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Mathematics & Statistics

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Anatomy of the Lorenz Attractor

Divakar Viswanath

University of Michigan

3:00pm-4:00pm

LIF 261

Arunava Mukherjea and J. S. Ratti

**Abstract**

The butterfly-like Lorenz attractor is one of the best known images of chaos. Although trajectories shuttle unpredictably between the two wings of the butterfly, it is possible to systematically break up the attractor into periodic orbits. There are a total of \(111011\) periodic orbits whose symbol sequence is of length \(20\) or less. The Cantor structure of the Lorenz attractor is closely related to symbolic dynamics. Finally, there are periodic orbits arbitrarily close to any given point on the Lorenz attractor, and I will show a method to compute them. This method gives an algorithmic realization of one of the basic assumptions of hyperbolicity theory.

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Noncommutative Geometry and its Applications

Jerry Kaminker

Indiana University-Purdue University at Indianapolis

3:00pm-4:00pm

PHY 108

Mohamed Elhamdadi

**Abstract**

Noncommutative geometry, as introduced by Alain Connes, has its origins in the study of elliptic differential operators. It has proved to be a useful tool in geometry, analysis and physics. In this talk we will give an introduction to the basic ideas of this theory and describe some applications to solid state physics.

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Nonlinear Analysis and the Magic of Degree Theory

Athanassios Kartsatos

3:00pm-4:00pm

PHY 118

**Abstract**

We start with a continuously differentiable function on the closure \(cB\) of an open and bounded set \(B\) in \(Rn\). We define the degree mapping \(d(f,B,p)\) for points \(p\) that are not in \(f(bB)\) and are not in \(f(Qf)\), where \(bB\) is the boundary of \(B\) and \(Qf\) is the set of critical points of \(f\) in \(B\). \(d(f,B,p)\) is an integer-valued function with 4 basic properties. Using Sard's Lemma and the Weierstrass theorem, we extend this degree mapping to an arbitrary continuous mapping \(f\). Using the fact that every compact operator in a Banach space can be uniformly approximated by compact operators of finite-dimensional range, we extend the degree mapping to a mapping \(d(I-T,B,p)\), where \(T:cB\) to \(X\) is a compact operator on the closure \(cB\) of an open and bounded subset \(B\) of \(X\) with boundary \(bB\), and \(p\) is in \(X\) but not in \((I-T)(bB)\). The operator \(T\) may be nonlinear. The 4 properties of the degree above are maintained. Three of these fundamental properties are:

- (degree of the identity) \(d(I,B,p)=1\) if \(p\) is in \(B\), and \(d(I,B,p)=0\) if \(p\) is not in \(cB\);
- (homotopy invariance) \(d(I-H(t,.),B,p)=\) constant for all \(t\) in \([0,1]\), where \(H\) is compact;
- (solvability) if \(d(I-T,B,0)\) is well-defined and \(d(I-T,B,0)\neq 0\), then \((I-T)(x)=0\), for some \(x\) in \(B\).

Thus, \(Tx=x\), i.e. \(T\) has a fixed point in \(B\). Beautiful!

Now, let's take this to skies!

Where is the “MAGIC?” Come to the talk and find out!

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Finite and Artinian Chain Rings

Youssef Alkhamees

King Saud University

Riyadh, Saudi Arabia

3:00pm-4:00pm

PHY 013

W. Edwin Clark

**Abstract**

We will cover some results involving the structure, the enumeration, and the group of automorphisms of finite and Artinian chain rings using coefficient rings and distinguished basis.

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A Very Asymmetric Function on the Reals

Peter Komjath

Eotvos University

Budapest, Hungary

3:00pm-4:00pm

PHY 120

Vilmos Totik

**Abstract**

We consider the existence of a function \(f:R\to R\) with \(\lim\limits_{h\to 0}\max(f(x-h),f(x+h))=\infty\) for every \(x\in R\). Under the continuum hypothesis we prove the existence. If the negation of the continuum hypothesis is assumed, then there are models where such functions exist and there are models where they do not exist.

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Convolutions for Orthogonal Polynomials and Lie Algebra Representations

Erik Koelink

Technische Universiteit Delft

The Netherlands

3:00pm-4:00pm

PHY 108

Mourad Ismail

**Abstract**

A number of the classical orthogonal polynomials, such as Charlier, Hermite, Meixner, Meixner-Pollaczek polynomials, satisfy convolution identities that can usually be derived from a generating function. These identities are special cases of more general convolution identities that can be obtained from tensor product representations of the Lie algebra \(\mathrm{sl}(2)\). The focus will be on the Meixner-Pollaczek polynomials, but this is just a simple example of the results that can be obtained in general.

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Monge-Kantorovich Mass Transfer and Variational Principle for Gas Dynamics

Chaocheng Huang

Wright State University

3:00pm-4:00pm

PHY 013

Yuncheng You

**Abstract**

The original mass transfer problem, proposed by Monge in the 1780's, asks how to find the cheapest way to move a pile of soil or rubber into an excavation. Mathematically, given two Radon measures (mass distributions) \(\mu\) and \(\nu\) with the same total mass and the cost function (the unit cost of moving mass at \(x\) to \(y\)) \(c(x,y)\), one looks for the optimal mapping (shipping plan) \(T\) that minimizes the total cost $$ I(T)=\int c(x,T(x))\,d\mu $$ over all mappings \(T\) that preserve the measures. No major progress was made until 1940 when Kantrovich introduced a dual problem and a relaxed variant of Monge's cost functional that remarkably transforms into a linear problem.

In this talk, I shall briefly introduce recent developments on the mass transfer problem for the distance function \(c(x,y)=|x-y|^p\) and its applications to kinetic equations raised in the gas dynamics, for instance, the Kramers system and Vlasov-Poisson-Fokker-Planck (VPFP) system. In particular, I shall show how to use the Monge-Kantorovich cost functional to establish a semi-discrete variational principle. The variational principle demonstrates an interesting phenomenon: VPFP dynamics may be viewed as the steepest descent of the total energy with respect to the Monge-Kantorovich functional.

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Crystallographic Invariants

David Rabson

Department of Physics, USF

3:00pm-4:00pm

PHY 108

Mohamed Elhamdadi

**Abstract**

We describe three invariants that come out of the homological description of crystallography. The first corresponds simply to the systematic extinctions present in all but two of the 157 periodic non-symmorphic space groups. The second, while perhaps less familiar, has been noted before: the two exceptional non-symmorphic periodic space groups (as well as others) exhibit a necessary electronic degeneracy, or “band sticking”, at defined points in the Brillouin zone. The third invariant, present for example in a rank-five, tetragonal modulated crystal, is new; we will discuss its physical implications.

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Groups of Automata and Their Geometry

Zoran Sunik

Research Assistant Professor

University of Nebraska, Lincoln

3:00pm-4:00pm

PHY 108

Nataša Jonoska

**Abstract**

We introduce several examples of groups that can be realized by automata and explore their connections to geometric group theory, dynamical systems, random walks, spectra and other areas of mathematics, thus demonstrating once again the unity and the beauty of the subject.

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Generators and Relations for the Mapping Class Groups of Surfaces

Susumu Hirose

Michigan State University

3:00pm-4:00pm

PHY 013

Masahiko Saito

**Abstract**

Two homeomorphisms are called isotopic if we can deform one to the other. The set of isotopy classes of homeomorphisms on a surface has a group structure defined by composition of homeomorphisms. We call this group “the mapping class group of the surface”. This group is one of the central subject of low dimensional topology, because this group has a deep relationship with the classification of 3-manifolds. In this talk, I will introduce a set of generators of the mapping class group, a presentation of this group, and a method to obtain this presentation.

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Optimization of Linear Error Correcting Codes

Larry Dunning

Professor of Computer Science

Bowling Green State University

3:00pm-4:00pm

PHY 108

W. Edwin Clark

**Abstract**

Suppose that a given linear block code is to be used for error-correction/error-detection. It is well known that such a code can be placed in systematic form where the message bits appear in the codeword. However, other encodings may provide better performance when the error rates for the message bits are considered individually. The greedy algorithm of matroid theory can be applied in this situation to obtain encodings that are optimal with respect to a number of different evaluation measures. In particular, the probability of message bit error and a generalization of the Hamming metric will be considered in detail.

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Best Approximation in Sobolev Spaces

Xin Li

University of Central Florida

3:00pm-4:00pm

PHY 108

Vilmos Totik

**Abstract**

For numerical solutions of differential equations, approximation with respect to a Sobolev norm (a norm involving both the function and its derivatives) is more appropriate. I will illustrate the use of best polynomial and rational approximation in Sobolev spaces, demostrate some basic properties of polynomials orthogonal in Sobolev-Laguerre and Sobolev-Legendre spaces, and discuss a general framework on orthogonal rational functions in Sobolev spaces.

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Asymptotic Properties of a Simple TCP Model

Goran Högnäs

Åbo Akademi University

Finland

3:00pm-4:00pm

PHY 120

Arunava Mukherjea

**Abstract**

We examine a simple discrete time Markov chain model of TCP congestion control and prove that it has a unique invariant measure. We also show that if the process is scaled by a factor \(\sqrt{p}\) (where \(p\) is an error probability), then the invariant measures converge to a limit as \(p\) tends to \(0\). If the scaled process is transformed in a suitable way we show that it converges to a piecewise linear limit process. The unique invariant measure of the limit process coincides with the limit of the invariant measures above and can be easily computed.

Joint work with graduate student Niclas Carlsson.

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Finding Meromorphic Solutions by Nevanlinna Theory

Yik-Man Chiang

Department of Mathematics

Hong Kong University of Science & Technology

3:00pm-4:00pm

PHY 013

Mourad Ismail

**Abstract**

We discuss how to use the classical Nevanlinna theory of meromorphic functions in the complex plane to find meromorphic solutions of certain ordinary algebraic differential equations with constant or polynomial coefficients. The method will combine with local series analysis to solve explicitly a subclass of certain ODEs. The idea behind is connected with Kowalevskaya's solution (1880's) to describing the mass, centre of mass, and moment of inertia of a spinning top that finding meromorphic solutions can be a useful tool of integrability.