Advances in Pure and Applied Mathematics

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In this paper, having analyzed the previously obtained results devoted to the root vectors series expansion in the Abel-Lidskii sense, we come to the conclusion that the concept can be formulated in the classical terms of the spectral theorem. Though, the spectral theorem for a sectorial operator has not been formulated even in the m-sectorial case, we can consider from this point of view a most simplified case related to the sectorial operator with a discrete spectrum. Thus, in accordance with the terms of the spectral theorem, we naturally arrive at the functional calculus for sectorial operators which is the main focus of this paper. Due to the functional calculus methods, we construct the operator class with the asymptotics more subtle then one of the power type.

The main purpose of this paper is to study cohomology and develop a deformation theory of restricted Lie algebras in positive characteristic $$$p$$$ > $$$0$$$. In the case $$$p\geq3$$$, it is shown that the deformations of restricted Lie algebras are controlled by the restricted cohomology introduced by Evans and Fuchs. Moreover, we introduce a new cohomology that controls the deformations of restricted morphisms of restricted Lie algebras. In the case $$$p=2$$$, we provide a full restricted cohomology complex with values in a restricted module and investigate its connections with formal deformations. Furthermore, we introduce a full deformation cohomology that controls deformations of restricted morphisms of restricted Lie algebras in characteristic $$$2$$$. As example, we discuss restricted cohomology with adjoint coefficients of restricted Heisenberg Lie algebras in characteristic $$$p\geq 2$$$.

In this paper, we study a new class of operators, so called $$$A(n,m)$$$-iso-contra-expansive operators. These new families of operators are considered as a generalization that combines the $$$m$$$-expansive operators as well as m-contractive operators and the classes of $$$(A,m)$$$-expansive and $$$(A,m)$$$-contractive operators and we recover the notion of $$$n$$$-quasi-$$$(A,m)$$$-isometric operators. Some spectral properties of these kind of operators are provided, we derive also a conditions to have the single-valued extension property (SVEP) and we finish by an application of Toeplitz operators on Bergman spaces.

Let $$$G$$$ be a permutation group on a set $$$\Omega$$$ with no fixed points in $$$\Omega$$$, and let $$$m$$$ be a positive integer. If for each subset $$$\Gamma$$$ of $$$\Omega$$$ the size $$$\Gamma^{g}-\Gamma|$$$ is bounded, for $$$g\in G$$$, the movement of $$$g$$$ is defined as move $$$(g):=\max{|\Gamma^{g}-\Gamma|}$$$ over all subsets $$$\Gamma$$$ of $$$\Omega$$$, and move $$$(G)$$$ is defined as the maximum of move $$$(g)$$$ over all non-identity elements of $$$g\in G$$$. Suppose that $$$G$$$ is not a 2-group. It was shown by Praeger that $$$|\Omega|\leqslant\lceil\frac{2mp}{p-1}\rceil+t-1$$$, where $$$t$$$ is the number of $$$G$$$-orbits on $$$\Omega$$$ and $$$p$$$ is the least odd prime dividing $$$|G|$$$. In this paper, we classify all permutation groups with maximum possible degree $$$|\Omega|=\lceil\frac{2mp}{p-1}\rceil+t-1$$$ for $$$t=2$$$, in which every non-identity element has constant movement $$$m$$$.

Let $$$f$$$ be a degree $$$d$$$ form in $$$n+1$$$ variables $$$x_0,\dots ,x_n$$$. Any additive decomposition of $$$f$$$ is associated to a finite set $$$A\subset ℙ^n$$$ with $$$\#A$$$ the number of non-proportional addenda. We study the index of regularity $$$\rho(A)$$$ of $$$A$$$, i.e. the first integer $$$t$$$ such that $$$h^1(\mathcal{I}_A(t)) = 0$$$, of the finite subset $$$A\subset ℙ^n$$$ associated to the additive decompositions of degree $$$d$$$ forms in $$$n+1$$$ variables. Obviously $$$\rho(A)\le d$$$. We prove that $$$\rho(A)\ge d-k$$$ if $$$A$$$ spans $$$ℙ^n$$$ and $$$k$$$ is the maximal integer such that $$$x_0^k$$$ divides at least one monomial of $$$f$$$. If $$$f$$$ essentially depends on less variables, but $$$A$$$ spans $$$ℙ^n$$$, then $$$\rho(A)=d$$$. We give examples (but with $$$\#A$$$ bigger that the rank of $$$f$$$) in which we have $$$\rho(A)=d$$$.

In this paper, we reconsider and slightly generalize various classes of Weyl almost automorphic functions ([29], [33]). More precisely, we consider here various classes of metrically Weyl almost automorphic functions of the form $$$F : {\mathbb R}^{n} \times X \rightarrow Y$$$ and metrically Weyl almost automorphic sequences of the form $$$F : {\mathbb Z}^{n} \times X \rightarrow Y$$$, where $$$X$$$ and $$$Y$$$ are complex Banach spaces. The main structural characterizations for the introduced classes of metrically Weyl almost automorphic functions and sequences are established. In addition to the above, we provide several illustrative examples, useful remarks and applications of the theoretical results.

This paper complements the description of finite-dimensional Jordan symplectic metric superalgebras on algebraically closed fields of characteristic zero. We discuss the graduation of the metric and the symplectic structures and use a new type of generalized double extension by two-dimensional Jordan superalgebras.