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# Kumaraswamy odd Burr G family of distributions with applications to reliability data

Authors: Arslan Nasir, Hassan S. Bakouch and Farrukh Jamal

We exhibit a general family of distributions named “Kumaraswamy odd Burr G family of distributions” with four additional parameters to generalize any existing baseline distribution. Some statistical properties of the family are derived, including r th moments, m th incomplete moments, moment generating function and entropies. The parameters of the family are estimated by the maximum likelihood (ML) method for complete sam- ples as well as censored samples. Some sub-models of the family are considered and it is noted that their density functions can be symmetric, left-skewed, right-skewed, unimodal, bimodal and their hazard rate functions can be increasing, decreasing, bathtub, upside- down bathtub and J-shaped. Simulation is carried out for one of the sub-models to check the asymptotic behavior of the ML estimates. Applications to reliability (complete and censored) data are carried out to check the usefulness of some sub-models of the family.

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# New limit results related to the coupon collector’s problem

Authors: Lenka Glavaš and Pavle Mladenović

We study point processes associated with coupon collector’s problem, that are defined as follows. We draw with replacement from the set of the first n positive integers until all elements are sampled, assuming that all elements have equal probability of being drawn. The point process we are interested in is determined by ordinal numbers of drawing elements that didn’t appear before. The set of real numbers is considered as the state space. We prove that the point process obtained after a suitable linear transformation of the state space converges weakly to the limiting Poisson random measure whose mean measure is determined.

We also consider rates of convergence in certain limit theorems for the problem of collecting pairs.

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# On certain lattices of varieties of completely regular semigroups

Author: Mario Petrich

The class CR of completely regular semigroups considered as algebras with binary multiplication and unary operation of inversion forms a variety. Kernel, trace, local and core relations, denoted by K, T, L and C, respectively, are quite useful in studying the structure of the lattice L(CR) of subvarieties of CR. They are equivalence relations whose classes are intervals. Their ends are used for defining operators on L(CR).

Starting with a few band varieties, we repeatedly apply operators induced by upper ends of classes of these relations and characterize corresponding classes up to certain variety low in the lattice L(CR). We consider only varieties whose origin are “central” band varieties, that is those in the middle column of the lattice L(B) of band varieties. Several diagrams represent the (semi)lattices studied.

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# On trigonometric sums with random frequencies

Authors: Alina Bazarova, István Berkes and Marko Raseta

We prove that if I k are disjoint blocks of positive integers and n k are independent random variables on some probability space (Ω,F,P) such that n k is uniformly distributed on I k, then $N−1/2∑k=1N(sin2πnkx−E(sin2πnkx))$ has, with P-probability 1, a mixed Gaussian limit distribution relative to the probability space ((0, 1),B, λ), where B is the Borel σ-algebra and λ is the Lebesgue measure. We also investigate the case when n k have continuous uniform distribution on disjoint intervals I k on the positive axis.

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# Approximation in weighted generalized grand smirnov classes

Authors: Daniyal M. Israfilov and Ahmet Testici

Let G be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂G. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes εp),θ(G,ω) and $εp),θ(G−,ω)$, 1 < p < ∞, in the term of the rth, r = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained.

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# A characterization of signed discrete infinitely divisible distributions

Authors: Huiming Zhang, Bo Li and G. Jay Kerns

In this article, we give some reviews concerning negative probabilities model and quasi-infinitely divisible at the beginning. We next extend Feller’s characterization of discrete infinitely divisible distributions to signed discrete infinitely divisible distributions, which are discrete pseudo compound Poisson (DPCP) distributions with connections to the Lévy-Wiener theorem. This is a special case of an open problem which is proposed by Sato (2014), Chaumont and Yor (2012). An analogous result involving characteristic functions is shown for signed integer-valued infinitely divisible distributions. We show that many distributions are DPCP by the non-zero p.g.f. property, such as the mixed Poisson distribution and fractional Poisson process. DPCP has some bizarre properties, and one is that the parameter λ in the DPCP class cannot be arbitrarily small.

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# The drazin invertibility of an anti-triangular matrix over a ring

Authors: Honglin Zou, Jianlong Chen and Dijana Mosić

Let R be a ring. The purpose of this paper is to study the existence and the representation for the anti-triangular matrix $[abc0]$ under some conditions, where a, b, cR. The results extend recent works given in the literature.

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# Generalized q-starlike functions

Authors: Khalida Inayat Noor and Sadia Riaz

In this paper, we introduce a new concept of q-bounded radius rotation and define the class R*m(q), m ≥ 2, q ∈ (0, 1). The class R*2(q) coincides with S*q which consists of q-starlike functions defined in the open unit disc. Distortion theorems, coefficient result and radius problem are studied. Relevant connections to various known results are pointed out.

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# Infinitely many solutions for semilinear Δλ-Laplace equations with sign-changing potential and nonlinearity

Authors: Jianhua Chen, Xianhua Tang and Zu Gao

In this paper, we prove the existence of infinitely many solutions for the following class of boundary value elliptic problems ${−Δλu+V(x)u=f(x,u),x∈Ω,u=0,x∈∂Ω,$ where Ω is a bounded domain in RN (N ≥ 2), Δλ is a strongly degenerate elliptic operator, V (x) is allowing to be sign-changing and f is a function with a more general super-quadratic growth, which is weaker than the Ambrosetti-Rabinowitz type condition.

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# On some differential subordinations

Authors: Mamoru Nunokawa and Janusz Sokół

The purpose of this work is to present a new geometric approach to some problems in differential subordination theory. We also discuss the new results closely related to the generalized Briot-Bouquet differential subordination.

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