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Mathematics & Statistics

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Moment Analysis of Distributions

Dr. Jordan Stoyanov

School of Mathematics & Statistics

University of Newcastle upon Tyne

UNITED KINGDOM

2:00pm-3:00pm

PHY 130

George Yanev

**Abstract**

We start with a brief discussion on the importance of the moments when analyzing probability/statistical distributions from both theoretical and applied points of view. Then we turn to the question of how, in terms of the moments, to characterize a distribution as being unique or non-unique. In particular:

*Under what conditions can a distribution be determined uniquely by its moments?*

This is the classical problem of moments originated in works by Chebyshev, Markov, Stieltjes with further contribution by Carleman, Cramer, …, Krein, Akhiezer, …, and more recently by Heyde, Berg, Slud, Lin, Pakes, Gut, …

We describe conditions under which a distribution is uniquely determined by its moments (M-det) and conditions when it is non-unique (M-indet). The illustrations involve distributions such as Normal, Log-normal, IG, Gamma, Poisson, etc. Many of the facts are new, hence not so well-known, and some of them even look surprising.

Of special interest in statistical applications are the so-called Box-Cox functional transformations of random data (coming from random variables or stochastic processes). What can we say about the moment determinacy or indeterminacy of the distributions involved? Some answers are based on classical results (Carleman and Cramer), however, we need new ideas and techniques (Krein, Slud, Lin, Pakes, Gut).

We show next that the non-uniqueness phenomenon can be efficiently studied by constructing Stieltjes classes (infinite families of distributions all having the same moments) and calculating their Index of dissimilarity.

More illustrations will be given to show the role of the moment determinacy when studying analytic properties of distributions (symmetry, unimodality and infinite divisibility) and also in applied areas such as aerosol sciences, stochastic financial modelling (Black-Scholes type models), identifiability for mixture distributions.

It will be clear that Moment Analysis of Distributions is a rich area of research with important practical implications. There are, however, challenging open questions and if time permits some of them will be briefly outlined.

The material will be presented in an understandable and (hopefully) attractive way, addressing the lecture not only to professionals in statistics, probability, mathematics (theory and/or applications), but also to doctoral and master students in these areas.

**References**

- Heyde, C. C. (1963). On a property of the lognormal distribution. J. Royal Statist. Society Ser. B 25, 392—393.
- Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed. Wiley, New York.
- Stoyanov, J. (1997). Counterexamples in Probability, 2nd ed. Wiley, Chichester-NY.
- Lin, G. D. (1997). On the moment problems. Statist. & Probab. Letters 35, 85—90.
- Stoyanov, J. (2000). Krein condition in probabilistic moment problems. Bernoulli 6, 939—949.
- Pakes, A. G., Hung, W.-L. and Wu, J.-W. (2001). Criteria for the unique determination of probability distributions by moments. Aust. N. Z. J. Stat. 43, 101—111.
- Gut, A. (2002). On the moment problem. Bernoulli 8, 407—421.
- Lin, G. D. and Stoyanov, J. (2002). On the moment determinacy of the distributions of compound geometric sums. J. Appl. Probab. 39, no. 3, 545—554.
- Stoyanov, J. (2004). Stieltjes classes for moment-indeterminate probability distributions. J. Appl. Probab. 41A, 281—294.
- Stoyanov, J. and Tolmatz, L. (2005). Method for constructing Stieltjes classes for \(M\)-indeterminate probability distributions. Appl. Math. & Comput. 165, 669—685.

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Relative difference sets fixed by inversion

Yuqing Chen

Wright State University

3:00pm-4:00pm

PHY 130

Xiang-Dong Hou

**Abstract**

If a group \(G\) acts on a divisible design regularly and transitively, the subsets in \(G\) that correspond to the blocks of the design are called relative difference sets relative to a subgroup \(N\) of \(G\). In this talk, I will discuss various techniques of constructing relatives difference sets that are closed under taking inversion.

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Walter, me and the odd order paper

John Thompson

Graduate Research Professor

University of Florida

3:00pm-4:00pm

PHY 141

Arunava Mukherjea

**Abstract**

In my lecture, I will outline some of the early results about groups of odd order, and then discuss in some detail the contribution of Walter Feit to the proof that groups of odd order are solvable, with special emphasis on his work on group characters. I won't give any proofs, since it is not that sort of lecture.

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The ropelengths of physical knots

Yuanan Diao

University of North Carolina

Charlotte, NC

2:00pm-3:00pm

LIF 266

Masahiko Saito

**Abstract**

The ropelength of a knot is the minimum amount of rope needed to tie the knot (assuming the rope has a unit thickness). An overview of the knot ropelength problem will be given in this talk. Topics will cover the global minimum ropelength of nontrivial knots, general lower and upper ropelength bounds for knots and links, and ropelength bounds for some knot and link classes.

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Boundary Value Problems Arising From Monge-Ampère Equations

Shouchuan Hu

Missouri State University

3:00pm-4:00pm

PHY 130

Yuncheng You

**Abstract**

In seeking radially symmetric convex solutions of a Monge-Ampère equation, we are led to a boundary value problem which can be studied by various methods. In this talk, we are going to use a topological approach to establish some results on the existence, multiplicity and nonexistence of radially symmetric convex solutions for the Monge-Ampère type equations.

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Extremal problems for non-vanishing functions in Bergman spaces

Catherine Bénéteau

Seton Hall University

3:15pm-4:15pm

PHY 108

Analysis Search Committee

**Abstract**

In this talk, I will give a brief survey of duality techniques introduced by S. Ya. Khavinson in 1949 and independently by Rogosinski and Shapiro in 1953. I will discuss how these techniques were then applied to general extremal problems for non-vanishing functions in analytic function spaces in the 50's and 60's. I will also examine the Bergman space context, where such techniques seem to fail. However, in a wide class of such problems, solutions exist and are unique, and we are able to obtain some regularity results with surprising twists. In particular, if we consider the specific problem of minimizing the norm of non-vanishing Bergman functions whose first two Taylor coefficients are given, the conjectured form of the extremal is not continuous in the closed disk! This most recent result is joint work with D. Aharonov, D. Khavinson, and H. Shapiro.

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Constant Mean Curvature Surfaces and Loop Groups

Hongyou Wu

Northern Illinois University

3:00pm-4:00pm

PHY 130

Wen-Xiu Ma

**Abstract**

Surfaces of non-zero constant mean curvature, i.e., soap bubbles, are fundamental surfaces and have many applications. We will discuss the dressing action of loop groups on these surfaces and present a Weierstrass type representation of the surfaces. We will also mention constructions of such surfaces with a non-trivial topology. Many examples of surfaces of non-zero constant mean curvature will be shown.

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The Hamiltonian Approach to Solving Differential Equations

Jin Yue

Dalhousie University

3:00pm-4:00pm

LIF 263

Analysis Search Committee

**Abstract**

Theory of differential equations as the mathematical community has known it since the 17th century (the term “differential equation“ was coined by Leibnitz in 1676) is a vast area of mathematics utilyzing various techniques from analysis, differential geometry, topology, etc.

In this talk I will discuss how recent developments in the area of Hamiltonian mechanics can be effectively employed to solve nonlinear ordinary differential equations. The approach is based on the invariant theory of Killing tensors that, in turn, subsumes the ideas of classical invariant theory and Cartan's geometry.

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Concentration Phenomenon in a Singularly Perturbed Quasi-linear Elliptic Problem

Chunshan Zhao

University of Iowa

3:00pm-4:00pm

LIF 266

Analysis Search Committee

**Abstract**

I will talk about the structure and concentration phenomenon of positive solutions to a singularly perturbed quasi-linear Neumann problem arising from a biological model.

In this talk I will first address questions regarding the existence, uniform boundedness, Harnack-type inequality and asymptotic behaviors of this quasi-linear Neumann problem. After that I will focus on a class of special solutions, so-called least-energy solutions and introduce new techniques involved intrinsic variation methods, exponential decay properties of ground states and the least-energy solutions. Spiky behaviors of the least-energy solutions and the location of peak(s) will be investigated.

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Generating Random Graphs With Given Degrees

Joseph Blitzstein

Stanford University

11:00am-12:00pm

PHY 109

Discrete Mathematics Search Committee

**Abstract**

Random graphs with a given degree sequence have recently been studied as an alternative to the Erdős-Renyi random graph model. We will discuss algorithms for generating such graphs, including an algorithm found in joint work with Persi Diaconis, which uses a combinatorial theorem of Erdős and Gallai to guarantee that the output has the desired degrees. The algorithm is easy to implement and use in tandem with sequential importance sampling. Applications such as studying a real-world food web and approximate enumeration of the number of graphs with a given degree sequence will be discussed.

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Asymptotic Results for Discrete Minimum Energy Problems

Sergiy Borodachov

Vanderbilt University

4:00pm-5:00pm

PHY 120

Analysis Search Committee

**Abstract**

We consider a generalization of the classical Thomson problem dealing with the final (equilibrium) positions of \(N\) electrons repelling each other on the surface of a sphere. The potential of the repelling force in our consideration is assumed, more generally, to be proportional to the reciprocal of the power \(s>0\) of the distance between points and the points (electrons) are constrained to a compact rectifiable set embedded in Euclidean space (for which the sphere is one example). For one-dimensional rectifiable sets, asymptotics for such minimal energy configurations were obtained by Rakhmanov *et al*. We will discuss the history of the question, recent progress for higher-dimensional rectifiable sets, applications, and related problems.

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From the Fundamental Theorem of Algebra to Astrophysics: a Harmonic Journey

Dmitry Khavinson

University of Arkansas

3:00pm-4:00pm

LIF 266

Analysis Search Committee

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Spherical Codes and Delsarte's Method

Florian Pfender

TU Berlin

Germany

10:00am-11:00am

PHY 109

Discrete Mathematics Search Committee

**Abstract**

In coding theory, a lot of energy is spend to find good codes. But very often it is hard to give good quality guarantees as upper bounds on sizes of codes are hard to come by. In the 70s, Phillippe Delsarte developed an approach which gives the best known bounds for many problems. In this talk we will take a closer look at this approach and see some recent improvements to the method. This leads to some new upper bounds for kissing numbers in several dimensions, the number of unit balls which can simultaneously touch one central ball.

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Multivariate Polynomial Spline Approximation

Tatiana Sorokina

University of Georgia

4:00pm-5:00pm

PHY 120

Analysis Search Committee

**Abstract**

The talk is designed to provide an overview of multivariate polynomial splines. We will discuss basic concepts, main research directions, open problems, and applications. Polynomial spline functions (in their simplest form, piecewise polynomials) are a well established computational tool. They are used for a variety of purposes, including approximating curves and surfaces, computer-aided geometric design, finite elements and solution of differential equations, image processing, biomedical imaging, etc. In contrast to the univariate case, the theory of multivariate polynomial splines is yet to be developed.

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Virtual Knot Theory

Louis H. Kauffman

University of Illinois at Chicago

3:00pm-4:00pm

PHY 130

Nagle Lecture Committee

**Abstract**

Virtual knot theory is a generalization of classical knot theory to knot diagrams that have crossings that are “not really there” just as an embedding of a non-planar graph must have special crossings when it is represented in the plane. Knot diagrams with such virtual crossings have a natural notion of extended topological deformation in the form of combinatorial rules for changing the diagrams.

This talk will discuss virtual knot theory showing how it can be interpreted in terms of stabilized embeddings of knots in thickened surfaces. We will discuss the Jones polynomial for virtual knots, and how this is related to conjectures about the Jones polynomial for classical knots. If time permits we will discuss how new algebra (biquandles) is related to virtual knots and how new invariants such as Khovanov homolgy are related to this theory. The talk will be self-contained.

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Natural Exponential Families and Approximation Operators

Mourad Ismail

University of Central Florida

3:00pm-4:00pm

LIF 260

Analysis Search Committee

**Abstract**

An exponential operator \(S_\lambda(f,t)\) has the form \(\int\limits_\mathbb{R} W_\lambda(x,t)f(x)\,dt\) where \(f\) is a continuous function with compact support. It is assumed that $$ \partial_t\int_{\mathbb R}W_\lambda(x,t)f(x)\,dt=\int_{\mathbb R}W_\lambda(x,t)\frac{u-t}{p(t)}\,f(x)\,dt $$ and mild assumptions on \(p\). It turns out that \(t\) is the first moment. This concept was introduced in the 1970's and rediscovered in a statistical context in the 1980's. We summarize the developments in this area and mention some open problems.

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Structure and self-assembly of viral capsids

Reidun Twarock

University of York

York, UNITED KINGDOM

2:00pm-3:00pm

PHY 130

Nataša Jonoska

**Abstract**

In a landmark paper Sir Aaron Klug and Don Caspar have established a theory that predicts the surface structures of viruses in terms of a family of polyhedra [1]. It is fundamental in virology and has a broad spectrum of applications, ranging from image analysis of experimental data to the construction of models for the self-assembly of viral capsids (i.e., of the protein shells encapsulating, and hence providing protection for, the viral genome). Despite its huge success, experimental results have provided evidence for the fact that this theory is incomplete, and in particular cannot account for the structure of viruses in the family of Papovaviridae, which are of particular interest for the public health sector because they contain cancer-causing viruses.

Based on group theory and tiling theory we have developed a theory that closes this gap [2, 3]. It leads to a new series of polyhedra, the triacontahedral series [4], that corresponds to the particles observed during self-assembly of the major capsid proteins of viruses in the family of Papovaviridae. Among others, it allows to classify the malformations that may occur during self-assembly (e.g., [5]). The new theory has opened up various areas of application. In this talk, we will focus in particular on our models for the self-assembly of viral capsids and the classification of crosslinking structures, which have been featured recently by Science News (Sept. 2005, Vol. 168, No. 10).

**(1) Assembly models:**

Our theory for the structural description of viruses encodes the locations of both the capsid proteins and the bonds (dimer- and trimer-interactions) between them, and hence predicts the local bonding structure in terms of the locations of the C-terminal arm extensions of the proteins. We use this information to derive graphs that encode the structure of the intermediate species occurring during self-assembly of the capsid proteins. These graphs are combinatorial objects that are used to derive quantities of interest such as the concentrations of the assembly intermediates, and they hence characterize the assembly process [6]. Moreover, they allow us to determine the dominant pathways of assembly and hence to develop strategies of interference with the assembly process [7].

**(2) Crosslinking structures:**

Crosslinking structures are additional covalent bonds that provide particular stability to the viral capsids. We have shown that our approach can be used to classify crosslinking structures, and that it provides a theoretical tool to probe whether crosslinking is possible for general types of viruses [8].

**References**

- Caspar, D. L. D and Klug, A. (1962) Physical Principles in the Construction of Regular Viruses. Cold Spring Harbor Symp. Quant. Biol. 27, pp. 1-24.
- Twarock, R. (2004), A tiling approach to virus capsid assembly explaining a structural puzzle in virology, J. Theoret. Biol. 226, no. 4, pp. 477-482.
- Twarock, R. (2005) The architecture of viral capsids based on tiling theory, J. Theor. Medicine 6, pp. 87-90.
- Keef, T. and Twarock, R. (2005) A novel family of polyhedra as blueprints for viral capsids in the family of
*Papovaviridae*, submitted to J. Math. Biol. - Twarock, R. (2005) Mathematical models for tubular structures in the family of
*Papovaviridae*, Bull. Math. Biol. 67, no. 5, pp. 973-987. - Keef, T., Taormina, A. and Twarock, R. (2005) Assembly Models for
*Papovaviridae*based on Tiling Theory, Phys. Biol. 2, pp. 175-188. - Keef, T., Micheletti, C. and Twarock, R. (2005) Master equation approach to the assembly of viral capsids, submitted to J. Theor. Biol.
- Twarock, R. and Hendrix, R. (2005) Crosslinking in Viral Capsids via Tiling Theory, to appear in J. Theor. Biol.

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Algebraic Aspects of Topological Quantum Computing

Eric Rowell

Indiana University, Bloomington

4:00pm-5:00pm

PHY 120

Discrete Mathematics Search Committee

**Abstract**

Quantum computers would take advantage of quantum mechanical phenomena to solve problems more efficiently than “classical” computers. The topological quantum computer (TQC) of Freedman/Kitaev uses topological degrees of freedom to achieve a higher error tolerence than the usual quantum circuit model. Moreover, their model can be described algebraically as modular categories. As such, many fundamental questions related to TQCs can be translated into algebraic problems (interesting in their own right).

In this talk I will give an overview of TQCs and their algebraic conterparts. As time permits, I will discuss a some problems and results underscoring the application of algebra to quantum computing.

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Physical Knot Theory

Jorge Calvo

Ave Maria University

Naples, FL

3:00pm-4:00pm

PHY 130

Mohamed Elhamdadi

**Abstract**

Knot theory is the study of simple closed loops embedded in 3-dimensional space and their deformations in this space. If a loop can be deformed into a perfectly round flat circle, then we say that loop is an unknot; otherwise, the loop is a knot. Any deformation is legal as long as it doesn't force one part of the knot through another (or pull the knot tight so it shrinks down to a single point). In the classical view of the theory, our knotted loops can be thought of as being made out of some sort of theoretical string which is infinitesimally thin and arbitrarily flexible. In this talk, we shall look at some knots which are made out of “real stuff”. In particular, we shall look at knots made out of straight rigid “sticks” and flexible pivot joints, and at the deformations which preserve this piecewise-linear structure. We will then examine how the mathematics change when we add this new level of rigidity.

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A Hypergraph Method in Arithmetic and Extremal Combinatorics

Brendan Nagle

University of Nevada, Reno

3:30pm-4:30pm

ENG 004

Discrete Mathematics Search Committee

**Abstract**

A so-called *Hypergraph Regularity Method* was established by V. Rödl, M. Schacht, J. Skokan and the speaker and, independently, by W. T. Gowers. This method provides an essential extension to hypergraphs of Szemerédi's Regularity Lemma for graphs, and yields alternative (and quantitative) proofs of well-known density theorems of Szemerédi and of Furstenberg and Katznelson. These proofs are, in fact, derived from the following corollary of the hypergraph regularity method (formerly a conjecture of Erdős, Frankl and Rödl (1986), now dubbed the ‘removal lemma’): Every \(k\)-graph \(\mathcal{H}_n^{(k)}\) (on \(n\) vertices) containing only \(o(n^t)\) copies of the clique \(K_t^{(k)}\) admits a \(K_t^{(k)}\)-free subhypergraph on only \(o(n^k)\) fewer edges.

In this talk, we give an overview of the hypergraph regularity method and some of its applications.

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The \(2\)-category associated to surfaces embedded in \(3\)-space

J. Scott Carter

University of South Alabama

3:00pm-4:00pm

PHY 130

Masahiko Saito

**Abstract**

A \(2\)-category has objects, \(1\)-morphisms, and \(2\)-morphisms. Many of the \(2\)-morphisms function as if they were identities among morphisms. What if they are not identities, but are invertible quantities that satisfy higher-order identities? In this talk, I describe in huristic terms a \(2\)-category whose objects are finite sets of points on a line, whose morphisms are arcs in the plane, and whose \(2\)-morphisms interpolate between a pair of families of arcs. I will describe how to decompose the topography of these things, and why they are categorically interesting.

The talk will be entirely self-contained, and no knowledge beyond calculus will be assumed.

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\(2\)-Groups: An introduction to higher-dimensional groups

Alissa Crans

Ohio State University/

Loyola Marymount University

2:00pm-3:00pm

PHY 130

Mohamed Elhamdadi

**Abstract**

Group theory plays a prominent role in many branches of science where symmetries appear. In many contexts where we are tempted to use groups, however, it is actually more natural to use a richer sort of structure, that of a higher-dimensional group, or ‘\(2\)-group’. A \(2\)-group blends together the notion of a group with that of a category.

Thus, in addition to group elements describing symmetries, a \(2\)-group also has isomorphisms between these, describing symmetries between symmetries. This talk consists of an introduction to higher-dimensional group theory in which we will examine examples of \(2\)-groups and address their contexts and motivation.