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## A note on isomorphism theorems of Krasner (m, n)-hyperrings

### Arabian Journal of Mathematics (2016-06-01) 5: 103-115 , June 01, 2016

Recently, Krasner (*m*, *n*)-hyperrings were introduced and analyzed by Davvaz et. al. This is a suitable generalization of Krasner hyperrings. In this research work, we consider that if *I* is a normal hyperideal of a Krasner (*m*, *n*)-hyperring *R*, then the quotient hyperring [*R* : *I**] is an (*m*, *n*)-ring. Moreover, we prove that if *R* is a multiplicative (*m*, *n*)-ary hyperring and *I* is a normal hyperideal of *R*, then [*R* : *I**] is an (*m*, *n*)-ring.

## Finitistic weak dimension of commutative arithmetical rings

### Arabian Journal of Mathematics (2012-04-01) 1: 63-67 , April 01, 2012

It is proven that each commutative arithmetical ring *R* has a finitistic weak dimension ≤ 2. More precisely, this dimension is 0 if *R* is locally IF, 1 if *R* is locally semicoherent and not IF, and 2 in the other cases.

## Irregular multiresolution analysis and associated wavelet

### Arabian Journal of Mathematics (2014-03-01) 3: 23-37 , March 01, 2014

We introduce two generalizations, the first of which generalizes the concept of multiresolution analysis. We define the irregular generalized multiresolution analysis (IGMRA). This structure is defined taking translations on sets that are not necessarily regular lattices, for which certain density requirements are required, and without using dilations, also allows each subspace of IGMRA to be generated by outer frames of translations of different functions. The second generalization concerns the concept of association of wavelets to these new structures. We take frames of translations of a countable set of functions, which we called *generalized wavelets*, and define the concept of association of these generalized wavelets to those previously defined IGMRA. In the next stage, we prove two existence theorems. In the first theorem, we prove existence of IGMRA, and in the second existence of generalized wavelets associated with it. In the latter, we show that we are able to associate frames of translations with optimal localization properties, to IGMRA. In the last section of this paper, concrete examples of these structures are presented for
and for
.

## Quasi generalized CR-lightlike submanifolds of indefinite nearly Sasakian manifolds

### Arabian Journal of Mathematics (2016-06-01) 5: 87-101 , June 01, 2016

In this paper, we introduce and study a new class of CR-lightlike submanifold of an indefinite nearly Sasakian manifold, called quasi generalized Cauchy–Riemann (QGCR) lightlike submanifold. We give some characterization theorems for the existence of QGCR-lightlike submanifolds and finally derive necessary and sufficient conditions for some distributions to be integrable.

## Certain properties of a new subclass of close-to-convex functions

### Arabian Journal of Mathematics (2012-09-01) 1: 309-317 , September 01, 2012

In the present paper we introduce and investigate an interesting subclass $${\mathcal{K}_{s}^{(k)}(\lambda,h)}$$ of analytic and close-to-convex functions in the open unit disk $${\mathbb{U}}$$ . For functions belonging to the class $${\mathcal{K}_{s}^{(k)}(\lambda,h)}$$ , we derive several properties as the inclusion relationships and distortion theorems. The various results presented here would generalize many known recent results.

## Adaptive synchronization of two complex networks with delayed and non-delayed coupling

### Arabian Journal of Mathematics (2012-06-01) 1: 219-226 , June 01, 2012

This paper studies the adaptive synchronization of two complex networks with non-delayed and delayed couplings, in which the coupling configuration matrices are not necessarily symmetric or irreducible. Considering the case of identical and nonidentical network topological structures, we obtain several criteria for synchronization of two complex networks based on the Lyapunov stability theory. Numerical simulations are presented to demonstrate the effectiveness of the proposed criteria.

## Ulam stability of a generalized reciprocal type functional equation in non-Archimedean fields

### Arabian Journal of Mathematics (2015-06-01) 4: 117-126 , June 01, 2015

In this paper, we obtain the solution of a new generalized reciprocal type functional equation in two variables and investigate its generalized Hyers–Ulam stability in non-Archimedean fields. We also present the pertinent stability results of Hyers–Ulam–Rassias stability, Ulam–Gavruta–Rassias stability and J. M. Rassias stability controlled by the mixed product-sum of powers of norms.

## On the probability of ruin in the compound Poisson risk model with potentially delayed claims

### Arabian Journal of Mathematics (2013-03-01) 2: 115-127 , March 01, 2013

In this paper, we consider the compound Poisson risk model involving two types of dependent claims, namely main claims and by-claims. The by-claim is induced by the main claim with a certain probability and the occurrence of a by-claim may be delayed depending on associated main claim amount. Using Rouché’s theorem, both of the survival probability with zero initial surplus and the Laplace transform of the survival probability are obtained from an integro-differential equations system. Then, using the Laplace transform, we derive a defective renewal equation satisfied by the survival probability. An exact representation for the solution of this equation is derived through an associated compound geometric distribution. For exponential claim sizes, we present an explicit formula for the survival probability. We also illustrate the influence of model parameters in the dependent risk model on the survival probability by numerical examples.

## Studying monoids is not enough to study multiplicative properties of rings: an elementary approach

### Arabian Journal of Mathematics (2015-03-01) 4: 29-34 , March 01, 2015

The aim of these notes is to indicate, using very simple examples, that not all results in ring theory can be derived from monoids and that there are results that deeply depend on the interplay between “ + ” and “·”.

## The set of prime divisors of generalized denominator ideals

### Arabian Journal of Mathematics (2013-12-01) 2: 333-343 , December 01, 2013

Let *R* be a Noetherian domain with quotient field *K*. Let *A* be an integral domain which contains *R* and whose elements are algebraic over *K*. We define
$${{\rm Eass}_{R}(A/R)}$$
to be the set of prime ideals
$${\mathfrak{p}}$$
’s of *R* such that
$${\mathfrak{p}}$$
is a prime divisor of a generalized denominator ideal *I*_{[β]} for some
$${\beta \in A}$$
. Assume that
$${A = R[\alpha_{1}, \ldots,\alpha_{n}]}$$
. We investigate the relation between
$${{\rm Eass}_{R}(A/R)}$$
and
$${\cup_{i = 1}^{n} {\rm Ass}_{R}(R/I_{[\alpha_{i}]})}$$
. Furthermore, for a finite subset *Δ* of
$${{\rm Eass}_{R}(A/R)}$$
, we construct the subring *A*_{Δ} of *A* such that *A*_{Δ} is the largest among those *B*’s with
.