The purpose of this paper is to give three new types of irresolute functions called, $\widetilde{g}-\theta$-irresolute functions, strongly $\widetilde{g}$-irresolute functions and quasi $\widetilde{g}$-irresolute functions. We obtain theirs characterizations and their basic properties.

We will revise the wreath sum of near-rings and establish some theorems of wreath sum of Near-rings and near-ring groups. Some related topics of wreath sum of near-rings will also be discussed. In particular,we will provide some examples to describe the notion of wreath sum of near-ring and near-ring groups. It is noted that in this context, we can explore (especially in the case of near-rings) some new areas and project the expectation that has been possibly reached by the author which he would like to claim. In contrast to the wreath product, in the structure of wreath sum of near rings, we observe an implication that the same set of wreath sum of near rings may have different explanation when the wreath product is defined in an innovative way. It is also noted that a subset of such a set of the so-called wreath sum with a specific algebraic structure need not be inherited in the other context. This observation leads to a new outlet to the fact that even in case of a particular near ring, we may have more than one near-ring group structure of the same wreath sum though the sets may not have the isomorphic algebraic structure. This important observation motivates us to extend the concept of wreath sum of near-rings to near-ring groups. This paper contains some other new results which are related to homomorphisms and the ideal structures of such systems. These results include the structure of wreath sum of near-ring groups, the non inheritance of Noetherian and Goldie characters.

In [7], we have introduced the concept of exact sequences of $R$-semimodules and $R$-homomorphisms. Here we give some interesting generalizations of exact sequences of modules and module homomorphisms over rings.

In order to extend the notion of convergence of sequences, statistical convergence of sequences was introduced by Fast [6] and Schoenberg [19]. The idea of constructing sequence spaces using Orlicz functions was introduced by Lindenstrauss and Tzafriri [15]. The main aim of this article is to study the concept of statistical convergence from some difference sequence spaces point of view which are defined over a real $2$-normed linear space by using an Orlicz function.

We first pose a Sudoku-type puzzle, involving lattice points in Pascal's Triangle, and lines passing through them. Then, through the use of an expanded notation for the binomial coefficient $\binom{n}{r}$, we exploit the geometry of Pascal's Triangle and produce a non-computational solution to the well-known Star of David pattern about products of two subsets of the 6 nearest neighbors to a given binomial coefficient. We generalize and obtain analogous patterns, with similar geometric proofs, for $2n$-gons where the entries at the lattice points of the symmetric (but not necessarily triangular) arrays are what we call separable functions (e.g., binomial coefficients, Leibnitz Harmonic coefficients, and Gaussian Polynomials, or $q$-analogues).

In [17], the concepts of $\sigma^{(A)}$-convergence of a bounded number sequence $x$ have been introduced. In this paper, we establish some inequalities by using a class of $\sigma^{(A)}$-conservative matrices, which are analogous to Knopp' s Core Theorem.

Wave packet systems are generated by the combined action of translations, modulations, and dilations on a finite family of functions. In this paper, we give necessary condition for wave packet system $\left\{D_{a_j}T_{bk}E_{c_m} \psi \right\}_{j,k,m \in \mathbb{Z}}$ to be a frame for $L^2(\mathbb R)$.

We show that, under certain conditions, the class of all left [right] regular semigroups is saturated. In particular all finite left [right] regular semigroups are saturated.

Applying weakly $s$-quasinormally embedded and $c$-supplemented subgroups, the following theorem is presented: Let $p$ be the smallest prime dividing the order of a group $|G|$ and $P$ a Sylow $p$-subgroup of $G$. If $G$ is $A_{4}$-free and every 2-maximal subgroup of $P$ is either weakly $s$-quasinormally embedded or $c$-supplemented in $G$, then $G$ is $p$-nilpotent. Some recent results are generalized and unified.

Let $x=(x_{jk})$ be a double sequence and $M$ be a bounded Orlicz function. We have proved that $x$ is statistically pre-Cauchy if and only if
$$\lim\limits_{m,n}\frac{1}{m^2n^2}\sum_{j,p\leq m}\sum_{k,q\leq n} M\bigg(\frac{|x_{jk}-x_{pq}|}{\rho}\bigg)=0$$
The main purpose of this paper is to extend the results of V.A. Khan and Q.M.D. Lohani [11] from single to double sequences.

V. Slavik [8] introduced the concept of ideal(filter) in a skew lattice. In this paper we introduce the concept of a filter in a skew lattice which includes the class of all filters in the sense of Slavik. Also we introduce the concept of prime filter in a skew lattice and study the distributive skew lattices and the space of prime filters of a distributive skew lattice with hull-kernel topology.

In this paper, it is proved that a matrix over a finite distributive lattice can be decomposed into the sum of some matrices over finite chains whose number of elements are of the same. As a result, both the decomposition theorem of a matrix over a finite distributive lattice recently stated by Z.T. Fan and the decomposition theorem of a matrix over a finite chain in a paper of X.Z. Zhao, Y.B. Jun and F. Ren are generalized and extended. Further, by the decomposition theorem, the convergence index and period index of a matrix over a distributive lattice are studied and some new results are obtained.

In this paper, the authors introduce and investigate the various properties and characteristics of the subclass $\mathcal{A}_{n,p}^*(m,q,\alpha,\gamma)$ of $p$-valent analytic functions with negative coefficients. In particular, some coefficient estimates and distortion theorems of the class functions belonging to this class are investigated. The radii of close-to-convexity, starlikeness and convexity of order $\delta(0 \leq\delta<1)$ for the subclass $\mathcal{A}_{n,p}^*(m,q,\alpha,\gamma)$ and the modified Hadamard product are also given.