Arabian Journal of Mathematics

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On the category of modules over some semisimple bialgebras

V. A. Artamonov, R. B. Mukhatov, R. Wisbauer

Arabian Journal of Mathematics > 2012 > 1 > 1 > 29-38

We study the tensor category of modules over a semisimple bialgebra H under the assumption that irreducible H-modules of the same dimension > 1 are isomorphic. We consider properties of Clebsch–Gordan coefficients showing multiplicities of occurrences of each irreducible H-module in a tensor product of irreducible ones. It is shown that, in general, these coefficients cannot have small values....

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On injective and divisible modules

Patrick F. Smith

Arabian Journal of Mathematics > 2012 > 1 > 1 > 127-137

Divisible modules over general rings are considered and a general notion of divisibility is defined. In order to study these divisible modules we generalize the notion of injectivity. One consequence is that rings for which every principal right ideal is projective can be characterized. In addition, a characterization is given of when a submodule of a projective module is projective.

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The subgroup graph of a group

David F. Anderson, Jodi Fasteen, John D. LaGrange

Arabian Journal of Mathematics > 2012 > 1 > 1 > 17-27

Given any subgroup H of a group G, let Γ_H(G) be the directed graph with vertex set G such that x is the initial vertex and y is the terminal vertex of an edge if and only if x ≠ y and $${xy\in H}$$ . Furthermore, if $${xy\in H}$$ and $${yx\in H}$$ for some $${x,y\in G}$$ with x ≠ y, then x and y will be regarded as being connected by a single undirected...

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Ascending the divided and going-down properties by absolute flatness

Gabriel Picavet

Arabian Journal of Mathematics > 2012 > 1 > 1 > 113-126

This paper aims to show that the “going-down ring” and the “divided ring” properties ascend along flat morphisms whose co-diagonal morphisms are flat, the so-called absolutely flat morphisms introduced by Olivier. But unibranchedness hypotheses are necessary as any henselization morphism shows. As a by-product, we get that the “unibranched divided ring” property is preserved by the formation of factor...

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Almost principal ideals in R[x]

Thomas G. Lucas

Arabian Journal of Mathematics > 2012 > 1 > 1 > 97-111

For an integral domain R with quotient field K, an upper-type ideal of R[x] is an ideal of the form $${I_f = f({\rm x})K[{\rm x}] \cap R[{\rm x}]}$$ for some polynomial $${f({\rm x}) \in K[{\rm x}] \backslash K}$$ . Clearly, I _f = I _rf for each nonzero $${r \in R}$$ . Hence one can always choose f (x) from R[x]. Such...

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Does Int $${(\mathbb {Z})}$$ have the stacked bases property?

Jean-Luc Chabert

Arabian Journal of Mathematics > 2012 > 1 > 1 > 47-52

It is known that a Prüfer domain either with dimension 1 or with finite character has the stacked bases property. Following Brewer and Klinger, some rings of integer-valued polynomials provide, for every n ≥ 2, examples of n-dimensional Prüfer domains without finite character which have the stacked bases property. But, the following question is still open: does the two-dimensional Prüfer domain ...

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Finitistic weak dimension of commutative arithmetical rings

François Couchot

Arabian Journal of Mathematics > 2012 > 1 > 1 > 63-67

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.

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A soft introduction to algebraic entropy

Anna Giordano Bruno, Luigi Salce

Arabian Journal of Mathematics > 2012 > 1 > 1 > 69-87

The goal of this mainly expository paper is to develop the theory of the algebraic entropy in the basic setting of vector spaces V over a field K. Many complications encountered in more general settings do not appear at this first level. We will prove the basic properties of the algebraic entropy of linear transformations $${\phi:V \to V}$$ of vector spaces and its characterization as the rank...

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Three frameworks for a general theory of factorization

D. D. Anderson, R. M. Ortiz-Albino

Arabian Journal of Mathematics > 2012 > 1 > 1 > 1-16

We discuss three different frameworks for a general theory of factorization in integral domains: τ-factorization, reduced τ-factorization and Γ-factorization. Let D be an integral domain, $${D^{\sharp}}$$ the non-zero, non-units of D, and τ a symmetric relation on $${D^{\sharp}}$$ . For $${a\in D^{\sharp}, a=\lambda a_{1}\cdots a_{n},\lambda}$$ a unit, $${a_{i}\in D^{\sharp}, n\geq1,}$$...

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On the delta set and the Betti elements of a BF-monoid

S. T. Chapman, P. A. García-Sánchez, D. Llena, A. Malyshev, more

Arabian Journal of Mathematics > 2012 > 1 > 1 > 53-61

We examine the Delta set of a cancellative and reduced atomic monoid S where every set of lengths of the factorizations of each element in S is bounded. In particular, we show the connection between the elements of Δ(S) and the Betti elements of S. We prove how the minimum and maximum element of Δ(S) can be determined using the Betti elements of S. This leads to a determination of when Δ(S) is a singleton...

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R-smash products of Hopf quasigroups

Tomasz Brzeziński, Zhengming Jiao

Arabian Journal of Mathematics > 2012 > 1 > 1 > 39-46

The theory of R-smash products for Hopf quasigroups is developed.

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Integral domains with Boolean t-class semigroup

S. Kabbaj, A. Mimouni

Arabian Journal of Mathematics > 2012 > 1 > 1 > 89-95

The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by t-multiplication. This paper investigates integral domains with Boolean t-class semigroup with an emphasis on the GCD and stability conditions. The main results establish t-analogues for well-known results on Prüfer domains and Bézout domains of finite character.

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The maximal correlation for the generalized order statistics and dual generalized order statistics

H. M. Barakat

Arabian Journal of Mathematics > 2012 > 1 > 2 > 149-158

In each of three exhaustive and distinct cases, it is found a distribution for which the correlation coefficient between the elements of the generalized order statistics (gos) is maximal. The corresponding result for the dual generalized order statistics (dgos) is derived for other three different distributions. Moreover, some interesting relations for the regression curves between the elements of...

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Numerical solutions of doubly perturbed stochastic delay differential equations driven by Lèvy process

Xiaotai Wu, Litan Yan

Arabian Journal of Mathematics > 2012 > 1 > 2 > 251-265

In this paper, the numerical solutions of doubly perturbed stochastic delay differential equations driven by Lèvy process are investigated. Using the Euler–Maruyama method, we define the numerical solutions, and show that the numerical solutions converge to the true solutions under the local Lipschitz condition. As a corollary, we give the order of convergence under the global Lipschtiz condition.

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Delay-dependent exponential stability results for uncertain stochastic Hopfield neural networks with interval time-varying delays

C. Pradeep, A. Vinodkumar, R. Rakkiyappan

Arabian Journal of Mathematics > 2012 > 1 > 2 > 227-239

This paper is concerned with stability analysis problem for uncertain stochastic neural networks with interval time-varying delays. The parameter uncertainties are assumed to be norm bounded and the delay is assumed to be time varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. Both the cases of the time-varying delays...

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

Ke Lou, Baotong Cui, Xiaojiao Zhang

Arabian Journal of Mathematics > 2012 > 1 > 2 > 219-226

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...

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A note on zero-divisors of commutative rings

M. Filipowicz, M. Kȩpczyk

Arabian Journal of Mathematics > 2012 > 1 > 2 > 191-194

In this paper we show that if a ring R has finite Goldie dimension, then every finitely generated ideal of R consisting of zero-divisors has non-zero annihilator. We also construct an example of a ring of infinite Goldie dimension such that above condition does not hold.

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Weak Hopf algebras corresponding to quantum algebras U _q(f (K, H))

Yanyong Hong, Fang Li

Arabian Journal of Mathematics > 2012 > 1 > 2 > 195-218

In this paper, we investigate weak Hopf algebras introduced in Li (J Algebra 208:72–100, 1998; Commun Math Phys 225:191–217, 2002) corresponding to quantum algebras U _q(f (K, H)) (see Wang et al. in Commun Algebra 30:2191–2211, 2002). A new class of algebras is defined, which is denoted by $${\mathfrak{w}U^{d}_q.}$$ For d = ((1, 1) | (1, 1)), denote $${\mathfrak{w}U^{d}_q}$$...

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Dimension of certain sets of regular and minus continued fractions with positive partial quotients

Dawoud Ahmadi Dastjerdi, Sanaz Lamei

Arabian Journal of Mathematics > 2012 > 1 > 2 > 139-148

An estimate for the Hausdorff dimension of $${x\, \in \, \mathbb{R}}$$ whose partial quotients of its regular continued fraction or minus continued fraction (MCF) are in $${E \, \subseteq \, \mathbb{N}}$$ is given. This enables us to give a new proof for the Texan conjecture on $${[0,\,\frac{1}{2}]}$$ which is valid for both regular and MCF. Also we show that if $${E \, \subseteq...

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Fundamental relations in simple and 0-simple semihypergroups of small size

Dario Fasino, Domenico Freni

Arabian Journal of Mathematics > 2012 > 1 > 2 > 175-190

We consider the fundamental relations β and γ in simple and 0-simple semihypergroups, especially in connection with certain minimal cardinality questions. In particular, we enumerate and exhibit all simple and 0-simple semihypergroups having order 3 where β is not transitive, apart of isomorphisms. Moreover, we show that the least order for which there exists a strongly simple semihypergroup where...

INFONA - science communication portal

Arabian Journal of Mathematics

On the category of modules over some semisimple bialgebras

On injective and divisible modules

The subgroup graph of a group

Ascending the divided and going-down properties by absolute flatness

Almost principal ideals in R[x]

Does Int $${(\mathbb {Z})}$$ have the stacked bases property?

Finitistic weak dimension of commutative arithmetical rings

A soft introduction to algebraic entropy

Three frameworks for a general theory of factorization

On the delta set and the Betti elements of a BF-monoid

R-smash products of Hopf quasigroups

Integral domains with Boolean t-class semigroup

The maximal correlation for the generalized order statistics and dual generalized order statistics

Numerical solutions of doubly perturbed stochastic delay differential equations driven by Lèvy process

Delay-dependent exponential stability results for uncertain stochastic Hopfield neural networks with interval time-varying delays

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

A note on zero-divisors of commutative rings

Weak Hopf algebras corresponding to quantum algebras U _q(f (K, H))

Dimension of certain sets of regular and minus continued fractions with positive partial quotients

Fundamental relations in simple and 0-simple semihypergroups of small size

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