Oct 2020, Volume 15 Issue 5
    

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  • RESEARCH ARTICLE
    Wenjing BI, Chunlei TANG

    We study the Schrödinger-KdV system

    {Δu+λ1(x)u=u3+βuv,uH1(N),Δv+λ2(x)v=12v2+β2u2,vH1(N),

    where N=1,2,3, λi(x)C(N,),lim|x|λi(x)=λi(), and λi(x)λi(),i= 1,2,a.e. xN.We obtain the existence of nontrivial ground state solutions for the above system by variational methods and the Nehari manifold.

  • RESEARCH ARTICLE
    Mu-Fa CHEN, Jin-Yu LI

    The first aim of the paper is to study the Hermitizability of secondorder differential operators, and then the corresponding isospectral operators. The explicit criteria for the Hermitizable or isospectral properties are presented. The second aim of the paper is to study a non-Hermitian model, which is now well known. In a regular sense, the model does not belong to the class of Hermitizable operators studied in this paper, but we will use the theory developed in the past years, to present an alternative and illustrated proof of the discreteness of its spectrum. The harmonic function plays a critical role in the study of spectrum. Two constructions of the function are presented. The required conclusion for the discrete spectrum is proved by some comparison technique.

  • RESEARCH ARTICLE
    Xiequan FAN, Haijuan HU, Quansheng LIU

    Let {Zn, n0}be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for log(Zn+n0/Zn0 ) uniformly in n0 ,which extend the corresponding results by I. Grama, Q. Liu, and M. Miqueu [Stochastic Process. Appl., 2017, 127: 1255–1281] established for n0= 0. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. are relaxed in our present setting. An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of log(Zn+n0/Zn0 ) and n.

  • RESEARCH ARTICLE
    Zhichao GAO, Shouhong QIAO, Huaguo SHI, Long MIAO

    Suppose that G is a finite group and H is a subgroup of G. H is said to be a p-CAP-subgroup of G if H either covers or avoids each pd-chief factor of G. We give some characterizations for a group G to be p-solvable under the assumption that some subgroups of G are p-CAP-subgroups of G.

  • RESEARCH ARTICLE
    Ze LI

    We prove the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by S. Cuccagna, V. Georgiev, and N. Visciglia [Comm. Pure Appl. Math., 2013, 6: 957–980] to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by G. Staffilani [Duke Math. J., 1997, 86(1): 109–142] to control the high moments of the solutions emerging from the vector fields method.

  • RESEARCH ARTICLE
    Yanmin NIU, Xiong LI

    We consider a pendulum type equation with p-Laplacian (ϕp(x))+Gx(t,x)=p(t), where ϕp(u)=|u|p2u,p>1,G(t,x) and p(t) are 1-periodic about every variable. The solutions of this equation present two interesting behaviors. On the one hand, by applying Moser's twist theorem, we find infinitely many invariant tori whenever 01p(t)dt=0, which yields the bounded-ness of all solutions and the existence of quasi-periodic solutions starting at t = 0 on the invariant tori. On the other hand, if p(t) = 0 and Gx(t,x) has some specific forms, we find a full symbolic dynamical system made by solutions which oscillate between any two different trivial solutions of the equation. Such chaotic solutions stay close to the trivial solutions in some fixed intervals, according to any prescribed coin-tossing sequence.

  • RESEARCH ARTICLE
    Jiangtao PENG, Yongke QU, Yuanlin LI

    Let G be a finite abelian group and S be a sequence with elements of G: We say that S is a regular sequence over G if |SH||H|1 holds for every proper subgroup H of G; where SH denotes the subsequence of S consisting of all terms of S contained in H: We say that S is a zero-sum free sequence over G if 0(S)0; where (S)G denotes the set of group elements which can be expressed as a sum of a nonempty subsequence of S: In this paper, we study the inverse problems associated with (S) when S is a regular sequence or a zero-sum free sequence over G=GpCp, where p is a prime.

  • RESEARCH ARTICLE
    Xiaoqian SUN, Xuelin YONG, Jianwei GAO

    Based on the Lie symmetry method, we derive the explicit optimal invest strategy for an investor who seeks to maximize the expected exponential (CARA) utility of the terminal wealth in a defined-contribution pension plan under a constant elasticity of variance model. We examine the point symmetries of the Hamilton-Jacobi-Bellman (HJB) equation associated with the portfolio optimization problem. The symmetries compatible with the terminal condition enable us to transform the (2+ 1)-dimensional HJB equation into a (1+ 1)-dimensional nonlinear equation which is linearized by its infinite-parameter Lie group of point transformations. Finally, the ansatz technique based on variables separation is applied to solve the linear equation and the optimal strategy is obtained. The algorithmic procedure of the Lie symmetry analysis method adopted here is quite general compared with conjectures used in the literature.

  • RESEARCH ARTICLE
    Hongbin WANG, Jingshi XU, Jian TAN

    We prove the boundedness for a class of multi-sublinear singular integral operators on the product of central Morrey spaces with variable exponents. Based on this result, we obtain the boundedness for the multilinear singular integral operators and two kinds of multilinear singular integral commutators on the above spaces.

  • RESEARCH ARTICLE
    Jun WANG, Jie FEI

    We study a superminimal surface M immersed into a hyperquadric Q2 in several cases classified by two global defined functions τX and τY, which were introduced by X. X. Jiao and J. Wang to study a minimal immersion f : MQ2. In case both τX and τY are not identically zero, it is proved that f is superminimal if and only if f is totally real or if:MP3 is also minimal, where i:Q2P3 is the standard inclusion map. In the rest case that τX0 or τY0, the minimal immersion f is automatically superminimal. As a consequence, all the superminimal two-spheres in Q2 are completely described.

  • RESEARCH ARTICLE
    Mengyan XIE, Qing-Wen WANG

    We establish necessary and sufficient conditions for the existence of the reducible solution to the quaternion tensor equation A*NX*NB=Cvia Einstein product using Moore-Penrose inverse, and present an expression of the reducible solution to the equation when it is solvable. Moreover, to have a general solution, we give the solvability conditions for the quaternion tensor equation A1*NX1*MB1+A1*NX2*MB2+A2*NX3*MB2=C, which plays a key role in investigating the reducible solution to A*NX*NB=C. The expression of such a solution is also presented when the consistency conditions are met. In addition, we show a numerical example to illustrate this result.

  • RESEARCH ARTICLE
    Zhao XU

    For a fixed even SL(2,) Hecke{Maass form f, we get an estimate for the second moment of L(s,φj×f) at special points, where φj runs over an orthogonal basis of Hecke{Maass cusp forms for SL3().