Many problems associated with the engineering technology field can be transformed into Fredholm integral equations of first kind to achieve problem-solving strategies. In this paper, iterative kernel technique was reformulated treat numerical solution for system degenerate kernel. Three new theorems have been proposed and proved. This programmed via Matlab achieved a good result.

In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarskỳ et al. [ESA 2013] proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by...

Closed-form algebraic formulae are derived for the local error incurred in the numerical solution of integral equations by iterated collocation methods, the analysis being illustrated by application to Fredholm integral equations of the second kind. The novel error analysis uses an asymptotic approach, in the small parameter of the numerical mesh size, applied to a finite-rank degenerate-kernel...

Approximation of solutions of integral equations by networks with kernel units is investigated theoretically. There are derived upper bounds on speed of decrease of errors in approximation of solutions of Fredholm integral equations by kernel networks with increasing numbers of units. The estimates are obtained for Gaussian and degenerate kernels.

A 3D stochastic Navier-Stokes equation with a suitable non degenerate additive noise is considered. The regularity in the initial conditions of every Markov transition kernel associated to the equation is studied by a simple direct approach. A by-product of the technique is the equivalence of all transition probabilities associated to every Markov transition kernel.

This thesis investigates some unaddressed problems in kernel nonparametrichypothesis testing. The contributions are grouped around three main themes:Wild Bootstrap for Degenerate Kernel Tests. A wild bootstrap method for non-parametric hypothesis tests based on kernel distribution embeddings is pro-posed. This bootstrap method is used to construct provably consistent teststh...