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Shalosh B. Ekhad

Publications and source records attributed to Shalosh B. Ekhad.

At least 55 records · Page 3Linked to original sources
arXiv2019

Using Symbolic Computation to analyze some Children's Board Games

In a delightful article that recently appeared in Mathematics Magazine, David and Lori Mccune analyze the board game "Count Your Chickens!", recommended to children three and up. Alas, they use the advanced theory of Markov chains, that presupposes a knowledge of linear algebra, that few three-years-olds are likely to understand. Here we present a much simpler, more intuitive, approach, that while unlikely to be understood by three-year-olds, will probably be understood by a smart 14-year-old. Moreover, our approach accomplishes much more, and is more efficient. It uses symbolic, rather than numeric computation.

math.HO↗
arXiv2019

Some Remarks on a recent article by J. -P. Allouche

In 1980 Otto G. Ruehr made some puzzling comments that a certain identity A, that he proved, is equivalent to another identity B, but he did not explain why they are equivalent. Recently J.-P. Allouche tried to explain why they are "equivalent", using about eight pages. We comment that it is extremely unlikely to be Ruehr's reasoning (he probably got mixed up with a different problem), but be that as it may, it is not worthwhile to try and deduce B from A, since both A and B are routinely provable using Wilf-Zeilberger algorithmic proof theory (implemented in Maple) and the Almkvist-Zeilberger algorithm.

math.CO↗
arXiv2019

A Detailed Analysis of Quicksort Running Time

One of the greatest algorithms of all time is Quicksort. Its average running time is famously O(nlog(n)), and its variance, less famously, is O(n^2) (hence its standard deviation is O(n)). But what about higher moments? Here we find explicit expressions for the first eight moments, their scaled limits, and we describe how to compute, approximately (but very accurately), percentiles of running time for any list-length

math.PR↗
arXiv2019

In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?

In Gert Almkvist's beautiful article, entitled "Invariants, mostly old ones", (that appeared in the Pacific Journal of Mathematics, vol. 86 (1980), pp. 1-13) he talked about a sequence of generating functions that came up in his work, that turned out to be the same as generating functions for the number of covariants of binary quadratic forms studied by Faa de Bruno, Cayley, Sylvester, and other 19th century savantes. Using a very simple-minded Maple program (that uses the partial-fraction decomposition of a rational function), we recompute them, and go all the way to degree 18. It turns out that the same method can be used to answer many other enumeration questions, including the one in the title.

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arXiv2019

The "Monkey Typing Shakespeare" Problem for Compositions

Suppose that your mother gave you n candies. You have to eat at least one candy each day. One possibility is to eat all n of them the first day. The other extreme is to make them last n days, and only eat one candy a day. Altogether, you have, famously, 2 to the power n-1 choices. If each such choice is equally likely, what is the probability that you never have three consecutive days, where in the first day you ate at least 2 candies, in the second day you ate at least 5 candies, and in the third day you ate at least 3 candies? This article describes algorithms, fully implemented in two Maple packages, to answer such important questions, and more general ones, of this kind.

math.CO↗
arXiv2018

D.H. Lehmer's Tridiagonal determinant: An Etude in (Andrews-Inspired) Experimental Mathematics

We use a "reverse engineering" method, pioneered by George Andrews, to discover an explicit expression for the determinant of a certain tridiagonal matrix discussed by Derrick Henry Lehmer in 1974, that lead to OEIS sequence A039924. Lehmer only did the infinite case, and here we also do the finite case, that immediately implies the former by taking the limit as n goes to infinity

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arXiv2018

How Many Rounds Should You Expect in Urn Solitaire?

A certain sampling process, concerning an urn with balls of two colors, proposed in 1965 by B.E. Oakley and R.L. Perry, and discussed by Peter Winkler and Martin Gardner, that has an extremely simple answer for the probability, namely the constant function 1/2, has a far more complicated expected duration, that we discover and sketch the proof of. So unlike, for example, the classical gambler's ruin problem, for which both `probability of winning' and `expected duration' have very simple expressions, in this case the expected number of rounds is extremely complicated, and beyond the scope of humans.

math.CO↗
arXiv2017

A Treatise on Sucker's Bets

In 1970, Statistics giant, Bradley Efron, amazed the world by coming up with a set of four dice, let's call them A,B,C,D, whose faces are marked with [0,0,4,4,4,4], [3,3,3,3,3,3],[2,2,2,2,6,6],[1,1,1,5,5,5] respectively, where die A beats die B, die B beats die C, die C beats die D, but, surprise surprise, die D beats die A! This was an amazing demonstration that "being more likely to win" is not a transitive relation. But that was only one example, and of course, instead of dice, we can use decks of cards, where they are called (by Martin Gardner, who popularized this way back in 1970) , "sucker's bets". Can you find all such examples, with a specified number of decks, and sizes? If you have a computer algebra system (in our case Maple), you sure can! Not only that, we can figure out how likely such sucker bets are, and derive, fully automatically, statistical information!

math.CO↗
arXiv2017

Automated Proofs of Many Conjectured Recurrences in the OEIS made by R.J. Mathar

The On-Line Encyclopedia Of Integer Sequences , that wonderful resource that most combinatorialists, and many other mathematicians and scientists, use at least once a day, is a treasure trove of mathematical information, and, one of its charms is that it contains many intriguing conjectures. But one should be on one's guard, because some of the conjectures are either already theorems, or can be routinely proved. In this case study we demonstrate, and actually fully implement (in an accompanying Maple package), how to turn many conjectures made in the OEIS by R.J. Mathar, regarding linear recurrences satisfies by a certain class of sequences, into fully rigorously-proved theorems, but we argue, that one should not emulate us, and while it is nice to have one example, of an algorithm to actually construct proofs (that are known to exist by a priori theoretical reasons), life is too short to do it in all cases, and hence we propose a new category for the OEIS, that of "provable conjecture", as opposed to "proved conjecture" (alias theorem).

math.HO↗
arXiv2017

Automated Generation of Anomalous Cancellations

If you cancel out the digit 6 from the ratio 16/64, you get the right answer by the wrong method. In 1979 R.P. Boas made an extensive study of such "anomalous cancellations" and generated many examples, in many bases. We continue his pioneering work, and generate many more examples, using the Extended Euclidean Algorithm.

math.HO↗
arXiv2016

Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences

Pisot sequences (sequences $a_n$ with initial terms $a_0=x, a_1=y$, and defined for $n>1$ by $a_n= \lfloor a_{n-1}^2/a_{n-2} + \frac{1}{2} \rfloor$) often satisfy linear recurrences with constant coefficients that are valid for all $n \geq 0$, but there are also cautionary examples where there is a linear recurrence that is valid for an initial range of values of $n$ but fails to be satisfied beyond that point, providing further illustrations of Richard Guy's celebrated "Strong Law of Small Numbers". In this paper we present a decision algorithm, fully implemented in an accompanying Maple program ({\tt Pisot.txt}), that first searches for a putative linear recurrence and then decides whether or not it holds for all values of $n$. We also explain why the failures happen (in some cases the `fake' linear recurrence may be valid for thousands of terms). We conclude by defining, and studying, higher-order analogs of Pisot sequences, and point out that similar phenomena occur there, albeit far less frequently. This article is dedicated to Richard K. Guy (b. Sept. 30, 1916) on his 100th birthday.

math.NT↗
arXiv2016

Going Back to Neil Sloane's FIRST LOVE (OEIS Sequence A435): On the Total Heights in Rooted Labeled Trees

In this tribute to Neil Sloane, we revisit the first sequence in the On-Line Encyclopedia of Integer Sequences, sequence A435 (1, 8, 78, 944, 13800, 237432, 4708144, 105822432, ...), that he encountered when he was a graduate student, and when normalized gives the average total height of rooted labeled trees. We state rigorously-computed explicit expressions for the first twelve moments of the random variable `total height' on rooted labeled trees, and pledge to donate to the OEIS 100 dollars in honor of the first to find an explicit expression for the probability density function of the limiting scaled probability distribution, as n goes to infinity.

math.CO↗
arXiv2016

Integrals Involving Rudin-Shapiro Polynomials and Sketch of a Proof of Saffari's Conjecture

Continuing pioneering work of Christophe Doche and Laurent Habsieger from 2004, we develop computer algebra algorithms, implemented in Maple, for finding the (necessarily rational) generating function for any integral of products, and in particular, moments, of Rudin-Shapiro polynomials. We generate a lot of output, and confirm again a conjecture of Saffari for the asymptotics for small (and not so small) powers. We also confirm, for small powers, a related, more general, conjecture, of Hugh Montgomery. Finally, we outline a proof of Saffari's full conjecture, that we believe can be turned into a full proof. [In this version we report that Brad Rodgers has independently found a (complete!) proof of Saffari's conjecture here http://arxiv.org/abs/1606.01637] .

math.CO↗
arXiv2016

On the number of Singular Vector Tuples of Hyper-Cubical Tensors

Shmuel Friedland and Giorgio Ottaviani's beautiful constant term expression for the number of singular vector tuples of generic tensors is used to derive a rational generating function for these numbers, that in turn, is used to obtain an asymptotic formula for the number of such tuples for n by n by n three-dimensional tensors, and to conjecture an asymptotic formula for the general d-dimensional case. A donation of 100 dollars, in honor of the first prover, will be made to the On-line Encyclopedia of Integer Sequences.

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arXiv2015

The C-finite Ansatz Meets the Holonomic Ansatz

We show how the continuous Almkvist-Zeilberger algorithm can be used to efficiently discover and prove differential equations satisfied by generating functions of sequences defined as integrals of powers of C-finite polynomial sequences (like the Chebyshev polynomials) from which one can automatically derive linear recurrences with polynomial coefficients for the sequences themselves. We fully implement this with a Maple package, CfiniteIntergal.txt .

math.CO↗
arXiv2015

Searching for Disjoint Covering Systems with Precisely One Repeated Modulus

A set of arithmetical sequences $$ a_1\, (\bmod{ \,\, m_1}) \quad, \quad a_2 \, (\bmod{\,\, m_2}) \quad, \quad \dots \quad , \quad a_k \, (\bmod{\,\,m_k}) \quad \quad , $$ with $$ m_1 \leq m_2 \leq \dots \leq m_k \quad \quad , $$ is called a {\it disjoint covering system} (alias {\it exact covering system}) if every positive integer belongs to {\bf exactly} one of the sequences. Mirski, Newman, Davenport and Rado famously proved that the moduli can't all be distinct. In fact the two largest moduli must be equal, i.e. $m_{k-1}=m_k$ This raises the natural question:"How close can you get to getting distinct moduli?", in other words, can you find all such systems where all the moduli are distinct except the largest, that is repeated $r$ times, for any, specific given $r$? It turns out (conjecturally, but almost certainly) that excluding the trivial case where the smallest modulus is 2, for any number of repeats $r$, there are only finitely many such systems. Marc Berger, Alexander Felzenbaum and Aviezri Fraenkel found them all for $r$ up to $9$, and Mekmamu Zeleke and Jamie Simpson extended the list for systems up to $12$ repeats. In the present article we continue the list up to $r=32$. All our systems are correct, but we did not bother to formally prove completeness, but we know for sure that the lists are complete if the largest modulus is $\leq 600$, and we are pretty sure that they are complete.

math.CO↗