RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR
Archive of Speakers and Talks --- 2005
Jump to Fall 2005
Spring 2005
Date: Thursday, January 27, 2005
Speaker: Robert Boyer, Drexel University
Title: Asymptotic Zero Distributions for Polynomials
Abstract:
Recently several researchers discovered that many natural sequences
of polynomials have interesting limiting sets of zeros in the
complex plane as their degrees go to infinity.
These observations have been both empirical as well as rigorous.
In recent work with Bill Goh, we have determined the limiting curve
and zero distribution for the Euler and Bernoulli polynomials.
This curve is related to the Szego curve, the limiting curve for the
Taylor polynomials of the exponential. As time permits, I will
also discuss open problems from combinatorics and graph theory.
Date: Thursday, February 3, 2005
Speaker: Pawel Hitczenko, Drexel University
Title: Some stochastic properties of random overpartitions
Abstract:
In this talk, I will explain how a desire to asymptotically evaluate an
alternating sum involving partition function led us to consider (and
prove) a local limit theorem for the weight of overlined parts, counted
with multiplicities, in random overpartitions of integers (all of these
notions will be explained). The talk is based on joint work with Sylvie
Corteel and Bill Goh.
Date: Thursday, February 10, 2005
Speaker: Yi Jin, Rutgers University
Title:Kalantari's bounds on zeros are asymptotically tight
Abstract: In this talk, we give a simple interpretation of a family of
bounds
on moduli of zeros of complex polynomials derived by Kalantari.
This gives rise to efficient algorithms to compute these bounds and leads
to
a proof of this family's asymptotic sharpness.
Date: Thursday, February 17, 2005
Speaker: Yuriy Gulak, Rutgers University
Title: Algebraic properties of some quadratic dynamical systems
Abstract: We discuss the interrelation between the dynamical
behavior and algebraic properties of elementary cellular automata and
quadratic differential equations.
Date: Thursday, February 24, 2005
TALK CANCELLED
Date: Thursday, March 3, 2005
Speaker: Mahendra Jani, William Patterson University
Title: Continued Fractions and Catalan Problems
Abstract: (Joint work with Dr. Robert G. Rieper)
We find a generating function expressed as a continued fraction that
enumerates
ordered trees by the number of vertices at different levels. Several Catalan
Problems are
mapped to an ordered-tree problem and their generating functions are also
expressed as a
continued fraction. Among these problems is the enumeration of (132)-pattern
avoiding
permutations that have a given number of increasing patterns of length k. This
extends
and illuminates a result of Robertson, Wilf and Zeilberger for the case k = 3.
Date: Thursday, March 10, 2005
Speaker: Drew Sills, Rutgers
Title: Disturbing the Dyson conjecture (in a "GOOD" way)
Abstract: Joint work with Doron Zeilberger. In 1962,
Freeman Dyson conjectured that the constant term in the Laurent polynomial
∏1≤i≠j≤n (1-xi/ xj)
aj (let us call this the "Dyson product") is
the multinomial coefficient (a1 + a2 + . . . + an
)!/ [a1! a2! . . . an! ].
Dyson's conjecture was first proved independently by Gunson and Wilson.
The most compact and elegant proof, however, was supplied by I.J. Good in 1970.
We present a case study in experimental yet rigorous mathematics
by describing an algorithm (which we have fully implemented in
the Mathematica and Maple packages "GoodDyson") that
automatically conjecture and then supply proofs
(inspired by Good's proof) of
closed form expressions for extensions of Dyson's conjecture to coefficients
beside the constant term in the Dyson product.
Click for the associated
preprint,
Mathematica package,
and Maple package.
Examples are available in the form of a
Mathematica 5.0 notebook, a
Maple 8 worksheet, and a
Maple 9 worksheet.
Date: Thursday, March 24, 2005
Speaker: Victor Moll, Tulane University
Title: Definite Integrals
Abstract: The question of evaluating definite integrals is as old as calculus itself.
In spite of that, there is no coherent theory that will tell us how to
proceed when confronted with a specific integrand.
In this talk I will discuss two aspects of this problem. The first one
deals with a series of transformations on the parameteres of a rational
integrand. This is a rational version of the classical transformation of
Landen, Gauss and Legendre for an elliptic integral that produced the
arithmetic-geometric mean.
The second problem illustrates the initial stages of
a homogeneity conjecture. We present a case study on the family
Ln = ∫01 lnn Gamma (q) dq.
Euler evaluated
L1 = ln (2 pi)½
and in joint work with O. Espinosa we obtained an evaluation of L2.
The evaluation of L3 is still open.
Note: This abstract has been edited due to the limitations of displaying
mathematics in HTML. To access the full abstract as a PDF file,
click here.
Date: Thursday, March 31, 2005
Speaker: Ileana Streinu, Smith College & UMass Amherst
Title: Pebble games for graph arboricity
Abstract:
The pebble game algorithm, invented by Hendrickson and Jacobs for the
study of 2-dimensional generically rigid graphs, is an elegant and
efficient paradigm underlying modern research on protein flexibility.
We analyze extensions of the original algorithm to a family of pebble
games induced by two parameters k and l. They characterize graphs which
can be turned into edge-disjoint unions of k spanning trees via the
addition of **ANY** l edges, and which satisfy hereditarily a linear
inequality of the form m <= k(n-1) - l between the number of vertices and
edges. They have well known matroidal properties for values of l in the
interval [0,k) but not otherwise.
In this setting, the "maximal components" are indicators of critical
subgraphs for a variety of tasks, and correspond to rigid components in
the protein flexibility applications. We show how to compute them
efficiently and how to use them to obtain new algorithms for Henneberg
sequences and edge-disjoint tree partitionings. As a side-effect we
improve some other existing algorithms involving pseudo-triangulations.
Along the way, the well known theorem of Tutte and Nash-Williams on
edge-disjoint tree decompositions will be extended with a characterization
via pebble games.
I will use a variety of graphical props to illustrate the algorithms
(presented indeed as "games") and some of their applications.
This is joint work with Audrey Lee and Louis Theran from University of
Massachusetts at Amherst.
Date: Thursday, April 7, 2005
Speaker: Vladimir Retakh, Rutgers
Title: Reduction formulas for multivariate hypergeometric series
Abstract: A hypergeometric series $F$ depends on of two sets: parameters
$A=(a_1, a_2,..., a_N)$ and variables $X=(x_1, x_2,..., x_n)$.
A classical problem: For which sets of parameters the series $F$ can
be written as a series $G$ with a lesser number of parameters
$B=(b_1, b_2,..., b_M)$ and variables $Y=(y_1, y_2,..., y_m)$ where
elements of $B$'s are rational expressions in $a_i$'s
and elements of $Y$ are rational expressions in $y_j$'s.
Such expressions are called reduction formulas.
A series of reduction formulas was constructed in 1993 by
Gelfand, Graev and Retakh as a by-product of a sophisticated
geometrical machine. An elementary proof of the simplest formula
of Gelfand, Graev and Retakh was given recently by C. Krathenthaler.
I would like to discuss a possibility of proving the reduction
formulas by Zeilberger-Wilf method.
Date: Thursday, April 14, 2005
Speaker: Sara Soffer, Rutgers
Title: New Measurement for the Clustering in Real Networks.
Abstract:
Real networks are graphs that describe the "real world". Being very
different from the usual Erdos-Renyi
random graphs, they inspire a new "graph theory". In the talk, I will
survey the main properties of real networks and present a new measurement
for the clustering coefficient, with an algorithm to find it.
Date: Thursday, April 21, 2005
Speaker: John F. Nash, Jr., Princeton University
Title: The 'Agencies Method' for Studying Cooperative Games and the
Resulting Mathematical Problem of Finding Equilibria in a Highly
Multi-Dimensional Context
Abstract:
Date: Thursday, April 28, 2005
Speaker: Bernard Beauzamy, Société de Calcul
Mathématique
Title: The use of probabilities in order to represent, store, exploit, the information given by physical measurements.
Abstract:
Any physical experience provides only a small number
of results, whereas a large number of parameters may vary. What is the "value"
of the information obtained this way, if, for instance, we have 300 measures,
where the whole experiment may involve 50 parameters, thus leading to possible
states, if each parameter can take 10 values? Can we, from this very limited
amount of information, predict the result of a new experience ?
We introduce a new concept, called Experimental Probabilistic
Hypersurface (in short EPH). It allows us to represent the information
obtained from any number of measures, in a physical experiment or in a
computational code. This information is stored as a density of probability,
above each point in the configuration space. If the experiment depend
upon parameters, the EPH consists in the collection of density functions.
These densities are built from the existing information (the measures that
have already been made). The existing information propagates
all over the space, with the following rule: the entropy should always be
maximal. The principle of maximal entropy thus governs the whole construction,
which allows us a construction with no artificial rules or probability laws.
If you are close to a place where the experiment has been performed, the
density will be more concentrated ; if you are far away, the density will be
less concentrated, because you know less.
The applications are multiple. The EPH is a storage of information,
which grows and becomes more precise when more and more experiments are
performed. It allows you to get immediately local results:
which regions or points are dangerous, which are safe, and so on.
The EPH is intended to replace both the deterministic methods (for
instance interpolation between existing values), which are artificial, and
the statistical methods, which are only global. The EPH gives local results,
but still keeps the global characteristics.
We are interested in developing joint research programs, from this concept,
including applications to various fields (risk analysis, epidemiology,
chemistry, and so on, just to mention a few).
Click to download Bernard
Beauzamy's article (pdf).
Date: Thursday, June 16, 2005,
Time: 3:00-3:50pm
Room: Hill 705
Speaker: Simone Severini, Univ. of York, UK
Title: X Rays of Permutations
Abstract: The X ray of a permutation is defined as the
sequence of diagonal sums in the associated permutation matrix.
This talk is an invitation to the study of the X rays of permutations.
Fall 2005
Date: Thursday, September 8, 2005
Speaker:
Greg Chaitin, IBM Research (Yorktown)
Title: Is Incompletness Serious?
Abstract:
The conventional view is that it is not. However,
I'll present an information-theoretic approach to
incompleteness that suggests that math and physics
are not that different and therefore experimental
math may be the only way to proceed if the road
is blocked by incompleteness. Discussion
welcome!
Date: Thursday, September 15, 2005
No EM seminar due to the
Kruskal conference
Date: Thursday, September 22, 2005
Speaker: Doron Zeilberger, Rutgers
Title: TEN
Abstract:
Ten who knows ten? Ten, I know Ten. There are 10 to the
power n ways of tossing a coin 5n times,
such that you win 3 dollars for Head and lose 2 dollars for a Tail,
breaking even, and every breaking-even streak that starts with a Head
may not be followed by a Tail.
(joint work with Nick Loehr, Greg Warrington, Bruce Sagan, Vince Vatter,
and most importantly, Shalosh B. Ekhad).
Date: Thursday, September 29, 2005
Speaker:
William Goh, Drexel University
Title: Asymptotic zero attractor
of the partition polynomials
(Joint work with Bob Boyer)
Many natural sequences of polynomials from combinatorics have interesting
limiting sets of zeros in the complex plane. We will discuss one of the
problems posed by Richard Stanley and Herb Wilf.
This is the sequence of partition polynomials
Fn (x) :=
∑ k=1 n pk(n)
xn,
where pk(n)
is the number of partitions of n with exactly k
parts. In Stanley's plot for degree 200, the zeros cluster around the unit
circle together with a sparse family inside the disc. Initially it was
unclear what role such zeros play in the limit. Using techniques from
analytic number theory and extensive numerical computation to degree 26,000,
we found that the zeros approach the unit circle as well as three curves
inside the disc given implicitly in terms of
fk(z) =
Re(Li2 (zk ) 1/2)/k,
where Li2 is the dilogarithm. We will outline our
attack on this problem in which we reduce the proof to a conjecture about
domination among the functions
fk which we verified numerically and in
special cases.
Date: Thursday, October 6, 2005
Speaker:
Tom Hales,
University of Pittsburgh
Title: Formal proofs, the four-color theorem, and the Kepler conjecture
for sphere packings
Abstract: The four-color theorem states that any map can be colored
with four colors in such a way that adjacent countries do not receive
the same color. This theorem is one of the most famous math problems
ever to be solved by computer. Last year, Georges Gonthier gave a
completely formal proof of the four-color theorem. This means the proof
has now been carefully checked at the level of fundamental axioms and
rules of inference of mathematics. I will describe Gonthier's project
and explain a number of interesting connections with my current work on
packing spheres.
Date: Thursday, October 13, 2005
No EM seminar in observance of
יום כפור (Yom Kippur)
Date: Thursday, October 20, 2005
Speaker:
Bruce Sagan,
Michigan State University
Title: Rational generating functions and compositions
Abstract:
(Joint work with Anders Björner)
A composition of the nonnegative integer n is a way of
writing n as an ordered sum. So the compositons of 3 are 1+1+1,
1+2, 2+1, and 3 itself. It is well-known and easy to prove that
if cn is the number of compositions of n
then cn = 2n - 1 for
n ≥ 1 and c0 = 1.
Equivalently, we have the generating function
∑n ≥ 0 cn xn
=
( 1 - x ) / ( 1 - 2x )
which is a rational function. We show that this is a special case of
a more general family of rational functions associated with
compositions. Our techniques include the use of formal languages.
Surprisingly, identities from the theory of hypergeometric series are
needed to do some of the computations.
Click for properly typeset abstract in pdf.
Date: Thursday, October 27, 2005
Speaker:
Drew Sills,
Rutgers University
Abstract: Let
Fn:=
Fn (x1, x2,
. . . , xn ;
a1, a2,
. . . , an) :=
∏1≤i≠j≤n (1-xi/ xj)
aj . In 1962, Freeman Dyson conjectured that
the constant term in the expansion of Fn is
the multinomial coefficient (a1 + a2
+ . . . + an)!/(a1!
a2! . . . an!).
In 1975, George Andrews extended Dyson's conjecture to a
q-analog.
A particularly elegant proof of Dyson's conjecture was given by
I. J. Good in 1970. Good's proof does not extend to the q-analog,
however, and the q-Dyson conjecture was not settled until
1985 when Zeilberger and Bressoud proved it combinatorially.
Last March in the Experimental Mathematics Seminar,
I demonstrated how a Maple package that I developed with Professor Zeilberger
could be used to automatically conjecture and prove closed
form expressions for coefficients in the expansion of
Fn besides the constant term, for fixed n.
The automated proofs are based on a generalization of Good's proof.
In this lecture, I will discuss more recent work, where the "disturbed
Dyson conjectures" and their proofs are extended to symbolic n, and
corresponding q-analogs are conjectured.
Date: Thursday, November 3, 2005
Speaker:
Noam Zeilberger,
Carnegie-Mellon University
Title: Why proofs count, and why to count proofs
Abstract:
Few people get excited by rigorous proofs of logical tautologies.
After this talk I hope you will be one of them. I will give an
introduction to structural proof theory and present a few of its
important results, stressing a view of proofs in logic as not just
confirmations of empty truths, but rather as mathematical objects with
deep computational significance and possessing elegant symmetries.
Then I will describe one of the products of modern proof theory --
linear logic -- and argue that linear logic proofs are particularly
interesting from the standpoint of combinatorics. No prior background
will be assumed.
Click to view the slides used.
Date: Thursday, November 10, 2005
Speaker:
Jonathan Borwein, Dalhousie University
Title: Computational Lists and Challenges in Mathematics
Abstract:
I aim to discuss Experimental
Mathodology, its
philosophy, history, current practice and proximate
future, and using concrete
accessible---entertaining I hope---examples, to explore implications for
mathematics and for mathematical
philosophy.
Thereby, to persuade you both of the power of
mathematical experiment and that
the traditional accounting of mathematical learning and research is
largely an ahistorical caricature.
I shall do so with a sample of material largely from the
2005 Clifford Lectures which I gave at
Tulane University in New Orleans in April 2005.
- Plausible Reasoning in the 21st Century, I is a general
introduction to Experimental
Mathematics, its Practice and its Philosophy.
It reprises the `Experimental methodology' that
David Bailey and I---among many others---have practiced over the
past two decades.
- Plausible Reasoning in the 21st Century, II
focusses on the differences between
Determining Truths and Proving Theorems.
It explores various of the tools available for deciding what
to believe in mathematics, and using accessible
examples---illustrates the rich experimental
tool-box mathematicians can now have access to.
- Ten Computational Challenge Problems
is a more advanced analysis of the themes
developed in Lectures 1 and 2. It discusses examples including
∫0∞ cos(2x)
∏n≥1 cos(x/n) dx
=? π/8.
- Apéry-Like Identities for ζ(n).
The final lecture comprises a research level case study of
generating functions for zeta functions.
The above is an edited version of the speaker's abstract.
Click for the full abstract.
Date: Thursday, November 17, 2005
Speaker:
Vince Vatter, University of St. Andrews
Title: Defence of Automatic (Symbolic!) Enumeration
Date: Thursday, December 1, 2005
Speaker:
Miklos Bona,
University of Florida (currently visiting University of Pennsylvania)
Title: New Records In Stanley-Wilf Limits
Abstract:
We provide examples for patterns q for which the value of the Stanley-Wilf
limit L(q) is smaller (resp. larger) than any previously known values.
The exact values of L(q) are computed. Generalizations are given.
Date: Thursday, December 8, 2005
Speaker:
Yury Grabovsky, Temple University
Title: A generalization of the Chandler Davis convexity theorem and applications
Abstract:
The classical theorem of Chandler Davis says that a rotationally
invariant function on symmetric matrices is convex if and only if it is
convex on the diagonal matrices. In this paper we uncover a simple
abstract mechanism behind this classical theorem. A motivation for the
present work comes from applied mathematics. In a problem from the
mathematical theory of composite materials it was important to
understand how to construct smallest convex and rotationally invariant
sets containing a given set. The new twist was that the action of the
rotation group and convexity happen in different coordinate systems.
Equivalently, we may consider a problem in a single coordinate system
but with a non-linear group action. The combination of convexity and
rotational invariance drew us to Davis's theorem, which we generalize
for non-linear group actions and infinite dimensional vector spaces.