In this paper, we study the unicity of transcendental entire functions concerning differences and small function and prove the following result: Let $f(z)$ be a transcendental entire functions of finite order with $\lambda(f-a)<\rho(f)$, where $a(z)$ is an entire small function of $f(z)$. Let $c$ be a nonzero constant and $n \ (n\ge 2)$ be a positive integer such that $\bigtriangleup_c^nf(z)\not\equiv0$. If $\bigtriangleup_c^nf(z)$ and $\bigtriangleup_c^{n-1}f(z)$ share $\bigtriangleup_c^{n-1}a(z)$ CM, then $f(z)=a(z)+Be^{Az}$, where $A,B$ are two nonzero constants and $a(z)$ is reduced to a polynomial with $\deg a \le n-2$. This result generalizes the previous theorem proved by He, Xiao and Fang in [15].

Let $(\mathcal{X},d,\mu)$ be a RD-space satisfying the doubling conditions and the reverse doubling conditions. In this setting, we obtain the more general definition of a generalized weighted Morrey space $\widetilde{M}^{u}_{p}(\omega)$, where the Lebesgue measurable function $u\in\mathbb{W}_{p}$, $p\in A_{p}(\mu)$ and $p\in[1,\infty)$. As applications, we show that a $\theta$-type Calder\'{o}n-Zygmund operator $T_{\theta}$ is bounded from generalized weighted Morrey spaces $\widetilde{M}^{u}_{p}(\omega)$ into spaces $\widetilde{M}^{u}_{p}(\omega)$, and also bounded from generalized weighted weak Morrey spaces $W\widetilde{M}^{u}_{1}(\omega)$ into spaces $\widetilde{M}^{u}_{1}(\omega)$; furthermore, the boundedness of the commutator $[b, T_{\theta}]$ formed by $b\in\mathrm{BMO}(\mu)$ and the $T_{\theta}$ on spaces $\widetilde{M}^{u}_{p}(\omega)$ is also obtained.

In this paper, we study planarity and outerplanarity of the iterated line graphs of the ideal-based zero divisor graphs when $R$ is a finite commutative ring. We give a complete characterization of all these graphs with respect to their planar and outerplanar indices. Also, we characterize the ring index of the graph $\Gamma_I(R)$.

In this paper, the concepts of multiplicative order of an element and multiplicatively finite elements in nearrings have been introduced to define semi-Baer and semi-quasi Baer $N$-groups which are the generalizations of Baer and quasi-Baer $N$-groups analogous to rings in [18]. Using these concepts, semi-Baerness of $N$-groups is characterized over reduced nearring. It is proved that the polynomial (including power series, Laurent polynomial and Laurent power series) extensions of $N$-groups preserves semi-Baerness. Further, it is shown that for a nearring $N$ and monoid $G$, if the monoid nearring $NG$ is semi-Baer then so is $N$.

In this paper, we consider a generalized finite Hilbert transform and establish some inequalities for differentiable functions whose derivative are either convex or, in the complex case, has the modulus convex. Applications for some particular instances of finite Hilbert transforms are given as well.

In this paper, we introduce a new subclass of uniformly convex functions with negative coefficients defined by Borel distribution series. We obtain the coefficient bounds, extreme points closure theorems and radii of starlikeness and convexity for functions belonging to the class. Furthermore, we discussed partial sums and neighbourhood results for this class.

Global hyperbolicity is the most important condition on causal structure of spacetime and also it is central concept in mathematical General Relativity. The basic aim of this review paper is to discuss the role of global hyperbolicity in spacetime geometry by introducing definitions, propositions and displaying diagrams. We also study the Cauchy hypersurfaces and some aspects of Lorentzian geometry which are relevant to global hyperbolicity. Then some recent developments regarding homotopy classes of causal curves and possible future scope of study on this area are discussed.

For a simple graph $G$, a vertex labeling $\phi:V(G) \rightarrow \{1, 2,\ldots,k\}$ is called $k$-labeling. The weight of an edge $xy$ in $G$, written $w_{\phi}(xy)$, is the sum of the labels of end vertices $x$ and $y$, i.e., $w_{\phi}(xy)=\phi(x)+\phi(y)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two different edges $e$ and $f$, $w_{\phi}(e) \neq w_{\phi}(f)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, written $es(G)$. In this note, we find the exact value of edge irregularity strength of Dandelion graph when $\Delta(G) \geq \lceil \frac{|E(G)|+1}{2} \rceil$; and determine the bounds when $\Delta(G) < \lceil \frac{|E(G)|+1}{2} \rceil $.

We introduce the notion of a twist product of semigroups. Our purpose is to describe a class of semigroups, namely {\it regular cyber-groups}. Regular cyber-groups in the class of abundant % semigroups are analogues of regular orthogroups in the class of completely regular semigroups. We give a structure theorem of a regular cyber-group in terms % of a twist product of semigroups, which generalizes the Yamada theorem for regular orthogroups. As an application of this structure theorem we give a description of regular cyber-groups using the spined product % of a left regular cyber-group and a right regular cyber-group with respect to a $C$-$a$ semigroup.

Let ${\smallerbf M}_{m,n}$ be the collection of all $m \times n$ real matrices. An $m \times n$ real matrix $D = [d_{ij} ] $ is called generalized substochastic (resp. row generalized substochastic) if $\sum_{j=1}^n|d_{ij}|\leq 1$ and $\sum_{i=1}^m|d_{ij}|\leq 1$ (resp. $\sum_{i=1}^m|d_{ij}|\leq 1$). For $X, Y \in {\smallerbf M}_{m,n}$, it is said that $X$ is $B$-Hadamard-majorized (resp. row $B$-Hadamard-majorized) by $Y$, denoted by $X \prec_H^B Y$ (resp. $X \prec_H^{rB} Y$), if there exists a generalized substochastic matrix $D\in {\smallerbf M}_{m,n}$ (resp. row generalized substochastic matrix $D\in {\smallerbf M}_{m,n}$) such that $X = D \circ Y$. In this paper, some properties of $\prec^B_H$ on ${\smallerbf M}_{m,n}$ are first obtained, and then, the (strong) linear preservers of $\prec^B_H$ and $\prec^{rB}_H$ on ${\smallerbf M}_{m,n}$ are characterized.

A graph $G$ with $p$ vertices and $q$ edges is said to have a Skolem difference odd mean labeling if there exists an injective function $f:V(G)\rightarrow\{1,2,3,\dots,4q-1\}$ such that the induced map $f^*:E(G)\rightarrow\{1,3,5,\dots, 2q-1\}$ defined by $f^*(uv)=\left\lceil\frac{|f(u)-f(v)|}{2}\right\rceil$ is a bijection. A graph that admits a Skolem difference odd mean labeling is called a Skolem difference odd mean graph. Here we investigate Skolem difference odd mean labeling of some star and path related graphs.