On Some Conditions for Complete Convergence for Arrays of Rowwise Negatively Dependent Random Variables

Abstract

This note contains some sufficient conditions for the complete convergence in the strong law of large numbers for arrays of rowwise negatively dependent random variables. Moreover, the rowwise sums are randomly indexed.     

Regular F-Abundant Semigroups

ABSTRACT

The investigation of regular F-abundant semigroups is initiated. In fact, F-abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F-abundant semigroups is obtained. In addition, we also prove that a regular F-abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F-inverse semigroups.     

On the Associated Primes of Matlis Duals of Top Local Cohomology Modules

After motivating the question, we prove various results about the set of associated primes of Matlis duals of top local cohomology modules. In some cases, we can calculate this set. An easy application of this theory is the well-known fact that Krull dimension can be expressed by the vanishing of local cohomology modules.     

Regular Action in Rings

Let R be a ring with identity, X(R) the set of all nonzero non-units of R and G(R) the group of all units of R. By considering left and right regular actions of G(R) on X(R), the following are investigated: (1) For a local ring R such that X(R) is a union of n distinct orbits under the left (or right) regular action of G(R) on X(R), if J ⁿ  ≠ 0 = J ⁿ⁺¹ where J is the Jacobson radical of R, then the set of all the distinct ideals of R is exactly {R, J, J ²,…, J ⁿ , 0}, and each orbit under the left regular action is equal to the one under the right regular action. (2) Such a ring R is left (and right) duo ring. (3) For the full matrix ring S of n × n matrices over a commutative ring R, the number of orbits under left regular action of G(S) on X(S) is equal to the number of orbits under right regular action of G(S) on X(S); the result also holds for the ring of n × n upper triangular matrices over R.     

The Join of Split Graphs Whose Regular Endomorphisms Form a Monoid

In this paper, the regular endomorphisms of the join of split graphs are investigated. We give a condition under which the regular endomorphisms of the join of split graphs form a monoid.     

A Family of Graded Modules Associated to a Module

In this article we introduce a certain family of graded modules associated to a given module. These modules provide a natural extension of the notion of the associated graded ring of an ideal. We will investigate their properties. In particular, we will try to extend Ree' theorem on the associated graded ring of an ideal generated by a regular sequence to this context.     

Limit theorems for a class of critical superprocesses with stable branching

We consider a critical superprocess $\{X;{\mathbf{P}}_{\mu }\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index ${\gamma }_0>1$. We first show that, under some conditions, ${\mathbf{P}}_{\mu }(\left|X_t\right|\ne 0)$ converges to 0 as $t\to \infty$ and is regularly varying with index ${({\gamma }_0-1)}^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\{X_t\left(f\right);{\mathbf{P}}_{\mu }\left(\cdot \right|6X_{t}6\ne 0)\}$, after appropriate rescaling, converges weakly to a positive random variable ${\mathbf{z}}^{({\gamma }_0-1)}$ with Laplace transform $E\left[e^{-u{\mathbf{z}}^{({\gamma }_0-1)}}\right]=1-{(1+u^{-({\gamma }_0-1)})}^{-1∕({\gamma }_0-1)}.$