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Let {F n}n≥0 be the sequence of Fibonacci numbers. The aim of this paper is to give linear independence results over (5) for the infinite series n=1χj(n)/Fn with certain nonprincipal real Dirichlet characters χ j. 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

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

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

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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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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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

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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In [14] we investigated some Vilenkin—Nörlund means with non-increasing coefficients. In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy space H 1/(1+α) to the space weak-L 1/(1+α), (0 < α ≦ 1). In this paper we construct a martingale in the space H 1/(1+α), which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin—Nörlund means with non-increasing coefficients are not bounded from the Hardy space H 1/(1+α) to the space L 1/(1+α). In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.

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We present here characterizations of the most recently introduced continuous univariate distributions based on: (i) a simple relationship between two truncated moments; (ii) truncated moments of certain functions of the 1th order statistic; (iii) truncated moments of certain functions of the n th order statistic; (iv) truncated moment of certain function of the random variable. We like to mention that the characterization (i) which is expressed in terms of the ratio of truncated moments is stable in the sense of weak convergence. We will also point out that some of these distributions are infinitely divisible via Bondesson’s 1979 classifications.

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On M p -embedded primary subgroups of finite groups

A subgroup H of G is called M p-embedded in G, if there exists a p-nilpotent subgroup B of G such that H p ∈ Sylp(B) and B is M p-supplemented in G. In this paper, we use M p-embedded subgroups to study the structure of finite groups.

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