USF Home > College of Arts and Sciences > Department of Mathematics & Statistics

Mathematics & Statistics

(Leader: Prof. Arunava Mukherjea)

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**Speaker**

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Self-assembling DNA Model and related problems

Ana Staninska

4:00pm-5:00pm

PHY 118

**Abstract**

I will describe a theoretical model of self-assembly inspired by DNA nano-technology and DNA computing, and introduce related mathematical problems. This model consists of tiles that assemble into graph-like complexes, which assembled "properly" can represent a solution to a given problem. It can be shown that the computational power is equivalent to solving NP complete problems.

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Stopping Times and the Evolution of Random Structures

Greg McColm

4:00pm-5:00pm

PHY 118

**Abstract**

One increasingly popular area of applied probability to combinatorics is the evolution of random structures, especially of random graphs. Such “evolutions” can be used to study the behavior of assembly, accretion, and development. One of the fundamental questions is *when* an important threshhold is crossed. This is a stopping time problem. We look at some of the basic notions in this field.

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Products of Random Circulant Matrices, Part II

Edgardo Cureg

4:00pm-5:00pm

PHY 118

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Products of Random Circulant Matrices, Part I

Edgardo Cureg

4:00pm-5:00pm

PHY 118

**Abstract**

An \(n\times n\) matrix of the form $$ \begin{matrix} a(0) & a(1) & a(2) & \dotsm & a(n-1) \\ a(n-1) & a(0) & a(1) & \dotsm & a(n-2) \\ \dotsm & \dotsm & \dotsm & \dotsm & \dotsm \\ \dotsm & \dotsm & \dotsm & \dotsm & \dotsm \\ a(1) & a(2) & a(3) & \dotsm & a(0) \end{matrix} $$ is called a circulant matrix. Such matrices have been studied in the context of random walks, BCH codes, smoothing of data, analysis of random number generators, etc. (See P. Diaconis, Patterned Matrices, Proc. of Symposium of Appl. Math. 40, AMS, 37-58, 1990).

In this talk we discuss some basic properties of such matrices and consider the problem of convergence in distribution of products of i.i.d. circulants. Orthogonal matrices play a key role in our solution.

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Weak and weak\(^*\)-convergence, Part II

Arunava Mukherjea

4:00pm-5:00pm

PHY 118

**Abstract**

To continue last week's discussion, I'll prove my old result that in a non-compact group, the random walk escapes to infinity.

In other words, if \(G\) is a locally compact Hausdorff non-compact group containing the support \(S(P)\) of a probability measure \(P\) such that no compact subgroup of \(G\) contains \(S(P)\), then for any compact subset \(K\) of \(G\), \(\Pr(Z(n)\in K\)) tends to zero as \(n\) tends to infinity, where \(Z(n)\) is the random walk induced by \(P\).

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Weak and weak\(^*\)-convergence, Part I

Arunava Mukherjea

4:00pm-5:00pm

PHY 118

**Abstract**

This will be mostly an introductory talk. New results in this context will be presented by others later in the semester.

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Planar Graphs, Random Walks and Heat Content

Patrick McDonald

New College at Sarasota

4:00pm-5:00pm

PHY 118

**Abstract**

There is a well-known and well-studied relationship between Brownian motion, boundary value problems and the geometry of Euclidean domains. This relationship gives rise to discrete analogs relating random walks, problems for discrete difference operators and the geometry of graphs embedded in Euclidean spaces. In this talk we survey the discrete material, developing techniques for moving between categories and using these techniques to discuss recent results. In particular, we will construct a pair of isospectral graphs and prove that these graphs are distinguished by their heat content.

The talk is aimed at a general mathematical audience and is reasonably self-contained. In particular, we develop those probabilistic and geometric tools which we will require.

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Local Limit Theorems for Random Integer Partitions

Ljuben Mutafchiev

4:00pm-5:00pm

PHY 118

**Abstract**

Certain power series expansions will be used to prove a local limit theorem for the length of the side of a Durfee square in a random partition of a positive integer \(n\) as \(n\) tends to infinity.

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Hypergroups: Examples, Idempotent and Invariant Probability Measures, Part II

Norbert Youmbi

4:00pm-5:00pm

PHY 118

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Hypergroups: Examples, Idempotent and Invariant Probability Measures, Part I

Norbert Youmbi

4:00pm-5:00pm

PHY 118

**Abstract**

A semihypergroup (Hypergroup) is a locally compact space on which the vector space of finite regular Borel measures has a convolution structure preserving the probability measures. The class of semihypergroups (Hypergroups) includes the class of locally compact topological semigroups (Groups). Hypergroups generalizes in many aspects locally compsc groups. Many \(n\)-dimensional hypergroups are obtained from orthogonal polynomials on spaces on which no structure of a group could be defined. We will give some practical examples of hypergroups as well as presenting some results on invariants and idempotent probability measures on semihypergroups.

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Random Fibonacci Sequences

Edgardo Cureg

4:00pm-5:00pm

PHY 118

**Abstract**

Viswanath's determination of the rate of growth of \((|x_n|)\), where \(x_{n+1}=\pm\;x_n+x_{n-1}\), \(n\ge 1\), \(x_0=x_1=1\), and the \(+\) and \(-\) signs each occur with probability \(1/2\).

The techniques involved in the solution illustrate an interplay between the theory of random matrix products, the Stern-Brocot tree, fractal measures, and computer simulations. We also present some generalizations of the random Fibonacci sequence.

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When convergence in distribution of products of \(d\times d\) i.i.d. matrices is determined essentially by their skeletons

Arunava Mukherjea

4:00pm-5:00pm

PHY 118

**Abstract**

Two nonnegative matrices \(A\) and \(B\) have the same skeleton if \(A(i,j)>0\) whenever \(B(i,j)>0\) and conversely.