45 research outputs found
The Number of Same-Sex Marriages in a Perfectly Bisexual Population is Asymptotically Normal
Why bother with fully rigorous proofs when one can very quickly get
semi-rigorous ones? Yes, yes, we know how to get a "rigorous" proof of the
result stated in the title of this article. One way is the boring, human one,
citing some heavy guns of theorems that already exist in the literature. We
also know how to get a fully rigorous proof automatically, using the methods in
this http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/georgy.htm neat
article (but it would be a little more complicated, since the probability
generating polynomial is not "closed form" but satisfies a second-order
recurrence gotten from the Zeilberger algorithm), otherwise the same method
would work, alas, it is not yet implemented.
Instead, we chose to use the great Maple package
http://www.math.rutgers.edu/~zeilberg/tokhniot/HISTABRUT">HISTABRUT(in fact, a
very tiny part of it, procedure AlphaSeq), explained in this other
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/histabrut.html">neat
article, and get a semi-rigorous proof. We also needed the nice little Maple
package http://www.math.rutgers.edu/~zeilberg/tokhniot/GuessRat">GuessRat, to
do the guessing of rational functions.
Equipped with these two packages, Zeilberger wrote a short Maple program
http://www.math.rutgers.edu/~zeilberg/tokhniot/SameSexMarriages">SameSexMarriages
that enabled the author to generate this paper.Comment: 3 page
Two Triple binomial sum supercongruences
In a recent article, Apagodu and Zeilberger
(http://arxiv.org/abs/1606.03351)discuss some applications of an algorithm for
finding and proving congruence identities (modulo primes) of indefinite sums of
many combinatorial sequence. At the end, they propose some supercongruences as
conjectures. Here we prove one of them, including a new companion enumerating
abelian squares, and we leave some remarks for the others.Comment: 17 page
Automatic Enumeration of Generalized Menage Numbers
I describe an empirical-yet-rigorous, algorithm, based on Riordan's rook
polynomials and the so-called C-finite ansatz, fully implemented in the
accompanying Maple package
(http://www.math.rutgers.edu/~zeilberg/tokhniot/MENAGES ), MENAGES, that
reproduces in a few seconds, rigorously-proved enumeration theorems on
permutations with restricted positions, previously proved by quite a few
illustrious human mathematicians, and that can go far beyond any human
attempts.Comment: 15 pages. An extended version of the last of three invited talks
given by the author at the 71th Seminaire Lotharingien de Combinatoire, that
took place in Bertinoro, Italy, Sept. 16-18, 201
Computer algebra tools for Feynman integrals and related multi-sums
In perturbative calculations, e.g., in the setting of Quantum Chromodynamics
(QCD) one aims at the evaluation of Feynman integrals. Here one is often faced
with the problem to simplify multiple nested integrals or sums to expressions
in terms of indefinite nested integrals or sums. Furthermore, one seeks for
solutions of coupled systems of linear differential equations, that can be
represented in terms of indefinite nested sums (or integrals). In this article
we elaborate the main tools and the corresponding packages, that we have
developed and intensively used within the last 10 years in the course of our
QCD-calculations
The 1958 Pekeris-Accad-WEIZAC Ground-Breaking Collaboration that Computed Ground States of Two-Electron Atoms (and its 2010 Redux)
In order to appreciate how well off we mathematicians and scientists are
today, with extremely fast hardware and lots and lots of memory, as well as
with powerful software, both for numeric and symbolic computation, it may be a
good idea to go back to the early days of electronic computers and compare how
things went then. We have chosen, as a case study, a problem that was
considered a huge challenge at the time. Namely, we looked at C.L. Pekeris's
seminal 1958 work on the ground state energies of two-electron atoms. We went
through all the computations ab initio with today's software and hardware, with
a special emphasis on the symbolic computations which in 1958 had to be made by
hand, and which nowadays can be automated and generalized.Comment: 8 pages, 2 photos, final version as it appeared in the journa