Acta Mathematica Scientia (Series B)
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Mathematics, CAS, China
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Scientia
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ISSN 0252-9602
CN 　42-1227/O
25 April 2020, Volume 40 Issue 2
Articles
 INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY $_2F_1(\tfrac12)$-SUMS Wenchang CHU Acta mathematica scientia,Series B. 2020, 40 (2):  293-315.  DOI: 10.1007/s10473-020-0201-y The unified Ω-series of the Gauss and Bailey $_2F_1(\tfrac12)$-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts. Several remarkable transformation theorems for the Ω-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type, including a couple of beautiful expressions for π and the Catalan constant discovered by Guillera (2008).
 MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY Lun GUO, Tingxi HU Acta mathematica scientia,Series B. 2020, 40 (2):  316-340.  DOI: 10.1007/s10473-020-0202-x In this article, we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity \begin{equation*} -\Delta u+(\lambda a(x)+1)u=\Big(\frac{1}{|x|^{\alpha}}\ast F(u)\Big)f(u) \ \ \text{in}\ \ \mathbb{R}^{N}, \end{equation*} where $N\geq 3$, $0<\alpha< \min\{N,4\}$, $\lambda$ is a positive parameter and the nonnegative potential function $a(x)$ is continuous. Using variational methods, we prove that if the potential well int$(a^{-1}(0))$ consists of $k$ disjoint components, then there exist at least $2^k-1$ multi-bump solutions. The asymptotic behavior of these solutions is also analyzed as $\lambda\to +\infty$.
 ASYMPTOTIC BEHAVIOR OF SOLUTION BRANCHES OF NONLOCAL BOUNDARY VALUE PROBLEMS Xian XU, Baoxia QIN, Zhen WANG Acta mathematica scientia,Series B. 2020, 40 (2):  341-354.  DOI: 10.1007/s10473-020-0203-9 In this article, by employing an oscillatory condition on the nonlinear term, a result is proved for the existence of connected component of solutions set of a nonlocal boundary value problem, which bifurcates from infinity and asymptotically oscillates over an interval of parameter values. An interesting and immediate consequence of such oscillation property of the connected component is the existence of infinitely many solutions of the nonlinear problem for all parameter values in that interval.
 ASYMPTOTIC DISTRIBUTION IN DIRECTED FINITE WEIGHTED RANDOM GRAPHS WITH AN INCREASING BI-DEGREE SEQUENCE Jing LUO, Hong QIN, Zhenghong WANG Acta mathematica scientia,Series B. 2020, 40 (2):  355-368.  DOI: 10.1007/s10473-020-0204-8 The asymptotic normality of the fixed number of the maximum likelihood estimators (MLEs) in the directed finite weighted network models with an increasing bi-degree sequence has been established recently. In this article, we further derive the central limit theorem for linear combinations of all the MLEs with an increasing dimension when the edges take finite discrete weight. Simulation studies are provided to illustrate the asymptotic results.
 SOME METRIC AND TOPOLOGICAL PROPERTIES OF NEARLY STRONGLY AND NEARLY VERY CONVEX SPACES Zihou ZHANG, Vicente MONTESINOS, Chunyan LIU Acta mathematica scientia,Series B. 2020, 40 (2):  369-378.  DOI: 10.1007/s10473-020-0205-7 We obtain characterizations of nearly strong convexity and nearly very convexity by using the dual concept of S and WS points, related to the so-called Rolewicz's property (α). We give a characterization of those points in terms of continuity properties of the identity mapping. The connection between these two geometric properties is established, and finally an application to approximative compactness is given.
 ON THE EXISTENCE OF SOLUTIONS TO A BI-PLANAR MONGE-AMPÈRE EQUATION Ibrokhimbek AKRAMOV, Marcel OLIVER Acta mathematica scientia,Series B. 2020, 40 (2):  379-388.  DOI: 10.1007/s10473-020-0206-6 In this article, we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampère operators acting in different two-dimensional coordinate sections. This equation is elliptic, for example, in the class of convex functions. We show that the notion of Monge-Ampère measures and Aleksandrov generalized solutions extends to this equation, subject to a weaker notion of convexity which we call bi-planar convexity. While the equation is also elliptic in the class of bi-planar convex functions, the contrary is not necessarily true. This is a substantial difference compared to the classical Monge-Ampère equation where ellipticity and convexity coincide. We provide explicit counter-examples: classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced. We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.
 INFINITELY MANY SOLUTIONS WITH PEAKS FOR A FRACTIONAL SYSTEM IN $\mathbb{R}^{N}$ Qihan HE, Yanfang PENG Acta mathematica scientia,Series B. 2020, 40 (2):  389-411.  DOI: 10.1007/s10473-020-0207-5 In this article, we consider the following coupled fractional nonlinear Schrödinger system in $\mathbb{R}^{N}$ $\left\{ \begin{array}{l} {\left( { - \Delta } \right)^s}u + P\left( x \right)u = {\mu _1}{\left| u \right|^{2p - 2}}u + \beta {\left| u \right|^p}{\left| u \right|^{p - 2}}u,\;\;\;x \in {{\mathbb{R}}^N},\\{\left( { - \Delta } \right)^s}v + Q\left( x \right)v = {\mu _2}{\left| v \right|^{2p - 2}}v + \beta {\left| v \right|^p}{\left| v \right|^{p - 2}}v,\;\;\;\;\;x \in {{\mathbb{R}}^N},\\u,\;\;v \in {H^s}\left( {{{\mathbb{R}}^N}} \right),\end{array} \right.$ where $N≥2, 0 < s < 1, 1 < p < \frac{N}{N-2s},\mu_1>0, \mu_2>0$ and $\beta \in \mathbb{R}$ is a coupling constant. We prove that it has infinitely many non-radial positive solutions under some additional conditions on $P(x), Q(x), p$ and $\beta$. More precisely, we will show that for the attractive case, it has infinitely many non-radial positive synchronized vector solutions, and for the repulsive case, infinitely many non-radial positive segregated vector solutions can be found, where we assume that $P(x)$ and $Q(x)$ satisfy some algebraic decay at infinity.
 APPROXIMATE CONTROLLABILITY RESULTS FOR INTEGRO-QUASILINEAR EVOLUTION EQUATIONS VIA TRAJECTORY REACHABLE SETS A. Vinodkumar, C. Loganathan, S. Vijay Acta mathematica scientia,Series B. 2020, 40 (2):  412-424.  DOI: 10.1007/s10473-020-0208-4 In this article, we study the approximate controllability results for an integro-quasilinear evolution equation with random impulsive moments under sufficient conditions. The results are obtained by the theory of C0 semigroup of bounded linear operators on evolution equations and using trajectory reachable sets. Finally, we generalize the results too with and without fixed type impulsive moments.
 ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE CHAFEE-INFANTE EQUATION Haochuan HUANG, Rui HUANG Acta mathematica scientia,Series B. 2020, 40 (2):  425-441.  DOI: 10.1007/s10473-020-0209-3 In higher dimension, there are many interesting and challenging problems about the dynamics of non-autonomous Chafee-Infante equation. This article is concerned with the asymptotic behavior of solutions for the non-autonomous Chafee-Infante equation $\frac{\partial u}{\partial t}- \Delta u =\lambda(t) (u -u^{3})$ in higher dimension, where $\lambda(t)\in C^{1}[0,T]$ and $\lambda(t)$ is a positive, periodic function. We denote $\lambda_{1}$ as the first eigenvalue of $-\Delta \varphi = \lambda \varphi, \; x \in \Omega; \;\; \varphi=0, \; x \in \partial \Omega.$ For any spatial dimension $N\geq1$, we prove that if $\lambda(t)\leq\lambda_{1}$, then the nontrivial solutions converge to zero, namely, $\underset{t\rightarrow+\infty }{\lim} u(x,t) =0, \; x\in\Omega$; if $\lambda(t)>\lambda_{1}$ as $t\rightarrow +\infty$, then the positive solutions are attracted'' by positive periodic solutions. Specially, if $\lambda(t)$ is independent of $t$, then the positive solutions converge to positive solutions of $- \Delta U =\lambda(U -U^{3})$. Furthermore, numerical simulations are presented to verify our results.
 LOCAL WELL-POSEDNESS OF STRONG SOLUTIONS FOR THE NONHOMOGENEOUS MHD EQUATIONS WITH A SLIP BOUNDARY CONDITIONS Hongmin LI, Yuelong XIAO Acta mathematica scientia,Series B. 2020, 40 (2):  442-456.  DOI: 10.1007/s10473-020-0210-x This article is concerned with the 3D nonhomogeneous incompressible magnetohydrodynamics equations with a slip boundary conditions in bounded domain. We obtain weighted estimates of the velocity and magnetic field, and address the issue of local existence and uniqueness of strong solutions with the weaker initial data which contains vacuum states.
 $L^0$-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN $L^0(\mathcal{F},B)$ Tiexin GUO, Erxin ZHANG, Yachao WANG, George YUAN Acta mathematica scientia,Series B. 2020, 40 (2):  457-469.  DOI: 10.1007/s10473-020-0211-9 Let $(B,\|\cdot\|)$ be a Banach space, $(\Omega,\mathcal{F},P)$ a probability space, and $L^0(\mathcal{F},B)$ the set of equivalence classes of strong random elements (or strongly measurable functions) from $(\Omega,\mathcal{F},P)$ to $(B,\|\cdot\|)$. It is well known that $L^0(\mathcal{F},B)$ becomes a complete random normed module, which has played an important role in the process of applications of random normed modules to the theory of Lebesgue-Bochner function spaces and random operator theory. Let $V$ be a closed convex subset of $B$ and $L^0(\mathcal{F},V)$ the set of equivalence classes of strong random elements from $(\Omega,\mathcal{F},P)$ to $V$. The central purpose of this article is to prove the following two results: (1) $L^0(\mathcal{F},V)$ is $L^0$-convexly compact if and only if $V$ is weakly compact; (2) $L^0(\mathcal{F},V)$ has random normal structure if $V$ is weakly compact and has normal structure. As an application, a general random fixed point theorem for a strong random nonexpansive operator is given, which generalizes and improves several well known results. We hope that our new method, namely skillfully combining measurable selection theorems, the theory of random normed modules, and Banach space techniques, can be applied in the other related aspects.
 STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS Hairong YUAN, Qin ZHAO Acta mathematica scientia,Series B. 2020, 40 (2):  470-502.  DOI: 10.1007/s10473-020-0212-8 Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets. For the three-dimensional steady non-isentropic compressible Euler system with frictions, we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections. In this article, we are devoted to proving rigorously that a large class of these transonic shock solutions are stable, under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces. This manifests that frictions have a stabilization effect on transonic shocks in ducts, in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts. Except its implications to applications, because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system, we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.
 ON THE AREAS OF THE MINIMAL TRIANGLES IN VEECH SURFACES Yumin ZHONG Acta mathematica scientia,Series B. 2020, 40 (2):  503-514.  DOI: 10.1007/s10473-020-0213-7 Smillie and Weiss proved that the set of the areas of the minimal triangles of Veech surfaces with area 1 can be arranged as a strictly decreasing sequence $\{a_n\}$. And each $a_n$ in the sequence corresponds to finitely many affine equivalent classes of Veech surfaces with area 1. In this article, we give an algorithm for calculating the area of the minimal triangles in a Veech surface and prove that the first element of $\{a_n\}$ which corresponds to non arithmetic Veech surfaces is $(5-\sqrt{5})/20$, which is uniquely realized by the area of the minimal triangles of the normalized golden $L$-shaped translation surface up to affine equivalence.
 STRONG INSTABILITY OF STANDING WAVES FOR A SYSTEM NLS WITH QUADRATIC INTERACTION Van Duong DINH Acta mathematica scientia,Series B. 2020, 40 (2):  515-528.  DOI: 10.1007/s10473-020-0214-6 We study the strong instability of standing waves for a system of nonlinear Schrödinger equations with quadratic interaction under the mass resonance condition in dimension d=5.
 INITIAL BOUNDARY VALUE PROBLEM FOR THE 3D MAGNETIC-CURVATURE-DRIVEN RAYLEIGH-TAYLOR MODEL Xueke PU, Boling GUO Acta mathematica scientia,Series B. 2020, 40 (2):  529-542.  DOI: 10.1007/s10473-020-0215-5 This article studies the initial-boundary value problem for a three dimensional magnetic-curvature-driven Rayleigh-Taylor model. We first obtain the global existence of weak solutions for the full model equation by employing the Galerkin's approximation method. Secondly, for a slightly simplified model, we show the existence and uniqueness of global strong solutions via the Banach's fixed point theorem and vanishing viscosity method.
 MAXIMUM TEST FOR A SEQUENCE OF QUADRATIC FORM STATISTICS ABOUT SCORE TEST IN LOGISTIC REGRESSION MODEL Qing YANG, Jiayan ZHU, Zhengbang LI Acta mathematica scientia,Series B. 2020, 40 (2):  543-556.  DOI: 10.1007/s10473-020-0216-4 This article proposes the maximum test for a sequence of quadratic form statistics about score test in logistic regression model which can be applied to genetic and medicine fields. Theoretical properties about the maximum test are derived. Extensive simulation studies are conducted to testify powers robustness of the maximum test compared to other two existed test. We also apply the maximum test to a real dataset about multiple gene variables association analysis.
 PROPERTIES ON MEROMORPHIC SOLUTIONS OF COMPOSITE FUNCTIONAL-DIFFERENTIAL EQUATIONS Manli LIU, Lingyun GAO Acta mathematica scientia,Series B. 2020, 40 (2):  557-571.  DOI: 10.1007/s10473-020-0217-3 With the aid of Nevanlinna value distribution theory, differential equation theory and difference equation theory, we estimate the non-integrated counting function of meromorphic solutions on composite functional-differential equations under proper conditions.We also get the form of meromorphic solutions on a type of system of composite functional equations. Examples are constructed to show that our results are accurate.
 EXPANDABLE PARALLEL FINITE ELEMENT METHODS FOR LINEAR ELLIPTIC PROBLEMS Guangzhi DU Acta mathematica scientia,Series B. 2020, 40 (2):  572-588.  DOI: 10.1007/s10473-020-0218-2 In this article, two kinds of expandable parallel finite element methods, based on two-grid discretizations, are given to solve the linear elliptic problems. Compared with the classical local and parallel finite element methods, there are two attractive features of the methods shown in this article: 1) a partition of unity is used to generate a series of local and independent subproblems to guarantee the final approximation globally continuous; 2) the computational domain of each local subproblem is contained in a ball with radius of $O(H)$ ($H$ is the coarse mesh parameter), which means methods in this article are more suitable for parallel computing in a large parallel computer system. Some a priori error estimation are obtained and optimal error bounds in both $H^1$-normal and $L^2$-normal are derived. Finally, numerical results are reported to test and verify the feasibility and validity of our methods.
 ULAM-HYERS-RASSIAS STABILITY AND EXISTENCE OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENCE EQUATIONS WITH MULTIPOINT SUMMATION BOUNDARY CONDITION Syed Sabyel HAIDER, Mujeeb Ur REHMAN Acta mathematica scientia,Series B. 2020, 40 (2):  589-602.  DOI: 10.1007/s10473-020-0219-1 The purpose of this study is to acquire some conditions that reveal existence and stability for solutions to a class of difference equations with non-integer order μ∈ (1, 2]. The required conditions are obtained by applying the technique of contraction principle for uniqueness and Schauder's fixed point theorem for existence. Also, we establish some conditions under which the solution of the considered class of difference equations is generalized Ulam-Hyers-Rassias stable. Example for the illustration of results is given.