Throughout, X := {X ï¿¿ } ï¿¿≥0 denotes a Lévy process on R ï¿¿ with triple (ï¿¿ ï¿¿ σï¿¿ ï¿¿), and exponent Ψ. And from now on, we let {ï¿¿ ï¿¿ } ï¿¿≥0 denote the natural filtration of X, all the time remembering that, in accord with our earlier convention, {ï¿¿ ï¿¿ } ï¿¿≥0 satisfies the usual conditions. Definition 1. The transition measures of X are the probability measures P ï¿¿ (ï¿¿ ï¿¿ A)...

Various consistency proofs for the kernel density estimator have been developed over the last few decades. Important milestones are the pointwise consistency and almost sure uniform convergence with a fixed bandwidth on the one hand and the rate of convergence with a fixed or even a variable bandwidth on the other hand. While considering global properties of the empirical distribution functions...

let $gamma$ be a normal subgroup of the full automorphism group $aut(g)$ of a group $g$‎, ‎and assume that $inn(g)leq gamma$‎. ‎an endomorphism $sigma$ of $g$ is said to be {it $gamma$-central} if $sigma$ induces the the identity on the factor group $g/c_g(gamma)$‎. ‎clearly‎, ‎if $gamma=inn(g)$‎, ‎then a $gamma$-central endomorphism is a {it central} endomorphism‎. ‎in this article the conditi...

Let Mk be a right k-module with endomorphism ring E = End(Mk). We prove that if E is an Abelian exchange ring then Mk has the full exchange property. We also give an extension of this result in the case E is regular.

The problem of computing the kernel of a finite monoid, popularized under the name “Rhodes type II conjecture”, led Ash to the proof of a strong property (the hyperdecidability) of the pseudovariety G of all finite groups. This paper is a survey of recent results on the study of some related properties of pseudovarieties of groups.

In this paper, we introduce and investigate the notion of projection invariant semisimple modules. Some structural properties aforementioned class modules are studied. We obtain indecomposable decompositions former under some module theoretical conditions. Moreover, explore when finite exchange property implies full for Finally, that endomorphism ring a is ?- Baer ring.