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# The countable type properties in free paratopological groups

Studia Scientiarum Mathematicarum Hungarica
Authors: Fucai Lin, Chuan Liu, and Kexiu Zhang

A space X is of countable type (resp. subcountable type) if every compact subspace F of X is contained in a compact subspace K that is of countable character (resp. countable pseudocharacter) in X. In this paper, we mainly show that: (1) For a functionally Hausdorff space X, the free paratopological group FP(X)and the free abelian paratopological group AP(X) are of countable type if and only if X is discrete; (2) For a functionally Hausdorff space X, if the free abelian paratopological group AP(X) is of subcountable type then X has countable pseudocharacter. Moreover, we also show that, for an arbitrary Hausdorff μ-space X, if AP 2(X) or FP 2(X) is locally compact, then X is a topological sum of a compact space and a discrete space.

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# Existence of three solutions for impulsive perturbed elastic beam fourth-order equations of Kirchhoff-type

Studia Scientiarum Mathematicarum Hungarica
Authors: Shapour Heidarkhani and Amjad Salari

In this paper, we study the existence of multiple solutions for a class of impulsive perturbed elastic beam equations of Kirchhoff-type. We give a new criteria for guaranteeing that the impulsive perturbed elastic beam equations of Kirchhoff-type have at least three generalized solutions by using a variational method and a critical points theorem of B. Ricceri.

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# Homoclinic orbits of 2nth-order difference equations involving p-Laplacian

Studia Scientiarum Mathematicarum Hungarica
Authors: Haiping Shi, Xia Liu, and Yuanbiao Zhang

By making use of the critical point theory, we establish some new existence criteria to guarantee that a 2nth-order nonlinear difference equation containing both advance and retardation with p-Laplacian has a nontrivial homoclinic orbit. Our conditions on the potential are rather relaxed, and some existing results in the literature are improved.

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# Linear independence results for the reciprocal sums of Fibonacci numbers associated with Dirichlet characters

Studia Scientiarum Mathematicarum Hungarica
Authors: Hiromi Ei, Florian Luca, and Yohei Tachiya

Let {Fn}n≥0 be the sequence of Fibonacci numbers. The aim of this paper is to give linear independence results over $ℚ(5)$ for the infinite series $∑n=1∞χj(n)/Fn$ with certain nonprincipal real Dirichlet characters χj. We also deduce the irrationality results for the special principal Dirichlet characters and for other multiplicative functions.

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# On multiple Borsuk numbers in normed spaces

Studia Scientiarum Mathematicarum Hungarica
Authors: Zsolt Lángi and Márton Naszódi

Hujter and Lángi defined the k-fold Borsuk number of a set S in Euclidean n-space of diameter d > 0 as the smallest cardinality of a family F of subsets of S, of diameters strictly less than d, such that every point of S belongs to at least k members of F.

We investigate whether a k-fold Borsuk covering of a set S in a finite dimensional real normed space can be extended to a completion of S. Furthermore, we determine the k-fold Borsuk number of sets in not angled normed planes, and give a partial characterization for sets in angled planes.

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# On rings with annihilator condition

Studia Scientiarum Mathematicarum Hungarica

In this paper we study rings R with the property that every finitely generated ideal of R consisting entirely of zero divisors has a nonzero annihilator. The class of commutative rings with this property is quite large; for example, noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose classical ring of quotients are von Neumann regular. We continue to study conditions under which right mininjective rings, right FP-injective rings, right weakly continuous rings, right extending rings, one sided duo rings, semiregular rings and semiperfect rings have this property.

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# On two conjectures on b-coloring of graph products

Studia Scientiarum Mathematicarum Hungarica
Authors: S. Francis Raj and T. Kavaskar

A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. It was conjectured in [10], that for any two graphs G and H, b(G[H]) ≦ b(G) − 1|V (H)| + Δ(H) + 1 and b(GH) ≦ max {b(G)(Δ(H) + 1), b(H) Δ(G) + 1)}, where G[H] and GH denotes the lexicographic product and the strong product of G and H, respectively. In this paper, we disprove both conjectures.

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# Walsh-Lebesgue points and restricted convergence of multi-dimensional Walsh-Fourier series

Studia Scientiarum Mathematicarum Hungarica
Author: Ferenc Weisz

A new concept of Walsh-Lebesgue points is introduced for higher dimensions and it is proved that almost every point is a modified Walsh-Lebesgue point of an integrable function. It is shown that the Walsh-Fejér means σ n f of a function fL 1[0, 1)d converge to f at each modified Walsh-Lebesgue point, whenever n→∞ and n is in a cone. The same is proved for other summability means, such as for the Weierstrass, Abel, Picard, Bessel, Cesàro, de La Vallée-Poussin, Rogosinski and Riesz summations.

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# Extended arcsine distribution to proportional data: Properties and applications

Studia Scientiarum Mathematicarum Hungarica
Authors: Gauss M. Cordeiro, Artur J. Lemonte, and Ana K. Campelo

We propose a new two-parameter continuous model called the extended arcsine distribution restricted to the unit interval. It is a very competitive model to the beta and Kumaraswamy distributions for modeling percentages, rates, fractions and proportions. We provide a mathematical treatment of the new distribution including explicit expressions for the ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating and quantile functions, Shannon entropy and order statistics. Maximum likelihood is used to estimate the model parameters and the expected information matrix is determined. We demonstrate by means of two applications to proportional data that it can give consistently a better fit than other important statistical models.

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# Livšic’s theorem for q-Sturm—Liouville operators

Studia Scientiarum Mathematicarum Hungarica
Authors: Hüseyin Tuna and Aytekin Eryilmaz

In this paper, we study dissipative q-Sturm—Liouville operators in Weyl’s limit circle case. We describe all maximal dissipative, maximal accretive, self adjoint extensions of q-Sturm—Liouville operators. Using Livšic’s theorems, we prove a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative q-Sturm—Liouville operators.

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