
https://read.somethingorotherwhatever.com/entry/Deuber2004
Deuber2004Sat, 26 Jan 2013 00:00:00 +0000Deuber, W A and Simonovits, M and Os, V T SIncorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality
https://read.somethingorotherwhatever.com/entry/Fiore2013
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.Fiore2013Mon, 21 Jan 2013 00:00:00 +0000Fiore, Thomas M. and Noll, Thomas and Satyendra, RamonDelay can stabilize: Love affairs dynamics
https://read.somethingorotherwhatever.com/entry/Bielczyk2012
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.Bielczyk2012Sat, 19 Jan 2013 00:00:00 +0000Bielczyk, Natalia and Bodnar, Marek and Foryś, UrszulaAlgorithmic self-assembly of DNA Sierpinski triangles.
https://read.somethingorotherwhatever.com/entry/Rothemund2004
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.Rothemund2004Fri, 18 Jan 2013 00:00:00 +0000Rothemund, Paul W K and Papadakis, Nick and Winfree, ErikInvited commentary: the perils of birth weight--a lesson from directed acyclic graphs.
https://read.somethingorotherwhatever.com/entry/Wilcox2006
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, Hernández-Díaz 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 Hernández-Díaz 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.Wilcox2006Wed, 02 Jan 2013 00:00:00 +0000Wilcox, Allen JThe paramagnetic and glass transitions in sudoku
https://read.somethingorotherwhatever.com/entry/Williams2012
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.Williams2012Sat, 29 Dec 2012 00:00:00 +0000Williams, Alex and Ackland, Graeme . J.Figures for "Impossible fractals"
https://read.somethingorotherwhatever.com/entry/item27
item27Tue, 18 Dec 2012 00:00:00 +0000Cameron BrowneBiologically Unavoidable Sequences
https://read.somethingorotherwhatever.com/entry/Alexander2012
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.Alexander2012Sun, 16 Dec 2012 00:00:00 +0000Alexander, SamuelHow to eat 4/9 of a pizza
https://read.somethingorotherwhatever.com/entry/Knauer2011
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.Knauer2011Wed, 12 Dec 2012 00:00:00 +0000Knauer, Kolja and Micek, Piotr and Ueckerdt, TorstenA stratification of the space of all $k$-planes in $\mathbb{C}_n$
https://read.somethingorotherwhatever.com/entry/Knutson2012
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 symmetryKnutson2012Mon, 19 Nov 2012 00:00:00 +0000Allen KnutsonConway's Wizards
https://read.somethingorotherwhatever.com/entry/Khovanova2012
I present and discuss a puzzle about wizards invented by John H. Conway.Khovanova2012Sat, 03 Nov 2012 00:00:00 +0000Khovanova, TanyaPicture-Hanging Puzzles
https://read.somethingorotherwhatever.com/entry/Demaine2012
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.Demaine2012Sat, 03 Nov 2012 00:00:00 +0000Demaine, Erik D. and Demaine, Martin L. and Minsky, Yair N. and Mitchell, Joseph S. B. and Rivest, Ronald L. and Patrascu, MihaiPapy's Minicomputer
https://read.somethingorotherwhatever.com/entry/Papy1970
Papy1970Wed, 31 Oct 2012 00:00:00 +0000Papy, FThe lost squares of Dr. Franklin: Ben Franklin's missing squares and the secret of the magic circle
https://read.somethingorotherwhatever.com/entry/Pasles2001
Pasles2001Tue, 30 Oct 2012 00:00:00 +0000Pasles, PCOn sphere-filling ropes
https://read.somethingorotherwhatever.com/entry/Gerlach2010
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.Gerlach2010Mon, 29 Oct 2012 00:00:00 +0000Gerlach, Henryk and von der Mosel, HeikoAlgebraic theory of Penrose's non-periodic tilings of the plane
https://read.somethingorotherwhatever.com/entry/Bruijn1981
Bruijn1981Sat, 13 Oct 2012 00:00:00 +0000Bruijn, NG DeEarliest Uses of Symbols of Calculus
https://read.somethingorotherwhatever.com/entry/item26
item26Tue, 09 Oct 2012 00:00:00 +0000Jeff MillerThe topology of the minimal regular cover of the Archimedean tessellations
https://read.somethingorotherwhatever.com/entry/Coulbois2012
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.Coulbois2012Fri, 05 Oct 2012 00:00:00 +0000Coulbois, Thierry and Pellicer, Daniel and Raggi, Miguel and Ramírez, Camilo and Valdez, FerránTwin Towers of Hanoi
https://read.somethingorotherwhatever.com/entry/Sunic2011
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.Sunic2011Fri, 28 Sep 2012 00:00:00 +0000Sunic, ZoranOnline Dating Recommender Systems: The Split-complex Number Approach
https://read.somethingorotherwhatever.com/entry/DatingRecommenderSystems
DatingRecommenderSystemsThu, 13 Sep 2012 00:00:00 +0000Jérôme KunegisMagic: the Gathering is Turing Complete
https://read.somethingorotherwhatever.com/entry/item24
We always knew Magic: the Gathering was a complex game. But now it's proven: you could assemble a computer out of Magic cards.item24Tue, 11 Sep 2012 00:00:00 +0000Alex ChurchillHow Java's floating-point hurts everyone everywhere
https://read.somethingorotherwhatever.com/entry/Kahan1998
Kahan1998Wed, 05 Sep 2012 00:00:00 +0000Kahan, W and Darcy, JDBeastly Numbers
https://read.somethingorotherwhatever.com/entry/Kahan
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.KahanWed, 05 Sep 2012 00:00:00 +0000Kahan, WModiﬁed Pascal Triangle and Pascal Surfaces
https://read.somethingorotherwhatever.com/entry/item23
item23Wed, 05 Sep 2012 00:00:00 +0000Rely Pellicer and David AlvoA Hamiltonian circuit for Rubik's Cube
https://read.somethingorotherwhatever.com/entry/item22
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.item22Tue, 04 Sep 2012 00:00:00 +0000cuBerBruceMastermind is NP-Complete
https://read.somethingorotherwhatever.com/entry/Stuckman2005
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.Stuckman2005Sun, 02 Sep 2012 00:00:00 +0000Stuckman, Jeff and Zhang, Guo-QiangVIP-club phenomenon: emergence of elites and masterminds in social networks
https://read.somethingorotherwhatever.com/entry/Masuda2005
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.Masuda2005Sun, 02 Sep 2012 00:00:00 +0000Masuda, Naoki and Konno, NorioHow far can Tarzan jump?
https://read.somethingorotherwhatever.com/entry/Shima2012
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.Shima2012Sun, 02 Sep 2012 00:00:00 +0000Shima, HiroyukiThe Canonical Basis of $\dot{\mathbf{U}}$ for Type $A_{2}$
https://read.somethingorotherwhatever.com/entry/Cui2012
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.Cui2012Wed, 29 Aug 2012 00:00:00 +0000Cui, WeidengThe Fastest and Shortest Algorithm for All Well-Defined Problems
https://read.somethingorotherwhatever.com/entry/Hutter2002
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.
Hutter2002Tue, 14 Aug 2012 00:00:00 +0000Hutter, MarcusA New Rose : The First Simple Symmetric 11-Venn Diagram
https://read.somethingorotherwhatever.com/entry/Mamakani2012
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.Mamakani2012Thu, 09 Aug 2012 00:00:00 +0000Mamakani, Khalegh and Ruskey, FrankThe usefulness of useless knowledge
https://read.somethingorotherwhatever.com/entry/TheUsefulnessOfUselessKnowledge
TheUsefulnessOfUselessKnowledgeMon, 16 Jul 2012 00:00:00 +0000Flexner, AbrahamSeven Staggering Sequences
https://read.somethingorotherwhatever.com/entry/Sloane2006
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.Sloane2006Sat, 14 Jul 2012 00:00:00 +0000Sloane, N J AThe top ten prime numbers
https://read.somethingorotherwhatever.com/entry/Dubner2001
Dubner2001Thu, 28 Jun 2012 00:00:00 +0000Dubner, HTrain Sets
https://read.somethingorotherwhatever.com/entry/Chalcraft
ChalcraftMon, 25 Jun 2012 00:00:00 +0000Chalcraft, Adam and Greene, MichaelEquilibrium solution to the lowest unique positive integer game
https://read.somethingorotherwhatever.com/entry/Baek2010
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.Baek2010Fri, 22 Jun 2012 00:00:00 +0000Baek, Seung Ki and Bernhardsson, SebastianThe wobbly garden table
https://read.somethingorotherwhatever.com/entry/Kraft2001
Kraft2001Sun, 17 Jun 2012 00:00:00 +0000Kraft, HanspeterA cohomological viewpoint on elementary school arithmetic
https://read.somethingorotherwhatever.com/entry/Isaksen2002
Isaksen2002Thu, 14 Jun 2012 00:00:00 +0000Isaksen, DCOn distributions computable by random walks on graphs
https://read.somethingorotherwhatever.com/entry/Kindler2004
Kindler2004Thu, 07 Jun 2012 00:00:00 +0000Kindler, GTo Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction
https://read.somethingorotherwhatever.com/entry/item20
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.item20Sun, 03 Jun 2012 00:00:00 +0000David C KeenanTopology Explains Why Automobile Sunshades Fold Oddly
https://read.somethingorotherwhatever.com/entry/Feist2012
We use braids and linking number to explain why automobile shades fold into an odd number of loops.Feist2012Wed, 23 May 2012 00:00:00 +0000Feist, Curtis and Naimi, RaminOn an error in the star puzzle by Henry E. Dudeney
https://read.somethingorotherwhatever.com/entry/Ravsky2012
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.Ravsky2012Tue, 22 May 2012 00:00:00 +0000Ravsky, AlexHow to recognise a 4-ball when you see one
https://read.somethingorotherwhatever.com/entry/Geiges2011
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.Geiges2011Sat, 19 May 2012 00:00:00 +0000Geiges, Hansjörg and Zehmisch, KaiG2 and the Rolling Ball
https://read.somethingorotherwhatever.com/entry/Baez2012
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.Baez2012Mon, 14 May 2012 00:00:00 +0000Baez, John C and Huerta, JohnLectures on lost mathematics
https://read.somethingorotherwhatever.com/entry/LostMathematics
LostMathematicsThu, 10 May 2012 00:00:00 +0000Branko GrünbaumEstimating the Effect of the Red Card in Soccer
https://read.somethingorotherwhatever.com/entry/Vecer2009
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.Vecer2009Wed, 09 May 2012 00:00:00 +0000Vecer, Jan and Kopriva, FrantisekA categorical foundation for Bayesian probability
https://read.somethingorotherwhatever.com/entry/Culbertson2012
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.Culbertson2012Tue, 08 May 2012 00:00:00 +0000Culbertson, Jared and Sturtz, KirkSurvey on fusible numbers
https://read.somethingorotherwhatever.com/entry/Xu2012
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.Xu2012Wed, 02 May 2012 00:00:00 +0000Xu, JunyanCardinal arithmetic for skeptics
https://read.somethingorotherwhatever.com/entry/Shelah1992
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.Shelah1992Wed, 02 May 2012 00:00:00 +0000Shelah, SaharonA mathematician's survival guide
https://read.somethingorotherwhatever.com/entry/Casazza
CasazzaSun, 29 Apr 2012 00:00:00 +0000Casazza, Peter GCalculus Made Easy
https://read.somethingorotherwhatever.com/entry/Thompson1914
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 CALCULUSThompson1914Thu, 26 Apr 2012 00:00:00 +0000Thompson, Silvanus PLong finite sequences
https://read.somethingorotherwhatever.com/entry/Friedman1998
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).Friedman1998Tue, 24 Apr 2012 00:00:00 +0000Friedman, Harvey MComputer analysis of Sprouts with nimbers
https://read.somethingorotherwhatever.com/entry/Lemoine2010
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.Lemoine2010Sat, 21 Apr 2012 00:00:00 +0000Lemoine, Julien and Viennot, SimonNim multiplication
https://read.somethingorotherwhatever.com/entry/item18
item18Sat, 21 Apr 2012 00:00:00 +0000H. W. Lenstra, Jr.Theory and History of Geometric Models
https://read.somethingorotherwhatever.com/entry/Poloblanco2007
Poloblanco2007Sun, 15 Apr 2012 00:00:00 +0000Polo-blanco, IreneRobust Soldier Crab Ball Gate
https://read.somethingorotherwhatever.com/entry/Gunji2011
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).Gunji2011Sun, 15 Apr 2012 00:00:00 +0000Gunji, YP and Nishiyama, YStatistical Modeling of Gang Violence in Los Angeles
https://read.somethingorotherwhatever.com/entry/Fathauer
FathauerFri, 13 Apr 2012 00:00:00 +0000Fathauer, ChrisHigh Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams
https://read.somethingorotherwhatever.com/entry/Williams2004
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.Williams2004Thu, 12 Apr 2012 00:00:00 +0000Williams, Hugh C. and Poorten, A. J. Van Der and Stein, AndreasCellular automata in the hyperbolic plane: proposal for a new environment
https://read.somethingorotherwhatever.com/entry/Chelghoum2004
Chelghoum2004Sun, 08 Apr 2012 00:00:00 +0000Chelghoum, Kamel and Margenstern, Maurice and Martin, Beno\^itLight reflecting off Christmas-tree balls
https://read.somethingorotherwhatever.com/entry/item17
'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?item17Sun, 08 Apr 2012 00:00:00 +0000Joseph O'RourkeCake Cutting Mechanisms
https://read.somethingorotherwhatever.com/entry/Ianovski2012
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.Ianovski2012Sat, 07 Apr 2012 00:00:00 +0000Ianovski, EgorThe Euler spiral: a mathematical history
https://read.somethingorotherwhatever.com/entry/Levien2008
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.Levien2008Sat, 07 Apr 2012 00:00:00 +0000Levien, RaphNavigating Hyperbolic Space with Fibonacci Trees
https://read.somethingorotherwhatever.com/entry/item16
item16Sat, 07 Apr 2012 00:00:00 +0000monikerUndecidable problems: a sampler
https://read.somethingorotherwhatever.com/entry/Poonen2012
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.Poonen2012Wed, 04 Apr 2012 00:00:00 +0000Poonen, BjornThe mate-in-n problem of infinite chess is decidable
https://read.somethingorotherwhatever.com/entry/Brumleve2012
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.Brumleve2012Wed, 28 Mar 2012 00:00:00 +0000Brumleve, Dan and Hamkins, Joel David and Schlicht, PhilippStatistical Laws Governing Fluctuations in Word Use from Word Birth to Word Death
https://read.somethingorotherwhatever.com/entry/Petersen2011
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.Petersen2011Tue, 27 Mar 2012 00:00:00 +0000Petersen, Alexander M and Tenenbaum, Joel and Havlin, Shlomo and Stanley, H EugeneThe bitangent sphere problem
https://read.somethingorotherwhatever.com/entry/Giblin1990
Giblin1990Sat, 24 Mar 2012 00:00:00 +0000Giblin, PJGaussian prime spirals
https://read.somethingorotherwhatever.com/entry/item15
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 $90^{\circ}$.item15Sat, 24 Mar 2012 00:00:00 +0000Joseph O'RourkeTopic-based vector space model
https://read.somethingorotherwhatever.com/entry/Becker2003
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.Becker2003Sat, 24 Mar 2012 00:00:00 +0000Jörg Becker and Dominik KuropkaA New Approximation to $\pi$ (Conclusion)
https://read.somethingorotherwhatever.com/entry/Ferguson
FergusonWed, 14 Mar 2012 00:00:00 +0000Ferguson, D. F. and Wrench, John WDoc, What Are My Chances?
https://read.somethingorotherwhatever.com/entry/Marasco2011
Marasco2011Sun, 11 Mar 2012 00:00:00 +0000Marasco, Joe and Doerfler, Ron and Roschier, LeifRandom walks reaching against all odds the other side of the quarter plane
https://read.somethingorotherwhatever.com/entry/VanLeeuwaarden2011
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.VanLeeuwaarden2011Thu, 01 Mar 2012 00:00:00 +0000van Leeuwaarden, Johan S. H. and Raschel, KilianQuotients Homophones des Groupes Libres Homophonic Quotients of Free Groups
https://read.somethingorotherwhatever.com/entry/Washington1986
Washington1986Mon, 27 Feb 2012 00:00:00 +0000Washington, Lawrence and Zagier, DonPoe, E.: Near A Raven
https://read.somethingorotherwhatever.com/entry/item14
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.item14Sat, 25 Feb 2012 00:00:00 +0000Mike KeithBarcodes: the persistent topology of data
https://read.somethingorotherwhatever.com/entry/Ghrist2008
Ghrist2008Thu, 23 Feb 2012 00:00:00 +0000Ghrist, RobertFurther evidence for addition and numerical competence by a Grey parrot (Psittacus erithacus)
https://read.somethingorotherwhatever.com/entry/Pepperberg2012
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.Pepperberg2012Tue, 21 Feb 2012 00:00:00 +0000Pepperberg, Irene M.Passage to the limit in Proposition I, Book I of Newton's Principia
https://read.somethingorotherwhatever.com/entry/Erlichson2003
Erlichson2003Wed, 15 Feb 2012 00:00:00 +0000Erlichson, HermanOrange Peels and Fresnel Integrals
https://read.somethingorotherwhatever.com/entry/Bartholdi2012
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.Bartholdi2012Wed, 15 Feb 2012 00:00:00 +0000Bartholdi, Laurent and Henriques, André G.Tropical Mathematics
https://read.somethingorotherwhatever.com/entry/Speyer2004
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.Speyer2004Sun, 12 Feb 2012 00:00:00 +0000Speyer, David and Sturmfels, BerndFractions without Quotients: Arithmetic of Repeating Decimals
https://read.somethingorotherwhatever.com/entry/Plagge1978
Plagge1978Fri, 10 Feb 2012 00:00:00 +0000Plagge, RichardPantologia. A new (cabinet) cyclopædia, by J.M. Good, O. Gregory, and N. Bosworth assisted by other gentlemen of eminence
https://read.somethingorotherwhatever.com/entry/Good1819
Good1819Fri, 10 Feb 2012 00:00:00 +0000Good, John Mason and Gregory, Olinthus GilbertNineteen dubious ways to compute the exponential of a matrix, twenty-five years later
https://read.somethingorotherwhatever.com/entry/Moler2003
Moler2003Thu, 09 Feb 2012 00:00:00 +0000Moler, Cleve and Van Loan, C.Good stories, pity they're not true
https://read.somethingorotherwhatever.com/entry/Devlin
DevlinWed, 08 Feb 2012 00:00:00 +0000Devlin, KeithFibonacci determinants-a combinatorial approach
https://read.somethingorotherwhatever.com/entry/Benjamin2007
Benjamin2007Wed, 08 Feb 2012 00:00:00 +0000Benjamin, A.T. and Cameron, N.T. and Quinn, J.J.Benjamin Peirce and the Howland will
https://read.somethingorotherwhatever.com/entry/Meier1980
Meier1980Tue, 07 Feb 2012 00:00:00 +0000Meier, Paul and Zabell, SandyMapping an unfriendly subway system
https://read.somethingorotherwhatever.com/entry/Flocchini2010
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.Flocchini2010Tue, 07 Feb 2012 00:00:00 +0000Flocchini, Paola and Kellett, Matthew and Mason, P.Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles
https://read.somethingorotherwhatever.com/entry/Gradwohl2007
Gradwohl2007Tue, 07 Feb 2012 00:00:00 +0000Gradwohl, Ronen and Naor, M. and Pinkas, Benny and Rothblum, G.In retrospect: On the Six-Cornered Snowflake
https://read.somethingorotherwhatever.com/entry/Ball2011
Ball2011Mon, 06 Feb 2012 00:00:00 +0000Ball, PhilipTropical Arithmetic and Tropical Matrix Algebra
https://read.somethingorotherwhatever.com/entry/Izhakian2005
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.Izhakian2005Fri, 03 Feb 2012 00:00:00 +0000Izhakian, ZurGerrymandering and Convexity
https://read.somethingorotherwhatever.com/entry/Hodge2010
Hodge2010Fri, 03 Feb 2012 00:00:00 +0000Hodge, Jonathan K. and Marshall, Emily and Patterson, GeoffContinued fractions constructed from prime numbers
https://read.somethingorotherwhatever.com/entry/Wolf2010
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.Wolf2010Wed, 01 Feb 2012 00:00:00 +0000Wolf, MarekGaming is a hard job, but someone has to do it!
https://read.somethingorotherwhatever.com/entry/Viglietta2012
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.Viglietta2012Fri, 27 Jan 2012 00:00:00 +0000Viglietta, GiovanniCompositional Reasoning Using Intervals and Time Reversal
https://read.somethingorotherwhatever.com/entry/Moszkowski2011
Moszkowski2011Fri, 27 Jan 2012 00:00:00 +0000Moszkowski, BenThe hardness of the Lemmings game, or Oh no, more NP-completeness proofs
https://read.somethingorotherwhatever.com/entry/Cormode2004
Cormode2004Fri, 27 Jan 2012 00:00:00 +0000Cormode, GrahamLondon Calling Philosophy and Engineering: WPE 2008
https://read.somethingorotherwhatever.com/entry/item13
item13Tue, 24 Jan 2012 00:00:00 +0000Glen MillerThe Snowblower Problem
https://read.somethingorotherwhatever.com/entry/Arkin2006
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.Arkin2006Sun, 22 Jan 2012 00:00:00 +0000Arkin, Esther M. and Bender, Michael A. and Mitchell, Joseph S. B. and Polishchuk, ValentinScooping the Loop Snooper
https://read.somethingorotherwhatever.com/entry/ScoopingLoopSnooper
ScoopingLoopSnooperFri, 20 Jan 2012 00:00:00 +0000Geoffrey K. PullumEthnomathematics as a new research field , illustrated by studies of mathematical ideas in African history
https://read.somethingorotherwhatever.com/entry/Gerdes2000
Gerdes2000Thu, 19 Jan 2012 00:00:00 +0000Gerdes, PaulusDrawings from Angola: living mathematics
https://read.somethingorotherwhatever.com/entry/item11
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.item11Thu, 19 Jan 2012 00:00:00 +0000Paulus GerdesUnderstanding Monads With JavaScript
https://read.somethingorotherwhatever.com/entry/item10
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.item10Wed, 18 Jan 2012 00:00:00 +0000Ionuț G. StanThe Collatz Fractal
https://read.somethingorotherwhatever.com/entry/Henderson
HendersonSun, 15 Jan 2012 00:00:00 +0000Henderson, XanderAnalysis of Casino Shelf Shuffling Machines
https://read.somethingorotherwhatever.com/entry/Diaconis2011
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.Diaconis2011Wed, 11 Jan 2012 00:00:00 +0000Diaconis, Persi and Fulman, Jason and Holmes, SusanA zero-knowledge Poker protocol that achieves confidentiality of the players' strategy or How to achieve an electronic Poker face
https://read.somethingorotherwhatever.com/entry/Crepeau1987
Crepeau1987Tue, 10 Jan 2012 00:00:00 +0000Crépeau, C.A Generalized Fibonacci LSB Data Hiding Technique
https://read.somethingorotherwhatever.com/entry/Battisti2006
Battisti2006Tue, 10 Jan 2012 00:00:00 +0000Battisti, F and Carli, M and Neri, A and Egiaziarian, KComputer evolution of buildable objects
https://read.somethingorotherwhatever.com/entry/Funes1999
Funes1999Mon, 09 Jan 2012 00:00:00 +0000Funes, Pablo and Pollack, JordanRandom Walks on Finite Groups
https://read.somethingorotherwhatever.com/entry/Saloffcoste
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.SaloffcosteThu, 05 Jan 2012 00:00:00 +0000Saloff-coste, LaurentComplexity of Langton's ant
https://read.somethingorotherwhatever.com/entry/Gajardo2002
Gajardo2002Wed, 04 Jan 2012 00:00:00 +0000Gajardo, A and Moreira, A and Goles, EOn badly approximable numbers and certain games
https://read.somethingorotherwhatever.com/entry/Schmidt1966
Schmidt1966Wed, 04 Jan 2012 00:00:00 +0000Schmidt, WMThree-dimensional finite point groups and the symmetry of beaded beads
https://read.somethingorotherwhatever.com/entry/Fisher2007
Fisher2007Wed, 04 Jan 2012 00:00:00 +0000Fisher, GL and Mellor, B.Carrots for dessert
https://read.somethingorotherwhatever.com/entry/Petersen2010
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.Petersen2010Mon, 02 Jan 2012 00:00:00 +0000Petersen, Carsten Lunde and Roesch, PascaleHow to Gamble If You're In a Hurry
https://read.somethingorotherwhatever.com/entry/Ekhad2011
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).Ekhad2011Thu, 15 Dec 2011 00:00:00 +0000Ekhad, Shalosh B and Georgiadis, Evangelos and Zeilberger, DoronOrigami Burrs and Woven Polyhedra
https://read.somethingorotherwhatever.com/entry/Lang2000
Lang2000Wed, 14 Dec 2011 00:00:00 +0000Lang, Robert JDeobfuscation is in NP
https://read.somethingorotherwhatever.com/entry/Appel2002
Appel2002Mon, 12 Dec 2011 00:00:00 +0000Appel, Andrew WTie knots, random walks and topology
https://read.somethingorotherwhatever.com/entry/Fink2000
Fink2000Fri, 09 Dec 2011 00:00:00 +0000Fink, T and Mao, YWhat Are the Odds?
https://read.somethingorotherwhatever.com/entry/Lane
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 OddsLaneFri, 09 Dec 2011 00:00:00 +0000Lane, F.C.Designing tie knots by random walks
https://read.somethingorotherwhatever.com/entry/Fink1999
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.Fink1999Fri, 09 Dec 2011 00:00:00 +0000Thomas M. Fink and Yong MaoAsymptotic statistics of the n-sided planar Poisson–Voronoi cell: I. Exact results
https://read.somethingorotherwhatever.com/entry/Hilhorst2005
Hilhorst2005Thu, 08 Dec 2011 00:00:00 +0000Hilhorst, H.J.Laying train tracks
https://read.somethingorotherwhatever.com/entry/item9
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.item9Sat, 03 Dec 2011 00:00:00 +0000Danny CalegariTetris is Hard, Even to Approximate
https://read.somethingorotherwhatever.com/entry/Demaine2008
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.Demaine2008Wed, 23 Nov 2011 00:00:00 +0000Demaine, Erik D and Hohenberger, Susan and Liben-Nowell, DavidRemainder Wheels and Group Theory
https://read.somethingorotherwhatever.com/entry/Brenton2008
Brenton2008Wed, 02 Nov 2011 00:00:00 +0000Brenton, LawrenceChalk: Materials and Concepts in Mathematics Research
https://read.somethingorotherwhatever.com/entry/Barany2011
Barany2011Tue, 01 Nov 2011 00:00:00 +0000Barany, Michael J and Mackenzie, DonaldThe experimental effectiveness of mathematical proof
https://read.somethingorotherwhatever.com/entry/Miquel2007
Miquel2007Sun, 30 Oct 2011 00:00:00 +0000Miquel, AlexandreScholarly communication in transition: The use of question marks in the titles of scientific articles in medicine, life sciences and physics 1966–2005
https://read.somethingorotherwhatever.com/entry/Ball2009
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.Ball2009Fri, 14 Oct 2011 00:00:00 +0000Ball, RafaelBaron Munchhausen Redeems Himself : Bounds for a Coin-Weighing Puzzle Background
https://read.somethingorotherwhatever.com/entry/Khovanova2010
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.Khovanova2010Fri, 14 Oct 2011 00:00:00 +0000Khovanova, Tanya and Lewis, Joel BrewsterCool irrational numbers and their rather cool rational approximations
https://read.somethingorotherwhatever.com/entry/Calogero2003
Calogero2003Thu, 13 Oct 2011 00:00:00 +0000Calogero, FrancescoA linear programming approach for aircraft boarding strategy
https://read.somethingorotherwhatever.com/entry/Bazargan2007
Bazargan2007Tue, 04 Oct 2011 00:00:00 +0000Bazargan, MThe elasto-plastic indentation of a half-space by a rigid sphere
https://read.somethingorotherwhatever.com/entry/Hardy1971
Hardy1971Sun, 02 Oct 2011 00:00:00 +0000Hardy, C. and Baronet, C. N. and Tordion, G. V.Gödel's Second Incompleteness Theorem Explained in Words of One Syllable
https://read.somethingorotherwhatever.com/entry/Boolos1994
Boolos1994Sat, 24 Sep 2011 00:00:00 +0000Boolos, GeorgeFusible Numbers
https://read.somethingorotherwhatever.com/entry/Erickson
EricksonWed, 21 Sep 2011 00:00:00 +0000Erickson, JeffThere is no "Uspensky's method"
https://read.somethingorotherwhatever.com/entry/Akritas1986
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.Akritas1986Thu, 15 Sep 2011 00:00:00 +0000Akritas, AGMad Abel : A card game for 2 + players
https://read.somethingorotherwhatever.com/entry/Mccarthy2006
Mccarthy2006Tue, 13 Sep 2011 00:00:00 +0000Mccarthy, SmáriShamos's Catalog of the Real Numbers
https://read.somethingorotherwhatever.com/entry/Shamos2011
Shamos2011Thu, 08 Sep 2011 00:00:00 +0000Shamos, Michael IanDoes Quantum Interference exist in Twitter?
https://read.somethingorotherwhatever.com/entry/Shuai
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.ShuaiThu, 08 Sep 2011 00:00:00 +0000Shuai, Xin and Ding, Ying and Busemeyer, Jerome and Sun, Yuyin and Chen, Shanshan and Tang, JiePacking circles and spheres on surfaces
https://read.somethingorotherwhatever.com/entry/item8
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.item8Sun, 04 Sep 2011 00:00:00 +0000Alexander Schiftner and Mathias Höbinger and Johannes Wallner and Helmut PottmannIrrationality from the book
https://read.somethingorotherwhatever.com/entry/Miller2009
Miller2009Mon, 29 Aug 2011 00:00:00 +0000Miller, Steven J and Montague, DavidAgainst Conditionalization
https://read.somethingorotherwhatever.com/entry/Bacchus1995
Bacchus1995Sun, 28 Aug 2011 00:00:00 +0000Bacchus, FahiemInvestigations of Game of Life cellular automata rules on Penrose Tilings : lifetime and ash statistics
https://read.somethingorotherwhatever.com/entry/Owens
OwensSun, 14 Aug 2011 00:00:00 +0000Owens, Nick and Stepney, SusanDoubly-, triply-, quadruply- and quintuply-innervated crustacean muscles
https://read.somethingorotherwhatever.com/entry/VanHarreveld1939
VanHarreveld1939Tue, 09 Aug 2011 00:00:00 +0000van Harreveld, A.Gödel's incompleteness theorem
https://read.somethingorotherwhatever.com/entry/Uspensky1994
Uspensky1994Wed, 01 Jun 2011 00:00:00 +0000Uspensky, VPenrose's Godelian argument
https://read.somethingorotherwhatever.com/entry/Feferman
FefermanMon, 16 May 2011 00:00:00 +0000Feferman, SolomonDeriving Uniform Polyhedra with Wythoff's Construction
https://read.somethingorotherwhatever.com/entry/Romano
RomanoTue, 10 May 2011 00:00:00 +0000Romano, DonTesting Petri Nets for Mobile Robots Using Gröbner Bases
https://read.somethingorotherwhatever.com/entry/Chandler
ChandlerMon, 09 May 2011 00:00:00 +0000Chandler, Angie and Heyworth, Anne and Blair, Lynne and Seward, DerekAccurate estimation of forward path geometry using two-clothoid road model
https://read.somethingorotherwhatever.com/entry/Khosla2002
Khosla2002Tue, 12 Apr 2011 00:00:00 +0000Khosla, DThe 1-Hyperbolic Projection for User Interfaces
https://read.somethingorotherwhatever.com/entry/Kolliopoulos
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.KolliopoulosTue, 12 Apr 2011 00:00:00 +0000Kolliopoulos, AlexanderDrawing and Labeling High-Quality Metro Maps by Mixed-Integer Programming.
https://read.somethingorotherwhatever.com/entry/Nollenburg2010
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.Nollenburg2010Thu, 07 Apr 2011 00:00:00 +0000Nöllenburg, Martin and Wolff, AlexanderA Paradoxical Property of the Monkey Book
https://read.somethingorotherwhatever.com/entry/Bernhardsson2011
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.Bernhardsson2011Sun, 03 Apr 2011 00:00:00 +0000Bernhardsson, Sebastian and Baek, Seung Ki and Minnhagen, PetterAn example of a computable absolutely normal number
https://read.somethingorotherwhatever.com/entry/Figueira2002
Figueira2002Mon, 28 Mar 2011 00:00:00 +0000Figueira, SantiagoA Note on Approximating the Normal Distribution Function
https://read.somethingorotherwhatever.com/entry/Aludaat2008
Aludaat2008Mon, 28 Mar 2011 00:00:00 +0000Aludaat, K M and Alodat, M TTheoretical Computer Science Cheat Sheet
https://read.somethingorotherwhatever.com/entry/CSCheatSheet
CSCheatSheetSat, 26 Mar 2011 00:00:00 +0000Steve SeidenTree automata techniques and applications
https://read.somethingorotherwhatever.com/entry/Comon1997
Comon1997Fri, 25 Mar 2011 00:00:00 +0000Comon, Hubert and Dauchet, M and Gilleron, RAutomatic calculation of plane loci using Grobner bases and integration into a Dynamic Geometry System
https://read.somethingorotherwhatever.com/entry/Gerh2010
Gerh2010Thu, 24 Mar 2011 00:00:00 +0000Gerh, MichaelJuggling Probabilities
https://read.somethingorotherwhatever.com/entry/Warrington2009
The act of a person juggling can be viewed as a Markov process if we assume that the juggler throws to random heights. I make this association for the simplest reasonable model of random juggling and compute the steady state probabilities in terms of the Stirling numbers of the second kind. I also explore several alternate models of juggling. Warrington2009Sun, 20 Mar 2011 00:00:00 +0000Warrington, Gregory S.The isoperimetric problem
https://read.somethingorotherwhatever.com/entry/Blasjo
BlasjoMon, 14 Mar 2011 00:00:00 +0000Blasjo, ViktorHierarchical Position Based Dynamics
https://read.somethingorotherwhatever.com/entry/Faure2008
Faure2008Fri, 11 Mar 2011 00:00:00 +0000Faure, F. and Teschner, M.James Garfield's Proof of the Pythagorean Theorem
https://read.somethingorotherwhatever.com/entry/Ellermeyer2008
Ellermeyer2008Fri, 18 Feb 2011 00:00:00 +0000Ellermeyer, S FSurreal Numbers – An Introduction
https://read.somethingorotherwhatever.com/entry/Tondering2005
Tondering2005Wed, 16 Feb 2011 00:00:00 +0000Tøndering, ClausA discursive grammar for customizing mass housing: the case of Siza's houses at Malagueira
https://read.somethingorotherwhatever.com/entry/Duarte2005
Duarte2005Tue, 15 Feb 2011 00:00:00 +0000Duarte, JHypercomputation: computing more than the Turing machine
https://read.somethingorotherwhatever.com/entry/Ord2002
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.Ord2002Wed, 09 Feb 2011 00:00:00 +0000Ord, TobyComparison of geometric figures
https://read.somethingorotherwhatever.com/entry/Glenis2008
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. Glenis2008Thu, 03 Feb 2011 00:00:00 +0000Glenis, 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 OthersOn a curious property of 3435
https://read.somethingorotherwhatever.com/entry/Berkel2009
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.Berkel2009Wed, 02 Feb 2011 00:00:00 +0000Berkel, Daan VanBetter approximations to cumulative normal functions
https://read.somethingorotherwhatever.com/entry/West2002
West2002Sat, 22 Jan 2011 00:00:00 +0000West, GraemeContinued fraction algorithms, functional operators, and structure constants
https://read.somethingorotherwhatever.com/entry/Flajolet1998
Flajolet1998Thu, 20 Jan 2011 00:00:00 +0000Flajolet, P. and Vallée, B.A classification for shaggy dog stories
https://read.somethingorotherwhatever.com/entry/Brunvand1963
Brunvand1963Wed, 12 Jan 2011 00:00:00 +0000Brunvand, J.H.On Buffon Machines and Numbers
https://read.somethingorotherwhatever.com/entry/Flajolet2011
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.Flajolet2011Wed, 12 Jan 2011 00:00:00 +0000Flajolet, PhilippeFour questions about fuzzy rankings
https://read.somethingorotherwhatever.com/entry/item5
item5Wed, 12 Jan 2011 00:00:00 +0000Brian HayesA history of mathematical notations
https://read.somethingorotherwhatever.com/entry/CajoriNotations
CajoriNotationsWed, 12 Jan 2011 00:00:00 +0000Florian CajoriInterpolating Solid Orientations with a $C^2$ -Continuous B-Spline Quaternion Curve
https://read.somethingorotherwhatever.com/entry/Ge2007
Ge2007Wed, 12 Jan 2011 00:00:00 +0000Ge, Wenbing and Huang, Zhangjin and Wang, GuopingCan One Hear the Shape of a Drum?
https://read.somethingorotherwhatever.com/entry/Kac
KacWed, 12 Jan 2011 00:00:00 +0000Kac, MarkAnimating rotation with quaternion curves
https://read.somethingorotherwhatever.com/entry/Shoemake1985
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.Shoemake1985Wed, 12 Jan 2011 00:00:00 +0000Shoemake, KenSpontaneous knotting of an agitated string.
https://read.somethingorotherwhatever.com/entry/Raymer2007
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 Möbius 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.Raymer2007Wed, 12 Jan 2011 00:00:00 +0000Raymer, Dorian M and Smith, Douglas EZaphod Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle
https://read.somethingorotherwhatever.com/entry/item6
item6Sun, 31 Oct 2010 00:00:00 +0000Andrew GranvilleImplications of the Turing Completeness of Reaction-Diffusion Models, informed by GPGPU simulations on an XBox 360: Cardiac Arrythmias, Re-entry and the Halting Problem
https://read.somethingorotherwhatever.com/entry/Scarle2008
Scarle2008Thu, 30 Sep 2010 00:00:00 +0000Scarle, Sopenttd logic gates
https://read.somethingorotherwhatever.com/entry/item3
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.item3Thu, 30 Sep 2010 00:00:00 +0000Heikki KallasjokiPush-pull LEGO logic gates
https://read.somethingorotherwhatever.com/entry/item4
item4Thu, 30 Sep 2010 00:00:00 +0000RandomwraithMisconceptions about the Golden Ratio
https://read.somethingorotherwhatever.com/entry/Markowsky1992
Markowsky1992Wed, 29 Sep 2010 00:00:00 +0000Markowsky, GeorgeOn Furstenberg's Proof of the Infinitude of Primes
https://read.somethingorotherwhatever.com/entry/Mercer2009
Mercer2009Thu, 09 Sep 2010 00:00:00 +0000Mercer, Idris DWhat symmetry groups are present in the Alhambra?
https://read.somethingorotherwhatever.com/entry/Grunbaum2006
Grunbaum2006Mon, 06 Sep 2010 00:00:00 +0000Grünbaum, BrankoThe Origin of Chemical Elements
https://read.somethingorotherwhatever.com/entry/Alpher1948
Alpher1948Fri, 03 Sep 2010 00:00:00 +0000Alpher, R. and Bethe, H. and Gamow, G.The role of instrumental and relational understanding in proofs about group isomorphisms
https://read.somethingorotherwhatever.com/entry/Weber2002
Weber2002Fri, 03 Sep 2010 00:00:00 +0000Weber, K.A Note on Boolos' Proof of the Incompleteness Theorem
https://read.somethingorotherwhatever.com/entry/Kikuchi1994
We give a proof of Gödel's first incompleteness theorem based on Berry's paradox, and from it we also derive the second incompleteness theorem model-theoretically.Kikuchi1994Fri, 03 Sep 2010 00:00:00 +0000Kikuchi, MakotoA Closed-Form Algorithm for Converting Hilbert Space-Filling Curve Indices
https://read.somethingorotherwhatever.com/entry/Chen2010
Chen2010Wed, 01 Sep 2010 00:00:00 +0000Chen, Chih-sheng and Lin, Shen-yi and Fan, Min-hsuan and Huang, Chua-huangHilbert R-tree: An improved R-tree using fractals
https://read.somethingorotherwhatever.com/entry/Kamel1994
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.Kamel1994Tue, 31 Aug 2010 00:00:00 +0000Kamel, Ibrahim and Faloutsos, ChristosDigital halftoning with space filling curves
https://read.somethingorotherwhatever.com/entry/item2
This paper introduces a new digital halftoning technique that uses space filling curves to generate aperiodic patterns of clustered dots. This method allows the parameterization of the size of pixel clusters, which can vary in one pixel steps. The algorithm unifies, in this way, the dispersed and clustered-dot dithering techniques.item2Tue, 31 Aug 2010 00:00:00 +0000Luiz Velho and Jonas de Miranda GomesA game for budding knot theorists
https://read.somethingorotherwhatever.com/entry/Entanglement
EntanglementWed, 25 Aug 2010 00:00:00 +0000Dave RichesonOn Mathematics and Mathematicians
https://read.somethingorotherwhatever.com/entry/Moritz2008
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...Moritz2008Tue, 24 Aug 2010 00:00:00 +0000Moritz, Robert EdowardDetection of transposition errors in decimal numbers
https://read.somethingorotherwhatever.com/entry/Freeman1967
Freeman1967Fri, 20 Aug 2010 00:00:00 +0000Freeman, HCircle Packing for Origami Design Is Hard
https://read.somethingorotherwhatever.com/entry/Demaine2010
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.Demaine2010Fri, 13 Aug 2010 00:00:00 +0000Demaine, E.D. and Fekete, S.P. and Lang, R.J.What Sequential Games , the Tychonoff Theorem and the Double-Negation Shift have in Common
https://read.somethingorotherwhatever.com/entry/Oliva2010
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)Oliva2010Wed, 04 Aug 2010 00:00:00 +0000Oliva, Paulo and Escardo, MartinThe Mathematics of Musical Instruments
https://read.somethingorotherwhatever.com/entry/Hall2001
Hall2001Sat, 24 Jul 2010 00:00:00 +0000Hall, Rachel W. and Josic, KresimirA combinatorial approach to sums of two squares and related problems
https://read.somethingorotherwhatever.com/entry/Elsholtz2002
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 .Elsholtz2002Fri, 16 Jul 2010 00:00:00 +0000Elsholtz, ChristianConstructive gem: juggling exponentials
https://read.somethingorotherwhatever.com/entry/Bauer
BauerThu, 15 Jul 2010 00:00:00 +0000Bauer, Andrej‘Knowable' As ‘Known After an Announcement'
https://read.somethingorotherwhatever.com/entry/Balbiani2008
Balbiani2008Sat, 26 Jun 2010 00:00:00 +0000Balbiani, Philippe and Baltag, Alexandru and Ditmarsch, Hans Van and Herzig, Andreas and Hoshi, Tomohiro and De Lima, TiagoFoolproof : A Sampling of Mathematical Folk Humor
https://read.somethingorotherwhatever.com/entry/Renteln
RentelnFri, 18 Jun 2010 00:00:00 +0000Renteln, Paul and Dundes, AlanA formal system for Euclid's Elements
https://read.somethingorotherwhatever.com/entry/Avigad2008
We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.Avigad2008Sun, 04 Apr 2010 00:00:00 +0000Avigad, Jeremy and Dean, Edward and Mumma, JohnAn aperiodic hexagonal tile
https://read.somethingorotherwhatever.com/entry/Socolar2010
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.Socolar2010Fri, 26 Mar 2010 00:00:00 +0000Socolar, Joshua E. S. and Taylor, Joan M.How to explain zero-knowledge protocols to your children
https://read.somethingorotherwhatever.com/entry/Quisquater
QuisquaterWed, 24 Mar 2010 00:00:00 +0000Quisquater, JJ and Quisquater, MPlane recursive trees, Stirling permutations and an urn model
https://read.somethingorotherwhatever.com/entry/Janson2008
Janson2008Thu, 25 Feb 2010 00:00:00 +0000Janson, SvanteUnbounded spigot algorithms for the digits of pi
https://read.somethingorotherwhatever.com/entry/Gibbons2006
Gibbons2006Tue, 05 Jan 2010 00:00:00 +0000Gibbons, J.