Deborah Tepper Haimo

EXPERIMENTATION AND CONJECTURE ARE NOT ENOUGH

This is an exciting time in mathematics. Its various special areas are coming together to emphasize the disciplines unity. In addition, there is general growing recognition that good teaching cannot be separated from research, and that we must be more successful in communicating the tenets of our field to the broader community. In underscoring the beauty of mathematics as well as its relevance, it is of utmost importance that we educate those naturally gifted and interested, as well as others who may have latent ability which should not be ignored, and the many who need to learn at least the basic concepts and to gain some appreciation and understanding of the fields vitality. It is heartening to see thatwwe are beginning to acknowledge our responsibility to become involved in raising the understanding of the mathematical knowledge of all our citizenry. We need

The dual Weierstrass-Laguerre transform

An inversion algorithm is derived for the dual Weierstrass-Laguerre transform 1O g<,(x, y; 1)9( V)yae ¢/(a + l)dy, where the function g<x(x, y. t) is associated with the source solution of the Laguerre differential heat equation xuv Y ( x, t) = (a + 1 x) u ( x, t) = u, (x, t). Correspondingly, sufficient conditions are established for a function to be represented by a Weierstrass-Laguerre Stieltjes transform 1O ga!(x, y; 1) d,8(y) of a nondecreasing function ,B.

Inversion of an integral transform related to a general form of heat equation

Inversion algorithms have been derived by D. V. Widder for the Weierstrass transform and by D. T. Haimo for the dual Weierstrass-Laguerre transform. For the genera1 form of heat equation - L,u = u,, L, a self adjoint operator, an integral transform is introduced with kernel related to the fundamental solution of the equation. An inversion formula for the transform is derived which includes the preViOUS reSUkS as special cases. Cf 1988 Academic Press, Inc.

Inversion of integral transforms associated with a class of perturbed heat equations

Abstract Let Lx be the Sturm-Liouville differential operator L x = −d 2 dx 2 + q(x); x ϵ (0, ∞) . We assume that Lx has either a purely discrete spectrum that is bounded from below by zero or a continuous spectrum that fills up the interval (0, ∞) with, possibly, a finite number of negative eigenvalues. The W-transform ψ of ψ(y) ϵ L2(0, ∞) is defined by ϵ (x) = ∫ 0 ∞ ϵ(y) g(x,y;1)dy , where g(x, y; t) is a function associated with the fundamental solution of the perturbed equation −L x u(x, t) = ∂u(x, t) ∂t . The main purpose of this paper is to derive an inversion formula for the W-transform. This inversion formula generalizes the known inversion formulae for the Weierstrass, Weierstrass-Hankel convolution, and Weierstrass-Laguerre transforms. The results of this paper are easily extended to the case where Lx is considered over the entire line (−∞, ∞).

Representations for Laguerre temperatures

Abstract The Laguerre difference heat equation ∇ n u(n,t)= ∂ ∂ u(n,t), where ∇ n ƒ(n)=(n+1)−(2n+a+1)ƒ(n)+(n+a)ƒ(n−1)a⩾0, provides an important discrete analogue of the classical heat equation ∂2u⧸∂x2=∂t. Conditions are established for solutions of this equation to be represented by integral and series transforms.