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## Abstract

To a branched cover *f* between orientable surfaces one can associate a certain *branch datum*
*f* such that *how many* these *f*'s exist, but one must of course decide what restrictions one puts on such *f*’s, and choose an equivalence relation up to which one regards them. As it turns out, quite a few natural choices for this relation are possible. In this short note we carefully analyze all these choices and show that the number of actually distinct ones is only three. To see that these three choices are indeed different from each other we employ Grothendieck's *dessins d'enfant.*

## Abstract

The aim of this paper is to study the congruences on abundant semigroups with quasi-ideal adequate transversals. The good congruences on an abundant semigroup with a quasi-ideal adequate transversal *S*° are described by the equivalence triple abstractly which consists of equivalences on the structure component parts *I, S*° and Λ. Also, it is shown that the set of all good congruences on this kind of semigroup forms a complete lattice.

## Abstract

We prove that in the category of firm acts over a firm semigroup monomorphisms co-incide with regular monomorphisms and we give an example of a non-injective monomorphism in this category. We also study conditions under which monomorphisms are injective and we prove that the lattice of subobjects of a firm act over a firm semigroup is isomorphic to the lattice unitary subacts of that act.

## Abstract

Let {*P _{n}*}

_{n≥0}be the sequence of Padovan numbers defined by

*P*

_{0}= 0,

*P*

_{1}= 1,

*P*

_{2}= 1, and

*P*

_{n}_{+3}=

*P*

_{n}_{+1}+

*P*for all

_{n}*n*≥ 0. In this paper, we find all integers

*c*admitting at least two representations as a difference between a Padovan number and a power of 3.

## Abstract

*α*and type

*β*. We also employ an entirely new method to generalize and extend the results of Theorems 1, 2 and 3 in . Finally, some inequalities concerning the norm of the pre-Schwarzian derivative for Dziok-Srivastava operator are also considered.

## Abstract

For *β* an ordinal, let PEA_{β} (SetPEA_{β}) denote the class of polyadic equality (set) algebras of dimension *β*. We show that for any infinite ordinal *α*, if *n* <* ω*, the *n*-neat reduct of _{n} (regardless of the representability of *f _{a}*(

*a*) ≠ 0 and

_{∞,ω}. Various notions of representability (such as ‘satisfying the Lyndon conditions’, weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CA

_{n}s satisfying the Lyndon conditions coincides with the class of atomic algebras in

**ElS**

_{c}

**Nr**

_{n}

**CA**

_{ω}, where

**El**denotes ‘elementary closure’ and

**S**

_{c}is the operation of forming complete subalgebras.

## Abstract

*X*:

_{n}*n*≧ 1} be a sequence of dependent random variables and let {

*w*: 1 ≦

_{nk}*k*≦

*n, n*≧ 1} be a triangular array of real numbers. We prove the almost sure version of the CLT proved by Peligrad and Utev [7] for weighted partial sums of mixing and associated sequences of random variables, i.e.

## Abstract

It is shown that if *N*(*R*) is a Lie ideal of *R* (respectively Jordan ideal and *R* is 2-torsion-free), then *N*(*R*) is an ideal. Also, it is presented a characterization of Noetherian NR rings with central idempotents (respectively with the commutative set of nilpotent elements, the Abelian unit group, the commutative commutator set).

## Abstract

In this paper we establish the boundedness of commutators of sublinear operators in weighted grand Morrey spaces. The sublinear operators under consideration contain integral operators such as Hardy-Littlewood and fractional maximal operators, Calderón-Zygmund operators, potential operators etc. The operators and spaces are defined on quasi-metric measure spaces with doubling measure.

## Abstract

In this article, the eigenvalues and eigenvectors of positive binomial operators are presented. The results generalize the previously obtained ones related to Bernstein operators. Illustrative examples are supplied.