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Advances in Pure and Applied Mathematics

List of Articles

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 $$$\rho(A)=d$$$.

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.

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.

In this paper we consider a class of nonlocal parabolic equations without uniqueness using a new framework developed by Cheskidov and Lu which called evolutionary system. We first prove the existence of weak solutions by using the compactness method. However, the Cauchy problem can be non-unique and we also give a sufficient condition for uniqueness. Then we use the theory of evolutionary system to investigate the asymptotic behavior of weak solutions via attractors and its properties. The novelty is that our results extend and improve the previous results and it seems to be the first results for this kind of system via using evolutionary systems.

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 $$$m$$$.

It is shown that the collection of all topologies on a given set $$$X$$$ coincide with the set of subsemirings of the power set $$$\mathcal{P}(X)$$$ (equipped with union and intersection) if and only if $$$X$$$ is finite. Furthermore, given a topological space $$$(X, \mathcal{T})$$$ and a subset $$$A$$$ of $$$X$$$, we characterize when the subspace topology $$$\mathcal{T}_A$$$ is a maximal (resp., a prime) ideal of the semiring $$$\mathcal{T}$$$. As applications, we provide an algebraic characterization of the one-point compactification of a noncompact, Tychonoff space. Moreover, we describe explicitly the semiring homomorphisms from $$$\mathcal{P}(X)$$$ into $$$\mathcal{P}(Y)$$$ in case $$$X$$$ is a finite set and $$$Y$$$ is an arbitrary nonempty set.

In a preceding paper we introduced a notion of compatibility between a Jacobi structure and a Riemannian structure on a smooth manifold. We proved that in the case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, (1/2)-Kenmotsu structures and locally conformally Kähler structures. In this paper we are generalizing this work to the framework of Lie algebroids.