Mathematics Department Colloquium (2007-2008)

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.

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.

Speaker: Shahla Nasserasr, College of William and Mary,  

Title: Primitive Digraphs with Smallest Large Exponent

Abstract.

November 16, Friday,  3pm in  Jones 131 

Speaker: Evelyn Sander, George Mason University,  

This is actually a CSUMS Lecture.

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

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.