Following the concept of porosity of subsets of natural numbers at infinity, we define notions of porosity-limit and porosity-cluster points for real valued sequences. These new two concepts, which are not equivalent, are compared with the usual limit points of sequences. Also, some topological properties of the sets of all porosity limit (or cluster) points of sequences are investigated.

Let $(G, +)$ be a finite group, written additively with identity $0$, but not necessarily abelian. Let $\Gamma = \{H_i\}$ be a nonempty collection of nonzero, proper subgroups of $G$. Then $N = \{f : G \to G\ |\ f(0) = 0 \ \hbox{and}\ f(H_i) \subseteq H_i\ \hbox{for all}\ i\}$ is a nearring under pointwise addition and function composition. We determine when $N$ is abelian and distributive, identify the center of $N$, and find necessary and sufficient conditions for the center to be a subnearring of $N$.

Let $R$ be a prime ring of characteristic different from $2$, $U$ its Utumi ring of quotients and $C$ its extended centroid, $f(x_1,\dots, x_n)$ a multilinear polynomial over $C$ that is noncentral-valued on $R$, $I$ a nonzero ideal of $R$ and $F$ a nonzero generalized derivation of $R$. If for some $0\neq a\in R$,

In this paper, a novel of convergence named the fuzzy u-uniform convergence is defined on the basis of the fuzzy order convergence in fuzzy Riesz spaces. The relation between the fuzzy u-uniform convergence and the fuzzy order convergence in fuzzy Archimedean Riesz spaces is studied. We also discuss the existence of the fuzzy u-uniform limit for the fuzzy u-uniform Cauchy sequence in fuzzy Archimedean Riesz Spaces. We give the characterization of fuzzy order completeness in fuzzy Riesz spaces.

In this paper, the authors prove that bilinear pseudo-differential operator $T_{\sigma}$ is bounded from the product of homogeneous Herz spaces $\dot{K}^{\alpha_{1},p_{1}}_{q_{1}}(\mathbb{R}^{n})\times \dot{K}^{\alpha_{2},p_{2}}_{q_{2}}(\mathbb{R}^{n})$ to homogeneous Herz space $\dot{K}^{\alpha,p}_{q}(\mathbb{R}^{n})$, and bounded from the product of homogenous Morrey-Herz spaces $M\dot{K}^{\alpha_{1},\lambda_{1}}_{p_{1},q_{1}}(\mathbb{R}^{n})\times M\dot{K}^{\alpha_{2},\lambda_{2}}_{p_{2},q_{2}}(\mathbb{R}^{n})$ to homogeneous Morrey-Herz space $M\dot{K}^{\alpha,\lambda}_{p,q}(\mathbb{R}^{n})$. Furthermore, via establishing the sharp maximal estimate for the commutator $[b_{1},b_{2},T_{\sigma}]$ which is generated by $T_{\sigma}$ and $b_{1},b_{2}\in\mathrm{BMO}(\mathbb{R}^{n})$, the boundedness of $[b_{1},b_{2},T_{\sigma}]$ on spaces $\dot{K}^{\alpha,p}_{q}(\mathbb{R}^{n})$ and on spaces $M\dot{K}^{\alpha,\lambda}_{p,q}(\mathbb{R}^{n})$ is also obtained.

We prove a special case of the Banach-Steinhaus theorem for Sargent spaces, making use of the Hahn-Banach theorem and by-passing the proof by Sargent which requires the Baire category theorem.

In this paper, we introduce $k$-directional Darboux curves of a given timelike curve $\alpha $ lying on a timelike surface in Minkowski $3$-space $\mathbb{E}_{1}^{3}$ as the integral curves of some vector fields generated by the Darboux frame along the curve. The relationships of such associate curves are given including the relationship of their curvatures. By using these curves, we give a new approach to classifying some special curves such as helices, and slant helices.

In [3], we defined and studied the order prime divisor graph $\mathscr{PD}(G)$ of a finite group $G$. In this paper, we also study several properties of the order prime divisor graph of a finite group. Also, we discuss the order prime divisor graph for direct product of finite groups. We characterize some graph theoretic properties of $\mathscr{PD}(\mathbb{Z}_{n})$, $\mathscr{PD}(Q_{4n})$, $\mathscr{PD}(Q_{2^{n+1}})$, $\mathscr{PD}(SD_{2^n})$.

The primary goal of this study is to propose the concept of an operation $\iota$ on $\delta-$preopen subsets in topological spaces. Here we defined the concepts of $\iota-$open sets, $\iota-$regular spaces and some topological properties such as $\iota-$closure, $\iota-$interior, $closure_\iota$ of sub sets and $\iota-$derived were analysed. $\iota-g.$closed sets and $\iota-T_{1/2}$ spaces were studied. And a new set called $(\iota)-$open sets are studied along with its properties.

We define a semi-Brouwerian almost distributive lattice as a description of a semi-Brouwerian algebra in the category of almost distributive lattices and derive a number of algebraic properties on them. We obtain a few equivalent conditions for an almost distributive lattice to become a semi-Brouwerian almost distributive lattice. Finally we identify a class of pseudo-complemented lattices and a class of semi-Brouwerian algebras in a semi-Brouwerian almost distributive lattice.

This paper focuses on three kinds of graphs built from semigroups: power graphs, enhanced power graphs and commuting graphs. Let $S$ be a semigroup constructed in a prescribed way from a semigroup $T$. We show in particular instances that pairs of these graphs built over $S$ coincide precisely when the same pairs built over $T$ coincide. This is true, for example, when $S$ is a completely simple semigroup and $T$ is a maximal subgroup. We give several other instances of this phenomenon. In addition, we give counterexamples to show that equality for an arbitrary pair of these three graphs does not always pass between semigroups and their maximal subgroups.