Journal of Mathematical Extension
Volume:17 Issue: 7, Jul 2023

The purpose of this paper is to introduce the concept of soft multiplicative linear functional in soft Banach algebras. Then the notions of soft spectrum and soft spectral radius of an element in soft Banach algebras are introduced. Furthermore we discuss some important characterizations and theorems of them.

‎In this paper‎, the Lie group of point symmetries for a‎ ‎kind of Benjamin-Bona-Mahoney (BBM) equation is obtained by applying the classical Lie‎ symmetry method‎. ‎An optimal system of sub-algebras of‎ ‎dimension one for the BBM equation is deduced by classifying the‎ ‎adjoint representation orbits of the Lie symmetry group‎. Then‎, ‎for any infinitesimal symmetry generators of the Lie group‎, ‎the related similarity reductions are generated‎. ‎Also‎, ‎new conservation laws for this equation are constructed by‎ ‎the method of scaling‎. ‎The conservation laws densities is calculated by using the concept‎ ‎of variables weight‎, ‎scaling symmetry and Euler operator‎ ‎and their fluxes is computed by applying the homotopy operator‎.

In this paper, we consider homogeneous Finsler spaces with generalized m-Kropina metric. First, we study the formula for the Ricci curvature of a homogeneous Finsler space with aforesaid metric. Further, we derive the formula of the Ricci curvature for generalized m-Kropina metric having vanishing S-curvature. Finally, we derive the formula for the projective Ricci curvature of generalized m-Kropina Finsler metric.

Let R be a commutative Noetherian ring and M an Rmodule. We begin by some results about the connection between the general local cohomology modules with respect to a system of ideals of R and an arbitrary Serre subcategory of R-modules. Let n = dim R and a be an ideal of R. We show that SuppR(H j a (M)) ⊆ A ∗ (a) ∪ ( ∪n−j i=1 Suppj+i R (H j a (M))) for all j ≥ 0. As a consequence, if R is a semi-local ring and M is a minimax R-module of dimension at most three, then the R-modules H j a (M) have finite sets of associated prime ideals for all j ≥ 0. Moreover, we give some results on the finiteness of the Bass numbers and the Betti numbers and cofiniteness of the ordinary local cohomology modules.

A subset $A$ of a topological space $X$ is called locally closed if it is open in $\ov{A}$; $X$ is called submaximal if every subset of $X$ is locally closed. In this paper, we show that if $\beta X$, the Stone-\v{C}ech compactification of $X$, is a submaximal space, then $X$ is a compact space and hence $\beta X=X$. We observe that every submaximal Hausdorff space is a $ncd$-space (a space in which does not have a nonempty compact and dense in itself subset). It turns out that every dense in itself Hausdorff space is pseudo-finite if and only if it is a $(cei,f)$-space (a space in which every compact subspace of $X$ with empty interior is finite). A new characterization for submaximal spaces is given. Given a topological space $(X,{\mathcal T})$, the collection of all locally closed subsets of $X$ forms a base for a topology on $X$ which is denotes by ${\mathcal T_l}$. We study some topological properties between $(X,{\mathcal T})$ and $(X,{\mathcal T_l})$, such as we show that $(X,{\mathcal T})$ is a locally indiscrete space if and only if ${\mathcal T}={\mathcal T_l}$. Finally, we prove that every clopen subspace of an lc-compact space is lc-compact. \end{abstract}

Let $X‎, ‎Y\in \Bbb R^n‎, ‎X,Y>0$‎, ‎we say $X$ {\it logarithm majorized} by $Y$‎, ‎written $X\prec_{log} Y$ if $\log X\prec \log Y$‎. ‎Let $M_{nm}^+$ be the collection of matrices with positive entries‎. ‎For $X,Y\in M_{nm}^+$‎, ‎it is said that $X$ is {\it logarithm column (row) majorized} by $Y$‎, ‎and is denoted as $X\prec_{log}^{column} Y (X\prec_{log}^{row}Y)$‎, ‎if $X_{j}\prec_{log} Y_{j} (X_{i}\prec_{log} Y_{i}) $ for all $j=1,2,\cdots m (i=1,2,\cdots n)$‎, ‎where $X_{j}$ and $Y_{j}$ ($X_{i}$ and $Y_{i}$) are the ith column (row) of $X$ and $Y$ respectively‎. ‎In the present paper‎, ‎the relations column (row) logarithm majorization on $M_{nm}^+$ are studied and also all linear operators $T:M_{nm}^+\longrightarrow M_{nm}^+$ preserving column (row) logarithm majorization will be characterized‎.