Let [Formula: see text] be a (unital) extension of (commutative) rings. We say that [Formula: see text] is a maximal non-quasi-local (respectively, non-PVD) subring of [Formula: see text] if [Formula: see text] is not quasi-local (respectively, PVD) and each subring of [Formula: see text] properly containing [Formula: see text] is quasi-local (respectively, PVD). The aim of this paper is to study this kind of ring extensions and to investigate the structure of the intermediate rings between [Formula: see text] and [Formula: see text].
This article investigates the approximate controllability of non-linear fractional stochastic differential inclusions with non-local conditions. We establish a set of sufficient conditions for their approximate controllability and provide results in terms of controllability for the fractional stochastic control system. Our approach relies on using fractional calculus and the fixed-point theorem for multiple-valued operators. Finally, we present an illustrative example to support our findings.
The focus of the research presented in this paper is on a new generalized family of degenerate three-variable Hermite–Appell polynomials defined here using a fractional derivative. The research was motivated by the investigations on the degenerate three-variable Hermite-based Appell polynomials introduced by R. Alyosuf. We show in the paper that, for certain values, the well-known degenerate Hermite–Appell polynomials, three-variable Hermite–Appell polynomials and Appell polynomials are seen as particular cases for this new family. As new results of the investigation, the operational rule for this new generalized family is introduced and the explicit summation formula is established. Furthermore, using the determinant formulation of the Appell polynomials, the determinant form for the new generalized family is obtained and the recurrence relations are also determined considering the generating expression of the polynomials contained in the new generalized family. Certain applications of the generalized three-variable Hermite–Appell polynomials are also presented showing the connection with the equivalent results for the degenerate Hermite–Bernoulli and Hermite–Euler polynomials with three variables.
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