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Counting surface branched covers

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 3
DOI:
https://doi.org/10.1556/012.2019.56.3.1431
Authors: Carlo Petronio and Filippo Sarti

Abstract

To a branched cover f between orientable surfaces one can associate a certain branch datum D(f), that encodes the combinatorics of the cover. This D(f) satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible branch datum D there exists a branched cover f such that D(f)=D. One can actually refine this problem and ask 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.

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Good congruences on abundant semigroups with quasi-ideal adequate transversals

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 3
DOI:
https://doi.org/10.1556/012.2019.56.3.1434
Author: Aifa Wang

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.

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Monomorphisms in categories of firm acts

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 3
DOI:
https://doi.org/10.1556/012.2019.56.3.1432
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

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 3
DOI:
https://doi.org/10.1556/012.2019.56.3.1435
Author: Mahadi Ddamulira

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

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 3
DOI:
https://doi.org/10.1556/012.2019.56.3.1429
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

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 3
DOI:
https://doi.org/10.1556/012.2019.56.3.1436
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

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 2
DOI:
https://doi.org/10.1556/012.2019.56.2.1426
Author: Khurelbaatar Gonchigdanzan

Abstract

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

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 2
DOI:
https://doi.org/10.1556/012.2019.56.2.1428
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

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 2
DOI:
https://doi.org/10.1556/012.2019.56.2.1425
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

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 56: Issue 2
DOI:
https://doi.org/10.1556/012.2019.56.2.1423
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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