| January 21 |
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| January 28 | Organizational Meeting |
| February 4 |
Hongguo Xu
University of Kansas Singular value-like decomposition for matrix triples and structured canonical forms Abstract It it will-known that a singular value decomposition (SVD) A=U&Sigma V* gives the spectral decompositions of the Hermitian positive semidefinite matrices AA* and A*A. We show that the SVD can be generalized with U and V in certain groups and &Sigma in certain upper triangular forms. The generalized SVDs give the spectral decompositions of algebraically structured matrices associated with A and A* (AT). This is a joint work with C. Mehl and V. Mehrmann. |
| February 11 |
Ford Ballantyne IV
Dept. of Ecology and Evolutionary Biology, Kansas Biological Survey, University of Kansas In search of keys while picking locks; using mathematics to ask ecological questions Abstract I will provide a brief overview of a few patterns in ecology and how I've attempted to understand them using differential equations and probability. I will also try to illustrate some additional questions ripe for addressing and potentially low hanging, for more skilled mathematicians. |
| February 18 |
Erik Van Vleck
University of Kansas Exponential Dichotomy, Matrix Decompositions, and Newton's Method Abstract In this talk we develop new methods for determining exponential dichotomies of linear time varying differential equations. The techniques are based upon continuous matrix factorizations (SVD and QR) and rely upon having integral separation or more generally stable Lyapunov exponents. A perturbation theory is developed by formulating a zero finding problem and applying a version of the classical Newton-Kantorovich Theorem. |
| February 25 |
Milena Stanislavova
University of Kansas Linear and nonlinear (in)stability of coherent structures Abstract In this talk we will be interested in the stability properties of special solutions of a fairly general class of reaction-diffusion system. More precisely, these will feature linearly unstable steady states, which are nevertheless physically interesting objects. We will consider a class of parabolic equations with (nonlinearly) unstable steady states. We will construct in a fairly explicit way a stable manifold of finite co-dimension. In particular, we will provide a precise estimate of the speed of the convergence of the solutions to the manifold. |
| March 4 |
Anna Ghazaryan
University of Kansas Nonlinear convective stability of traveling fronts near Turing and Hopf instabilities Abstract We analyze the instability of a front in a reaction- di?usion system when the instability is caused by the rest state behind the front undergoing a supercritical Turing or Hopf bifurcation. On the linear level there exists an exponentially weighted norm that stabilizes the front, i.e., the instability of the front is convective. It will be very restrictive to assume that the nonlinearity will be well behaved in that particular norm. For example, this is not true for polynomial nonlinearities. Therefore the nonlinear stability cannot be simply inferred from the linear stability in the weighted norm. In a joint work with M. Beck and B. Sandstede, we show that the amplitude of any emerging pattern can be controlled in terms of the bifurcation parameter, and then, using the interplay of norms with and without weight, we prove that, in the coordinate frame that moves together with the front, the pattern is pushed away from the interface of the front. The result implies the convective character of the instability on the nonlinear level. This is a joint work with Bjorn Sandstede and Margaret Beck |
| March 11 |
Weishi Liu
University of Kansas A maximal sub-to-super transonic wave of gas flow through a contracting-expanding nozzle and its linear stability. Abstract We will discuss the standing asymptotic states of an isentropic viscous compressible gas flow through a contracting-expanding nozzle. Particularly interesting is the existence of a maximal sub-to-super transonic wave and its role in the formation of other transonic waves consisting of a sub-to-super portion. The linear stability of this maximal sub-to-super transonic wave will also be discussed. |
| March 18 | Spring Break |
| March 25 |
Weizhang Huang
University of Kansas Understanding anisotropic mesh adaptation from the perspective of uniform meshes in a metric space: Theory and applications Abstract Anisotropic mesh adaptation has proven to be a useful tool for enhancing accuracy and efficacy in the numerical solution of partial differential equations, especially those exhibiting anisotropic features in their solutions and/or structures. On the other hand, the mathematical characterization of anisotropic meshes has not been well understood. In this talk I will present an approach with which anisotropic meshes are viewed as uniform ones in some metric space. An advantage of this approach is that the simple characterization of uniform meshes will lead to a clear mathematical characterization of adaptive, often non-uniform meshes. As a result, two conditions, the well-known equidistribution condition and a less-known alignment one, are shown to be able to characterize the size and the shape and orientation of elements of anisotropic meshes, respectively. Applications of the characterization to anisotropic mesh generation and error analysis in finite element computation will be discussed. Numerical examples and applications will be given. |
| April 1 |
Not scheduled
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| April 8 |
Mohamed Badawy
University of Kansas Lyapunov exponents: a survey of recent results and applications Abstract Abstract: Lyapunov exponents provide us with a very useful tool in the study of the stability of dynamical systems. The theory of Lyapunov exponents for finite dimensional dynamical systems is quite established. Several algorithms have been developed to compute these exponents along with their error analysis results. In the infinite dimensional case, however, there is still a lot of work that needs to be done. In this talk, we will give a brief survey of the major results and algorithms for the computation of these exponents in the finite dimensional case, and some applications. We will also talk about the possibility of generalizing the error analysis results, using the so called QR-method, in the case of infinite dimensional dynamical systems over a Hilbert space. |
| April 15 |
Jyoti Saraswat
University of Kansas A study of Vandermonde Matrix Systems Abstract Vandermonde matrices usually arise when considering systems of polynomials evaluated at specific points. These systems find use in many approximation and interpolation problems since solving the system of equations Vx=b for x with V as a mxn Vandermonde matrix implies finding coefficients a j of the polynomial P(x)=aj xj. Vandermonde matrix system are generally ill-conditioned systems. The talk focuses on results and algorithms for fast LU, QR factorization and inverse of Vandermonde matrices. The talk will also introduce Vandermonde like and confluent Vandermonde matrices and also Vandermonde matrices as Krylov matrices. |
| April 22 456 Snow |
Tim Dorn
University of Kansas On The Steady State Of A Liquid-Crystal Model For Friction Abstract A liquid-crystal is a phase of material between the solid and liquid phases, and is used often to model frictional phenomenon. In this talk we will consider one such model and present results on the steady state(s). |
| April 29 |
Myunghyun Oh
University of Kansas Instability of low density supersonic waves of a viscous isentropic gas flow through a nozzle Abstract In this work, we examine the stability of stationary non-transonic waves for viscous isentropic compressible flows through a nozzle with varying cross-section areas. The main result in this paper is, for small viscous strength, stationary supersonic waves with sufficiently low density are spectrally unstable; more precisely, we will establish the existence of positive eigenvalues for the linearization along such waves. The result is achieved via a center manifold reduction of the eigenvalue problem. The reduced eigenvalue problem is then studied in the framework of the Sturm-Liouville Theory. |
| May 6 |
Andrzej Swiech
Georgia Tech Maximum principles and weak Harnack inequality for fully nonlinear elliptic equations Abstract Aleksandrov-Bakelman-Pucci (ABP) maximum principle and Harnack inequality are classical tools in the theory of elliptic PDE and in the last two decades these topics have been revisited for fully nonlinear uniformly elliptic equations from the point of view of viscosity solutions. We will discuss old and new results and show when the ABP maximum principle is true for equations with superlinear growth in the gradient (in which case it has been known to fail in general). Moreover we will present recent improvements in weak Harnack inequality for fully nonlinear equations. As a consequence we will show how these techniques allow to obtain new results about solvability of nonlinear PDE, in particular of Pucci extremal equations. |
(Link to CAM seminar talks in Fall, 2008)
(Link to CAM seminar talks in Spring, 2008)
(Link to CAM seminar talks in Fall, 2007)
(Link to CAM seminar talks in Spring, 2007)
(Link to CAM seminar talks in Fall, 2006)
(Link to CAM seminar talks in other years.)
(Link to Numerical Analysis Group Webpage)