Mark C. Wilson, University of Auckland, New Zealand
"Higher order asymptotics from multivariate generating functions"
Time: 5:00 PM
Location: Hill 705
Abstract: For the past decade I have been working on a project with Robin Pemantle to systematize coefficient extraction from multivariate generating functions. I will give a sketchy overview of results so far and then concentrate on recent work on higher order terms. Emphasis is placed on concrete computation and interesting examples.
Thursday, November 5th
Roland van der Veen , University of Amsterdam
"Quantum spin network evaluations"
Time: 5:00 PM
Location: Hill 705
Abstract: In this talk we introduce spin networks, which are labeled graphs representing contractions of SU(2) invariant tensors, widely used in quantum theory, statistical mechanics and knot theory. We will show how to evaluate such networks in terms of hypergeometric multisums and how these sums can be encoded neatly in terms of an explicit rational generating function defined in terms of the network.
In knot theory one is led to study q-analogues of these spin networks, whose evaluations are q-hypergeometric multisums closely related to the Jones polynomial. We'll sketch how this works and how one can in many cases guess the correct q-evaluation of the spin network by looking at the classical formulas in the "right" way. Making this precise is a challenge for experimental mathematics since in some cases the guessed answers are known to be false. In the case of planar spin networks we will discuss how to resolve the matter in terms of a modified generating function.
Thursday, October 29th
Alexander Kister, Rutgers University
"Relationship between Amino Acids Sequences and Protein Structures: Folding Patterns and Sequence Patterns"
Time: 5:00 PM
Location: Hill 705
Abstract: A fundamental principle that governs sequence-structure relationship of proteins states that the native structure of a protein is determined by its amino acid sequence. This principle implies that similar sequences encode similar structures. Observations showed that proteins tend to share similar three-dimensional structures when their sequence identity exceeds 30%. This is an important observation because, first, it provides the threshold for structure prediction from amino acid sequences and, secondly, it suggests that a relatively small number of residues in a sequence are critical to structure formation, while others play a relatively minor structural role. Thus, even though each residue makes some contribution to 3D structure formation, the relative weights of the contributions vary greatly. The goal of this research is to develop the method for identification of residues, which play an essential role for structure formation.
Thursday, October 22nd
Luis Medina, Rutgers University
"p-regularity of the p-adic valuation of the Fibonacci sequence"
Time: 5:00 PM
Location: Hill 705
Abstract: In this talk we present the study of the p-adic valuation of the sequence F_n of Fibonacci numbers from the perspective of regular sequences. We establish that this sequence is p-regular for every prime p and give an upper bound on the rank for primes such that Wall's question has an affirmative answer. We also point out that for primes p = 1,4 mod 5 the p-adic valuation of F_n depends only on the p-adic valuation of n and on the sum modulo p-1 of the base-p digits of n --- not the digits themselves or their order.
Thursday, October 15th
Arthur DuPre, Rutgers University (Newark)
" Edge-matching - Where Are the Theorems?"
Time: 5:00 PM
Location: Hill 705
Abstract: MacMahon discovered early last century that the set of triangles whose edges are colored with at most 4 colors, 24 in all, could be arranged into a hexagon in which each pair of adjacent edges are the same color (see http://math.rutgers.edu/~dupre/puzzles/hextriangles.gif). Similarly, he also showed that the 24 possible squares colored with at most 3 colors could be arranged into either a 3x8 rectangle or a 4x6 one, edges matching as above. A few months ago, I began to use these triangles and squares and certain subsets to tile the surfaces of various polyhedra. My work has so far been by hand, but Jacques Haubrich of the Netherlands and Peter Esser in England have been kind enough to use their computer skills to answer some of the questions I have posed to them.
Most of the results in this subject have been gained by use of computer programs, but it is a challenging idea to see some order in all this seemingly chaotic landscape, and this talk will be a description of my tantalizingly unsuccessful attempts to do this.
Thursday, October 8th
Vladimir Retakh, Rutgers University
" Noncommutative Laurent Phenomenon"
Time: 5:00 PM
Location: Hill 705
Abstract: Let A be an algebra and S is a "small" subset of generators of A. Assume that A is embedded into an algebra B. We say that Laurent phenomenon is valid for the triple S,A,B if any element of A can be expressed in B as a Laurent polynomial of elements from S. Examples of such triples are provided by the theory of commutative cluster algebras A developed by Fomin and Zelevinsky and their followers. In my talk I will construct noncommutative examples of Laurent phenomenon.
Thursday, October 1st
Wesley Pegden, Rutgers University
"Sequence games: using a Local Lemma to show strategies for nonrepetitiveness"
Time: 5:00 PM
Location: Hill 705
Abstract: In 1906, Thue began the study of nonrepetitive sequences by showing the existence of a sequence 21020121012... over three symbols without any consecutive identical blocks. Since then, there has been much research on generalizations and modifications of this kind of nonrepetitiveness. In this talk, we will discuss techniques that can be used to show that many of these modifications and generalizations hold in the setting of a game, where the sequence is produced out of competition between two players (alternately selecting digits), in the sense that a player seeking nonrepetitiveness can achieve it in the face of an adversary trying to foil his plans. Our tool is a one-sided or "lefthanded" generalization of the Lovasz Local Lemma, which can be applied in this game setting in spite of the dependencies introduced by the unknown adversary's strategy.
Thursday, September 24th
Michael Somos, Georgetown University
"Rational Function Multiplicative Coefficients"
Time: 5:00 PM
Location: Hill 705
Abstract: See "essay" on Experimental Math Seminar page
Thursday, September 17th
Victor Moll, Tulane University
"The p-adic valuations of interesting sequences"
Time: 5:00 PM
Location: Hill 705
Abstract: Experimentally we have discovered many unsuspected patterns for the p-adic valuations of some sequences.
I will present some (mostly conjectured) results for Stirling numbers as well as the sequence counting Alternating Sign
Matrices.
Thursday, September 10th
Doron Zeilberger, Rutgers University
" In How Many Ways Can you Break up Many Russian Dolls?"
Time: 5:00 PM
Location: Hill 705
Abstract: If you only have one Russian Doll (Matryoshka), this is a Stirling question, and it rings a Bell (as Joel Spencer put it (private communication), ca. 1981). But if you have k identical such babushkas, then things become much more complicated. The case k=2 is in Sloane, and goes back to Comtet (1968), but k=3 and beyond is missing, and apparently the "formulas" previously suggested in the literature were too complicated to implement. Thotsaporn "Aek" Thanatipanonda and I found a (relatively) quick way to compute many terms for k=3,4,5, ...., by teaching the computer how to automatically generate the "evolution differential operator" (all by itself), and then using it to crank out as many terms as desired.
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