The unique solvability issues of the Cauchy problem with a fractional derivative is considered in case when coefficient at desired function continuous function. solution found an explicit form. uniqueness theorem proved. existence for to proved by reducing it Volterra equation second kind singularity kernel, and necessary sufficient conditions are obtained.

We connect shift-invariant characteristic kernels to infinitely divisible distributions on R. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures. The contribution of this paper is twofold. First, we show, using the Lévy–Khintchine formula, that any shift-invariant kernel given by a bounded, continuou...

This manuscript presents a method for the numerical solution of the Cauchy type singular integral equations of the first kind, over a finite segment which is bounded at the end points of the finite segment. The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density function. The force function is approximated by using the Chebys...

Let Ω = G/K be a bounded symmetric domain in a complex vector space V with the Lebesgue measure dm(z) and the Bergman reproducing kernel h(z, w). Let dμα(z) = h(z, z̄) dm(z), α > −1, be the weighted measure on Ω. The group G acts unitarily on the space L(Ω, μα) via change of variables together with a multiplier. We consider the discrete parts, also called the relative discrete series, in the irr...

We study entire solutions of the biharmonic heat equation on complete Riemannian manifolds without boundary. provide exponential decay estimates for kernel under assumptions lower bound Ricci curvature and noncollapsing unit balls. And we prove a uniqueness criteria Cauchy problem. As corollaries, conservation law uniform $$L^\infty $$ estimate starting with bounded initial data.

The crease set of an event horizon or a Cauchy horizon is an important object which determines qualitative properties of the horizon. In particular, it determines the possible topologies of the spatial sections of the horizon. By Fermat’s principle in geometric optics, we relate the crease set and the Maxwell set of a smooth function in the context of singularity theory. We thereby give a class...

This paper shows that least-square estimation (mean calculation) in a reproducing kernel Hilbert space (RKHS) F corresponds to different M-estimators the original depending on function associated with F. In particular, we present proof of correspondence mean an RKHS for Gaussian robust performed Welsch M-estimator. result is generalized other types M-estimators. generalization facilitates defin...