Brownian motion with drift on spaces with varying dimension

Many properties of Brownian motion on spaces with varying dimension (BMVD in abbreviation) have been explored in Chen and Lou (2018). In this paper, we study Brownian motion with drift on spaces with varying dimension (BMVD with drift in abbreviation). Such a process can be conveniently defined by a regular Dirichlet form that is not necessarily symmetric. Through the method of Duhamel’s principle, it is established in this paper that the transition density of BMVD with drift has the same type of two-sided Gaussian bounds as that for BMVD (without drift). As a corollary, we derive Green function estimate for BMVD with drift.     

Schatten properties,nuclearity and minimality of phase shift invariant spaces

We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces.We use these results to prove that the pseudo-differential operator $\mathrm{Op}(a)$ is a Schatten-q operator from $M^{\infty }$ to $M^p$ and r-nuclear operator from $M^{\infty }$ to $M^r$ when $a\in M^r$ for suitable p, q and r in $(0,\infty ]$.     

A variational approach for a generalized emden-fowler equation

Existence of positive solutions of singular boundary value problems related to Emden-Fowler equation is proved. A general minimization theorem in Sobolev spaces is applied.     

Asymptotically Constant Solutions of Functional Difference Systems

We consider the functional difference system ( A ) j x i ( n )= f i ( n ; X ), 1 h i h k , where X =( x 1 ,…, x k ) and f 1 (·; X ),…, f k (·; X ) are real-valued functionals of X , which may depend quite arbitrarily on values of X ( l ) for multiple values of l ] Z . We give sufficient conditions for ( A ) to have solutions that approach specified constant vectors as n M X . Some of the results guarantee only that the solutions are defined for n sufficiently large, while others are global. The proof of the main theorem is based on the Schauder-Tychonoff theorem. Applications to specific quasi-linear systems are included.     

Infinitely many singular interactions on noncompact manifolds

We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional non-compact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan–Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et al. (2004).     

D-admissible hybrid control of a class of singular systems

This paper mainly studies the problem of designing a hybrid state feedback D-admissible controller for a class of linear and nonlinear singular systems. Based on the relationship between singular discrete systems and singular delta operator systems, several necessary and sufficient conditions for a linear singular delta operator system to be D-admissible (i.e. regular, causal and all finite poles lie in a prescribed circular region) with different representations are derived. Then, the existence conditions and explicit expressions of a desirable D-admissible controller are given by means of matrix inequalities and strict linear matrix inequalities, respectively. We further extend the obtained results to singular delta operator systems with Lipschitz nonlinear perturbations, and the design methods of hybrid controller are presented for the nonlinear case as well. Finally, numerical examples as well as simulations are provided to illustrate the effectiveness of the theoretical outcomes obtained in the paper.     

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral <Emphasis Type="Italic">su</Emphasis>(2) WZNW Model

A zero modes’ Fock space is constructed for the extended chiral WZNW model. It gives room to a realization of the fusion ring of representations of the restricted quantum universal enveloping algebra at an even root of unity, and of its infinite dimensional extension by the Lusztig operators We provide a streamlined derivation of the characteristic equation for the Casimir invariant from the defining relations of A central result is the characterization of the Grothendieck ring of both and in Theorem 3.1. The properties of the fusion ring in are related to the braiding properties of correlation functions of primary fields of the conformal current algebra model.