Let Bc denote the real-valued functions continuous on the extended real line and vanishing at −∞. Let Br denote the functions that are left continuous, have a right limit at each point and vanish at −∞. Define Ac to be the space of tempered distributions that are the nth distributional derivative of a unique function in Bc. Similarly with A n r from Br. A type of integral is defined on distribu...

The problem of computing singular and hypersingular integrals involved in a large class of boundary value problems is considered. The method is based on Green’s theorem for calculating the diagonal elements of the resulting discretized matrix using the Nyström discretization method. The method is successfully applied to classical boundary value problems. Convergence of the method is also discus...

This is a complement to the comparison theorem for improper integrals in the textbook. The vanilla version presented in the textbook is good enough to solve some very easy examples and it becomes exponentially gory with the complexity of the integral. Fortunately it is not hard to refine the statement in the book, and turn it into a powerful tool to estimate the convergence of arbitrarily compl...

Back in 1955, I worked on extending the mean value theorem of Leifur Ásgeirsson [1937] for the ultrahyperbolic equation on Rn ×Rn to Riemannian homogeneous spaces G/K ×G/K. I was motivated by Godement’s generalization [1952] of the mean value theorem for Laplace’s equation Lu = 0 to the system Du = 0 for all G-invariant differential operators D (annihilating the constants) on G/K. At the time (...

Simple derivation of the dual conservation integrals in small strain elasticity is presented, without the aid of Noether’s theorem on invariant variational principles. The derived integrals are related to the release rates of the potential and complementary potential energy associated with the defect motion. The analysis corrects the errors in earlier derivation of the relationship between the ...