Applied Mathematics Commons^™

Convolution And Fourier-Feynman Transforms, Chull Park, David Skough

Department of Mathematics: Faculty Publications

In this paper, for a class of funtionals on Wiener space of the form F(x) = exp{∫^T₀ f(t, x(t)) dt}, we show that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms. This allows us to compute the transform of the convolution product without computing the convolution product.

The Exponential Stability Of A Coupled Hyperbolic/Parabolic System Arising In Structural Acoustics, George Avalos

Department of Mathematics: Faculty Publications

We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE’s which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the interior of a bounded domain Ω, coupled to a “parabolic–like” beam equation holding on ∂Ω, and wherein the coupling is accomplished through velocity terms on the boundary. Our result is an analog of a recent result by Lasiecka and Triggiani which shows the exponential stability of the wave equation via Neumann feedback control, and like that work, depends upon a trace regularity estimate for solutions of hyperbolic …

Generalized Geynman Integrals: The 𝓛(L2, L2) Theory, Chull Park, David Skough

Department of Mathematics: Faculty Publications

In this paper we develop an 𝓛(L₂(R), L₂(R)) theory for the Feynman integral of functionals of general stochastic processes.

Perturbations Of Certain Reflexive Algebras, David R. Pitts

Department of Mathematics: Faculty Publications

In this note we use cohomological techniques to prove that if there is a linear map between two CSL algebras which is close to the identity, then the two CSL algebras are similar. We use our result to show that if 2' is a purely atomic, hyperreflexive CSL with uniform infinite multiplicity which satisfies the 4-cycle interpolation condition, then there are constants d, C > 0 such that whenever L is another CSL such that d(Alg2' , AlgL) < d, then there is an invertible operator S such that S Alg2'S-1 = AlgL and IISII liS-III < 1 + Cd(AIg2' , AIgL).

The Proportion Of Fixed-Point-Free Elements Of A Transitive Permutation Group, Nigel Boston, Walter Dabrowski, Tuval Foguel, Paul J. Gies, Judy Leavitt Walker, David T. Ose, David A. Jackson

Department of Mathematics: Faculty Publications

In 1990 Hendrik W. Lenstra, Jr. asked the following question: if G is a transitive permutation group of degree n and A is the set of elements of G that move every letter, then can one find a lower bound (in terms of n) for f(G) = |A|/|G|? Shortly thereafter, Arjeh Cohen showed that 1/n is such a bound.

Lenstra’s problem arose from his work on the number field sieve. A simple example of how f(G) arises in number theory is the following: if h is an irreducible polynomial …

Rewriteability In Finite Groups, Judy Leavitt Walker, G. J. Sherman, Mark E. Walker

Department of Mathematics: Faculty Publications

What's the probability that two elements in a finite group commute? A formal answer,

Pr₂(G) = {(x, y) [element of] G² |xy = yx}| / |G|²

begs our next question. How many ordered pairs of elements of a finite group commute?

A Note On The Connectedness Problem For Nest Algebras, David R. Pitts

Department of Mathematics: Faculty Publications

It has been conjectured that a certain operator T belonging to the group ff of invertible elements of the algebra Alg Z of doubly infinite uppertriangular bounded matrices lies outside the connected component of the identity in W. In this note we show that T actually lies inside the connected component of the identity of & .

Homoclinic Bifurcations With Nonhyperbolic Equilibria, Bo Deng

Department of Mathematics: Faculty Publications

A general geometric approach is given for bifurcation problems with homoclinic orbits to nonhyperbolic equilbrium points of ordinary differential equations. It consists of a special normal form called admissible variables, exponential expansion, strong A-lemma, and Lyapnunov- Schmidt reduction for the Poincare maps under Sil'nikov variables. The method is based on the Center Manifold Theory, the contraction mapping principle, and the Implicit Function Theorem.

An Illustrative Model Describing The Refraction Of Long Water Waves By A Circular Island, Gregory A. Kriegsmann

Department of Mathematics: Faculty Publications

The refraction of small shallow water waves by an idealized island is studied in this paper. The island's shoal is modeled by a quartic polynomial in the radial variable. This particular model allows the explicit construction of the rays (wave orthogonals) and the determination of several important features of the wave motion. The various shortcomings of the particular profile are discussed.