PREFACESince the pUblication of Atkinson's book "Multiparameter Eigenvalue Problems, Vol. I" in 1972, multiparameter spectral theory has become a subject of growing interest. Several authors have made important contributions to the theory; see the references at the end of this book. There are also two monographs of Sleeman (1978a) and McGhee and Picard (1988) It is not assumed that the reader knows already some multiparameter spectral theory but it is supposed that the reader is familiar with the usual one-parameter eigenvalue problems for compact Hermitian operators and their inverses. Brouwer's degree of maps will be used in the first two chapters. Basic properties of the tensor product of linear spaces will be needed in Chapters 4,5,6.

This paper concerns two-parameter Sturm-Liouville problems of the form -(p(x)yl) + q(x)y (kr(x) + lz)y, a < x < b with self-adjoint boundary conditions at a and b. The set of (),/) 6 R for which there exists a nontrivial y satisfying the differential equation and the boundary conditions turns out to be a countable union of graphs of analytic functions. Our focus is on these graphs, which are termed eigencurves in the literature.Although eigencurves have been used in a variety of ways for about a century, they seem comparatively underdeveloped in their own fight. Our plan is to give motivation for the topic, elementary properties of eigencurves, illustrations on a simple example first studied by Richardson in 1918 (and since then by several authors), and sone natural questions which may whet the reader's appetite. Some of these questions lead to new types of inverse Sturm-Liouville problems.

It is shown that spectral properties of Sturm-Liouville eigenvalue problems with indefinite weights are related to integral inequalities studied by Everitt. A result of Beals on indefinite problems leads to a sufficient condition for the validity of such an inequality. A Baire category argument is used to show that, in general, the inequality under consideration does not hold.

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It is shown that a Gibbs phenomenon occurs in the wavelet expansion of a function with a jump discontinuity at 0 for a wide class of wavelets. Additional results are provided on the asymptotic behavior of the Gibbs splines and on methods to remove the Gibbs phenomenon.

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In previous work the authors found the asymptotic expansion of the L 2 -norm of the solution u ( t , x ) of the strongly damped wave equation u t t − Δ u t − Δ u = 0 and also of the L 2 -norm of the difference between u ( t , x ) and its asymptotic approximation ν ( t , x ). This was done in space dimension N ⩾ 3. In the present work results are extended to the exceptional cases N = 1 and N = 2. This extension is achieved by deriving new lemmas on the asymptotic expansion of some parameter dependent integrals.

A nonlocal continuum electrostatic model, defined as integro-differential equations, can significantly improve the classic Poisson dielectric model, but is too costly to be applied to large protein simulations. To sharply reduce the model's complexity, a modified nonlocal continuum electrostatic model is presented in this paper for a protein immersed in water solvent, and then transformed equivalently as a system of partial differential equations. By using this new differential equation system, analytical solutions are derived for three different nonlocal ionic Born models, where a monoatomic ion is treated as a dielectric continuum ball with point charge either in the center or uniformly distributed on the surface of the ball. These solutions are analytically verified to satisfy the original integro-differential equations, thereby, validating the new differential equation system.