We present a method to regularize first and second kind integral equations of Fredholm type with singular kernel. By appropriate application of the Poincaré-Bertrand formula we change such integral equations into a second kind Fredholm’s integral equation with at most weakly singular kernel. 2000 Mathematics Subject Classification. 45Exx, 45E05, 45B05.

We define a discretized Langevin equation in Stratonovich-type calculus. We show that a generating functional with a field-dependent kernel can be written in mid-point prescription only when we calculate in the calculus. Moreover we investigate whether supersymmetry of the stochastic action with field-dependent kernel exists or not. – 1 –

In this paper, we investigate pure jump symmetric processes in R whose jumping kernel is comparable to the one for truncated rotationally symmetric α-stable process where jumps of size larger than a fixed number κ > 0 is removed. We establish sharp two-sided heat kernel estimate and derive parabolic Harnack principle for such jump-type processes. Moreover, we show that bounded functions that ar...

The kernel method has become a useful trick and has been widely applied to various learning models to extend their nonlinear approximation and classification capabilities. Such extensions have also recently occurred to the Self-Organising Map (SOM). In this paper, two recently proposed kernel SOMs are reviewed, together with their link to an energy function. The Self-Organising Mixture Network ...

There are two key issues for the kernel-based regularization method: one is how to design a suitable kernel to embed in the kernel the prior knowledge of the LTI system to be identified, and the other one is how to tune the kernel such that the resulting regularized impulse response estimator can achieve a good bias-variance tradeoff. In this paper, we focus on the issue of kernel design. Depen...

A universal approach by Laplace transform to the variational iteration method for fractional derivatives with the nonsingular kernel is presented; in particular, the Caputo-Fabrizio fractional derivative and the Atangana-Baleanu fractional derivative with the non-singular kernel is considered. The analysis elaborated for both non-singular kernel derivatives is shown the necessity of considering...

We present a novel machine translation framework based on kernel regression techniques. In our model, the translation task is viewed as a string-to-string mapping, for which a regression type learning is employed with both the source and the target sentences embedded into their kernel induced feature spaces. We report the experiments on a French-English translation task showing encouraging resu...

We address the estimation of “extreme” conditional quantiles i.e. when their order converges to one as the sample size increases. Conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed kernel estimators. A Weissman-type estimator and kernel estimators of the conditional tailindex are derived, permitting to estimate extreme conditio...

In this paper we present a method of studying convolution operator under the Sonin conditions imposed on kernel. The particular case kernel is fractional integral Riemman–Liouville operator, other various types kernels are Bessel-type function, functions with power-logarithmic singularities at origin e.t.c. We pay special attention to study close power type functions. main our aim Sonin–Abel eq...