• Open Math. (IF 0.773) Pub Date : 2020-09-09
Yu Zhang; Xinsheng Liu; Yuncai Yu; Hongchang Hu

In this article, an errors-in-variables regression model in which the errors are negatively superadditive dependent (NSD) random variables is studied. First, the Marcinkiewicz-type strong law of large numbers for NSD random variables is established. Then, we use the strong law of large numbers to investigate the asymptotic normality of least square (LS) estimators for the unknown parameters. In addition

更新日期：2020-09-15
• Open Math. (IF 0.773) Pub Date : 2020-09-10
Huiwen Wang; Fang Li

In this article, we discuss the nonlinear boundary value problems involving both left Riemann-Liouville and right Caputo-type fractional derivatives. By using some new techniques and properties of the Mittag-Leffler functions, we introduce a formula of the solutions for the aforementioned problems, which can be regarded as a novelty item. Moreover, we obtain the existence result of solutions for the

更新日期：2020-09-15
• Open Math. (IF 0.773) Pub Date : 2020-09-03
Zhongbi Wang; Chao Qin; Heng Lv; Yanxiong Yan; Guiyun Chen

For a positive integer n and a prime p, let np denote the p-part of n. Let G be a group, cd(G) the set of all irreducible character degrees of G, ρ(G) the set of all prime divisors of integers in cd(G), V(G)=pep(G)|p∈ρ(G), where pep(G)=max{χ(1)p|χ∈Irr(G)}. In this article, it is proved that G≅L2(p2) if and only if |G|=|L2(p2)| and V(G)=V(L2(p2)).

更新日期：2020-09-10
• Open Math. (IF 0.773) Pub Date : 2020-08-24
Mikhail G. Katz; Tahl Nowik

The first paper in systolic geometry was published by Loewner’s student P. M. Pu over half a century ago. Pu proved an inequality relating the systole and the area of an arbitrary metric in the real projective plane. We prove a stronger version of Pu’s systolic inequality with a remainder term.

更新日期：2020-08-28
• Open Math. (IF 0.773) Pub Date : 2020-08-24
Liang Kong; Jianhua Zhang

In this article, we give a sufficient and necessary condition for every Jordan {g,h}-derivation to be a {g,h}-derivation on triangular algebras. As an application, we prove that every Jordan {g,h}-derivation on τ(N) is a {g,h}-derivation if and only if dim0+≠1 or dimH−⊥≠1, where N is a non-trivial nest on a complex separable Hilbert space H and τ(N) is the associated nest algebra.

更新日期：2020-08-28
• Open Math. (IF 0.773) Pub Date : 2020-08-24
Gülnaz Boruzanlı Ekinci; Csilla Bujtás

Let k be a positive integer and let G be a graph with vertex set V(G). A subset D⊆V(G) is a k-dominating set if every vertex outside D is adjacent to at least k vertices in D. The k-domination number γk(G) is the minimum cardinality of a k-dominating set in G. For any graph G, we know that γk(G)≥γ(G)+k−2 where Δ(G)≥k≥2 and this bound is sharp for every k≥2. In this paper, we characterize bipartite

更新日期：2020-08-28
• Open Math. (IF 0.773) Pub Date : 2020-08-04
Agata Chmielowska; Michał Różański; Barbara Smoleń; Ireneusz Sobstyl; Roman Wituła

The aim of this study was to present a simple method for finding the asymptotic relations for products of elements of some positive real sequences. The main reason to carry out this study was the result obtained by Alzer and Sandor concerning an estimation of a sequence of the product of the first k primes.

更新日期：2020-08-11
• Open Math. (IF 0.773) Pub Date : 2020-08-04
Imed Kedim; Maher Berzig; Ahdi Noomen Ajmi

Consider an ordered Banach space and f,g two self-operators defined on the interior of its positive cone. In this article, we prove that the equation f(X)=g(X) has a positive solution, whenever f is strictly α-concave g-monotone or strictly (−α)-convex g-antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear

更新日期：2020-08-11
• Open Math. (IF 0.773) Pub Date : 2020-08-04
Choonkil Park; Kandhasamy Tamilvanan; Ganapathy Balasubramanian; Batool Noori; Abbas Najati

In this article, we obtain the general solution and prove the Hyers-Ulam stability of the following quadratic-multiplicative functional equation: ϕ(st−uv)+ϕ(sv+tu)=[ϕ(s)+ϕ(u)][ϕ(t)+ϕ(v)]by using the direct method and the fixed point method.

更新日期：2020-08-11
• Open Math. (IF 0.773) Pub Date : 2020-07-22
Jing Zhao; Maojun Bin

In this article, we study the nonlinear and nonsmooth interval-valued optimization problems in the face of data uncertainty, which are called interval-valued robust optimization problems (IVROPs). We introduce the concept of nondominated solutions for the IVROP. If the interval-valued objective function f and constraint functions gi are nonsmooth on Banach space E, we establish a nonsmooth and robust

更新日期：2020-08-08
• Open Math. (IF 0.773) Pub Date : 2020-07-27
Linlin Wang

Let S=g1⋅…⋅gn be a sequence with elements gi from an additive finite abelian group G. S is called a tiny zero-sum sequence if S is non-empty, g1+…+gn=0 and k(S)≔∑i=1n1ord(gi)≤1. Let t(G) be the smallest integer t such that every sequence of t elements (repetition allowed) from G contains a tiny zero-sum sequence. In this article, we mainly focus on the explicit value of t(G) and compute this value

更新日期：2020-08-08
• Open Math. (IF 0.773) Pub Date : 2020-07-29

In this article, we study some topological properties of the multiplication operator on Orlicz-Cesáro mean sequence spaces equipped with the pre-quasi norm and the pre-quasi operator ideal constructed by this sequence space and s-numbers.

更新日期：2020-08-08
• Open Math. (IF 0.773) Pub Date : 2020-07-29

We extend a boundedness result for Marcinkiewicz integral operator. We find a new space of radial functions for which this class of singular integral operators remains Lp-bounded when its kernel satisfies only the sole integrability condition.

更新日期：2020-08-08
• Open Math. (IF 0.773) Pub Date : 2020-07-29
Fuxiao Li; Zhanshou Chen; Yanting Xiao

Change-point detection in categorical time series has recently gained attention as statistical models incorporating change-points are common in practice, especially in the area of biomedicine. In this article, we propose a sequential change-point detection procedure based on the partial likelihood score process for the detection of changes in the coefficients of multinomial logistic regression model

更新日期：2020-08-08
• Open Math. (IF 0.773) Pub Date : 2020-07-21
Xinhong Zhang; Ruijuan Li

Let G be a Hamiltonian graph. A nonempty vertex set X⊆V(G) is called a Hamiltonian cycle enforcing set (in short, an H-force set) of G if every X-cycle of G (i.e., a cycle of G containing all vertices of X) is a Hamiltonian cycle. For the graph G, h(G) (called the H-force number of G) is the smallest cardinality of an H-force set of G. Ore’s theorem states that an n-vertex graph G is Hamiltonian if

更新日期：2020-07-29
• Open Math. (IF 0.773) Pub Date : 2020-07-22
Jiangfeng Han; Pshtiwan Othman Mohammed; Huidan Zeng

The primary objective of this research is to establish the generalized fractional integral inequalities of Hermite-Hadamard-type for MT-convex functions and to explore some new Hermite-Hadamard-type inequalities in a form of Riemann-Liouville fractional integrals as well as classical integrals. It is worth mentioning that our work generalizes and extends the results appeared in the literature.

更新日期：2020-07-29
• Open Math. (IF 0.773) Pub Date : 2020-07-22
Ildefonso Castro; Ildefonso Castro-Infantes; Jesús Castro-Infantes

Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in L2 whose curvature depends on the Lorentzian pseudodistance from the origin, and those

更新日期：2020-07-29
• Open Math. (IF 0.773) Pub Date : 2020-07-22
Hijaz Ahmad; Tufail A. Khan; Shao-Wen Yao

The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results

更新日期：2020-07-29
• Open Math. (IF 0.773) Pub Date : 2020-07-01

In this study, Levinson-type inequalities are generalized by using new Green functions and Montgomery identity for the class of k-convex functions (k ≥ 3). Čebyšev-, Grüss- and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, a new functional is introduced based on f divergence and then some estimates for new functional are obtained. Some inequalities

更新日期：2020-07-21
• Open Math. (IF 0.773) Pub Date : 2020-07-02
Yonghong Ding; Yongxiang Li

This article deals with the exact controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space under the assumption that the semigroup generated by the linear part is noncompact. Our main results are obtained by utilizing stochastic analysis technique, measure of noncompactness and the Mönch fixed point theorem. In the end, an example

更新日期：2020-07-21
• Open Math. (IF 0.773) Pub Date : 2020-07-06
Yong Cai; Liqun Hu

In this article, we show that every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of one prime, one prime squares, two prime cubes, and 187 powers of 2.

更新日期：2020-07-21
• Open Math. (IF 0.773) Pub Date : 2020-07-06
Hongxing Wang; Xiaoyan Zhang

In this article, we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse: ||Mx−b||F=minsubjecttox∈ℛ(M),where M∈ℂnCM. We get the unique solution to the problem, provide two Cramer’s rules for the unique solution and establish two new expressions for the core inverse.

更新日期：2020-07-21
• Open Math. (IF 0.773) Pub Date : 2020-07-10
Javanshir J. Hasanov; Rabil Ayazoglu; Simten Bayrakci

更新日期：2020-07-21
• Open Math. (IF 0.773) Pub Date : 2020-07-17
Yongge Tian

This article brings together miscellaneous formulas and facts on matrix expressions that are composed by idempotent matrices in one place with cogent introduction and references for further study. The author will present the basic mathematical ideas and methodologies of the matrix analytic theory in a readable, up-to-date, and comprehensive manner, including constructions of various algebraic matrix

更新日期：2020-07-21
• Open Math. (IF 0.773) Pub Date : 2020-07-19
Carles Rovira

We give the rate of convergence to a Brownian sheet from a family of processes constructed starting from a set of independent standard Poisson processes. These processes have realizations that converge almost surely to the Brownian sheet, uniformly in the unit square.

更新日期：2020-07-21
• Open Math. (IF 0.773) Pub Date : 2020-06-10
Natalia Pavlovna Bondarenko

The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition

更新日期：2020-06-30
• Open Math. (IF 0.773) Pub Date : 2020-06-11
Jinxia Cen; Chao Min; Van Thien Nguyen; Guo-ji Tang

The goal of this paper is to discuss the well-posedness and the generalized well-posedness of a new kind of differential quasi-variational-hemivariational inequality (DQHVI) in Hilbert spaces. Employing these concepts, we explore the essential relation between metric characterizations and the well-posedness of DQHVI. Moreover, the compactness of the set of solutions for DQHVI is delivered, when problem

更新日期：2020-06-30
• Open Math. (IF 0.773) Pub Date : 2020-06-18
Fan-Liang Li

Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric

更新日期：2020-06-30
• Open Math. (IF 0.773) Pub Date : 2020-06-18
Liping Wei; Ruyun Ma

Let T>1 be an integer, T≔[1,T]Z={1,2,…,T},Tˆ≔{0,1,…,T+1}. In this article, we are concerned with the global structure of the set of sign-changing solutions of the discrete second-order boundary value problem {Δ2u(x−1)+λh(x)f(u(x))=0,x∈T,u(0)=u(T+1)=0,where λ>0 is a parameter, f∈C(ℝ,ℝ) satisfies f(0)=0,sf(s)>0 for all s≠0 and h:Tˆ→[0,+∞). By using the directions of a bifurcation, we obtain existence

更新日期：2020-06-30
• Open Math. (IF 0.773) Pub Date : 2020-06-20
Shuhua Su; Fuyao Liu; Shuqun Yang

This study pursues an investigation on L-semihypergroups equipped with an L-order. First, the concept of L-ordered L-semihypergroups is introduced by L-posets and L-semihypergroups, and some related results are obtained. Then, prime, weakly prime, and semiprime L-hyperideals of L-ordered L-semihypergroups are studied. Moreover, the relationships among the three types of L-hyperideals are established

更新日期：2020-06-30
• Open Math. (IF 0.773) Pub Date : 2020-06-22
Jiangmin Pan

A Cayley graph Γ on a group G is called a dual Cayley graph on G if the left regular representation of G is a subgroup of the automorphism group of Γ (note that the right regular representation of G is always an automorphism group of Γ). In this article, we study finite dual Cayley graphs regarding identification, construction, transitivity and such graphs with automorphism groups as small as possible

更新日期：2020-06-30
• Open Math. (IF 0.773) Pub Date : 2020-06-22

In this work, we show that the existence of fixed points of F-contraction mappings in function weighted metric spaces can be ensured without third condition (F3) imposed on Wardowski function F:(0, ∞)→ℜ. The present article investigates (common) fixed points of rational type F-contractions for single-valued mappings. The article employs Jleli and Samet’s perspective of a new generalization of a metric

更新日期：2020-06-30
• Open Math. (IF 0.773) Pub Date : 2020-06-10
Xianghu Liu

The aim of this study is to investigate the finite approximate controllability of certain Hilfer fractional evolution systems with nonlocal conditions. To achieve this, we first transform the mild solution of the Hilfer fractional evolution system into a fixed point problem for a condensing map. Then, by using the topological degree approach, we present sufficient conditions for the existence and uniqueness

更新日期：2020-06-10
• Open Math. (IF 0.773) Pub Date : 2020-06-04

In this article, we study the commutators of Hausdorff operators and establish their boundedness on the weighted Herz spaces in the setting of the Heisenberg group.

更新日期：2020-06-04
• Open Math. (IF 0.773) Pub Date : 2020-06-04
Erdal Karapınar; V. M. L. Hima Bindu

In this paper, we introduce a new contraction, namely, almost Z contraction with respect to ζ∈Z, in the setting of complete metric spaces. We proved that such contraction possesses a fixed point and the given theorem covers several existing results in the literature. We consider an example to illustrate our result.

更新日期：2020-06-04
• Open Math. (IF 0.773) Pub Date : 2020-06-04
Mohammad Abudayah; Omar Alomari; Torsten Sander

Given any digraph D, its non-negative spectrum (or N-spectrum, shortly) consists of the eigenvalues of the matrix AAT, where A is the adjacency matrix of D. In this study, we relate the classical spectrum of undirected graphs to the N-spectrum of their oriented counterparts, permitting us to derive spectral bounds. Moreover, we study the spectral effects caused by certain modifications of a given digraph

更新日期：2020-06-04
• Open Math. (IF 0.773) Pub Date : 2020-06-04
Shuangjian Guo; Xiaohui Zhang; Shengxiang Wang

The goal of this paper is to examine the structure of split involutive regular BiHom-Lie superalgebras, which can be viewed as the natural generalization of split involutive regular Hom-Lie algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular BiHom-Lie superalgebra L is of the form

更新日期：2020-06-04
• Open Math. (IF 0.773) Pub Date : 2020-06-04
Na Zhang; Yonggui Kao; Fengde Chen; Binfeng Xie; Shiyu Li

A predator-prey model interaction under fluctuating water level with non-selective harvesting is proposed and studied in this paper. Sufficient conditions for the permanence of two populations and the extinction of predator population are provided. The non-negative equilibrium points are given, and their stability is studied by using the Jacobian matrix. By constructing a suitable Lyapunov function

更新日期：2020-06-04
• Open Math. (IF 0.773) Pub Date : 2020-05-29
Takao Komatsu

In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first

更新日期：2020-05-29
• Open Math. (IF 0.773) Pub Date : 2020-05-29
Jingguo Liu

A semigroup is called an epigroup if some power of each element lies in a subgroup. Under the universal of epigroups, the aim of the paper is devoted to presenting elements in the groupoid together with the multiplication of Malcev products generated by classes of completely simple semigroups, nil-semigroups and semilattices. The information about the set inclusion relations among them is also provided

更新日期：2020-05-29
• Open Math. (IF 0.773) Pub Date : 2020-05-28
Gang Wang; Hua Mao

Using the notion of preconcept, we generalize Pawlak’s approximation operators from a one-dimensional space to a two-dimensional space in a formal context. In a formal context, we present two groups of approximation operators in a two-dimensional space: one is aided by an equivalence relation defined on the attribute set, and another is aided by the lattice theoretical property of the family of preconcepts

更新日期：2020-05-28
• Open Math. (IF 0.773) Pub Date : 2020-05-28
Qingdong Guo; Wenhua Wang

In this article, the authors establish the characterizations of a class of anisotropic Herz-type Hardy spaces with two variable exponents associated with a non-isotropic dilation on ℝn in terms of molecular decompositions. Using the molecular decompositions, the authors obtain the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy space with two variable exponents

更新日期：2020-05-28
• Open Math. (IF 0.773) Pub Date : 2020-05-28
Muhammad Nazam; Anam Arif; Hasan Mahmood; Choonkil Park

The self-mappings satisfying implicit relations were introduced in a previous study [Popa, Fixed point theorems for implicit contractive mappings, Stud. Cerc. St. Ser. Mat. Univ. Bacău 7 (1997), 129–133]. In this study, we introduce self-operators satisfying an ordered implicit relation and hence obtain their fixed points in the cone metric space under some additional conditions. We obtain a homotopy

更新日期：2020-05-28
• Open Math. (IF 0.773) Pub Date : 2020-05-26

In this study, we develop the concept of multivalued Suzuki-type θ-contractions via a gauge function and established two new related fixed point theorems on metric spaces. We also discuss an example to validate our results.

更新日期：2020-05-26
• Open Math. (IF 0.773) Pub Date : 2020-05-26
Slavko Simić; Sara Salem Alzaid; Hassen Aydi

In this study, we work with the relative divergence of type s,s∈ℝ, which includes the Kullback-Leibler divergence and the Hellinger and χ2 distances as particular cases. We study the symmetrized divergences in additive and multiplicative forms. Some basic properties such as symmetry, monotonicity and log-convexity are established. An important result from the convexity theory is also proved.

更新日期：2020-05-26
• Open Math. (IF 0.773) Pub Date : 2020-05-26
Zhien Li; Chao Wang

In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive

更新日期：2020-05-26
• Open Math. (IF 0.773) Pub Date : 2020-05-26
Junying Guo; Xiaojiang Guo

It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S]

更新日期：2020-05-26
• Open Math. (IF 0.773) Pub Date : 2020-05-22
Wasfi Shatanawi; Ishak Altun; Hacer Dağ

In this paper, we present some fixed point results for multivalued mappings with both closed values and proximinal values on left K-complete quasi metric spaces. We also provide a nontrivial example to illustrate our results.

更新日期：2020-05-22
• Open Math. (IF 0.773) Pub Date : 2020-05-21
Ruijuan Li; Juanjuan Liang

Sullivan stated the conjectures: (1) every oriented graph has a vertex x such that d++(x) ≥ d−(x) and (2) every oriented graph has a vertex x such that d++(x) + d+(x) ≥ 2d−(x). In this paper, we prove that these conjectures hold for local tournaments. In particular, for a local tournament D, there are at least two vertices satisfying (1) and either there exist two vertices satisfying (2) or there exists

更新日期：2020-05-21
• Open Math. (IF 0.773) Pub Date : 2020-05-21
Chao Ma; Hao Liang; Qimiao Xie; Pengcheng Wang

The eigenvalues and the spectral radius of nonnegative tensors have been extensively studied in recent years. In this paper, we investigate the analytic properties of nonnegative tensors and give some inequalities on the spectral radius.

更新日期：2020-05-21
• Open Math. (IF 0.773) Pub Date : 2020-05-21
Hüseyin Işık; Choonkil Park

The goal of this article is to prove some coupled common fixed point results by using weakly increasing mappings with two variables. Several examples indicating the usability are provided. Also, we use the results obtained to demonstrate the existence of a common solution to a system of integral equations.

更新日期：2020-05-21
• Open Math. (IF 0.773) Pub Date : 2020-05-14
Dorota Kuziak; Ismael G. Yero

A set W of vertices of a connected graph G strongly resolves two different vertices x, y ∉ W if either dG(x, W) = dG(x, y) + dG(y, W) or dG(y, W) = dG(y, x) + dG(x, W), where dG(x, W) = min{d(x,w): w ∈ W} and d(x,w) represents the length of a shortest x − w path. An ordered vertex partition Π = {U1, U2,…,Uk} of a graph G is a strong resolving partition for G, if every two different vertices of G belonging

更新日期：2020-05-14
• Open Math. (IF 0.773) Pub Date : 2020-05-11
Yu Zhang; Hai Bi; Yidu Yang

In this study, for the first time, we discuss the posteriori error estimates and adaptive algorithm for the non-self-adjoint Steklov eigenvalue problem in inverse scattering. The differential operator corresponding to this problem is non-self-adjoint and the associated weak formulation is not H1-elliptic. Based on the study of Armentano et al. [Appl. Numer. Math. 58 (2008), 593–601], we first introduce

更新日期：2020-05-11
• Open Math. (IF 0.773) Pub Date : 2020-05-10
Shengjiang Chen; Aizhu Xu

Let f(z) be an entire function of hyper order strictly less than 1. We prove that if f(z) and its nth exact difference Δcnf(z) share 0 CM and 1 IM, then Δcnf(z)≡f(z). Our result improves the related results of Zhang and Liao [Sci. China A, 2014] and Gao et al. [Anal. Math., 2019] by using a simple method.

更新日期：2020-05-10
• Open Math. (IF 0.773) Pub Date : 2020-03-26
Yumnam Rohen; Nabil Mlaiki

In this paper, we introduce a new type of contraction to seek the existence of tripled best proximity point results. Here, using the new contraction and P-property, we generalize and extend results of W. Shatanawi and A. Pitea and prove the existence and uniqueness of some tripled best proximity point results. Examples are also given to support our results.

更新日期：2020-03-26
• Open Math. (IF 0.773) Pub Date : 2020-03-26

In this article, more general types of fractional proportional integrals and derivatives are proposed. Some properties of these operators are discussed.

更新日期：2020-03-26
• Open Math. (IF 0.773) Pub Date : 2020-03-26
Vladimir Kanovei; Mikhail G. Katz; Tahl Nowik

We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by Do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric

更新日期：2020-03-26
• Open Math. (IF 0.773) Pub Date : 2020-03-20
Zaiyun Zhang; Limei Li; Chunhua Fang; Fan He; Chuangxia Huang; Wen Zhu

In this paper, we investigate the Cauchy problem for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion. Furthermore, we establish the blow-up result of the positive solutions in finite time under certain conditions on the initial datum. This result complements the early one in the literature, such as [E. Novruzov, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm

更新日期：2020-03-20
• Open Math. (IF 0.773) Pub Date : 2020-03-20
He Yuan; Liangyun Chen

Let R be a subset of a unital ring Q such that 0 ∈ R. Let us fix an element t ∈ Q. If R is a (t; d)-free subset of Q, then Tn(R) is a (t′; d)-free subset of Tn(Q), where t′ ∈ Tn(Q), tll′ = t, l = 1, 2, …, n, for any n ∈ N.

更新日期：2020-03-20
• Open Math. (IF 0.773) Pub Date : 2020-03-20
Oğuzhan Demirel

An n-sided hyperbolic polygon of type (ϵ, n) is a hyperbolic polygon with ordered interior angles π2 + ϵ, θ1, θ2, …, θn−2, π2 − ϵ, where 0 < ϵ < π2 and 0 < θi < π satisfying

更新日期：2020-03-20
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