Interesting EsotericaThu, 25 May 2017 05:25:46 -0700Cuckoo Filter: Simplification and Analysis
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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.Wed, 31 Dec 1969 16:33:37 -0800David EppsteinThe opaque square
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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.1Wed, 31 Dec 1969 16:33:37 -0800Adrian Dumitrescu and Minghui JiangHomotopy type theory: the logic of space
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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).2Wed, 31 Dec 1969 16:33:37 -0800Michael ShulmanOn Fibonacci Quaternions
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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.3Wed, 31 Dec 1969 16:33:37 -0800Serpil HaliciTropical totally positive matrices
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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.4Wed, 31 Dec 1969 16:33:37 -0800Stéphane Gaubert and Adi NivNo, This is not a Circle
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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.5Wed, 31 Dec 1969 16:33:37 -0800Zoltán KovácsPVC Polyhedra
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We describe how to construct a dodecahedron, tetrahedron, cube, and
octahedron out of pvc pipes using standard fittings.6Wed, 31 Dec 1969 16:33:37 -0800David GlickensteinThe number dictionary
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The purpose is to provide an opportunity to show properties of numbers.7Wed, 31 Dec 1969 16:33:37 -0800Ovals and Egg Curves
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8Wed, 31 Dec 1969 16:33:37 -0800Jürgen Köller Approval Voting in Product Societies
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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.9Wed, 31 Dec 1969 16:33:37 -0800Kristen Mazur and Mutiara Sondjaja and Matthew Wright and Carolyn YarnallPauli Pascal Pyramids, Pauli Fibonacci Numbers, and Pauli Jacobsthal Numbers
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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?10Wed, 31 Dec 1969 16:33:37 -0800Martin Erik HornJewish Problems
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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.11Wed, 31 Dec 1969 16:33:37 -0800Tanya Khovanova and Alexey RadulBest Laid Plans of Lions and Men
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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
$\varepsilon>0$.12Wed, 31 Dec 1969 16:33:37 -0800Mikkel Abrahamsen and Jacob Holm and Eva Rotenberg and Christian Wulff-NilsenBeyond Floating Point: Next-Generation Computer Arithmetic
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13Wed, 31 Dec 1969 16:33:37 -0800John L. GustafsonPAPAC-00, a Do-It-Yourself Paper Computer
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14Wed, 31 Dec 1969 16:33:37 -0800Rollin P. MayerCrazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9
/entry/15
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.15Wed, 31 Dec 1969 16:33:37 -0800Inder J. TanejaThe mathematics of lecture hall partitions
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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.16Wed, 31 Dec 1969 16:33:37 -0800Carla D. SavageStatistics Done Wrong
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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.17Wed, 31 Dec 1969 16:33:37 -0800Alex ReinhartThree Thoughts on “Prime Simplicity”
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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.18Wed, 31 Dec 1969 16:33:37 -0800Michael HardyMeaning in Classical Mathematics: Is it at Odds with Intuitionism?
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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.19Wed, 31 Dec 1969 16:33:37 -0800Karin Usadi Katz and Mikhail G. KatzPrime Simplicity
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20Wed, 31 Dec 1969 16:33:37 -0800Michael Hardy and Catherine WoodgoldPlane partitions in the work of Richard Stanley and his school
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These notes provide a survey of the theory of plane partitions, seen through
the glasses of the work of Richard Stanley and his school.21Wed, 31 Dec 1969 16:33:37 -0800C. KrattenthalerRandom Triangles and Polygons in the Plane
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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.22Wed, 31 Dec 1969 16:33:37 -0800Jason Cantarella and Tom Needham and Clayton Shonkwiler and Gavin StewartTransfinite Version of Welter's Game
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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.23Wed, 31 Dec 1969 16:33:37 -0800Tomoaki AbukuOn the Existence of Ordinary Triangles
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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
$\Omega(|P|)$.24Wed, 31 Dec 1969 16:33:37 -0800Radoslav Fulek and Hossein Nassajian Mojarrad and Márton Naszódi and József Solymosi and Sebastian U. Stich and May SzedlákAnalysis of Carries in Signed Digit Expansions
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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.25Wed, 31 Dec 1969 16:33:37 -0800Clemens Heuberger and Sara Kropf and Helmut ProdingerHunting Rabbits on the Hypercube
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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.26Wed, 31 Dec 1969 16:33:37 -0800Jessalyn Bolkema and Corbin GroothuisRules for Folding Polyminoes from One Level to Two Levels
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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.27Wed, 31 Dec 1969 16:33:37 -0800Julia Martin and Elizabeth WilcoxHuman Inferences about Sequences: A Minimal Transition Probability Model
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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.28Wed, 31 Dec 1969 16:33:37 -0800Florent Meyniel and Maxime Maheu and Stanislas DehaeneA Singular Mathematical Promenade
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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.29Wed, 31 Dec 1969 16:33:36 -0800Etienne GhysBalloon Polyhedra
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30Wed, 31 Dec 1969 16:33:36 -0800Erik D. Demaine and Martin L. Demaine and Vi HartTwo short proofs of the Perfect Forest Theorem
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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.31Wed, 31 Dec 1969 16:33:36 -0800Yair Caro and Josef Lauri and Christina ZarbEvery natural number is the sum of forty-nine palindromes
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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.32Wed, 31 Dec 1969 16:33:36 -0800William D. BanksSequences of consecutive \(n\)-Niven numbers
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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. 33Wed, 31 Dec 1969 16:33:36 -0800H.G. GrundmanDeveloping a Mathematical Model for Bobbin Lace
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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.34Wed, 31 Dec 1969 16:33:36 -0800Veronika Irvine and Frank RuskeyQuasipractical Numbers
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35Wed, 31 Dec 1969 16:33:36 -0800Harvey J. HindinCryptographic Protocols with Everyday Objects
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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.36Wed, 31 Dec 1969 16:33:36 -0800James Heather and Steve Schneider and Vanessa TeagueOn the interval containing at least one prime number
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37Wed, 31 Dec 1969 16:33:36 -0800Jitsuro NaguraOn subsets with intersections of even cardinality
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This paper solves a question by Paul Erdős38Wed, 31 Dec 1969 16:33:36 -0800E.R. BerlekampTwo remarks on even and oddtown problems
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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 $\not\equiv 0 \pmod \ell$.39Wed, 31 Dec 1969 16:33:36 -0800Benny Sudakov and Pedro VieiraA Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence
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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.40Wed, 31 Dec 1969 16:33:36 -0800Johan NilssonGeometric Mechanics of Curved Crease Origami
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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.41Wed, 31 Dec 1969 16:33:36 -0800Marcelo A. Dias and Levi H. Dudte and L. Mahadevan and Christian D. SantangeloA Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents
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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.42Wed, 31 Dec 1969 16:33:36 -0800Haris Aziz and Simon MackenzieAvoiding Squares and Overlaps Over the Natural Numbers
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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.43Wed, 31 Dec 1969 16:33:36 -0800Mathieu Guay-Paquet and Jeffrey ShallitCounting Cases in Marching Cubes: Towards a Generic Algorithm for Producing Substitopes
/entry/44
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.44Wed, 31 Dec 1969 16:33:36 -0800David C. Banks and Stephen LintonFractal geometry of a complex plumage trait reveals bird's quality
/entry/45
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.45Wed, 31 Dec 1969 16:33:36 -0800Lorenzo Pérez-Rodríguez and Roger Jovani and Fran\ccois MougeotProgramming quantum computers using 3-D puzzles, coffee cups, and doughnuts
/entry/46
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.46Wed, 31 Dec 1969 16:33:36 -0800Simon J. DevittThe Nesting and Roosting Habits of The Laddered Parenthesis
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47Wed, 31 Dec 1969 16:33:36 -0800R. K. Guy and J. L. SelfridgeHistorical methods for multiplication
/entry/48
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.48Wed, 31 Dec 1969 16:33:36 -0800Bjørn Smestad and Konstantinos NikolantonakisPonytail Motion
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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.49Wed, 31 Dec 1969 16:33:36 -0800Joseph B. KellerSeven Puzzles You Think You Must Not Have Heard Correctly
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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.50Wed, 31 Dec 1969 16:33:36 -0800Peter WinklerTopologically Distinct Sets of Non-intersecting Circles in the Plane
/entry/51
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.51Wed, 31 Dec 1969 16:33:36 -0800Richard J. MatharBeckett-Gray Codes
/entry/52
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.52Wed, 31 Dec 1969 16:33:36 -0800Mark Cooke and Chris North and Megan Dewar and Brett StevensThe general counterfeit coin problem
/entry/53
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.53Wed, 31 Dec 1969 16:33:36 -0800Lorenz Halbeisen and Norbert HungerbühlerSearching for generalized binary number systems
/entry/54
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.54Wed, 31 Dec 1969 16:33:36 -0800Attila KovácsThe denominators of convergents for continued fractions
/entry/55
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.55Wed, 31 Dec 1969 16:33:36 -0800Lulu Fang and Min Wu and Bing LiMatters Computational - Ideas, Algorithms, Source Code
/entry/56
This is the book "Matters Computational" (formerly titled "Algorithms for Programmers"), published with Springer.56Wed, 31 Dec 1969 16:33:36 -0800Jörg ArndtTen Lessons I Wish I Had Learned Before I Started Teaching Differential Equations
/entry/57
57Wed, 31 Dec 1969 16:33:36 -0800Giancarlo RotaBad groups in the sense of Cherlin
/entry/58
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 $K$.58Wed, 31 Dec 1969 16:33:36 -0800Olivier FréconRational Polynomials That Take Integer Values at the Fibonacci Numbers
/entry/59
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.59Wed, 31 Dec 1969 16:33:36 -0800Keith Johnson and Kira ScheibelhutAn Irrationality Measure for Regular Paperfolding Numbers
/entry/60
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.60Wed, 31 Dec 1969 16:33:36 -0800Michael Coons and Paul VrbikThe nesting and roosting habits of the laddered parenthesis
/entry/61
61Wed, 31 Dec 1969 16:33:36 -0800R.K. Guy and J. L. SelfridgeWhat is the smallest prime?
/entry/62
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.62Wed, 31 Dec 1969 16:33:36 -0800Chris K. Caldwell and Yeng XiongHow do you compute the midpoint of an interval?
/entry/63
63Wed, 31 Dec 1969 16:33:36 -0800Frédéric GoualardComplexity and Completeness of Finding Another solution and its Application to Puzzles
/entry/64
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.64Wed, 31 Dec 1969 16:33:36 -0800Takayushi YatoDividing by zero - how bad is it, really?
/entry/65
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.65Wed, 31 Dec 1969 16:33:36 -0800Takayuki Kihara and Arno PaulyFuzzy plane geometry I: Points and lines
/entry/66
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. 66Wed, 31 Dec 1969 16:33:36 -0800J.J. Buckley and E. AslamiContinued Logarithms And Associated Continued Fractions
/entry/67
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.67Wed, 31 Dec 1969 16:33:36 -0800Jonathan M. Borwein and Neil J. Calkin and Scott B. Lindstrom and Andrew MattinglyThe snail lemma
/entry/68
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.68Wed, 31 Dec 1969 16:33:36 -0800Enrico M. VitalePhotoelectric Number Sieve Machine ("Gear Machine")
/entry/69
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.69Wed, 31 Dec 1969 16:33:36 -0800D. H. Lehmer and Robert CanepaDr Mitchill and the Mathematical Tetrodon
/entry/70
70Wed, 31 Dec 1969 16:33:36 -0800Kevin DannNotes on the Fourth Dimension
/entry/71
Hyperspace, ghosts, and colourful cubes — Jon Crabb on the work of Charles Howard Hinton and the cultural history of higher dimensions.71Wed, 31 Dec 1969 16:33:36 -0800Jon Crabb Challenging mathematical problems with elementary solutions
/entry/72
72Wed, 31 Dec 1969 16:33:36 -0800A.M. Yaglom and I.M. YaglomOn Pellegrino's 20-Caps in $S_{4,3}$
/entry/73
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 $S_{5,3}$.73Wed, 31 Dec 1969 16:33:36 -0800R. HillCounting groups: gnus, moas and other exotica
/entry/74
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.74Wed, 31 Dec 1969 16:33:36 -0800John H. Conway and Heiko Dietrich and E.A. O’BrienTwo notes on notation
/entry/75
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.75Wed, 31 Dec 1969 16:33:36 -0800Donald E. KnuthDismal Arithmetic
/entry/76
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.76Wed, 31 Dec 1969 16:33:36 -0800David Applegate and Marc LeBrun and N. J. A. SloaneOn the Cookie Monster Problem
/entry/77
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.77Wed, 31 Dec 1969 16:33:36 -0800Leigh Marie Braswell and Tanya KhovanovaFibonacci Jigsaw Puzzle
/entry/78
78Wed, 31 Dec 1969 16:33:36 -0800Akio HizumePrime numbers in certain arithmetic progressions
/entry/79
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.79Wed, 31 Dec 1969 16:33:36 -0800Ram Murty and Nithum ThainDivision by zero
/entry/80
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.80Wed, 31 Dec 1969 16:33:36 -0800Je\vrábek, EmilTransposable integers in arbitrary bases
/entry/81
81Wed, 31 Dec 1969 16:33:36 -0800Anne L. LudingtonA Dozen Hat Problems
/entry/82
82Wed, 31 Dec 1969 16:33:36 -0800Ezra Brown and James TantonDe Bruijn's Combinatorics
/entry/83
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.83Wed, 31 Dec 1969 16:33:36 -0800Hung, J.W.Nienhuys (Ling-Ju and Eds.), Ton KloksOn gardeners, dukes and mathematical instruments
/entry/84
Postprint (author's final draft)84Wed, 31 Dec 1969 16:33:36 -0800Blanco Abellán, MónicaComparative kinetics of the snowball respect to other dynamical objects
/entry/85
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.85Wed, 31 Dec 1969 16:33:36 -0800Diaz, Rodolfo A. and Gonzalez, Diego L. and Marin, Francisco and Martinez, R.Area and Hausdorff Dimension of Julia Sets of Entire Functions
/entry/86
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.86Wed, 31 Dec 1969 16:33:35 -0800Curt McMullenApproaches to the Enumerative Theory of Meanders
/entry/87
87Wed, 31 Dec 1969 16:33:35 -0800Michael La CroixThe Theory of Heaps and the Cartier-Foata Monoid
/entry/88
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.88Wed, 31 Dec 1969 16:33:35 -0800C. KrattenthalerPlanar graph is on fire
/entry/89
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.89Wed, 31 Dec 1969 16:33:35 -0800Gordinowicz, PrzemysławOn the Number of Times an Integer Occurs as a Binomial Coefficient
/entry/90
90Wed, 31 Dec 1969 16:33:35 -0800H. L. Abbott and P. Erdős and D. HansonWhat to do when the trisector comes
/entry/91
91Wed, 31 Dec 1969 16:33:35 -0800Dudley, UnderwoodThe effective content of Reverse Nonstandard Mathematics and the nonstandard content of effective Reverse Mathematics
/entry/92
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.92Wed, 31 Dec 1969 16:33:35 -0800Sanders, SamWhen is .999... less than 1?
/entry/93
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.93Wed, 31 Dec 1969 16:33:35 -0800Katz, Karin Usadi and Katz, Mikhail G.Spiralling self-avoiding walks: an exact solution
/entry/94
94Wed, 31 Dec 1969 16:33:35 -0800Blote, H W J and Hilhorst, H JHaruspicy 3: The anisotropic generating function of directed bond-animals is not D-finite
/entry/95
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].95Wed, 31 Dec 1969 16:33:35 -0800Rechnitzer, AndrewHaruspicy and anisotropic generating functions
/entry/96
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.96Wed, 31 Dec 1969 16:33:35 -0800Rechnitzer, AndrewHow to Beat Your Wythoff Games' Opponent on Three Fronts
/entry/97
97Wed, 31 Dec 1969 16:33:35 -0800Aviezri S. FraenkelDice - Numericana
/entry/98
98Wed, 31 Dec 1969 16:33:35 -0800Gérard P. MichonProposal to Encode the Ganda Currency Mark for Bengali in ISO/IEC 10646
/entry/99
99Wed, 31 Dec 1969 16:33:35 -0800Anshuman PandeyAnother Proof of Segre's Theorem about Ovals
/entry/100
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 2.100Wed, 31 Dec 1969 16:33:35 -0800Müller, PeterFair Dice
/entry/101
101Wed, 31 Dec 1969 16:33:35 -0800Diaconis, Persi and Keller, Joseph BOn the Existence of Generalized Parking Spaces for Complex Reflection Groups
/entry/102
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.102Wed, 31 Dec 1969 16:33:35 -0800Ito, Yosuke and Okada, SoichiMind the Croc! Rationality Gaps vis-à-vis the Crocodile Paradox
/entry/103
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.103Wed, 31 Dec 1969 16:33:35 -0800Gerogiorgakis, StamatiosDenser Egyptian Fractions
/entry/104
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.104Wed, 31 Dec 1969 16:33:35 -0800Martin, GregReversible quantum cellular automata
/entry/105
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.105Wed, 31 Dec 1969 16:33:35 -0800Schumacher, B. and Werner, R. F.Representations of Palindromic, Prime and Number Patterns
/entry/106
106Wed, 31 Dec 1969 16:33:35 -0800Inder 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
/entry/107
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.107Wed, 31 Dec 1969 16:33:35 -0800Ferrie, Christopher and Kueng, RichardThe Lost Calculus (1637-1670): Tangency and Optimization without Limits
/entry/108
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.108Wed, 31 Dec 1969 16:33:35 -0800Jeff SuzukiThe accuracy of Buffon's needle: a rule of thumb used by ants to estimate area
/entry/109
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.109Wed, 31 Dec 1969 16:33:35 -0800Mugford, S. T.Maximum Matching and a Polyhedron With 0,1-Vertices
/entry/110
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.110Wed, 31 Dec 1969 16:33:35 -0800Jack EdmondsFinding long chains in kidney exchange using the traveling salesman problem
/entry/111
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.111Wed, 31 Dec 1969 16:33:35 -0800Anderson, Ross and Ashlagi, Itai and Gamarnik, David and Roth, Alvin E.Magic squares of seventh powers
/entry/112
112Wed, 31 Dec 1969 16:33:35 -0800Christian BoyerOn Legendre's Prime Number Formula
/entry/113
113Wed, 31 Dec 1969 16:33:35 -0800Janos PintzA Brief Critique of Pure Hypercomputation
/entry/114
114Wed, 31 Dec 1969 16:33:35 -0800Cotogno, PaoloComplexity and Algorithms for Graph and Hypergraph Sandwich Problems
/entry/115
115Wed, 31 Dec 1969 16:33:35 -0800Golumbic, Martin Charles and Wassermann, AmirA combinatorial theorem in plane geometry
/entry/116
116Wed, 31 Dec 1969 16:33:35 -0800Chvátal, VThe dying rabbit problem revisited
/entry/117
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.117Wed, 31 Dec 1969 16:33:35 -0800Oller, Antonio M.Rational approximations to $\pi$ and some other numbers
/entry/118
118Wed, 31 Dec 1969 16:33:35 -0800Hata, Masayoshi and Mignotte, M and Chudnovsky, G V and Beukers, FEfficient Algorithms for Zeckendorf Arithmetic
/entry/119
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.119Wed, 31 Dec 1969 16:33:35 -0800Ahlbach, Connor and Usatine, Jeremy and Pippenger, NicholasExact Approximations of Omega Numbers
/entry/120
120Wed, 31 Dec 1969 16:33:35 -0800Calude, C.S and Dinneen, MichaelEnumeration of symmetry classes of convex polyominoes on the honeycomb lattice
/entry/121
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 70.121Wed, 31 Dec 1969 16:33:35 -0800Gouyou-Beauchamps, Dominique and Leroux, PierreOn dice and coins: Models of computation for random generation
/entry/122
122Wed, 31 Dec 1969 16:33:35 -0800Feldman, D and Impagliazzo, R and Naor, MThis is the (co)end, my only (co)friend
/entry/123
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.123Wed, 31 Dec 1969 16:33:35 -0800Loregian, FoscoThe Eudoxus Real Numbers
/entry/124
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.124Wed, 31 Dec 1969 16:33:35 -0800Arthan, R. D.Bells, Motels and Permutation Groups
/entry/125
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 2003.125Wed, 31 Dec 1969 16:33:34 -0800McGuire, GaryMusic: a Mathematical Offering
/entry/126
126Wed, 31 Dec 1969 16:33:34 -0800Dave BensonIrrationality From The Book
/entry/127
We generalize Tennenbaum's geometric proof of the irrationality of sqrt(2) to
sqrt(n) for n = 3, 5, 6 and 10.127Wed, 31 Dec 1969 16:33:34 -0800Miller, Steven J. and Montague, DavidDivision by three
/entry/128
128Wed, 31 Dec 1969 16:33:34 -0800Doyle, Peter G. and Conway, John HortonHow not to prove the Poincaré conjecture
/entry/129
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!129Wed, 31 Dec 1969 16:33:34 -0800Stallings, JRSolving Triangular Peg Solitaire
/entry/130
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).130Wed, 31 Dec 1969 16:33:34 -0800Bell, George I.An Application of Elementary Group Theory to Central Solitaire
/entry/131
131Wed, 31 Dec 1969 16:33:34 -0800Bialostocki, ArieThe Super Patalan Numbers
/entry/132
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.132Wed, 31 Dec 1969 16:33:34 -0800Richardson, Thomas M.The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?
/entry/133
133Wed, 31 Dec 1969 16:33:34 -0800Durán, Antonio J and Pérez, Mario and Varona, Juan LProofs without syntax
/entry/134
134Wed, 31 Dec 1969 16:33:34 -0800Hughes, DJDMethods for studying coincidences
/entry/135
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.135Wed, 31 Dec 1969 16:33:34 -0800Diaconis, P and Mosteller, FrederickSudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes
/entry/136
136Wed, 31 Dec 1969 16:33:34 -0800Bailey, RAHow often should you clean your room?
/entry/137
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.137Wed, 31 Dec 1969 16:33:34 -0800Martin, Kimball and Shankar, KrishnanPondering an Artist's Perplexing Tribute to the Pythagorean Theorem
/entry/138
138Wed, 31 Dec 1969 16:33:34 -0800Ivars PetersonA Fresh Look at Peg Solitaire
/entry/139
139Wed, 31 Dec 1969 16:33:34 -0800George I. BellThe shape of a Mobius band
/entry/140
140Wed, 31 Dec 1969 16:33:34 -0800Mahadevan, L and Keller, JBFoldings and Meanders
/entry/141
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.141Wed, 31 Dec 1969 16:33:34 -0800Legendre, StéphaneMathematics and group theory in music
/entry/142
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.142Wed, 31 Dec 1969 16:33:34 -0800Papadopoulos, AthanaseAn arctic circle theorem for groves
/entry/143
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.143Wed, 31 Dec 1969 16:33:34 -0800Petersen, T. K. and Speyer, D.History-dependent random processes
/entry/144
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.144Wed, 31 Dec 1969 16:33:34 -0800Clifford, P. and Stirzaker, D.LIM is not slim
/entry/145
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.145Wed, 31 Dec 1969 16:33:34 -0800Fink, Alex and Fraenkel, Aviezri S. and Santos, CarlosThe Number-Pad Game
/entry/146
146Wed, 31 Dec 1969 16:33:34 -0800Alex Fink and Richard GuyNim Fractals
/entry/147
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.147Wed, 31 Dec 1969 16:33:34 -0800Khovanova, Tanya and Xiong, JoshuaUseful inequalities cheat sheet
/entry/148
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.148Wed, 31 Dec 1969 16:33:34 -0800László KozmaGeneralizing Zeckendorf's Theorem to f-decompositions
/entry/149
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.149Wed, 31 Dec 1969 16:33:34 -0800Demontigny, Philippe and Do, Thao and Kulkarni, Archit and Miller, Steven J. and Moon, David and Varma, UmangA Mathematical Coloring Book
/entry/150
150Wed, 31 Dec 1969 16:33:34 -0800Hampton, MarshallOn the diagram of 132-avoiding permutations
/entry/151
151Wed, 31 Dec 1969 16:33:34 -0800Reifegerste, AstridA number system with an irrational base
/entry/152
152Wed, 31 Dec 1969 16:33:34 -0800George BergmanEponymy in Mathematical Nomenclature: What's in a Name, and What Should Be?
/entry/153
153Wed, 31 Dec 1969 16:33:34 -0800Henwood, Mervyn R. and Rival, IvanTable for Fundamentals of Series : Part I : Basic Properties of Series and Products
/entry/154
154Wed, 31 Dec 1969 16:33:34 -0800Gould, Henry W.More ties than we thought
/entry/155
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.155Wed, 31 Dec 1969 16:33:34 -0800Hirsch, Dan and Patterson, Meredith L and Sandberg, Anders and Vejdemo-Johansson, MikaelMathematical Games
/entry/156
156Wed, 31 Dec 1969 16:33:34 -0800Silva, Jorge NunoRithmomachia
/entry/157
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.157Wed, 31 Dec 1969 16:33:34 -0800Daniel U. Thibault and Michel BoutinLinear recurrences through tilings and Markov chains
/entry/158
158Wed, 31 Dec 1969 16:33:34 -0800Benjamin, AT and Hanusa, CRH and Su, FEThe Stick Problem
/entry/159
159Wed, 31 Dec 1969 16:33:34 -0800Augustine BertagnolliCircular orbits on a warped spandex fabric
/entry/160
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.160Wed, 31 Dec 1969 16:33:34 -0800Middleton, Chad A. and Langston, MichaelThe topology of competitively constructed graphs
/entry/161
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.161Wed, 31 Dec 1969 16:33:33 -0800Frieze, Alan and Pegden, WesleyThe mathematics of Septoku
/entry/162
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.162Wed, 31 Dec 1969 16:33:33 -0800Bell, George I.Fair but irregular polyhedral dice
/entry/163
163Wed, 31 Dec 1969 16:33:33 -0800Joseph O'RourkeSolving Differential Equations by Symmetry Groups
/entry/164
164Wed, 31 Dec 1969 16:33:33 -0800John StarretA knowledge-based approach of connect-four
/entry/165
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.165Wed, 31 Dec 1969 16:33:33 -0800Allis, VictorWHAT IS Lehmer's number?
/entry/166
166Wed, 31 Dec 1969 16:33:33 -0800Eriko HironakaAnalyse algébrique d'un scrutin
/entry/167
167Wed, 31 Dec 1969 16:33:33 -0800Guilbaud, GT and Rosenstiehl, PLone Axes in Outer Space
/entry/168
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.168Wed, 31 Dec 1969 16:33:33 -0800Mosher, Lee and Pfaff, CatherineThe Maximum Throughput Rate for Each Hole on a Golf Course
/entry/169
169Wed, 31 Dec 1969 16:33:33 -0800Whitt, WardThe Math Encyclopedia of Smarandache Type Notions
/entry/170
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.170Wed, 31 Dec 1969 16:33:33 -0800Coman, MariusWolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
/entry/171
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
189.171Wed, 31 Dec 1969 16:33:33 -0800Mestrovic, Romeo2178 And All That
/entry/172
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.172Wed, 31 Dec 1969 16:33:33 -0800Sloane, NJAFibonacci numbers and Leonardo numbers
/entry/173
173Wed, 31 Dec 1969 16:33:33 -0800Dijkstra, E.W.Pancake Flipping is Hard
/entry/174
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.174Wed, 31 Dec 1969 16:33:33 -0800Bulteau, Laurent and Fertin, Guillaume and Rusu, IrenaGiuga Numbers and the arithmetic derivative
/entry/175
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.175Wed, 31 Dec 1969 16:33:33 -0800Grau, José María and Oller-Marcén, Antonio M.Swiss cheeses, rational approximation and universal plane curves
/entry/176
176Wed, 31 Dec 1969 16:33:33 -0800Feinstein, JF and Heath, MJPlaying pool with $\pi$ (the number $\pi$ from a billiard point of view)
/entry/177
177Wed, 31 Dec 1969 16:33:33 -0800Galperin, GRandom Structures from Lego Bricks and Analog Monte Carlo Procedures
/entry/178
178Wed, 31 Dec 1969 16:33:33 -0800Althöfer, IIs POPL Mathematics or Science?
/entry/179
179Wed, 31 Dec 1969 16:33:33 -0800Andrew W. AppelProofs by Descent
/entry/180
180Wed, 31 Dec 1969 16:33:33 -0800CONRAD, KPractical numbers
/entry/181
181Wed, 31 Dec 1969 16:33:33 -0800Srinivasan, A.K.How to differentiate a number
/entry/182
182Wed, 31 Dec 1969 16:33:33 -0800Ufnarovski, Victor and Åhlander, BThe Ubiquitous Thue-Morse Sequence
/entry/183
183Wed, 31 Dec 1969 16:33:33 -0800Jeffrey ShallitSloane's Gap: Do Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS?
/entry/184
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.184Wed, 31 Dec 1969 16:33:33 -0800Gauvrit, Nicolas and Delahaye, Jean-Paul and Zenil, HectorPerfect Matchings and the Octahedron Recurrence
/entry/185
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.185Wed, 31 Dec 1969 16:33:33 -0800Speyer, David EThe Strange and Surprising Saga of the Somos Sequences
/entry/186
186Wed, 31 Dec 1969 16:33:33 -0800Gale, DavidCookie Monster Devours Naccis
/entry/187
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.187Wed, 31 Dec 1969 16:33:33 -0800Braswell, Leigh Marie and Khovanova, TanyaAn Infinite Set of Heron Triangles with Two Rational Medians
/entry/188
188Wed, 31 Dec 1969 16:33:33 -0800Buchholz, Ralph H and Rathbun, Randall LThe Laurent phenomenon
/entry/189
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.189Wed, 31 Dec 1969 16:33:33 -0800Fomin, Sergey and Zelevinsky, AndreiOn n-Dimensional Polytope Schemes
/entry/190
190Wed, 31 Dec 1969 16:33:33 -0800Fouhey, David F and Maturana, DanielOnly problems, not solutions!
/entry/191
191Wed, 31 Dec 1969 16:33:33 -0800Smarandache, FlorentinFrom Unicode to Typography, a Case Study the Greek Script
/entry/192
192Wed, 31 Dec 1969 16:33:33 -0800Haralambous, YannisHalf of a coin: negative probabilities
/entry/193
193Wed, 31 Dec 1969 16:33:33 -0800Székely, GJSmooth neighbors
/entry/194
We give a new algorithm that quickly finds smooth neighbors.194Wed, 31 Dec 1969 16:33:33 -0800Conrey, Brian and Holmstrom, Mark and McLaughlin, TaraOn a problem of Störmer
/entry/195
195Wed, 31 Dec 1969 16:33:33 -0800Lehmer, DHMissing Data: Instrument-Level Heffalumps and Item-Level Woozles
/entry/196
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.196Wed, 31 Dec 1969 16:33:33 -0800Philip L. Roth and Fred S. Switzer IIIPascal's Pyramid Or Pascal's Tetrahedron
/entry/197
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.197Wed, 31 Dec 1969 16:33:33 -0800Jim NugentA Line of Sages
/entry/198
198Wed, 31 Dec 1969 16:33:33 -0800Khovanova, TanyaNon-sexist solution of the ménage problem
/entry/199
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.199Wed, 31 Dec 1969 16:33:33 -0800Kenneth P. BogartThe Ubiquitous Pi
/entry/200
200Wed, 31 Dec 1969 16:33:33 -0800Castellanos, DarioSix Ways to Sum a Series
/entry/201
A discussion of the sum of squares of the reciprocals of the positive integers with a review of several proofs.201Wed, 31 Dec 1969 16:33:33 -0800Kalman, DanUsing Monoidal Categories in the Transformational Study of Musical Time-Spans and Rhythms
/entry/202
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.202Wed, 31 Dec 1969 16:33:33 -0800Popoff, AlexandreKindergarten Quantum Mechanics
/entry/203
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.203Wed, 31 Dec 1969 16:33:33 -0800Coecke, BobCyclic twill-woven objects
/entry/204
204Wed, 31 Dec 1969 16:33:33 -0800Akleman, Ergun and Chen, Jianer and Chen, YenLin and Xing, Qing and Gross, Jonathan L.Division of labor in child care: A game-theoretic approach
/entry/205
205Wed, 31 Dec 1969 16:33:33 -0800Vierling-Claassen, a.A Do-It-Yourself Paper Digital Computer, 1959.
/entry/206
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).206Wed, 31 Dec 1969 16:33:33 -0800Ptak, John F.Familial sinistrals avoid exact numbers.
/entry/207
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.207Wed, 31 Dec 1969 16:33:33 -0800Sauerland, Uli and Gotzner, NicoleCircuitry in 3D chess
/entry/208
208Wed, 31 Dec 1969 16:33:33 -0800Goucher, AdamThe urinal problem
/entry/209
209Wed, 31 Dec 1969 16:33:33 -0800Kranakis, Evangelos and Krizanc, DannyA Smaller Sleeping Bag For A Baby Snake
/entry/210
210Wed, 31 Dec 1969 16:33:33 -0800Linusson, Svante and ASTLUND, JWConstructing the Tits ovoid from an elliptic quadric
/entry/211
211Wed, 31 Dec 1969 16:33:33 -0800Cherowitzo, WEDas 2: 3-Ei-ein praktikables Eimodell
/entry/212
212Wed, 31 Dec 1969 16:33:33 -0800Möller, HProblems to sharpen the young
/entry/213
An annotated translation of Propositiones ad acuendos juvenes, the oldest mathematical problem collection in Latin, attributed to Alcuin of York.213Wed, 31 Dec 1969 16:33:33 -0800Hadley, John and Singmaster, DavidReview of "Groups" by Georges Papy in New Scientist
/entry/214
214Wed, 31 Dec 1969 16:33:33 -0800T. H. O'BeirneThe Circle-Squaring Problem Decomposed
/entry/215
215Wed, 31 Dec 1969 16:33:33 -0800Pierce, Pamela and Ramsay, JohnZeroless Arithmetic: Representing Integers ONLY using ONE
/entry/216
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.216Wed, 31 Dec 1969 16:33:33 -0800Ghang, EK and Zeilberger, DoronCircular reasoning: who first proved that $C/d$ is a constant?
/entry/217
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 $\$emph{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.217Wed, 31 Dec 1969 16:33:33 -0800Richeson, DavidThe Muddy Children : A logic for public announcement
/entry/218
218Wed, 31 Dec 1969 16:33:33 -0800Hughes, JesseEmbedding countable groups in 2-generator groups
/entry/219
219Wed, 31 Dec 1969 16:33:33 -0800Galvin, FredWhat are some of the most ridiculous proofs in mathematics?
/entry/220
220Wed, 31 Dec 1969 16:33:33 -0800AnonymousMarkets are efficient if and only if P = NP
/entry/221
221Wed, 31 Dec 1969 16:33:33 -0800Maymin, PZConway's Rational Tangles
/entry/222
222Wed, 31 Dec 1969 16:33:33 -0800Davis, TomA note on paradoxical metric spaces
/entry/223
223Wed, 31 Dec 1969 16:33:33 -0800Deuber, W A and Simonovits, M and Os, V T SIncorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality
/entry/224
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.224Wed, 31 Dec 1969 16:33:33 -0800Fiore, Thomas M. and Noll, Thomas and Satyendra, RamonDelay can stabilize: Love affairs dynamics
/entry/225
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.225Wed, 31 Dec 1969 16:33:33 -0800Bielczyk, Natalia and Bodnar, Marek and Foryś, UrszulaAlgorithmic self-assembly of DNA Sierpinski triangles.
/entry/226
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.226Wed, 31 Dec 1969 16:33:33 -0800Rothemund, Paul W K and Papadakis, Nick and Winfree, ErikInvited commentary: the perils of birth weight--a lesson from directed acyclic graphs.
/entry/227
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.227Wed, 31 Dec 1969 16:33:33 -0800Wilcox, Allen JThe paramagnetic and glass transitions in sudoku
/entry/228
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.228Wed, 31 Dec 1969 16:33:32 -0800Williams, Alex and Ackland, Graeme . J.Figures for "Impossible fractals"
/entry/229
229Wed, 31 Dec 1969 16:33:32 -0800Cameron BrowneBiologically Unavoidable Sequences
/entry/230
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.230Wed, 31 Dec 1969 16:33:32 -0800Alexander, SamuelHow to eat 4/9 of a pizza
/entry/231
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.231Wed, 31 Dec 1969 16:33:32 -0800Knauer, Kolja and Micek, Piotr and Ueckerdt, TorstenA stratification of the space of all $k$-planes in $\mathbb{C}_n$
/entry/232
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 symmetry232Wed, 31 Dec 1969 16:33:32 -0800Knutson, APicture-Hanging Puzzles
/entry/233
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.233Wed, 31 Dec 1969 16:33:32 -0800Demaine, Erik D. and Demaine, Martin L. and Minsky, Yair N. and Mitchell, Joseph S. B. and Rivest, Ronald L. and Patrascu, MihaiConway's Wizards
/entry/234
I present and discuss a puzzle about wizards invented by John H. Conway.234Wed, 31 Dec 1969 16:33:32 -0800Khovanova, TanyaPapy's Minicomputer
/entry/235
235Wed, 31 Dec 1969 16:33:32 -0800Papy, FThe lost squares of Dr. Franklin: Ben Franklin's missing squares and the secret of the magic circle
/entry/236
236Wed, 31 Dec 1969 16:33:32 -0800Pasles, PCOn sphere-filling ropes
/entry/237
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.237Wed, 31 Dec 1969 16:33:32 -0800Gerlach, Henryk and von der Mosel, HeikoAlgebraic theory of Penrose's non-periodic tilings of the plane
/entry/238
238Wed, 31 Dec 1969 16:33:32 -0800Bruijn, NG DeEarliest Uses of Symbols of Calculus
/entry/239
239Wed, 31 Dec 1969 16:33:32 -0800Jeff MillerThe topology of the minimal regular cover of the Archimedean tessellations
/entry/240
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.240Wed, 31 Dec 1969 16:33:32 -0800Coulbois, Thierry and Pellicer, Daniel and Raggi, Miguel and Ramírez, Camilo and Valdez, FerránTwin Towers of Hanoi
/entry/241
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.241Wed, 31 Dec 1969 16:33:32 -0800Sunic, ZoranOnline Dating Recommender Systems: The Split-complex Number Approach
/entry/242
242Wed, 31 Dec 1969 16:33:32 -0800Jérôme KunegisMagic: the Gathering is Turing Complete
/entry/243
We always knew Magic: the Gathering was a complex game. But now it's proven: you could assemble a computer out of Magic cards.243Wed, 31 Dec 1969 16:33:32 -0800Alex ChurchillModiﬁed Pascal Triangle and Pascal Surfaces
/entry/244
244Wed, 31 Dec 1969 16:33:32 -0800David AlvoHow Java's floating-point hurts everyone everywhere
/entry/245
245Wed, 31 Dec 1969 16:33:32 -0800Kahan, W and Darcy, JDBeastly Numbers
/entry/246
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.246Wed, 31 Dec 1969 16:33:32 -0800Kahan, WA Hamiltonian circuit for Rubik's Cube
/entry/247
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.247Wed, 31 Dec 1969 16:33:32 -0800cuBerBruceHow far can Tarzan jump?
/entry/248
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.248Wed, 31 Dec 1969 16:33:32 -0800Shima, HiroyukiMastermind is NP-Complete
/entry/249
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.249Wed, 31 Dec 1969 16:33:32 -0800Stuckman, Jeff and Zhang, Guo-QiangVIP-club phenomenon: emergence of elites and masterminds in social networks
/entry/250
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.250Wed, 31 Dec 1969 16:33:32 -0800Masuda, Naoki and Konno, NorioThe Canonical Basis of $\dot{\mathbf{U}}$ for Type $A_{2}$
/entry/251
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.251Wed, 31 Dec 1969 16:33:32 -0800Cui, WeidengThe Fastest and Shortest Algorithm for All Well-Defined Problems
/entry/252
252Wed, 31 Dec 1969 16:33:32 -0800Hutter, MarcusCarcassonne and multivariate calculus
/entry/253
253Wed, 31 Dec 1969 16:33:32 -0800Douglas WeathersA New Rose : The First Simple Symmetric 11-Venn Diagram
/entry/254
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.254Wed, 31 Dec 1969 16:33:32 -0800Mamakani, Khalegh and Ruskey, FrankThe usefulness of useless knowledge
/entry/255
255Wed, 31 Dec 1969 16:33:32 -0800Flexner, AbrahamSeven Staggering Sequences
/entry/256
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.256Wed, 31 Dec 1969 16:33:32 -0800Sloane, N J AThe top ten prime numbers
/entry/257
257Wed, 31 Dec 1969 16:33:32 -0800Dubner, HTrain Sets
/entry/258
258Wed, 31 Dec 1969 16:33:32 -0800Chalcraft, Adam and Greene, MichaelEquilibrium solution to the lowest unique positive integer game
/entry/259
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.259Wed, 31 Dec 1969 16:33:32 -0800Baek, Seung Ki and Bernhardsson, SebastianThe wobbly garden table
/entry/260
260Wed, 31 Dec 1969 16:33:32 -0800Kraft, HanspeterA cohomological viewpoint on elementary school arithmetic
/entry/261
261Wed, 31 Dec 1969 16:33:32 -0800Isaksen, DCOn distributions computable by random walks on graphs
/entry/262
262Wed, 31 Dec 1969 16:33:32 -0800Kindler, GTo Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction
/entry/263
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.263Wed, 31 Dec 1969 16:33:32 -0800David C KeenanTopology Explains Why Automobile Sunshades Fold Oddly
/entry/264
We use braids and linking number to explain why automobile shades fold into an odd number of loops.264Wed, 31 Dec 1969 16:33:32 -0800Feist, Curtis and Naimi, RaminOn an error in the star puzzle by Henry E. Dudeney
/entry/265
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.265Wed, 31 Dec 1969 16:33:32 -0800Ravsky, AlexHow to recognise a 4-ball when you see one
/entry/266
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.266Wed, 31 Dec 1969 16:33:32 -0800Geiges, Hansjörg and Zehmisch, KaiG2 and the Rolling Ball
/entry/267
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.267Wed, 31 Dec 1969 16:33:32 -0800Baez, John C and Huerta, JohnLectures on lost mathematics
/entry/268
268Wed, 31 Dec 1969 16:33:32 -0800Branko GrünbaumEstimating the Effect of the Red Card in Soccer
/entry/269
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.269Wed, 31 Dec 1969 16:33:32 -0800Vecer, Jan and Kopriva, FrantisekA categorical foundation for Bayesian probability
/entry/270
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.270Wed, 31 Dec 1969 16:33:32 -0800Culbertson, Jared and Sturtz, KirkSurvey on fusible numbers
/entry/271
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.271Wed, 31 Dec 1969 16:33:32 -0800Xu, JunyanCardinal arithmetic for skeptics
/entry/272
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.272Wed, 31 Dec 1969 16:33:32 -0800Shelah, SaharonA mathematician's survival guide
/entry/273
273Wed, 31 Dec 1969 16:33:32 -0800Casazza, Peter GCalculus Made Easy
/entry/274
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 CALCULUS274Wed, 31 Dec 1969 16:33:32 -0800Thompson, Silvanus PLong finite sequences
/entry/275
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).275Wed, 31 Dec 1969 16:33:32 -0800Friedman, Harvey MComputer analysis of Sprouts with nimbers
/entry/276
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.276Wed, 31 Dec 1969 16:33:32 -0800Lemoine, Julien and Viennot, SimonNim multiplication
/entry/277
277Wed, 31 Dec 1969 16:33:32 -0800H. W. Lenstra, Jr.Robust Soldier Crab Ball Gate
/entry/278
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).278Wed, 31 Dec 1969 16:33:32 -0800Gunji, YP and Nishiyama, YTheory and History of Geometric Models
/entry/279
279Wed, 31 Dec 1969 16:33:32 -0800Polo-blanco, IreneStatistical Modeling of Gang Violence in Los Angeles
/entry/280
280Wed, 31 Dec 1969 16:33:32 -0800Fathauer, ChrisHigh Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams
/entry/281
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.281Wed, 31 Dec 1969 16:33:32 -0800Williams, Hugh C. and Poorten, A. J. Van Der and Stein, AndreasLight reflecting off Christmas-tree balls
/entry/282
'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?282Wed, 31 Dec 1969 16:33:32 -0800Joseph O'RourkeCellular automata in the hyperbolic plane: proposal for a new environment
/entry/283
283Wed, 31 Dec 1969 16:33:32 -0800Chelghoum, Kamel and Margenstern, Maurice and Martin, Beno\^itThe Euler spiral: a mathematical history
/entry/284
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.284Wed, 31 Dec 1969 16:33:32 -0800Levien, RaphNavigating Hyperbolic Space with Fibonacci Trees
/entry/285
285Wed, 31 Dec 1969 16:33:32 -0800monikerCake Cutting Mechanisms
/entry/286
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.286Wed, 31 Dec 1969 16:33:32 -0800Ianovski, EgorUndecidable problems: a sampler
/entry/287
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.287Wed, 31 Dec 1969 16:33:32 -0800Poonen, BjornThe mate-in-n problem of infinite chess is decidable
/entry/288
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.288Wed, 31 Dec 1969 16:33:32 -0800Brumleve, Dan and Hamkins, Joel David and Schlicht, PhilippStatistical Laws Governing Fluctuations in Word Use from Word Birth to Word Death
/entry/289
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.289Wed, 31 Dec 1969 16:33:32 -0800Petersen, Alexander M and Tenenbaum, Joel and Havlin, Shlomo and Stanley, H EugeneThe bitangent sphere problem
/entry/290
290Wed, 31 Dec 1969 16:33:32 -0800Giblin, PJGaussian prime spirals
/entry/291
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 $90^{\circ}$.291Wed, 31 Dec 1969 16:33:32 -0800Joseph O'RourkeTopic-based vector space model
/entry/292
292Wed, 31 Dec 1969 16:33:32 -0800Becker, JörgA New Approximation to $\pi$ (Conclusion)
/entry/293
293Wed, 31 Dec 1969 16:33:32 -0800Ferguson, D. F. and Wrench, John WDoc, What Are My Chances?
/entry/294
294Wed, 31 Dec 1969 16:33:32 -0800Marasco, Joe and Doerfler, Ron and Roschier, LeifRandom walks reaching against all odds the other side of the quarter plane
/entry/295
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.295Wed, 31 Dec 1969 16:33:32 -0800van Leeuwaarden, Johan S. H. and Raschel, KilianQuotients Homophones des Groupes Libres Homophonic Quotients of Free Groups
/entry/296
296Wed, 31 Dec 1969 16:33:32 -0800Washington, Lawrence and Zagier, DonPoe, E.: Near A Raven
/entry/297
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.297Wed, 31 Dec 1969 16:33:32 -0800Mike KeithBarcodes: the persistent topology of data
/entry/298
298Wed, 31 Dec 1969 16:33:32 -0800Ghrist, RobertFurther evidence for addition and numerical competence by a Grey parrot (Psittacus erithacus)
/entry/299
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.299Wed, 31 Dec 1969 16:33:32 -0800Pepperberg, Irene M.Passage to the limit in Proposition I, Book I of Newton's Principia
/entry/300
300Wed, 31 Dec 1969 16:33:32 -0800Erlichson, HermanOrange Peels and Fresnel Integrals
/entry/301
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.301Wed, 31 Dec 1969 16:33:32 -0800Bartholdi, Laurent and Henriques, André G.Tropical Mathematics
/entry/302
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.302Wed, 31 Dec 1969 16:33:32 -0800Speyer, David and Sturmfels, BerndPantologia. A new (cabinet) cyclopædia, by J.M. Good, O. Gregory, and N. Bosworth assisted by other gentlemen of eminence
/entry/303
303Wed, 31 Dec 1969 16:33:32 -0800Good, John Mason and Gregory, Olinthus GilbertFractions without Quotients: Arithmetic of Repeating Decimals
/entry/304
304Wed, 31 Dec 1969 16:33:32 -0800Plagge, RichardNineteen dubious ways to compute the exponential of a matrix, twenty-five years later
/entry/305
305Wed, 31 Dec 1969 16:33:32 -0800Moler, Cleve and Van Loan, C.Fibonacci determinants-a combinatorial approach
/entry/306
306Wed, 31 Dec 1969 16:33:32 -0800Benjamin, A.T. and Cameron, N.T. and Quinn, J.J.Good stories, pity they're not true
/entry/307
307Wed, 31 Dec 1969 16:33:32 -0800Devlin, KeithCryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles
/entry/308
308Wed, 31 Dec 1969 16:33:32 -0800Gradwohl, Ronen and Naor, M. and Pinkas, Benny and Rothblum, G.Benjamin Peirce and the Howland will
/entry/309
309Wed, 31 Dec 1969 16:33:32 -0800Meier, Paul and Zabell, SandyMapping an unfriendly subway system
/entry/310
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.310Wed, 31 Dec 1969 16:33:32 -0800Flocchini, Paola and Kellett, Matthew and Mason, P.In retrospect: On the Six-Cornered Snowflake
/entry/311
311Wed, 31 Dec 1969 16:33:32 -0800Ball, PhilipTropical Arithmetic and Tropical Matrix Algebra
/entry/312
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.312Wed, 31 Dec 1969 16:33:32 -0800Izhakian, ZurGerrymandering and Convexity
/entry/313
313Wed, 31 Dec 1969 16:33:32 -0800Hodge, Jonathan K. and Marshall, Emily and Patterson, GeoffContinued fractions constructed from prime numbers
/entry/314
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.314Wed, 31 Dec 1969 16:33:32 -0800Wolf, MarekThe hardness of the Lemmings game, or Oh no, more NP-completeness proofs
/entry/315
315Wed, 31 Dec 1969 16:33:32 -0800Cormode, GrahamGaming is a hard job, but someone has to do it!
/entry/316
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.316Wed, 31 Dec 1969 16:33:32 -0800Viglietta, GiovanniCompositional Reasoning Using Intervals and Time Reversal
/entry/317
317Wed, 31 Dec 1969 16:33:32 -0800Moszkowski, BenLondon Calling Philosophy and Engineering: WPE 2008
/entry/318
318Wed, 31 Dec 1969 16:33:32 -0800Glen MillerThe Snowblower Problem
/entry/319
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.319Wed, 31 Dec 1969 16:33:32 -0800Arkin, Esther M. and Bender, Michael A. and Mitchell, Joseph S. B. and Polishchuk, ValentinScooping the Loop Snooper
/entry/320
320Wed, 31 Dec 1969 16:33:32 -0800Geoffrey K. PullumEthnomathematics as a new research field , illustrated by studies of mathematical ideas in African history
/entry/321
321Wed, 31 Dec 1969 16:33:32 -0800Gerdes, PaulusDrawings from Angola: living mathematics
/entry/322
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.322Wed, 31 Dec 1969 16:33:32 -0800Paulus GerdesUnderstanding Monads With JavaScript
/entry/323
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.323Wed, 31 Dec 1969 16:33:32 -0800Ionuț G. StanThe Collatz Fractal
/entry/324
324Wed, 31 Dec 1969 16:33:32 -0800Henderson, XanderAnalysis of Casino Shelf Shuffling Machines
/entry/325
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.325Wed, 31 Dec 1969 16:33:32 -0800Diaconis, Persi and Fulman, Jason and Holmes, SusanA Generalized Fibonacci LSB Data Hiding Technique
/entry/326
326Wed, 31 Dec 1969 16:33:32 -0800Battisti, F and Carli, M and Neri, A and Egiaziarian, KA zero-knowledge Poker protocol that achieves confidentiality of the players' strategy or How to achieve an electronic Poker face
/entry/327
327Wed, 31 Dec 1969 16:33:32 -0800Crépeau, C.Computer evolution of buildable objects
/entry/328
328Wed, 31 Dec 1969 16:33:32 -0800Funes, Pablo and Pollack, JordanRandom Walks on Finite Groups
/entry/329
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.329Wed, 31 Dec 1969 16:33:32 -0800Saloff-coste, LaurentComplexity of Langton's ant
/entry/330
330Wed, 31 Dec 1969 16:33:32 -0800Gajardo, A and Moreira, A and Goles, EOn badly approximable numbers and certain games
/entry/331
331Wed, 31 Dec 1969 16:33:32 -0800Schmidt, WMThree-dimensional finite point groups and the symmetry of beaded beads
/entry/332
332Wed, 31 Dec 1969 16:33:32 -0800Fisher, GL and Mellor, B.Carrots for dessert
/entry/333
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.333Wed, 31 Dec 1969 16:33:32 -0800Petersen, Carsten Lunde and Roesch, PascaleHow to Gamble If You're In a Hurry
/entry/334
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).334Wed, 31 Dec 1969 16:33:31 -0800Ekhad, Shalosh B and Georgiadis, Evangelos and Zeilberger, DoronOrigami Burrs and Woven Polyhedra
/entry/335
335Wed, 31 Dec 1969 16:33:31 -0800Lang, Robert JDeobfuscation is in NP
/entry/336
336Wed, 31 Dec 1969 16:33:31 -0800Appel, Andrew WWhat Are the Odds?
/entry/337
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 Odds337Wed, 31 Dec 1969 16:33:31 -0800Lane, F.C.Tie knots, random walks and topology
/entry/338
338Wed, 31 Dec 1969 16:33:31 -0800Fink, T and Mao, YDesigning tie knots by random walks
/entry/339
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.339Wed, 31 Dec 1969 16:33:31 -0800Thomas M. Fink and Yong MaoAsymptotic statistics of the n-sided planar Poisson–Voronoi cell: I. Exact results
/entry/340
340Wed, 31 Dec 1969 16:33:31 -0800Hilhorst, H.J.Laying train tracks
/entry/341
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.341Wed, 31 Dec 1969 16:33:31 -0800Danny CalegariTetris is Hard, Even to Approximate
/entry/342
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.342Wed, 31 Dec 1969 16:33:31 -0800Demaine, Erik D and Hohenberger, Susan and Liben-Nowell, DavidRemainder Wheels and Group Theory
/entry/343
343Wed, 31 Dec 1969 16:33:31 -0800Brenton, LawrenceChalk : Materials and Concepts in Mathematics Chalk in Hand
/entry/344
344Wed, 31 Dec 1969 16:33:31 -0800Barany, Michael J and Mackenzie, DonaldThe experimental effectiveness of mathematical proof
/entry/345
345Wed, 31 Dec 1969 16:33:31 -0800Miquel, AlexandreScholarly communication in transition: The use of question marks in the titles of scientific articles in medicine, life sciences and physics 1966–2005
/entry/346
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.346Wed, 31 Dec 1969 16:33:31 -0800Ball, RafaelBaron Munchhausen Redeems Himself : Bounds for a Coin-Weighing Puzzle Background
/entry/347
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.347Wed, 31 Dec 1969 16:33:31 -0800Khovanova, Tanya and Lewis, Joel BrewsterCool irrational numbers and their rather cool rational approximations
/entry/348
348Wed, 31 Dec 1969 16:33:31 -0800Calogero, FrancescoA linear programming approach for aircraft boarding strategy
/entry/349
349Wed, 31 Dec 1969 16:33:31 -0800Bazargan, MThe elasto-plastic indentation of a half-space by a rigid sphere
/entry/350
350Wed, 31 Dec 1969 16:33:31 -0800Hardy, C. and Baronet, C. N. and Tordion, G. V.Gödel's Second Incompleteness Theorem Explained in Words of One Syllable
/entry/351
351Wed, 31 Dec 1969 16:33:31 -0800Boolos, GeorgeFusible Numbers
/entry/352
352Wed, 31 Dec 1969 16:33:31 -0800Erickson, JeffThere is no "Uspensky's method"
/entry/353
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.353Wed, 31 Dec 1969 16:33:31 -0800Akritas, AGMad Abel : A card game for 2 + players
/entry/354
354Wed, 31 Dec 1969 16:33:31 -0800Mccarthy, SmáriShamos's Catalog of the Real Numbers
/entry/355
355Wed, 31 Dec 1969 16:33:31 -0800Shamos, Michael IanDoes Quantum Interference exist in Twitter?
/entry/356
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.356Wed, 31 Dec 1969 16:33:31 -0800Shuai, Xin and Ding, Ying and Busemeyer, Jerome and Sun, Yuyin and Chen, Shanshan and Tang, JiePacking circles and spheres on surfaces
/entry/357
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.357Wed, 31 Dec 1969 16:33:31 -0800Alexander Schiftner and Mathias Höbinger and Johannes Wallner and Helmut PottmannIrrationality from the book
/entry/358
358Wed, 31 Dec 1969 16:33:31 -0800Miller, Steven J and Montague, DavidAgainst Conditionalization
/entry/359
359Wed, 31 Dec 1969 16:33:31 -0800Bacchus, FahiemInvestigations of Game of Life cellular automata rules on Penrose Tilings : lifetime and ash statistics
/entry/360
360Wed, 31 Dec 1969 16:33:31 -0800Owens, Nick and Stepney, SusanDoubly-, triply-, quadruply- and quintuply-innervated crustacean muscles
/entry/361
361Wed, 31 Dec 1969 16:33:31 -0800van Harreveld, A.Gödel's incompleteness theorem
/entry/362
362Wed, 31 Dec 1969 16:33:31 -0800Uspensky, VPenrose's Godelian argument
/entry/363
363Wed, 31 Dec 1969 16:33:31 -0800Feferman, SolomonDeriving Uniform Polyhedra with Wythoff's Construction
/entry/364
364Wed, 31 Dec 1969 16:33:31 -0800Romano, DonTesting Petri Nets for Mobile Robots Using Gröbner Bases
/entry/365
365Wed, 31 Dec 1969 16:33:31 -0800Chandler, Angie and Heyworth, Anne and Blair, Lynne and Seward, DerekAccurate estimation of forward path geometry using two-clothoid road model
/entry/366
366Wed, 31 Dec 1969 16:33:31 -0800Khosla, DThe 1-Hyperbolic Projection for User Interfaces
/entry/367
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.367Wed, 31 Dec 1969 16:33:31 -0800Kolliopoulos, AlexanderDrawing and Labeling High-Quality Metro Maps by Mixed-Integer Programming.
/entry/368
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.368Wed, 31 Dec 1969 16:33:31 -0800Nöllenburg, Martin and Wolff, AlexanderA Paradoxical Property of the Monkey Book
/entry/369
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.369Wed, 31 Dec 1969 16:33:31 -0800Bernhardsson, Sebastian and Baek, Seung Ki and Minnhagen, PetterAn example of a computable absolutely normal number
/entry/370
370Wed, 31 Dec 1969 16:33:31 -0800Figueira, SantiagoA Note on Approximating the Normal Distribution Function
/entry/371
371Wed, 31 Dec 1969 16:33:31 -0800Aludaat, K M and Alodat, M TTheoretical Computer Science Cheat Sheet
/entry/372
372Wed, 31 Dec 1969 16:33:31 -0800Steve SeidenTree automata techniques and applications
/entry/373
373Wed, 31 Dec 1969 16:33:31 -0800Comon, Hubert and Dauchet, M and Gilleron, RAutomatic calculation of plane loci using Grobner bases and integration into a Dynamic Geometry System
/entry/374
374Wed, 31 Dec 1969 16:33:31 -0800Gerh, MichaelJuggling Probabilities
/entry/375
375Wed, 31 Dec 1969 16:33:31 -0800Warrington, Gregory S.The isoperimetric problem
/entry/376
376Wed, 31 Dec 1969 16:33:31 -0800Blasjo, ViktorHierarchical Position Based Dynamics
/entry/377
377Wed, 31 Dec 1969 16:33:31 -0800Faure, F. and Teschner, M.James Garfield's Proof of the Pythagorean Theorem
/entry/378
378Wed, 31 Dec 1969 16:33:31 -0800Ellermeyer, S FSurreal Numbers – An Introduction
/entry/379
379Wed, 31 Dec 1969 16:33:31 -0800Tøndering, ClausA discursive grammar for customizing mass housing: the case of Siza's houses at Malagueira
/entry/380
380Wed, 31 Dec 1969 16:33:31 -0800Duarte, JHypercomputation: computing more than the Turing machine
/entry/381
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.381Wed, 31 Dec 1969 16:33:31 -0800Ord, TobyComparison of geometric figures
/entry/382
382Wed, 31 Dec 1969 16:33:31 -0800Glenis, 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
/entry/383
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.383Wed, 31 Dec 1969 16:33:31 -0800Berkel, Daan VanBetter approximations to cumulative normal functions
/entry/384
384Wed, 31 Dec 1969 16:33:31 -0800West, GraemeContinued fraction algorithms, functional operators, and structure constants
/entry/385
385Wed, 31 Dec 1969 16:33:31 -0800Flajolet, P. and Vallée, B.Can One Hear the Shape of a Drum?
/entry/386
386Wed, 31 Dec 1969 16:33:31 -0800Kac, MarkAnimating rotation with quaternion curves
/entry/387
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.387Wed, 31 Dec 1969 16:33:31 -0800Shoemake, KenA classification for shaggy dog stories
/entry/388
388Wed, 31 Dec 1969 16:33:31 -0800Brunvand, J.H.Spontaneous knotting of an agitated string.
/entry/389
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.389Wed, 31 Dec 1969 16:33:31 -0800Raymer, Dorian M and Smith, Douglas EOn Buffon Machines and Numbers
/entry/390
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.390Wed, 31 Dec 1969 16:33:31 -0800Flajolet, PhilippeA history of mathematical notations
/entry/391
391Wed, 31 Dec 1969 16:33:31 -0800Florian CajoriInterpolating Solid Orientations with a $C^2$ -Continuous B-Spline Quaternion Curve
/entry/392
392Wed, 31 Dec 1969 16:33:31 -0800Ge, Wenbing and Huang, Zhangjin and Wang, GuopingFour questions about fuzzy rankings
/entry/393
393Wed, 31 Dec 1969 16:33:31 -0800Brian HayesZaphod Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle
/entry/394
394Wed, 31 Dec 1969 16:33:30 -0800Andrew 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
/entry/395
395Wed, 31 Dec 1969 16:33:30 -0800Scarle, Sopenttd logic gates
/entry/396
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.396Wed, 31 Dec 1969 16:33:30 -0800Heikki KallasjokiPush-pull LEGO logic gates
/entry/397
397Wed, 31 Dec 1969 16:33:30 -0800RandomwraithMisconceptions about the Golden Ratio
/entry/398
398Wed, 31 Dec 1969 16:33:30 -0800Markowsky, GeorgeOn Furstenberg's Proof of the Infinitude of Primes
/entry/399
399Wed, 31 Dec 1969 16:33:30 -0800Mercer, Idris DWhat symmetry groups are present in the Alhambra?
/entry/400
400Wed, 31 Dec 1969 16:33:30 -0800Grünbaum, BrankoA Note on Boolos' Proof of the Incompleteness Theorem
/entry/401
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.401Wed, 31 Dec 1969 16:33:30 -0800Kikuchi, MakotoThe Origin of Chemical Elements
/entry/402
402Wed, 31 Dec 1969 16:33:30 -0800Alpher, R. and Bethe, H. and Gamow, G.The role of instrumental and relational understanding in proofs about group isomorphisms
/entry/403
403Wed, 31 Dec 1969 16:33:30 -0800Weber, K.A Closed-Form Algorithm for Converting Hilbert Space-Filling Curve Indices
/entry/404
404Wed, 31 Dec 1969 16:33:30 -0800Chen, Chih-sheng and Lin, Shen-yi and Fan, Min-hsuan and Huang, Chua-huangDigital halftoning space filling curves
/entry/405
405Wed, 31 Dec 1969 16:33:30 -0800Luiz C. Velho and Jonas M. GomesHilbert R-tree: An improved R-tree using fractals
/entry/406
406Wed, 31 Dec 1969 16:33:30 -0800Kamel, Ibrahim and Faloutsos, ChristosA game for budding knot theorists
/entry/407
407Wed, 31 Dec 1969 16:33:30 -0800Dave RichesonOn Mathematics and Mathematicians
/entry/408
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...408Wed, 31 Dec 1969 16:33:30 -0800Moritz, Robert EdowardARTIFICIAL NEURAL NETWORK MODELING OF APPLE DRYING PROCESS
/entry/409
409Wed, 31 Dec 1969 16:33:30 -0800KHOSHHAL, ABBAS and DAKHEL, ASGHAR ALIZADEH and ETEMADI, AHMAD and ZERESHKI, SINADetection of transposition errors in decimal numbers
/entry/410
410Wed, 31 Dec 1969 16:33:30 -0800Freeman, HCircle Packing for Origami Design Is Hard
/entry/411
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.411Wed, 31 Dec 1969 16:33:30 -0800Demaine, E.D. and Fekete, S.P. and Lang, R.J.What Sequential Games , the Tychonoff Theorem and the Double-Negation Shift have in Common
/entry/412
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)412Wed, 31 Dec 1969 16:33:30 -0800Oliva, Paulo and Escardo, MartinThe Mathematics of Musical Instruments
/entry/413
413Wed, 31 Dec 1969 16:33:30 -0800Hall, Rachel W. and Josic, KresimirThe liouville-heath-brown-zagier proof of the two squares theorem and generalizations
/entry/414
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 .414Wed, 31 Dec 1969 16:33:30 -0800Elsholtz, ChristianConstructive gem: juggling exponentials
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415Wed, 31 Dec 1969 16:33:30 -0800Bauer, Andrej‘Knowable' As ‘Known After an Announcement'
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416Wed, 31 Dec 1969 16:33:30 -0800Balbiani, Philippe and Baltag, Alexandru and Ditmarsch, Hans Van and Herzig, Andreas and Hoshi, Tomohiro and De Lima, TiagoFoolproof : A Sampling of Mathematical Folk Humor
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417Wed, 31 Dec 1969 16:33:30 -0800Renteln, Paul and Dundes, AlanA formal system for Euclid's Elements
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We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.418Wed, 31 Dec 1969 16:33:30 -0800Avigad, Jeremy and Dean, Edward and Mumma, JohnAn aperiodic hexagonal tile
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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.419Wed, 31 Dec 1969 16:33:30 -0800Socolar, Joshua E. S. and Taylor, Joan M.How to explain zero-knowledge protocols to your children
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420Wed, 31 Dec 1969 16:33:30 -0800Quisquater, JJ and Quisquater, MPlane recursive trees, Stirling permutations and an urn model
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421Wed, 31 Dec 1969 16:33:30 -0800Janson, SvanteUnbounded spigot algorithms for the digits of pi
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422Wed, 31 Dec 1969 16:33:30 -0800Gibbons, J.