Analysis and Applications

In this paper, we present several results concerning vector potentials and scalar potentials with data in Sobolev spaces with negative exponents, in a not necessarily simply-connected, three-dimensional domain. We then apply these results to Poincaré's theorem and to Korn's inequality.

This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on two aspects: vector valued feature maps and universal kernels. In particular, we characterize the structure of translation invariant kernels on abelian groups and we relate it to the universality problem.

It is well known that the scattered field of a z polarized plane wave incident on a dielectric circular cylinder can be expanded as an infinite series involving Hankel functions. From numerical calculations of this expansion, Lam and Yedlin [5] observed that the mean square measure, over all space, of the difference of the scattered fields from two or more distinct values of the dielectric constant of the cylinder can take very small values, thereby almost contradicting the uniqueness property. We investigate this phenomenon rigorously using uniform asymptotic expansions of Bessel functions, and from our analysis we determine the spurious values of the dielectric constant which lead to this quasi-nonuniqueness.

A new systematic approach to holonomic constraint is presented. By holonomic we mean a constraint problem in the calculus of variations/optimal control theory where the constraints are independent of the derivative of the dependent variable. It is seen that these new methods follow from a general theory of constraint optimization previously given by the author. A major, new emphasis of this work is the necessity of properly handling the boundary values of our introduced variables.

The author's previous theory allows the solution of a wide variety of general or anholonomic problems which include those identified with the areas of control, delay, stochastic, and partial differential equations in the sense that simpler constraint problems in these areas can be solved exactly by analytical means while more complex problems can be solved by efficient numerical algorithms with an a priori maximal error of O(h²), where h is the node size. Thus, our methods in this paper are immediately applicable to holonomic problems in these other diverse areas.

While our new results do not involve a significant formal mathematical jump from the author's previous theory of constraint optimization, the treatment of boundary values is significant and of note. For this reason we will give several examples to illustrate these ideas. Finally, we illustrate how our methods can be used to solve a wide variety of meaningful holonomic constraint or generalized coordinate problems in the study of dynamics/classical mechanics, using the simple pendulum problem as an example.

A new class of sequences convergent to Euler's constant is investigated. Special choices of parameters show that the class includes the original sequence defined by Euler, as well as more recently defined sequences due to DeTemple [1] and Vernescu [9]. It is shown how the rate of convergence of the sequences can be improved by computing optimal values of the parameters.