Abstract

In this paper, we show the convergence of matrix series and the conditions for their convergence by finding an upper bound for some specific matrix inequalities. Finally, we introduce a new form of arithmetic-geometric matrix series and analyze their convergence.

Keywords: Harmonic series; condition number of a matrix; Hilbert-Schmidt norm; norm inequality; positive definite matrix; singular value; spectral norm; matrix convergence.

MSC: 41A65, 47H17, 47H20

Abstract

Let $\frak F_{\lambda}^2$ be the space of tensor densities of degree $\lambda\in \mathbb{C}$ on the supercircle $S^{1|2}$. We consider the space $\mathfrak{D}_{\lambda,\mu}^{2,k}$ of $k$-th order linear differential operators from $\frak F_{\lambda}^2$ to $\frak F_{\mu}^2$ as a module over the superalgebra $\mathcal{K}(2)$ of contact vector fields on $S^{1|2}$ and we compute the superalgebra $\mathcal{K}_{\lambda,\mu}^{2,k}$ of endomorphisms on $\mathfrak{D}_{\lambda,\mu}^{2,k}$ commuting with the $\mathcal{K}(2)$-action. We prove that this algebra is trivial except for $\lambda= 0$.

Keywords: Contact structure; differential operators; densities.

MSC: 53D55

Abstract

We introduce golden STCR-lightlike submanifolds of golden semi-Riemannian manifolds. We find new conditions for the induced connection to be a metric connection. Moreover, we find some necessary and sufficient conditions for such submanifolds.

Keywords: Golden semi-Riemannian manifold; golden structure; lightlike submanifold; golden STCR-lightlike submanifold.

MSC: 53C15, 53C40, 53C50

Abstract

In this paper we study with statistical convergence in the sense of the power series method which is not comparable with statistical convergence. Using this notion, we introduce the concepts of $P_{p}$--statistical exhaustiveness and weak $P_{p}$--statistical exhaustiveness. Also, we study several types of convergence of sequences of functions between two metric spaces and we obtain more general results from the concepts of exhaustiveness and the strong uniform convergence on a bornology.

Keywords: Statistical convergence with respect to power series method; bornology; exhaustiveness; $P_{p}$--statistical exhaustiveness.

MSC: 53B40, 53C60

Abstract

Let $R$ be a commutative local ring and let $C$ be a semidualizing $R$-module. In [E. Tavasoli, M. Salimi, Relative Matlis duality with respect to a semidualizing module, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér., 66(114), 4 (2023), 433--444], the notion of relative Matlis duality with respect to $C$, and $C$-Matlis reflexive modules are introduced, which generalized the notions of Matlis duality and Matlis reflexive modules. In this paper, we investigate conditions under which the $R$-modules ${\rm Ext}^{i\geqslant 0}_R{(M,N)}$ and ${\rm Tor}^R_{i\geqslant 0}{(M,N)}$ become $C$-Matlis reflexive, where $M$ and $N$ are $R$-modules. In addition, we deal with the isomorphic modules to the relative Matlis duality of $R$-modules ${\rm Ext}^{i\geqslant 0}_R{(M,N)}$, and ${\rm Tor}^R_{i\geqslant 0}{(M,N)}$ in the case that $M$ and $N$ are Matlis reflexive modules over the complete ring $R$.

Keywords: Semidualizing; Matlis duality; Matlis reflexive.

MSC: 13D05, 13D45, 18G20

Abstract

This paper investigates the proper class of all metric spaces considered up to isometry, equipped with the Gromov--Hausdorff distance. There constructed a pair of complete metric spaces, $X$ and $Y$ such that they have no metric spaces at zero distance, no optimal correspondence between $X$ and $Y$, and therefore no linear geodesics joining them, but there exists a geodesic between them of a different type. There also described everywhere dense subclass of the Gromov--Hausdorff class such that any two points at finite distance within this subclass can be connected by a linear geodesic.

Keywords: Metric space; Gromov--Hausdorff space; geodesic; distance-preserving function.

MSC: 53C23

Abstract

We present new results about the factorization of $\Phi_p(M) \in \mathbb{F}_2[x]$, where $p$ is a prime number, $\Phi_p$ is the corresponding cyclotomic polynomial, and $M$ is a Mersenne prime polynomial. In particular, these results improve our understanding of the factorization of the sum of the divisors of $M^{2h}$ for a positive integer $h$. This is related to the fixed points of the sum of divisors function $\sigma$ on $\mathbb{F}_2[x]$. The factorization of composed polynomials over finite fields is not well understood, and classical results on cyclotomic polynomials primarily concern the special case where $M$ is replaced by $x$.

Keywords: Cyclotomic polynomials; sum of divisors; finite fields.

MSC: 11T55, 11T06