Strong regularity of smash products associated with G-set gradings

For a group G, a G-graded ring R and a finite left G-set A, we study the strong regularity of the smash product of R and A.

Strongly regular points of mappings

In this paper, we use a robust lower directional derivative and provide some sufficient conditions to ensure the strong regularity of a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound.

The number of representations of squares by integral quaternary quadratic forms

Let [Formula: see text] be a positive definite (non-classic) integral quaternary quadratic form. We say [Formula: see text] is strongly[Formula: see text]-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this paper, we show that there are exactly [Formula: see text] strongly [Formula: see text]-regular diagonal quaternary quadratic forms representing [Formula: see text] (see Table [Formula: see text]). In particular, we use eta-quotients to prove the strong [Formula: see text]-regularity of the quaternary quadratic form [Formula: see text], which is, in fact, of class number [Formula: see text] (see Lemma 4.5 and Proposition 4.6).

A New Symmetric Key Cryptographic Algorithm using Paley Graphs and ASCII Values

The main objective of the study conducted in this article is to introduce a new algorithm of encryption and decryption of a sensitive message after transforming it into a binary message. Our proposed encryption algorithm is based on the study of a particular graph constructed algebraically from the quadratic residues. We have exploited the Paley graph to introduce an abstract way of encryption of such message bit according to the other message bits by the intermidiate study of the neighborhood of a graph vertex. The strong regularity of the Paley graphs and the unknown behavior of the quadratic residues will play a very important role in the cryptanalysis part which allows to say that the brute force attack remains for the moment the only way to obtain the set of possible messages.

Morrey spaces and generalized Cheeger sets

We maximize the functional\frac{\int_{E}h(x)\,dx}{P(E)},where {E\subset\overline{\Omega}} is a set of finite perimeter, Ω is an open bounded set with Lipschitz boundary and h is nonnegative. Solutions to this problem are called generalized Cheeger sets in Ω. We show that the Morrey spaces {L^{1,\lambda}(\Omega)}, {\lambda\geq n-1}, are natural spaces to study this problem. We prove that if {h\in L^{1,\lambda}(\Omega)}, {\lambda>n-1}, then generalized Cheeger sets exist. We also study the embedding of Morrey spaces into {L^{p}} spaces. We show that, for any {0<\lambda<n}, the Morrey space {L^{1,\lambda}(\Omega)} is not contained in any {L^{q}(\Omega)}, {1<q<p=\frac{n}{n-\lambda}}. We also show that if {h\in L^{1,\lambda}(\Omega)}, {\lambda>n-1}, then the reduced boundary in Ω of a generalized Cheeger set is {C^{1,\alpha}} and the singular set has Hausdorff dimension at most {n-8} (empty if {n\leq 7}). For the critical case {h\in L^{1,n-1}(\Omega)}, we demonstrate that this strong regularity fails. We prove that a bounded generalized Cheeger set E in {\mathbb{R}^{n}} with {h\in L^{1}(\mathbb{R}^{n})} is always pseudoconvex, and any pseudoconvex set is a generalized Cheeger set for some h.

Stable difference scheme for a nonlocal boundary value heat conduction problem

In this paper, a new finite difference method to solve nonlocal boundary value problems for the heat equation is proposed. The most important feature of these problems is the non-self-adjointness. Because of the non-self-adjointness, major difficulties occur when applying analytical and numerical solution techniques. Moreover, problems with boundary conditions that do not possess strong regularity are less studied. The scope of the present paper is to justify possibility of building a stable difference scheme with weights for mentioned type of problems above.