Band 73, Heft 4

ABSTRACT We show that there is a set of 2 2 κ ultrafilters incomparable in the Rudin-Frolík order of βκ \ κ , where κ is regular, for which no subset with more than one element has an infimum.

ABSTRACT Our goal in this paper is to compute the 2-rank of the class group of the fields K=ℚ(−2,q,d) \[\mathbb{K}=\mathbb{Q}\left( \sqrt{-2},\sqrt{q},\sqrt{d} \right)\] for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2, 2), (2, 4) or (2, 2, 2).

ABSTRACT Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set [ n ] = {1, 2, … n }, this paper introduces the deranged partition as a free-fixed-block permutation of its blocks, then defines the deranged Bell numbers that count the total number of the deranged partitions of [ n ]. At first, we study the classical properties of these numbers (generating function, explicit formula, convolutions, etc.), we then present an asymptotic behavior of the deranged Bell numbers. Finally, we briefly review some results regarding the r -extension of these numbers by considering that the r first elements must be in distinct blocks.

ABSTRACT Let K be a number field generated by a root θ of a monic irreducible trinomial F(x)=xn+axm+b∈ℤ[x] . In this paper, we study the problem of monogenity of K . More precisely, we provide some explicit conditions on a, b, n , and m for which K is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree n = 2 r · 3 k with r and k two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this paper.

ABSTRACT Let k ≥ 2 be a fixed integer. The k -generalized Lucas sequence { Ln(k) }n≥0 \[{{\left\{ L_{n}^{\left( k \right)} \right\}}_{n\ge 0}}\] starts with the positive integer initial values k , 1, 3, …, 2 k −1 – 1, and each term afterward is the sum of the k consecutive preceding elements. An integer n is said to be close to a positive integer m if n satisfies | n−m |<m \[\left| n-m \right|<\sqrt{m}\] . In this paper, we combine these two concepts. We solve completely the diophantine inequality | Ln(k)−2m |<2m/2 \[\left| L_{n}^{\left( k \right)}-{{2}^{m}} \right|<{{2}^{m/2}}\] in the non-negative integers k , n , and m . This problem is equivalent to the resolution of the equation Ln(k)=2m+t \[L_{n}^{\left( k \right)}={{2}^{m}}+t\] with the condition | t | < 2 m /2 , t∈ℤ \[t\in \mathbb{Z}\] . We also discovered a new formula for Ln(k) \[L_{n}^{\left( k \right)}\] which was very useful in the investigation of one particular case of the problem.

ABSTRACT In this note, for any integers m, n ≥ 2, we find a condition on a positive integer c under which there exists a monic polynomial f∈ℤ[x] of degree n for which f ( x ) m – c has mn integral roots counting with multiplicities. This is the case if and only if m = 2 and c is a constant that comes from a solution of the Prouhet-Tarry-Escott problem of size n . For example, the smallest positive integer c for which there exists a monic degree 7 polynomial f∈ℤ[x] such that f ( x ) 2 – c has 14 integral roots is c = 6620176679276160000.

ABSTRACT We present a remarkably simple and surprisingly natural interpretation of the values of zeta functions at negative integers and zero. Namely, we are able to relate these values to areas related to partial sums of powers. We apply these results to further interpretations of values of L -functions at negative integers. We hint in a very brief way at some expected connections of this work with other current efforts to understand the mysteries of the values of zeta functions at integers.

ABSTRACT The classical memoryless property is well-known to induce the geometric distribution for discrete random variables and the exponential distribution for continuous random variables. When the range of the random variable is just an arbitrary closed subset of the real line, significant difficulties arise as to the interpretation of the memoryless property itself. Multiple proposals have been made, and we explore their consequences. In particular, we consider whether a given definition of “memoryless” will induce a specific distribution.

ABSTRACT In this paper, we study the higher-order Schwarzian derivative S n ( f ) proposed by H. Tamanoi [ Higher Schwarzian operators and combinatorics of the Schwarzian derivative , Math. Ann. 305 (1996), 127–151]. For the strongly starlike functions of order α and strongly convex functions of order α , the sharp bound of | S 3 ( f )(0)| is obtained. When n ∈ [2, 7], we prove that the higher Bers maps induced by S n ( f ) on Weil-Petersson Teichmüller space and BMO-Teichmüller space are holomorphic.

ABSTRACT We prove a lower bound estimate for Hajłasz-Besov capacity in metric spaces in terms of Netrusov-Hausdorff content. We also prove a similar estimate for Hajłasz-Triebel-Lizorkin capacity in terms of Hausdoroff content. These results are improvements of the earlier results obtained by Nuutinen in 2016 and the first author in 2020.

ABSTRACT In this paper new oscillatory criteria for odd order linear functional differential equations of the type y(n)(t)+p(t)y(τ(t))=0 \[{{y}^{(n)}}(t)+p(t)y(\tau (t))=0\] have been established. Deviating argument τ ( t ) is supposed to have dominating delay part.

ABSTRACT Here, the solvability of the following inclusion elliptic problem −ΔHn,pu∈F(ξ,u)in  Ω,u=0on  ∂Ω, $$\left\{ \begin{array}{lll} -\Delta _{\mathbb{H}^n, p}u\in \mathcal{F}(\xi, u)& \text{in} \ \Omega, \\ u=0 & \text{on} \ \partial \Omega, \end{array} \right.$$ is proved, via variational technique, where Ω is a Korányi ball in the Heisenberg Lie group ℍ n and 𝓕 : Ω × ℝ → 𝒫(ℝ) is a real set-valued mapping.

ABSTRACT The Dirichlet boundary value problem for elliptic equations involving the ( p ( x ), q ( x ))-Laplacian operator with a reaction term depending on the gradient and on two real parameters is considered in this paper. Using the topological degree theory for a class of demicontinuous operators of generalized ( S + ) and the theory of the variable exponent Sobolev spaces, we prove the existence of at least one weak solution of such problem.

ABSTRACT In a way different from traditional methods, recently, new series spaces have been established by using absolute summability methods. Among these absolute summability methods, the absolute generalized Nörlund summability method | N , p n , q n , θ n | k is extremely important as it includes several summability methods such as absolute Nörlund, weighted, Cesàro and Euler summability methods. Now, it is natural to ask behavior of the space derived by the absolute summability of generalized Nörlund means. The purpose of this study is to derive a more general series space including some well known spaces such as the spaces |N˜qθ|k ${\left| {\tilde N_q^\theta } \right|_k}$ , | C α | k , | C α , β | k , |Npθ|k ${\left| {N_p^\theta } \right|_k}$ and |Erθ|k ${\left| {E_r^\theta } \right|_k}$ , and to investigate its topological structure, α-, β- and γ-duals. Beside this, some matrix transformations related to these spaces have been characterized.

ABSTRACT In [ACG+1] and [ACG+2], it is shown that a complement of a neighborhood of a partially smoothed total toric divisor of a closed toric symplectic 4-manifold is a Weinstein domain. In this article, we extend the family of Weinstein domains that can be realized as such complements by producing an infinite family of new examples.

ABSTRACT Kathryn F. Porter wrote a nice paper about several definitions of local homogeneity [ Local homogeneity , JP Journal of Geometry and Topology 9 (2009), 129–136]. In this paper, she mentions that G. S. Ungar defined a uniformly locally homogeneous space [ Local homogeneity , Duke Math. J. 34 (1967), 693–700]. We realized that this notion is very similar to what we call the uniform property of Effros [ On Jones’ set function 𝒯 and the property of Kelley for Hausdorff continua , Topology Appl. 226 (2017), 51–65]. Here, we compare the uniform property of Effros with the uniform local homogeneity. We also consider other definitions of local homogeneity given in Porter’s paper and compare them with the uniform property of Effros. We show that in the presence of compactness, the uniform property of Effros is equivalent to uniform local homogeneity and the local homogeneity according to Ho. With this result, we can change the hypothesis of the uniform property of Effros in Jones’ and Prajs’ decomposition theorems to uniform local homogeneity and local homogeneity according to Ho. We add to these two results the fact that the corresponding quotient space also has the uniform property of Effros.

ABSTRACT In this work, suppose that { X n ; n ≥ 1}is a sequence of asymptotically negatively associated random variables and { a ni ; 1 ≤ i ≤ n , n ≥ 1} is an array of real numbers such that ∑i=1n|ani|q=O(n) for some q > max {αp−1α−1/2,2} with αp > 1 and α>12 . Let l ( x ) > 0 be a slowly varying function at infinity. We establish some equivalent conditions of the complete convergence for weighted sums of this form ∑n=1∞nαp−2l(n)P(max1≤j≤n|∑i=1janiXi|>εnα)<∞ for all ε > 0. As applications, some strong laws of large numbers for weighted sums of asymptotically negatively associated random variables are also obtained.

ABSTRACT This paper proposes a regression model for the continuous responses bounded to the unit interval which is based on the unit-Gompertz distribution as an alternative to quantile regression models based on the unit-Birnbaum-Saunders, unit-Weibull, L-Logistic, Kumaraswamy and Johnson SB distributions. Re-parameterizing the unit-Gompertz distribution as a function of its quantile allows us to model the effect of covariates across the entire response distribution, rather than only at the mean. Our proposal sometimes outperforms the other distributions available in the literature. These discoveries are provided by Monte Carlo simulations and one application using a real data set. An R package, including parameter estimation, model checking as well as density, cumulative distribution, quantile and random number generating functions of the unit-Gompertz distribution are developed and can be readily used in applications.

ABSTRACT In this research paper, we introduce an extension of the Gamma-Lindley distribution using a particular exponentiation of its cumulative distribution function, which offers a more flexible model for lifetime data. Another attractive feature of this extension is that it has several particular cases: Weibull, generalized Pareto, Lindley, exponential and Gamma-Lindley distributions. Different statistical properties of this distribution are explored, such as the density, failure rate and the r th moments. We attempt to prove that the extended Gamma-Lindley distribution is characterized by its truncated moment of order statistics. One of the main merits of this work lies in the fact that it provides the ability of this characterization to simulate the new distribution compared with the inverse transform sampling method. Estimation of the parameters using the tailed regression method is investigated. The COVID-19 real data evolution in Tunisia illustrates the performance of the EGL distribution compared with its special cases through some criteria such as the Kolmogorov-Smirnov test, Mean Squared error and Kullback-Leibler divergence. The predictive ability of the extended Gamma-Lindley distribution proved to provide a better fit to expect the number of deaths by Corona-virus in Tunisia in the future period.

ABSTRACT Modeling bivariate data with different marginals is an important problem and have numerous applications in diverse disciplines. This paper introduces a new family of bivariate generalized linear exponential Weibull distribution having generalized linear and exponentiated Weibull distributions as marginals. Some important quantities like conditional distributions, conditional moments, product moments and bivariate quantile functions are derived. Concepts of reliability and measures of dependence are also discussed. The methods of maximum likelihood and Bayesian estimation are considered to estimate model parameters. Monte Carlo simulation experiments are performed to demonstrate the performance of the estimators. Finally, a real data application is also discussed to demonstrate the usefulness of the proposed distribution in real-life situations.