Physical Sciences and Mathematics Commons^™

Error Bounds For The Lanczos Methods For Approximating Matrix Exponentials, Qiang Ye

Mathematics Faculty Publications

In this paper, we present new error bounds for the Lanczos method and the shift-and-invert Lanczos method for computing e−^τAv for a large sparse symmetric positive semidefinite matrix A. Compared with the existing error analysis for these methods, our bounds relate the convergence to the condition numbers of the matrix that generates the Krylov subspace. In particular, we show that the Lanczos method will converge rapidly if the matrix A is well-conditioned, regardless of what the norm of τA is. Numerical examples are given to demonstrate the theoretical bounds.

Toric Varieties And Cobordism, Andrew Wilfong

Theses and Dissertations--Mathematics

A long-standing problem in cobordism theory has been to find convenient manifolds to represent cobordism classes. For example, in the late 1950's, Hirzebruch asked which complex cobordism classes can be represented by smooth connected algebraic varieties. This question is still open. Progress can be made on this and related problems by studying certain convenient connected algebraic varieties, namely smooth projective toric varieties. The primary focus of this dissertation is to determine which complex cobordism classes can be represented by smooth projective toric varieties. A complete answer is given up to dimension six, and a partial answer is described in dimension …

On A Paley-Wiener Theorem For The Zs-Akns Scattering Transform, Ryan D. Walker

Theses and Dissertations--Mathematics

In this thesis, we establish an analog of the Paley-Wiener Theorem for the ZS-AKNS scattering transform on a set of real potentials. We also demonstrate one application of our techniques to the study of an inverse spectral problem for a half-line Miura potential Schroedinger equation.

Regularity And Uniqueness Of Some Geometric Heat Flows And It's Applications, Tao Huang

Theses and Dissertations--Mathematics

This manuscript demonstrates the regularity and uniqueness of some geometric heat flows with critical nonlinearity.

First, under the assumption of smallness of renormalized energy, several issues of the regularity and uniqueness of heat flow of harmonic maps into a unit sphere or a compact Riemannian homogeneous manifold without boundary are established.

For a class of heat flow of harmonic maps to any compact Riemannian manifold without boundary, satisfying the Serrin's condition,

the regularity and uniqueness is also established.

As an application, the hydrodynamic flow of nematic liquid crystals in Serrin's class is proved to be regular and unique.

The natural …

Relative Perturbation Theory For Diagonally Dominant Matrices, Megan Dailey

Theses and Dissertations--Mathematics

Diagonally dominant matrices arise in many applications. In this work, we exploit the structure of diagonally dominant matrices to provide sharp entrywise relative perturbation bounds. We first generalize the results of Dopico and Koev to provide relative perturbation bounds for the LDU factorization with a well conditioned L factor. We then establish relative perturbation bounds for the inverse that are entrywise and independent of the condition number. This allows us to also present relative perturbation bounds for the linear system Ax=b that are independent of the condition number. Lastly, we continue the work of Ye to provide relative perturbation bounds …