Definition 1.1. A Gaussian process {Xt }t∈T indexed by a set T is a family of (real-valued) random variables Xt , all defined on the same probability space, such that for any finite subset F ⊂ T the random vector XF := {Xt }t∈F has a (possibly degenerate) Gaussian distribution; if these finitedimensional distributions are all non-degenerate then the Gaussian process is said to be nondegenerate....

In this paper we focus on the validity of some fundamental estimates for time-degenerate Schr\"{o}dinger-type operators. On one hand derive global homogeneous smoothing operators any order by means suitable comparison principles (that shall obtain here). other hand, prove weighted Strichartz-type Scrh\"{o}dinger and apply them to local well-posedness semilinear Cauchy problem. Most our results ...

A real polynomial system with support W ⊂ Zn is called extremal if all its complex solutions are positive solutions. A support W having n + 2 elements is called a circuit. We previously showed that the number of nondegenerate positive solutions of a system supported on a circuit W ⊂ Zn is at most m(W)+1, where m(W) ≤ n is the degeneracy index of W. We prove that if a circuit W ⊂ Zn supports an ...

In this paper, we are concerned with the study of mathematical analysis for an optimal control a nonlocal degenerate aggregation model. This model describes organisms such as pedestrian movements, chemotaxis, animal swarming. We establish wellposedness (existence and uniqueness) weak solution direct problem by means auxiliary nondegenerate equation, Faedo–Galerkin method (for existence result) ...

Abstract The current article is concerned with the traveling fronts for a class of double degenerate equations delay. We first show that decay algebraically at one end, while those may exponentially or other which depend on wave speed fronts. Based asymptotical behavior, uniqueness and stability are then proved. Of particular interest effect lower order term higher critical speed. mention that,...

The compartmentalization of distinct templates in protocells and the exchange of templates between them (migration) are key elements of a modern scenario for prebiotic evolution. Here we use the diffusion approximation of population genetics to study analytically the steady-state properties of such a prebiotic scenario. The coexistence of distinct template types inside a protocell is achieved b...

We present nonlinear optical four-wave mixing in a silicon nanodisk dimer metasurface. Under the oblique incident plane waves, designed metasurface exhibits multiresonant feature with simultaneous excitations of three quasi-bound states continuum (BIC). Through employing these quasi-BICs maximizing electric field energy at input bump wavelengths, significant enhancement third-order processes in...

Four-wave mixing (FWM) in silicon waveguides is considered to be a promising effect to realize the wavelength conversion function for wavelength-division-multiplexing optical communication systems. Compared to the degenerate FWM with a single pump, the nondegenerate FWM with two pumps shows more flexibility in phase-matching condition and has more opportunities to acquire broader conversion ban...