Archive of Colloquiums 2007 - 2008
Talks in Spring 2008
April 25, Friday, 1pm in Jones 131
Speaker: Zlatko Drmac, University of Zagreb
Title: Subspace gap residuals for Rayleigh--Ritz approximations
Abstract: Large scale eigenvalue and singular value computations are usually based on extracting information from a compression of the matrix to suitably chosen low dimensional subspaces. This paper introduces new a posteriori relative error bounds based on a residual expressed using the largest principal angle (gap) between relevant subspaces. The eigenvector approximations are estimated using subspace gaps and relative separation of the eigenvalues.
April 21, Monday, 3pm in Jones 131
Speaker: David Shoikhet, ORT College Braude, Israel,
Title: Old and New in Complex Dynamics
Abstract: Historically, complex dynamics and geometrical function theory have been intensively developed from the beginning of the twentieth century. They provide the foundations for broad areas of mathematics. In the last …fifty years the theory of holomorphic mappings on complex spaces has been studied by many mathematicians with many applications to nonlinear analysis, functional analysis, di¤erential equations, classical and quantum mechanics. The laws of dynamics are usually presented as equations of motion which are written in the abstract form of a dynamical system: dx/dt + f(x) = 0, where x is a variable describing the state of the system under study, and f is a vector-function of x. The study of such systems when f is a monotone or an accretive (generally non-linear) operator on the underlying space has been recently the subject of much research by analysts working on quite a variety of interesting topics, including boundary value problems, integral equations and evolution problems .
There is a long history associated with the problem on iterating holomorphic mappings and their …xed points, the work of G. Julia, J. Wol¤ and C. Carathéodory being among the most important.
In this talk we give a brief description of the classical statements which combine celebrated Julia’s Theorem in 1920 , Carathéodory’s contribution in 1929 and Wol¤’s boundary version of the Schwarz Lemma in 1926 and their modern interpretations. Also we present some applications of complex dynamical systems to geometry of domains in complex spaces and operator theory.
April 7, Monday, 3pm in Jones 131
Speaker: Rika Hagihara College of William and Mary,
Title: Complex Dynamics Lacking Period 2 Orbits
Abstract: Most polynomials and rational functions can be easily solved for periodic points of all periods. Periodic points play an important role in the theory of dynamical systems. In this talk we will study quadratic rational maps that are missing period 2 orbits. We will introduce the parameter space of such maps, and investigate how different kinds of dynamics are reflected in it. We will also compare the parameter space of quadratic rational maps lacking period 2 orbits with the Mandelbrot set, the parameter space for all quadratic polynomials, to see the similarities and differences.
Talks in Fall 2007
November 30, Friday, 3pm in Jones 131
Speaker: Shahla Nasserasr, College of William and Mary,
Title: Primitive Digraphs with Smallest Large Exponent
Abstract: A digraph D is primitive if there exists an integer m such that any two not necessarily distinct vertices u and v are joined by a directed path of length m. The smallest such m is called the exponent of D and denoted by °(D).
A primitive digraph D has large exponent if °(D) satisfices ®n = bwn=2c+2 · °(D) · wn, where wn = (n ¡ 1)2 + 1. It is shown that the minimum number of arcs in a primitive digraph D on n vertices with exponent equal to ®n is either n+1 or n+2. For given n ¸ 8, there always exists a primitive digraph on n vertices with exponent ®n and n+2 arcs. Such digraphs for the cases n even and n odd are di®erent. An algorithm determines for a given n, whether the minimum number of arcs in such digraphs is n + 1 or n + 2. This is joint work with G. MacGillivray, D. D. Olesky and P. van den Driessche.
November 16, Friday, 3pm in Jones 131
Speaker: Evelyn Sander, George Mason University,
Abstract:
In this talk, I will discuss results for pattern formation in reaction-diffusion models for chemical dynamics which often arise in the field of biology. In particular, a 1952 discovery of Turing showed that a near-equilibrium mixture of chemical reactants could have diffusion-driven pattern formation. I will describe the mathematical nature of this pattern formation, as well as showing direct numerical simulation results and numerical bifurcation results for reaction-diffusion equations.
4pm
Speaker: Zlatko Drmac, University of Zagreb
Title: Computing matrix spectral decompositions in finite precision arithmetic
Abstract: Many problems in numerical linear algebra can be considered completely solved from the purely algebraic (or analytic) point of view. However, in the real world applications, we have somewhat different picture. The initial matrices are usually given with uncertainties (as measurement errors, errors from previous computation) and further computation of a matrix function (such as e.g. the rank, the inverse, or the eigenvalues) is carried out over a finite number of rationals (machine numbers) using a finite precision machine arithmetic (in which adding three numbers accurately represents a nontrivial challenge). We will address these issues in numerical computation of the spectral and the SVD decompositions, and present some results of the recent exciting development, with contributions of several researches (Barlow, Demmel, Drmac, Gu, Koev, Eisenstat, Veselic). Our approach will follow the following paradigm:
i) Use perturbation theory to describe classes of matrices together with classes of admissible perturbations for which computation with high relative accuracy is possible. This, in some cases, means changing the usual matrix representation.
ii) Develop an algorithm capable of achieving the theoretical accuracy. This may require different approach for each separate class of matrices. However, some unifying principles are developed. We show that a new variant of the Jacobi method can be used as a core routine for all cases.
iii)Implement the algorithms in reliable mathematical software and prove that the implementation has the required accuracy properties. This can be tedious and not satisfactory because it involves technical details, hardware and compiler issues. As an example how tricky this can be, we show how a subtle numerical bug (dating back to LINPACK, 1971.) survived 36 years in all numerical libraries before being detected.
November 9, Friday, 3pm in Jones 131
Speaker: Jianjun Paul Tian, College of William and Mary
Title: Spin representations of Artin's braid group
Abstract: Motivated by distinguishing two opposite orientations of a knot, we construct new linear representations of Artin's braid group, spin representations and multi-parameter Burau representations, by smoothing a representation variety. We will work on unitary matrices SU(2,C) for explicit computation purpose. The talk is based on the following paper.
November 2, Friday, 3pm in Jones 131
Speaker: Roberto Costas, College of William and Mary
Title: Extensions of discrete classical orthogonal polynomials beyond the orthogonality
Abstract: It is well known that the family of Hahn polynomials (Hn(x;N)) is orthogonal with respect to a certain weight function up to N. In this talk we present a factorization for Hahn polynomials
Abstract: for a degree higher than N and we prove that these polynomials can be characterized by a discrete Sobolev orthogonality.
October 26, Friday, 3pm in Jones 131
Speaker: Carsten Collon, TU-Dresden
Title: Invariant tracking control of the kinematic car
Abstract: Influencing the behavior of a dynamical system by determing inputs from system outputs via a feedback law is an important subject in control theory. This talk considers an invariant parameterization of control laws w.r.t. Lie transformation groups. The well-known example of the kinematic car serves as motivation for reviewing the choice of coordinates used to design the control law. The example is followed by a presentation of a "normalization procedure" as an approach to compute a complete set of invariants for the action of a given Lie transformation group. These invariants can be used to obtain an invariant parameterization of the control law.
October 19, Friday, 3pm in Jones 131
Speaker: Bill Kalies, Florida Atlantic University
Title: Computational dynamics from a topological point of view
Abstract: In this talk, we will review some recently developed techniques for computing global dynamics, including changes with respect to parameters, and some applications. The talk will focus on computing recurrence in the iteration of a continuous map by reducing the dynamics to a finite directed graph. Then we give results that allow the information from the reduced system to be lifted back to information about the original map. These methods can identify regions of recurrent an non-recurrent behavior and also provide rigorous computer-assisted proofs of the existence of various types of dynamical structures such as periodic points, connecting orbits, and chaotic behavior.
October 5, Friday, 3pm in Jones 131
Speaker: J. Brown, College of William and Mary
Title: Pseudotrees, Suslin Trees, and Cardinal Functions
Abstract: A tree is a partially ordered set (T;·) such that for every t 2 T, the set T # t = fr 2 T : r · tg is well-ordered. A pseudo-tree is a generalization of a tree: a partially ordered set (T;·) such that the sets T # t = fr 2 T : r · tg are only required to be linearly ordered. Trees and pseudotrees are both a source of interesting examples and a useful tool in set theory and Boolean algebra, and they can have some surprising features. For example, consistently there exist Suslin trees { uncountable trees having no uncountable branch and no uncountable antichain. In this talk we will provide connections between the \allowable" structure, or shape, of pseudotrees and the existence of Suslin trees via a Boolean-algebraic cardinal function.
September 28, Friday, 3pm in Jones 131
Speaker: C.-K. Li, College of William and Mary
Title: Numerical ranges and dilations of operators
Abstract: The numerical range of an operator A acting on a Hilbert space H is the set
W(A) = {(Ax,x): x in H, (x,x) = 1}.
We say that D is a dilation of A if A can be viewed as a compression of the operator D. In this talk, we will discuss how one can use the numerical range of A to find a dilation of A with simple structure. Extension of the result to the joint numerical range and higher rank numerical range will also be mentioned.
September 21, Friday, 3pm in Jones 131
Speaker: Roberto S. Costas Santos, College of William and Mary
Title: Classical orthogonal polynomial. A general difference calculus approach
Abstract: It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this talk we present the essential part of the study of classical orthogonal polynomials in a more general context by using the differential (or difference) calculus and Operator Theory, and, in such a way, we obtain a unified representation of them.
September 14, Friday, 3pm in Jones 131
Speaker: Katarzyna Filipiak, Agricultural University of Poznan, Poland
Title: Connectedness and optimality of block deigns under an interference model
Abstract: We consider experiments in which interplot interference may occur. Our aim is to characterize connected and optimal designs under an interference model with neighbor effects. The conditions of connectedness and optimality of designs can be formulated using the properties of information matrices. The information matrix can be expressed as the Schur complement of some matrices and it has such properties as symmetry, nonnegativedefiniteness and zero row and column sums. We study such properties of information matrices as maximality of rank, complete symmetry and maximality of the trace. We are interested in determining designs in which:
- the sum of eigenvalues of the inverse of information matrix is minimal,
- the product of nonzero eigenvalues of the information matrix is maximal, and
- the minimal nonzero eigenvalue of information matrix is maximal.
