Volume 71, Issue 6

In 2012, the second author introduced, and initiated the investigations into, the variety 𝓘 of implication zroupoids that generalize De Morgan algebras and ∨-semilattices with 0. An algebra A = 〈 A , →, 0 〉, where → is binary and 0 is a constant, is called an implication zroupoid (𝓘-zroupoid, for short) if A satisfies: ( x → y ) → z ≈ [( z ′ → x ) → ( y → z )′]′, where x ′ := x → 0, and 0″ ≈ 0. Let 𝓘 denote the variety of implication zroupoids and A ∈ 𝓘. For x , y ∈ A , let x ∧ y := ( x → y ′)′ and x ∨ y := ( x ′ ∧ y ′)′. In an earlier paper, we had proved that if A ∈ 𝓘, then the algebra A mj = 〈 A , ∨, ∧〉 is a bisemigroup. The purpose of this paper is two-fold: First, we generalize the notion of semidistributivity from lattices to bisemigroups and prove that, for every A ∈ 𝓘, the bisemigroup A mj is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety 𝓜𝓔𝓙 of 𝓘, defined by the identity: x ∧ y ≈ x ∨ y , satisfies the Whitman Property. We conclude the paper with two open problems.

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

We introduce the concept of Z-symmetric rings. In fact, the classes of all semicommutative rings, nil rings, reduced rings, Artinian rings and eversible rings are Z-symmetric rings. In order to sustain our assertion, we provide a number of examples of Z-symmetric and non Z-symmetric rings. We observe that the class of Z-symmetric rings lies strictly between the classes of eversible rings and the Dedekind finite rings. In particular, we consider the extensions of Z-symmetric rings. Finally, some new results between the Z-symmetric rings and Armendariz rings will be explored and investigated.

Let F denote the factorable matrix and X ∈ { ℓ p , c 0 , c , ℓ ∞ }. In this study, we introduce the domains X ( F ) of the factorable matrix in the spaces X . Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X ( F ). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes ( ℓ p ( F ), ℓ ∞ ), ( ℓ p ( F ), f ) and ( X , Y ( F )) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F . Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

In a very recent paper, Wang and Bulut [ A note on the coefficient estimates of bi-close-to-convex functions , C. R. Acad. Sci. Paris, Ser. I 355 (2017), 876–880] determined the estimates for the general Taylor-Maclaurin coefficients of functions belonging to the bi-close-to-convex function class. In this study, we introduce the class of Libera type bi-close-to-convex functions and obtain the upper bounds for the coefficients of functions belonging to this class. Our results generalize the results in the above mentioned paper.

A class of third order differential equations with several sublinear neutral terms of the type (a(t)(b(t)(x(t)+∑ j=1npj(t)xα j(τ j(t)))′)′)′+∑ i=1mfi(t,x(σ i(t)))=0,t≥ t0> -->0 $$\begin{array}{} \displaystyle \bigg( a(t)\Big( b(t)\Big(x(t)+\sum\limits_{j=1}^{n}p_{j}(t)x^{\alpha _{j}}(\tau _{j}(t))\Big)'\Big)'\bigg)' +\sum\limits_{i=1}^{m}f_{i}(t,x(\sigma _{i}(t)))=0,\qquad t\geq t_{0} \gt 0 \end{array}$$ is considered. Some oscillation criteria are presented to improve and complement those in the literature. Two examples are established to illustrate the main results.

The aim of this article is to examine the solutions of the boundary value problem of the nonlinear elliptic equation ε 2 △ u = f ( u ). We describe the asymptotic behavior as ε tends to zero of the solutions on a spherical crown C of R N , ( N ≥ 2) in a direct non-classical formulation which suggests easy proofs. We propose to look for interesting solutions in the case where the condition at the edge of the crown is a constant function. Our results are formulated in classical mathematics.Their proofs use the stroboscopic method which is a tool of the nonstandard asymptotic theory of differential equations.

This paper deals with a class of fourth order elliptic equations of Kirchhoff type with variable exponent Δ p(x)2u− M(∫ Ω 1p(x)|∇ u|p(x)dx)Δ p(x)u+|u|p(x)− 2u=λ f(x,u)+μ g(x,u) in Ω ,u=Δ u=0 on ∂ Ω , $$\begin{array}{} \left\{\begin{array}{} \Delta^2_{p(x)}u-M\bigg(\displaystyle\int\limits_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\,\text{d} x \bigg)\Delta_{p(x)} u + |u|^{p(x)-2}u = \lambda f(x,u)+\mu g(x,u) \quad \text{ in }\Omega,\\ u=\Delta u = 0 \quad \text{ on } \partial\Omega, \end{array}\right. \end{array}$$ where p− :=infx∈ Ω ¯ p(x)> -->max1,N2,λ > -->0 $\begin{array}{} \displaystyle p^{-}:=\inf_{x \in \overline{\Omega}} p(x) \gt \max\left\{1, \frac{N}{2}\right\}, \lambda \gt 0 \end{array}$ and μ ≥ 0 are real numbers, Ω ⊂ ℝ N ( N ≥ 1) is a smooth bounded domain, Δ p(x)2u=Δ (|Δ u|p(x)− 2Δ u) $\begin{array}{} \displaystyle \Delta_{p(x)}^2u=\Delta (|\Delta u|^{p(x)-2} \Delta u) \end{array}$ is the operator of fourth order called the p ( x )-biharmonic operator, Δ p ( x ) u = div(|∇ u | p ( x )–2 ∇ u ) is the p ( x )-Laplacian, p : Ω → ℝ is a log-Hölder continuous function, M : [0, +∞) → ℝ is a continuous function and f , g : Ω × ℝ → ℝ are two L 1 -Carathéodory functions satisfying some certain conditions. Using two kinds of three critical point theorems, we establish the existence of at least three weak solutions for the problem in an appropriate space of functions.

In this paper, we are concerned with the existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities. By using the upper and lower solution method and Schauder’s fixed point theorem, we establish the existence of traveling wave solutions. To illustrate our results, the existence of traveling wave solutions for a nonlocal delayed higher-dimensional lattice cooperative system with two species are considered.

In a recent paper, [Sarigöl, M. A.: Characterization of summability methods with high indices , Math. Slovaca 63 (2013), 1053–1058], the equivalence ∣ C , 0∣ k ⟺ ∣ R , p n ∣ k , k ≥ 1, was characterized for infinite single series. In the present paper, this result is extended to doubly summability a different approach.

We investigate topological conditions on a compact Hausdorff space Y , such that any lattice isomorphism φ : C ( X , I ) → C ( Y , I ), where X is a compact Hausdorff space and I is the unit interval [0, 1], is continuous. It is shown that in either of cases that the set of G δ points of Y has a dense pseudocompact subset or Y does not contain the Stone-Čech compactification of ℕ, such a lattice isomorphism is a homeomorphism.

This paper presents new fixed point theorems for 2 × 2 block operator matrix with countably condensing or countably 𝓓-set-contraction multi-valued inputs. Our theory will then be used to establish some new existence theorems for coupled system of functional differential inclusions in general Banach spaces under weak topology. Our results generalize, improve and complement a number of earlier works.

In this article, we prove some new common fixed point results under the generalized contraction condition using w -distance and weak altering distance functions. Also, the validity of the results is demonstrated by an example along with numerical experiment for approximating the common fixed point. Later, as applications, the unique common solutions for the system of nonlinear Fredholm integral equations, nonlinear Volterra integral equations and nonlinear fractional differential equations of Caputo type are derived.

A representation formula for locally defined operators acting between Banach spaces of continuous functions of bounded variation in the Waterman sense is presented. Moreover, the Nemytskij composition operators will be investigated and some consequences for locally bounded as well as uniformly bounded local operators will be given.

In this paper, we characterize almost co-Kähler manifolds with a conformal vector field. It is proven that if an almost co-Kähler manifold has a conformal vector field that is collinear with the Reeb vector field, then the manifold is a K -almost co-Kähler manifold. It is also shown that if a ( κ , μ )-almost co-Kähler manifold admits a Killing vector field V , then either the manifold is K -almost co-Kähler or the vector field V is an infinitesimal strict contact transformation, provided that the (1,1) tensor h remains invariant under the Killing vector field.

In this paper, we give the complete classification of real hypersurfaces in a nonflat complex space form from the viewpoint of the η -parallelism of the tensor field h (= (1/2)𝓛 ξ ϕ ). In addition we investigate real hypersurfaces whose tensor h is either Killing type or transversally Killing tensor. In particular, we shall determine Hopf hypersurfaces whose tensor h is transversally Killing tensor by using an application of the classification of real hypersurfaces admitting η -parallelism with respect to the tensor h .

In this paper, we present a new parametric class of distributions based on the log-alpha-power distribution, which contains the well-known log-normal distribution as a special case. This new family is useful to deal with unimodal as well as bimodal data with asymmetry and kurtosis coefficients ranging far from that expected based on the log-normal distribution. The usual approach is considered to perform inferences, and the traditional maximum likelihood method is employed to estimate the unknown parameters. Monte Carlo simulation results indicate that the maximum likelihood approach is quite effective to estimate the model parameters. We also derive the observed and expected Fisher information matrices. As a byproduct of such study, it is shown that the Fisher information matrix is nonsingular throughout the sample space. Empirical applications of the proposed family of distributions to real data are provided for illustrative purposes.

The main aim of this paper is to introduce a new class of continuous generalized exponential distributions, both for the univariate and bivariate cases. This new class of distributions contains some newly developed distributions as special cases, such as the univariate and also bivariate geometric generalized exponential distribution and the exponential-discrete generalized exponential distribution. Several properties of the proposed univariate and bivariate distributions, and their physical interpretations, are investigated. The univariate distribution has four parameters, whereas the bivariate distribution has five parameters. We propose to use an EM algorithm to estimate the unknown parameters. According to extensive simulation studies, we see that the effectiveness of the proposed algorithm, and the performance is quite satisfactory. A bivariate data set is analyzed and it is observed that the proposed models and the EM algorithm work quite well in practice.

In this paper, we consider the solvability of the Pellian equation x2− (d2+1)y2=− m, $$\begin{array}{} \displaystyle x^2-(d^2+1)y^2 = -m, \end{array} $$ in cases d = n k , m = n 2 l −1 , where k , l are positive integers, n is a composite positive integer and d = pq , m = pq 2 , p , q are primes. We use the obtained results to prove results on the extendibility of some D (−1)-pairs to quadruples in the ring Z[− t] $\begin{array}{} {\mathbb{Z}}[\sqrt{-t}] \end{array} $, with t > 0.