: The present paper may be considered as a sequel to our previous paper in the Proceedings of the Cambridge Philosophical Society, Theorie generale de noyaux reproduisants-Premiere partie (vol 39 (1944)) which was written in 1942-1943 In the introduction to this paper we outlined the plan of papers which were to follow In the meantime, however, the general theory has been developed in many directions, and our original plans have had to be changed Due to wartime conditions we were not able, at the time of writing the first paper, to take into account all the earlier investigations which, although sometimes of quite a different character, were, nevertheless, related to our subject Our investigation is concerned with kernels of a special type which have been used under different names and in different ways in many domains of mathematical research We shall therefore begin our present paper with a short historical introduction in which we shall attempt to indicate the different manners in which these kernels have been used by various investigators, and to clarify the terminology We shall also discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter (KAR) P 2

/pdf/theory-of-reproducing-kernels-2rpdlfjhn3.pdf

Theory of Reproducing Kernels.

Extended Theories of Gravity

Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

Linear elasticity is one of the more successful theories of mathematical physics Its pragmatic success in describing the small deformations of many materials is uncontested The origins of the three-dimensional theory go back to the beginning of the 19th century and the derivation of the basic equations by Cauchy, Navier, and Poisson The theoretical development of the subject continued at a brisk pace until the early 20th century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff, Lame, Saint-Venant, Somigliana, Stokes, and others These authors established the basic theorems of the theory, namely compatibility, reciprocity, and uniqueness, and deduced important general solutions of the underlying field equations In the 20th century the emphasis shifted to the solution of boundary-value problems, and the theory itself remained relatively dormant until the middle of the century when new results appeared concerning, among other things, Saint-Venant’s principle, stress functions, variational principles, and uniqueness

The Linear Theory of Elasticity

A new improved energy-momentum tensor

Introduction to General Relativity

Kernel Functions and Elliptic Differential Equations in Mathematical Physics

Functionals of Finite Riemann Surfaces

: A rigorous theory is developed for variation of domain functions in a space of 3 dimensions as well as in the plane. The classical Hadamard variational formulas in space are discussed, and the so-called interior variational method is generalized to 3 dimensions. Interior variations of a 3- dimensional domain D are defined by means of differential mappings of D which depend on a small parameter Epsilon. The first-order shifts in terms of Epsilon of the Green's function, Neumann's function, and eigen-values, which result from this variation of D, are calculated rigorously by referring all varied quantities back to the original D through the infinitesimal mappings. Since proof is possible that the varied domain functions can be expanded in powers of Epsilon, the perturbation method is employed to calculate the second variations of these functions. Second variation expressions are obtained for the capacity, virtual mass, and eigenvalues corresponding to various particular ways in which D can be shifted. The variational theory is specialized to the case of 2 independent variables to show the existence of vortex sheets in axially symmetric, irrotational flow of an incompressible fluid. An external characterization of vortex sheets in 3-dimensional space without symmetry of any kind is sketched heuristically. In a study of the eigen functions and eigenvalues of the vibrating membrane, the second variation is used to show that under certain conformal mappings of a domain, which depend on a suitable parameter, the inverse square of the principal frequency of the domain becomes a convex function of the parameter.

Convexity of domain functionals

https://projecteuclid.org/download/pdf_1/euclid.acta/1485889463

Menahem Schiffer

Papers

Introduction to General Relativity

Kernel Functions and Elliptic Differential Equations in Mathematical Physics

Functionals of Finite Riemann Surfaces

Convexity of domain functionals

Half-order differentials on Riemann surfaces