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Mathematics & Statistics

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Non-equilibrium statistical mechanics of cluster-cluster aggregation

Colm Connaughton

Warwick Mathematics Institute and Warwick Centre for Complexity Science

3:00pm-4:00pm

CMC 130

Razvan Teodorescu

**Abstract**

Consider a large cloud of particles which are moved around in space by a random transport process such as diffusion. If these particles are “sticky” so that they clump together irreversibly upon contact, then the resulting distribution of cluster sizes evolves in time as smaller clusters stick to each other to produce larger ones. The statistical dynamics of such sticky particles has applications in surface physics, colloids, granular materials, bio-physics and atmospheric science. It also provides a rich variety of non-equilibrium phenomena for theoretical analysis. One of the most striking of these phenomena is the so-called gelation transition which, roughly speaking, corresponds to the generation of clusters of infinite size in a finite time. In this talk, I will discuss the scaling theory of cluster aggregation at the level of mean field theory and explain the meaning of the gelation transition. At the end I will discuss the somewhat mysterious phenomenon of “instantaneous” gelation and its relation to some problems in cloud physics.

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A universal syntactic invariant of flow equivalence of symbolic dynamical systems

Benjamin Steinberg

Department of Mathematics

City College and CUNY Graduate Center

New York, NY

3:00pm-4:00pm

CMC 130

Dmytro Savchuk

**Abstract**

A symbolic dynamical system (or shift space) is a shift-invariant closed subspace of a Cantor space \(A^{\mathbb Z}\). Flow equivalence is a classical coarsening of the notion of conjugacy (or isomorphism). Since \(A^{\mathbb Z}\) is the space of all bi-infinite words over the alphabet \(A\), shift spaces are determined by their languages of finite factors, and flow equivalence can be described syntactically in terms of sliding block codes and symbol expansions, it is natural to try to derive syntactic invariants of flow equivalence from their associated languages.

In this talk we introduce a new syntactic invariant, called the Karoubi envelope, of a symbolic dynamical system. We indicate how under mild hypotheses it classifies the Markov-Dyck and Markov-Motzkin shifts associated to graphs by W. Krieger. It turns out to be a finer invariant than essentially all syntactic invariants that we are aware of in the literature (including some that were only known to be conjugacy invariants). We also have that the Karoubi envelope is a universal syntactic invariant for sofic shifts in the sense that any flow equivalence invariant for sofic shifts that agrees on all sofic shifts with isomorphic syntactic semigroups automatically agrees on all shifts with equivalent Karoubi envelopes.

I will not assume for this talk the audience has any prior contact with symbolic dynamics (I am an algebraist so all this is relatively new to me). All notions will be introduced from scratch and worked with combinatorially. I hope the talk will be accessible to graduate students.

This is a report on joint work with Alfredo Costa (Coimbra).

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Highly Effective Evaluation and Reconstruction of High Degree Spherical Polynomials

Kamen Ivanov

University of South Carolina

3:00pm-4:00pm

CMC 130

Vilmos Totik

**Abstract**

Methods for effective solution of two intimately related problems on the unit \(2\)-d sphere are presented: The first problem is the evaluation of high degree (\(>2000\)) spherical polynomials (band-limited functions) at many scattered points on the sphere, and the second is the reconstruction of band-limited functions on the sphere from their values at irregular sampling points. Our methods rely on the sub-exponential localization of the spherical father needlets and their compatibility with spherical harmonics. They are fast, local, memory efficient, numerically stable and with guaranteed (prescribed) accuracy. Software realization of the algorithms and numerical experiments are also presented. Targeted applications of these algorithms are to Geopotential Modeling and, in particular, to fast and memory efficient evaluation of the geoid undulation, the disturbing potential and their derivatives.

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Gravitational Lensing: Connecting Mathematics to Cosmology

Charles Keeton

Department of Physics & Astronomy

Rutgers University

3:00pm-4:00pm

CMC 130

Nagle Lecture Series Committee

**Abstract**

Gravity's action on light is simple in concept yet rich in detail. The physical phenomenon gives cosmologists a variety of ways to study stars, black holes, dark matter, and much more. Those applications are supported, in turn, by mathematical topics ranging from differential geometry to catastrophe theory to probability theory and stochastic processes. I will use gravitational lensing to discuss some of the exciting synergies between mathematics and cosmology. This talk serves as a complement to the Nagle Lecture, filling in some of the formal foundations.

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Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations

Willy Hereman

Department of Applied Mathematics and Statistics

Colorado School of Mines

Golden, CO

3:00pm-4:00pm

CMC 130

Wen-Xiu Ma

**Abstract**

A method will be presented for the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs) involving multiple space variables and time.

Using the scaling symmetries of the PDE, the conserved densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative is used to compute the undetermined coefficients. The homotopy operator is used to invert the divergence operator, leading to the analytic expression of the flux vector.

The method is algorithmic and has been implemented in the syntax of the computer algebra system Mathematica. The software is being used to compute conservation laws of nonlinear PDEs occurring in the applied sciences and engineering.

The software package will be demonstrated for PDEs that model shallow water waves, ion-acoustic waves in plasmas, sound waves in nonlinear media, and transonic gas flow. The featured equations include the Korteweg-de Vries, Kadomtsev-Petviashvili, Zakharov-Kuznetsov, and Khoklov-Zabolotskaya equations.

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Hyperbolic divided differences and Nevanlinna-Pick interpolation

Line Baribeau

Department of Mathematics & Statistics

Laval University

Québec, Canada

4:00pm-5:00pm

CMC 130

Dmitry Khavinson

**Abstract**

I will dene hyperbolic divided differences, which allows us to solve Nevanlinna-Pick interpolation problems by a procedure analogous to the Newton algorithm for polynomial interpolation. I will then discuss the more difficult problem of interpolation into the spectral unit ball.

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Spectra and pseudospectra

Thomas Ransford

Department of Mathematics & Statistics

Laval University

Québec, Canada

3:00pm-4:00pm

CMC 130

Dmitry Khavinson

**Abstract**

Eigenvalues are amongst the most useful tools of mathematics: they permit diagonalization of matrices, they describe asymptotics and stability, they give a matrix personality. However, when the matrix in question is not normal,
standard eigenvalue analysis is only partially applicable and can even be misleading. This talk will be an introduction to the theory of pseudospectra, a refinement of standard spectral theory which has proved successful in applications concerning non-normal matrices. In particular, I shall focus on the question: *do pseudospectra determine matrix behavior*?

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A Comparison of Models of Self-Assembly: Single Tile Attachments vs. Hierarchical Assembly

Matthew Patitz

University of Arkansas

3:00pm-4:00pm

CMC 130

Nataša Jonoska

**Abstract**

Theoretical models of self-assembly have provided a great foundation for research into fundamental properties of self-assembling systems and their building blocks. However, the profusion of natural and biological systems which utilize self-assembly, as well as modern artificial systems (using molecular building blocks such as novel DNA-based structures) exhibit great variety in a number of physical characteristics. In order to model this diversity, as well as to help guide experimental efforts to improve laboratory implementations, researchers have developed a number of self-assembly models which each attempt to incorporate unique sets of properties. In this talk, we will present two such models: the abstract Tile Assembly Model (aTAM), and the 2-Handed Assembly Model (2HAM). Both have as their fundamental components square “tiles” which are able to attach to each other based on “glues” on each edge. However, in the aTAM every system begins growth from a special “seed” and all growth of an assembly is performed by the attachment of a single tile at a time. Alternatively, in the 2HAM every tile is considered a valid seed from which growth can begin, and arbitrarily large assemblies are allowed to bind two at a time. These differences between the models give rise to a wide array of fundamentally different powers (e.g., what can be built and how efficiently, which systems can simulate others, etc.), and we will present and discuss several such differences.

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Uniform attractors for non-autonomous strongly damped wave equations

Hongyan Li

Shanghai University of Science and Engineering

Shanghai, People's Republic of China

3:00pm-4:00pm

CMC 130

Yuncheng You

**Abstract**

The long-time behavior of solutions for non-autonomous strongly damped wave equations is studied. The existence of uniform attractors for the equations is proved and then an upper estimate for the Kolmogorov ε-entropy of the uniform attractor is obtained.

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The space-time finite element method for evolution equations

Hong Li

Inner Mongolia University

Inner Mongolia, China

2:00-3:00pm

CMC 130

Yuncheng You

**Abstract**

Space-time finite element method (STFEM) is widely used in approximately solving evolution equations. Different from traditional FEM, the STFEM unifies the spatial and temporal variables in obtaining the weak formulation. STFEM concludes three kinds of discrete forms: continuous in both spatial and temporal variables, continuous in space but discontinuous in time, and discontinuous in both space and time. We mainly discuss the second method: discontinuous in time but continuous in space. The basic conceptions and formulations are given by a parabolic model problem. The development and applied foreground are discussed. Numerical simulations for some classical model problems are presented to illustrate the efficiency and reliability of the discontinuous space-time finite element method.

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Congruences for algebraic sequences

Reem Yassawi

Trent University

Peterborough, Ontario

Canada

3:00pm-4:00pm

CMC 130

Nataša Jonoska

**Abstract**

A sequence of integers \(\left(a_k\right)_{k\in\mathbb N}\) is *algebraic* if its generating function \(y=\sum_k a_k x^k\) is the root of a polynomial \(P(x,y)\) with integer coefficients. Many illustrious combinatorial sequences, such as the Motzkin numbers or the Fibonacci numbers, are algebraic. A result of Christol, and also Denef and Lipshitz, tells us that given any prime \(p\) and natural number \(m\), the sequence \(\left(a_k\,\mathrm{mod} \,p^m\right)_{k\in\mathbb N}\) is a \(p\)-*automatic* sequence: it is generated by a finite state machine. Another way to say this (using Cobham's theorem) is that the sequence \(\left(a_k\,\mathrm{mod}\,p^m\right)_{k\in\mathbb N}\) is the letter to letter projection of a fixed point of a constant length \(p\) *substitution*. We apply this result to show that, for any such algebraic sequence \(\left(a_k\right)\), and any \(p\) and \(m\), there is a constructive procedure to compute this sequence modulo \(p^m\). We compute several examples, reproving, in a unified way, several disparate results in the combinatorics literature. We also compute new congruences, such as for the Apéry numbers, which are not algebraic, but which are “diagonals” of higher dimensional algebraic arrays. We also discuss how these algebraic sequences naturally lead to a definition of a constant length substitution and corresponding dynamical system, on infinitely many letters. This research is joint work with Eric Rowland, and the talk is accessible to a mathematical audience.

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Large equilibrium configurations of two-dimensional fluid vortices

Robert Buckingham

University of Cincinnatti

3:00pm-4:00pm

CMC 130

Seung-Yeop Lee

**Abstract**

The point-vortex equations, a discretization of the Euler equations, describe the motion of collections of two-dimensional fluid vortices. The poles and zeros of rational solutions to the Painlevé II equation describe equilibrium configurations of vortices of the same strength and mixed rotation directions. There is an infinite sequence of such rational solutions with an increasing number of poles and zeros. In joint work with P. Miller (Michigan), we compute detailed asymptotic behavior of these rational functions with error estimates. Our results include the limiting density of vortices for these configurations. We will also describe how these rational functions, as well as other Painlevé-type functions, arise in the study of critical phenomena in the solution of nonlinear wave equations.