RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR
Archive of Speakers and Talks --- 2006
Jump to Fall 2006
Spring 2006
Date: Thursday, January 19, 2006
Speaker:
Eric Rowland, Rutgers
Title: Nested Structures in Cellular Automata
Abstract:
Among one-dimensional cellular automata, one finds beautiful examples of
global nested ("fractal") patterns; the 2-color nearest-neighbor rule
numbers 60, 90, 105, 150 all generate global nested structures when begun
from a single black cell. Additionally, some cellular automata exhibit
local nested structures on one side. The notoriously intractable rule
30 is such an example -- less obvious and more interesting than its global
counterparts. In general, rules with local nested structure are
characterized by positional bijectivity in one of the rule components.
That is, under certain natural conditions, these rules are reversible in
time, so that it makes sense to talk about the infinite history of the
automaton as well as its future.
We will prove a general theorem accounting for nested structure in a right
bijective, k-color rule, and we will explore a new class of integer
sequences suggested by these investigations. A computer will be on hand
to show us nice pictures of what we are talking about.
Date: Thursday, January 26, 2006
Speaker: Arvind Ayyer, Rutgers (Physics)
Title:
Some Combinatorial Identities from Noncommutative Field Theory
Abstract:
tarting from a particular Noncommutative Field Theory in six
dimensions, we motivate and prove two sets of combinatorial identities,
one hypergeometric and the other more general. I would like to show how
the more general case works in one nontrivial case. The paper is posted on
the e-print arXiv at hep-th/0508014.
Date: Thursday, February 2, 2006
Speaker:
Ilias Kotsireas
, Wilfrid Laurier University
Title:
Algorithms in Combinatorial Design Theory
Abstract:
Many existence problems in Combinatorial Design Theory are directly
amenable to algebraic formulations that enable the application of powerful
computational algebra and metaheuristic methods. The application of
computational algebra methods, often aided by supercomputing, reveals
important combinatorial and group theoretic structural information on the
varieties associated with combinatorial designs. The application of
metaheuristic methods, such as genetic algorithms for instance, allows the
construction of designs of large orders.
These ideas lead to some interesting consequences in Coding Theory, such
as the construction of new self-dual codes with larger minimal distances.
One of the practical outcomes of these projects is the deployment of
several combinatorial designs databases, integrated in the Magma Computer
Algebra System developed in Sydney. In this talk I will discuss specific
examples of applications and new results, as well as future research
perspectives.
Date: Thursday, February 9, 2006
Speaker:
Sarah Mason,
University of Pennsylvania
Title: Nonsymmetric Schur functions and an analogue of the RSK
algorithm
Abstract:
The Schur functions, sμ, form a basis for the ring of
symmetric functions. Macdonald polynomials are symmetric
functions Pμ(x;q,t)
in variables x= x1, x2,
... , with
coefficients which are rational functions of two parameters q and
t. The Schur functions are obtained from Macdonald polynomials by
setting q = t = 0. Recently Haglund, Haiman, and Loehr derived a
combinatorial formula for nonsymmetric Macdonald polynomials, which
gives a new decomposition of the Macdonald polynomial into
nonsymmetric components and provides a combinatorial description of
the
nonsymmetric
Schur functions, NSλ.
Letting q = t = 0 in this identity implies
sμ(x) =
∑λNSλ(x),
where the sum is over all
rearrangements λ of the partition μ. We exhibit a
weight-preserving bijection between
semi-standard Young tableaux and semi-standard skyline fillings to give a
combinatorial proof of the formula. The bijection involves an
analogue of the Robinson-Schensted-Knuth Algorithm. We also provide a
non-recursive combinatorial interpretation of the standard bases of
Lascoux and Schützenberger.
Date: Thursday, February 16, 2006
Speaker:
Amy Novick-Cohen, Technion---Israel Institute of Technology and
Courant Institute, NYU
Title: An Allen-Cahn/Cahn-Hilliard System
Abstract:
An Allen-Cahn/Cahn-Hilliard system was derived in 1994 to describe
simulateous phase separation and ordering,
in isothermal systems. When concentration dependence of the mobility is
taken into account,
a degenerate coupled fourth and second order parabolic system is obtained.
At low temperatures
and at near 50%-50% stiochiometry, formal asymptotics predict coupled
surface diffusion and motion
by mean curvature. This limiting system is important in and of itself in
many contexts in
materials science, and its relationship to various questions regarding
wetting and spreading is also of interest. In terms
of the regularity properites of the system, while existence, uniqueness,
and long time behavior have been analyzed in depth in the constant mobility
case, the regularity
properties for the degenerate system are far more elusive. Indeed, so far
for the degenerate system,
existence has only been proven in one-dimension and the possibility of
nonuniqueness has been
demonstrated. It can also be shown to have many features in common with
the thin film equations, and techniques developed in that context are
relevant here.
Date: Thursday, February 23, 2006
Speaker:
Neil Sloane, AT&T
Title: The influence of the On-Line Encyclopedia of Integer Sequences on research
Abstract:
The On-Line Encyclopedia of Integer Sequences (OEIS) is a
database of over 100000 number sequences.
It is freely available on the Web and is widely used.
There are several ways in which it benefits research.
1. It serves as a dictionary, to tell the user what is
known about a particular sequence. There are hundreds of papers
which thank the OEIS for assistance in this way.
2. The associated Sequence Fans mailing list is a worldwide network
which has evolved into a powerful machine for tackling new problems.
3. As a direct source of new theorems, when a sequence arises
in two different contexts.
4. As a source of new research, when one sees a sequence
in the OEIS that cries out to be analyzed.
The talk will be illustrated with numerous examples, emphasizing
new sequences that have arrived in the past few months.
Many open problems will be mentioned.
Date: Thursday, March 2, 2006
Speaker:
William Goh, Drexel University
Title: Limiting distributions for extreme branch sizes for random recursive trees
Abstract:
A recursive tree can be viewed as a labeled tree rooted at 1. Let D be
the
set of nodes of the first generation. A subtree with its root in D is
called a branch, which is also a recursive tree if all labels in it are
shifted by appropriate constants. The total number of nodes in a branch is
called the size of the branch. Let Xn and
X~n be the
random variables marking the minimum branch size and maximum branch size of
the tree, respectively. In this talk I will explain how the limiting
distribution for Xn and
X~n are obtained. It turns out
that we proved the following theorems.
Theorem 1 Let Xn be the minimum branch size in a
random recursive tree of order n. The asymptotic distribution of
Xn is given by
limn->∞
Pr( Xn = m ) = exp(-Hm-1)
(1 - exp(-1/m)), m = 1,2, . . . ,
where Hk:= ∑j=1k 1/j
is the k-th harmonic number with H0 = 0.
Theorem 2 Let X~n be the maximum
branch size in a random recursive tree of order n. The asymptotic
distribution of Yn:= X~n/n is:
For 0 < ξ ≤ 1,
limn->∞
Pr( Yn ≤ ξ ) =
ξ exp( γ - a(ξ) ) / (2 π )
∫-∞∞
exp( ∫0ξ(a(ξ)+iθ)
[exp(t) - 1]/t dt - iθ ) dθ,
where γ denotes Euler's constant and a(ξ) is the positive
function that satisfies the functional equation
exp(ξ a(ξ)) = a(ξ) + 1
This is a joint work with Professor Chun Su and Mr. Qunqiang Feng of
Department of Statistics and Finance, University of Science and Technology
of China.
Date: Thursday, March 9, 2006
Speaker:
Yi Jin,
J. P. Morgan
Title: Finding roots of a particular multiplicity in high style
Abstract:
Many high order iterative root-finding algorithms such as Newton's
method and Halley's method degrade to linear convergent algorithms when
the root is of multiplicity greater than 1 (such a root is often called a
multiple root). In this talk we show how to construct an algorithm of
arbitrary integral order m≥2 for roots of a particular multiplicity
gracefully, i.e., through an algebraic combinatorial approach.
(joint work with Bahman Kalantari)
Date: Thursday, March 16, 2006
No seminar due to Spring Break
Date: Thursday, March 23, 2006
Speaker:
Feng Luo, Rutgers
Title: Some applications of the cosine law in low-dimensional
geometry
Abstract:
In the discrete approach to smooth metrics
on surfaces, the basic building blocks are sometimes
taken to be triangles in constant curvature spaces. In
this setting edge lengths and inner angles of
triangles correspond to the metrics and its
curvatures. For triangles in hyperbolic, spherical and
Euclidean geometries, edge lengths and inner angles
are related by the cosine law. Thus cosine law should
be considered as the metric-curvature relation. From
this point of view, the derivative of the cosine law
is an analogy of the Bianchi identity in Riemannian
geometry. This talk is focus on many applications of
the derivative cosine law. It provides a unification
of many known approaches of constructing constant
curvature metrics on surfaces. These include the work
of Colin de Vedierer, Greg Leibon, Bragger, Chow and
myself on variational approaches and discrete
curvature flows on triangulated surfaces.
Date: Thursday, March 30, 2006
Speaker:
Melkamu Zeleke, William Patterson University
Title: Generalized Motzkin Numbers & k-Trees
Abstract:
(Joint work with Mahendra Jani)
We use k-trees to generalize the sequence
of Motzkin numbers
Mn = [zn]
{ (1 - z - (1- 2 z - 3 z2)½)
/ (2 z2) }
and use this generalization to show that Baxter's generalization of
Temperley-Lieb operators is a special case of generalized Motzkin
numbers. We also show that the terms of 3-Motzkin numbers are
palindromic sums and
investigate some asymptotic properties of the terms of
k-Motzkin numbers.
Date: Thursday, April 6, 2006
Speaker:Adi
Ben-Israel, Rutgers University (RUTCOR)
Title:A Newton-Bracketing Method for Convex Minimization and its
Application to Location Problems
Slides:
Click for PDF.
Abstract:
An iterative method for the minimization of convex functions f: R^n -->
R, called a Newton Bracketing (NB) method, is presented, [2].
The NB method proceeds by using Newton iterations to improve upper and
lower bounds on the minimum value. Each iteration of the NB method does
one step of the Directional Newton Method, [1].
An advantage of the NB method (over say gradient methods) is a natural
stopping rule (the bracket size.)
The NB method is applied to large scale Fermat-Weber location problems,
where it works better the larger is the problem size. This anomaly is due
to an interesting and unexpected "experimental factor".
(Joint work with Yuri Levin of Queens University, Canada.)
References:
[1] Y. Levin and A. B-I, Directional Newton Methods in n Variables ,
Math. of Computation 71(2002), 237-250
[2] Y. Levin and A. B-I, The Newton Bracketing Method for Convex
Minimization , Comput. Optimiz. and Appl. 21(2002), 213-229
Date: Thursday, April 13, 2006
Speaker: Eliyahu Hanavi
(אליהוּ הנביא),
Tishbe Univ.
Title:
Abstract:
Date: Thursday, April 20, 2006
Speaker:
Saber Elaydi, Trinity University
Title: Periodic Discrete Dynamical Systems
Abstract: The dynamics of nonautonomous
discrete dynamical systems or difference
equations
can be better understood in the setting of skew-product dynamical systems. We
will
survey the recent developments in the theory of periodic discrete
semi-dynamical
systems from stability to chaos. This includes, among other things, the various
extensions of Sharkovsky's theorem, and Elaydi-Yakubu-Sacker theorem.
The basic tools in this area are combinatorics, and number theory.
Date: Thursday, April 27, 2006
Speaker:
Lou Shapiro,
Howard University
Title: Random walks and the lurking Riordan group
Abstract:
The Riordan group is a simple concept which nonetheless provides a
framework for investigating combinatorial structures and decomposing them
into simpler components. The Riordan graoup also pops up in unexpected
places. In this talk we will run through some of the basic definitions and
examples, then go to some of the less usual settings, and also discuss
several open problems. This talk is designed to be accessible to graduate
students.
Fall 2006
Date: Thursday, September 14, 2006
Speaker:
Herbert Wilf, University of Pennsylvania
Title: Mathematics: An Experimental Science
Abstract:
We'll discuss some recent results that were conjectured via
computer experiments and then proved. The examples will be drawn from
number theory, Young tableaux, hypergeometric determinant evaluation,
theory of matrices of 0's and 1's, etc. Several unsolved problems that
also resulted from these studies will be presented.
Date: Thursday, September 21, 2006
Speaker:
Greg Chaitin, IBM Research
and Rutgers
Title: The Halting Probability Omega
Abstract:
I explain various ways to define the Omega
number and outline the proof that it is computationally
and logically irreducible.
Date: Thursday, September 28, 2006
Speaker:
Adi Ben-Israel, Rutgers University (RUTCOR)
Title: Two Tricks Related to Newton's Method
Abstract: Two
tricks, based on the geometry of Newton's method, are presented.
1) Trick 1: An iterative method for solving scalar equations f(x) = 0,
that uses the same data as Newton's method (values of f and f'), has
convergence order 1+sqrt(2), and coincides with Halley's method (cubic
convergence) for quadratics, see [1].
2) Trick 2: An iterative method for the minimization of convex functions
f: R^n --> R, called a Newton Bracketing (NB) method, see [3].
The NB method uses brackets of lower and upper bounds on the minimum
value. Each NB iteration does one iteration of the Directional Newton
Method, [2], and the bracket is reduced, by a factor of 1/2 on the
average, per iteration.
An advantage of the NB method (over say gradient methods) is a natural
stopping rule (the bracket size).
The NB method is applied to large scale Fermat-Weber location problems,
where it works better the larger is the problem size. This anomaly is due
to an interesting and unexpected "experimental" fact.
(Joint work with Yuri Levin of Queens University, Canada.)
References:
[1] A. B-I, Newton's Method with Modified Functions, Contemporary Math
204(1997), 39-50
[2] Y. Levin and A. B-I, Directional Newton Methods in n Variables ,
Math. of Computation 71(2002), 237-250
[3] Y. Levin and A. B-I, The Newton Bracketing Method for Convex
Minimization , Comput. Optimiz. and Appl. 21(2002), 213-229
Slides
Date: Thursday, October 5, 2006
Speaker: Tung-Shan Fu, National Pingtung Inst. (Taiwan) and DIMACS
Title: Area of Catalan Paths on a Checkerboard
Abstract:
In this talk, a bijection between the area under all Catalan
paths of length n and the set of inversions of all 321-avoiding
permutations of length n+1 is established. As intermediate stages, a
number of interesting bijective results that pave the way to the
requested bijection are presented.
Date: Thursday, October 12, 2006
Speaker: Arvind Ayyer, Rutgers (Physics)
Title: Experimental Mathematics to the Aid of Theoretical Physics
Abstract:
Statistical Mechanics is nothing but weighted enumerative
combinatorics, that is amenable to be studied by experimental mathematics.
Date: Thursday, October 19, 2006
Speaker:
Jacques Carette, McMaster University
Title: Holonomic programs
Abstract:
We want to add a third member to the holonomic family: after sequences
and functions, programs. First, we review simple ways to transform
recurrences into programs to compute terms, and thence from differential
equations to programs to approximation solutions via Taylor series.
These programs have a particularly simple structure. We then look at
the inverse problem of taking programs from this family and find the
appropriate 'holonomic object' to which it corresponds.
One intriguing aspect of this work is that programs compute certain
constants (such as exp(1)) fit within our scheme, so that from a way to
compute exp(1) we can uncover which holonomic object it corresponds to,
in a completely deterministic fashion. The underlying theory and 'live'
examples of "reverse engineering" of holonomic information from programs
will be shown.
Date: Thursday, October 26, 2006
Speaker:
Dylan Thurston, Columbia University (Barnard College)
Title: A random tunnel number one 3-manifold does not fiber over the circle
Abstract:
There are two natural notions of "random" for tunnel number
one 3-manifolds (that is, a manifold obtained by attaching a disk to a
genus 2 handlebody). With respect to both notions of random,
experiments show that a random manifold does not fiber over S^1 when
the manifold is large enough. We prove it with respect to one notion.
The question is motivated by the virtual fibration conjecture. We use
techniques of Brown to turn the question into a group theory question
and techniques of Agol, Hass, and W. Thurston to study the question for
such large manifolds.
This is joint work with Nathan Dunfield.
Date: Thursday, November 2, 2006
Speaker:
James McLaughlin, West Chester University
Title: Continued Fractions, and Generalizations, with Many Limits
Abstract:
Consider the following two examples of infinite
processes:
(1) Let the sequence {x0}n≥0 be defined by
x0 =4/3, xn+1 =
4/3 - 1/xn, n≥ 1.
What can be said about the sequence {xn}?
(2) Let α and β be points on the unit circle, let
|q|<1, and define the continued fraction
G(q,α, β):=
-αβ -αβ
-αβ
----- ------
------
α+β+q + α+β+q2 + . . .
+ α+β+qn + . . .
What can be said about the convergence or divergence of
G(q,α β)?
What difference, if any, does it make if α and
β are roots of unity?
It turns out that both the sequence and the continued fraction
diverge, but diverge in quite interesting ways.
There are several infinite processes (matrix products, continued
fractions, (r,s)-matrix continued fractions, recurrence
sequences) which, under certain circumstances, do not converge but
instead diverge in a very predictable way.
This talk will present a survey of results in the area,
concentrating on recent results by Douglas Bowman and the speaker.
Date: Thursday, November 9, 2006
SEMINAR CANCELLED due to anticipated football-related mayhem.
Date: Thursday, November 16, 2006
Speaker:
Jim Haglund, University of Pennsylvania
Title: A Combinatorial Formula for Nonsymmetric Macdonald Polynomials
Abstract:
Nonsymmetric Macdonald polynomials are multivariate polynomials
which
satisfy an orthogonality condition. They were introduced by Macdonald, after
which Cherednik, Knop, Sahi and others developed the theory further. In this
talk I will discuss how these functions form natural counterparts in many ways
to the more well-known symmetric Macdonald polynomials. Then I will show how
combinatorial results of a few years ago, joint with M. Haiman and N. Loehr, on
the symmetric Macdonald polynomials, together with past results of Knop and
Sahi
on nonsymmetric Jack polynomials, led experimentally to a conjectured
combinatorial formula for nonsymmetric poloynomials, which Haiman, Loehr and I
have since proved.
Date: Thursday, November 30, 2006
Speaker:
Timothy Chow, Center for Communication Research (Princeton)
Title: Chess tableaux and chess problems
Abstract:
A chess tableau is a standard Young tableau (SYT) in which orthogonally
adjacent entries have opposite parity. Remarkably, the number of 3xn
chess tableaux is the same as the number of 3x(n-1) nonconsecutive
tableaux (SYT in which i and i+1 never appear in the same row), the
Charney-Davis statistic of a 3xn shape, and the number of Baxter
permutations of n. A WZ-style proof of at least some of these facts
should be possible in principle, but the naive approaches we have tried
have crashed the computer. In the last part of the talk we present two
chess problems that are related to chess tableaux.
Date: Thursday, December 7, 2006
Speaker:
Saul Schleimer, Rutgers
Title:
Playing dots and boxes to WIN!
Abstract:
Are you losing pen-and-paper games at conferences? Suffering defeat from
the hands of sneering physicists and engineers? Or worse, tired of the
gibes after being beaten by barely adolescent kids at Yahoo Games(TM)
Online?
You can TURN THAT AROUND after attending only ONE of our exclusive
EXPERIMENTAL MATHEMATICS seminars! The secrets of the Hard-Hearted
Handout, the Parity Rule, Nimstring, and MUCH MORE are available ONLY at
this one-time affair!
(Supplies limited. Void where prohibited. To avoid delays in delivery
bring own pen and paper. Professors Berlekamp, Conway, and Guy are not
responsible for, affiliated with, or in any way connected to this offer.)