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# Monomorphisms in categories of firm acts

Authors: Valdis Laan and Ülo Reimaa

## 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.

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# On the problem of Pillai with Padovan numbers and powers of 3

## Abstract

Let {P n}n≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = 1, P 2 = 1, and Pn +3 = Pn +1 + Pn for all 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.

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# Pre-Schwarzian norm for linear operators of uniformly convex functions of order α and type β

Authors: Jacek Dziok and Hanaa M. Zayed

## Abstract

By making use of the pre-Schwarzian norm given by

$‖f‖ = supz∈U (1−|z|2) |f′′(z)f′(z)|,$
we obtain such norm estimates for Hohlov operator of functions belonging to the class of uniformly convex functions of order 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.

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# Various notions of represetability for cylindric and polyadic algebras

Author: Tarek Sayed Ahmed

## Abstract

For β an ordinal, let PEAβ (SetPEAβ) denote the class of polyadic equality (set) algebras of dimension β. We show that for any infinite ordinal α, if $A ∈PEAα$ is atomic, then for any n < ω, the n-neat reduct of $A$, in symbols $ℜrnA→B$, is a completely representable PEAn (regardless of the representability of $A$). That is to say, for all non-zero $a∈ℜrnA$, there is a $Ba∈SetPEAn$ and a homomorphism $fa:ℜrnA→B$ such that fa(a) ≠ 0 and $fa(∑X) =∪x∈Xfa(x)$ for any $X ⊂= A$ for which $∑X$ exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under ≡∞,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 CAns 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.

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# Almost sure central limit theorems for weighted dependent sequences

Author: Khurelbaatar Gonchigdanzan

## Abstract

Let {Xn: n ≧ 1} be a sequence of dependent random variables and let {wnk: 1 ≦ kn, 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.

$limn→∞1log n∑k=1n1kI(∑i=1kwkiXi≦x)=12π∫−∞xe−12t2dt a.s..$

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# Associative rings in which nilpotents form an ideal

Author: Orest D. Artemovych

## 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).

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# Commutators of sublinear operators in grand morrey spaces

Authors: Vakhtang Kokilashvili, Alexander Meskhi and Humberto Rafeiro

## 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.

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# Eigenstructure for binomial operators

Authors: Shifen Wang and Chungou Zhang

## 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.

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# Groups with restrictions on proper uncountable subgroups

Authors: Francesco De Giovanni and Marco Trombetti

## Abstract

A group G is called metahamiltonian if all its non-abelian subgroups are normal. The aim of this paper is to investigate the structure of uncountable groups of cardinality ℵ in which all proper subgroups of cardinality ℵ are metahamiltonian. It is proved that such a group is metahamiltonian, provided that it has no simple homomorphic images of cardinality ℵ. Furthermore, the behaviour of elements of finite order in uncountable groups is studied in the second part of the paper.

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# The odd Nadarajah-Haghighi family of distributions: properties and applications

Authors: Abraão D. C. Nascimento, Kássio F. Silva, Gauss M. Cordeiro, Morad Alizadeh, Haitham M. Yousof and G. G. Hamedani

## Abstract

We study some mathematical properties of a new generator of continuous distributions called the Odd Nadarajah-Haghighi (ONH) family. In particular, three special models in this family are investigated, namely the ONH gamma, beta and Weibull distributions. The family density function is given as a linear combination of exponentiated densities. Further, we propose a bivariate extension and various characterization results of the new family. We determine the maximum likelihood estimates of ONH parameters for complete and censored data. We provide a simulation study to verify the precision of these estimates. We illustrate the performance of the new family by means of a real data set.

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