This work is concerned with the construction and analysis of high order product integration methods for a class of Volterra integral equations with logarithmic singular kernel. Sufficient conditions for the methods to be convergent are derived and it is shown that optimal convergence orders are attained if the exact solution is sufficiently smooth. The case of non-smooth solutions is dealt with...

For surfaces of revolution B in R, we investigate the limit distribution of minimum energy point masses on B that interact according to the logarithmic potential log(1/r), where r is the Euclidean distance between points. We show that such limit distributions are supported only on the “out-most” portion of the surface (e.g., for a torus, only on that portion of the surface with positive curvatu...

In a recent paper [3], Y. Cao and Y. Xu established the Galerkin method for weakly singular Fredholm integral equations that preserves the singularity of the solution. Their Galerkin method provides a numerical solution that is a linear combination of a certain class of basis functions which includes elements that reflect the singularity of the solution. The purpose of this paper is to extend t...

We begin by studying certain semigroup estimates which are more singular than those implied by a Sobolev embedding theorem but which are equivalent to certain logarithmic Sobolev inequalities. We then give a method for proving that such log–Sobolev inequalities hold for Euclidean regions which satisfy a particular Hardy–type inequality. Our main application is to show that domains which have ex...

In this paper, we consider convex quadratic semidefinite optimization problems and provide a primal-dual Interior Point Method (IPM) based on a new kernel function with a trigonometric barrier term. Iteration complexity of the algorithm is analyzed using some easy to check and mild conditions. Although our proposed kernel function is neither a Self-Regular (SR) fun...

In the present paper, we introduce some singular integral operators, singular quadrature operators and discretization matrices of singular integral equations with Hilbert kernel. These results both improve the classical theory of singular integral equations and develop the theory of singular quadrature with Hilbert kernel. Then by using them a unified framework for various collocation methods o...

Function values are, in some sense, “almost as good” general linear information for L2-approximation (optimal recovery, data assimilation) of functions from a reproducing kernel Hilbert space. This was recently proved by new upper bounds on the sampling numbers under assumption that singular embedding this space into L2 are square-summable. Here we mainly prove lower bounds. In particular behav...

The main results of the article are short time estimates and asymptotic for first two order derivatives logarithmic heat kernel a complete Riemannian manifold. We remove all curvature restrictions also develop several techniques. A basic tool developed here is intrinsic stochastic variations with prescribed second covariant differentials, allowing to obtain path integration representation semig...