We are concerned with the three-component generalized Gross-Pitaevskii Equations (GGPE) in the spinor Bose-Einstein condensates with trapping potentials. Under certain conditions on trapping potentials and parameters, we establish the existence and uniqueness of corresponding standing wave solutions with behaviors tending to zero at infinity. The same issues for the corresponding Dirichlet boundary value problems on the ball centered at the origin are also concluded. Additionally, we provide a classification of various types of solutions to radially symmetric situations.

In this paper, we prove the existence and uniqueness of solitary wave solutions of time-dependent nonlinear Schrödinger equations with behaviors tending to zero at infinity under certain conditions on trapping potentials and parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established. Particularly, for constant trapping potentials, we conclude that there is at most one radial ground state under certain conditions on parameters.

以近几年贵州省高中数学学业水平考试试卷中几个优秀题为题例,发掘其培养师范本科生创新思维能力的解题教学价值.先给出几个题例,解析其在解题教学中培养创新思维能力的亮点,最后给出这些题例在解题教学实践中培养师范生创新思维能力的策略方法和教学模式.

In this paper, we consider the nonlinear equations arising from the self-dual Maxwell-Chern-Simons gauged \begin{document}$ O(3) $\end{document} sigma model on (2+1)-dimensional Minkowski space \begin{document}$ {\bf R^{2,1}} $\end{document} with the metric \begin{document}$ {\mathrm {diag}}(1,-1,-1) $\end{document}. We establish the asymptotic behavior of multivortex solutions corresponding to their flux and find the range of the flux for non-topological solutions. Moreover, we prove the radial symmetry property under certain conditions in one vortex point case.

For Guizhou Province high school math examination questions in December 2021, 43, gives its multiple solution method, such as Lagrange multiplier method, discriminant method, average inequality method, cauchy inequality method, gradient method and so on, as well as the solutions of the these analysis, then the problem of general quadratic constraints under the condition of the existence of the solution of linear target optimization problem, And a variety of general solution methods are discussed.

In this paper, we consider the nonlinear equation arising from the Chern–Simons theory of planar matter fields interacting with the Chern–Simons gauge field in a CP(1) invariant fashion. Then, we establish the sharp region of flux for non-topological solutions and prove the classification of solutions of all types in the case of one vortex point. Moreover, we also give the complete result of Theorem 1.3 in the work by Choe et al. [J. Differ. Equations, 255, 2136 (2013)] from Theorem 1.4(ii) as follows.

在不具有线性结构的T-凸空间中,利用经典的集值分析方法和KKM方法,证明且建立了FanKy不等式定理,并借助该定理获得了分别与弱不动点和不动点相关的两个重要结果;作为对该定理的应用,将著名的FanKy截口定理和Fan-Browder不动点定理推广到T-凸空间中.

证明了满足广义KKM映射条件与H0条件的抽象凸度量空间的任意两个邻近对子集之间的集值映射的有限交性质.进一步,在紧性假设条件下,得到了该类集值映射的无限交性质.最后,证明了满足H0条件的抽象凸度量空间的任意两个邻近对子集之间的集值映射的最佳邻近点的存在性.

[目的]为了在不具有线性结构的T-凸空间中得到弱于H0-条件的GH0-条件下的KKM引理.[方法]利用非线性分析中关于度量以及从属于紧集的有限覆盖的单位分解构造辅助函数,并基于GH0-条件的特征,构造复合函数.[结果]在弱于H0-条件的GH0-条件下建立了T-凸度量空间和T-凸拓扑空间中的KKM引理.[结论]在空间结构和凸结构条件较弱的情形下推广了KKM引理.

罗尔中值定理是微积分解题需要用到的基本定理之一，是研究函数及其导函数关系的重要工具。本文首先对罗尔中值定理进行论述，然后通过例题对其应具体应用进行分析，重点阐述应用罗尔中值定理构造辅助函数的一般方法，使学生能够掌握罗尔中值定理在微积分解题中的使用技巧。

分析了2019年12月贵州省高中数学毕业会考试卷最后一道关于解析几何的解答题所涵盖的核心知识、思想方法和答题情况,并给出该题的部分多种解法,提出关于解析几何的几点教学建议以供教学参考.

In this paper, by constructing a family of approximation solutions and applying a specific version of the Implicit Function Theorem (please see, e.g. [18]), we prove the existence of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory.

充分利用KKM方法和经典的集值分析方法,在不具有线性结构的T-凸空间中,建立并证明了弱Fan Ky点存在引理,并借助该引理获得了两个Fan Ky点存在定理;最后,作为对Fan Ky点存在定理的应用,在T-凸策略空间中建立并证明了n人非合作博弈Nash平衡点存在定理.

本文分析了2019年贵州省高中数学毕业会考试卷最后一道关于数列的解答题所涵盖的核心知识、思想方法和答题情况,给出了该题的多种解法,并提出关于各种数学中数列的一些教学建议以供教学参考.

AbstractIn this paper, we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear Schrödinger equation. We establish the existence and uniqueness of solutions to the Dirichlet problem on the ball and entire space as the parameters within certain regions. In addition, a complete structure of different types of solutions for the radial case is also provided.

在局部T-凸空间框架下,不依赖KKM技巧,建立了两个新的不动点定理.分别以局部H-凸空间中的Schauder不动点定理和Browder不动点定理为其特例,将Schauder不动点定理和Browder不动点定理推广到T-凸空间.

In this paper, we prove the uniqueness of stationary standing wave solutions of an optical model generated by Type II Second Harmonic Generation (SHG) with behaviors tending to zero at infinity under certain conditions on parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established.

Neuromuscular impairments in Parkinson's disease (PD) alter the mechanics and control of the body, leading to an increased risk of falling during more challenging functional tasks such as obstacle-crossing. However, little is known about these changes. The current study aimed to bridge the gap by quantifying the motion of the body's center of mass (COM) relative to the center of pressure (COP) in terms of COM-COP inclination angles (IA) and their rate of change (RCIA) in fifteen older adults with mild PD and fifteen healthy controls when crossing obstacles of heights of 10, 20 and 30% leg length. There were no between-group differences for either the leading or trailing toe clearances (p > 0.05). With the unaffected limb leading, the PD subjects significantly increased the crossing sagittal and frontal IA, crossing sagittal RCIA, the peak RCIA, and average sagittal and frontal RCIAs during double-limb support for all obstacle heights when compared to those with the affected limb leading and those of the Controls (p < 0.05). The poor balance control in the mediolateral direction during obstacle-crossing in PD indicated an increased risk of falling. The differences in the crossing patterns between leading with the affected or unaffected limb suggest that patients with PD should lead with the affected limb when crossing obstacles. The current findings suggest that early dynamic balance training is important in the management of patients with PD.

In this paper, we prove the uniqueness of topological solutions for the self-dual Maxwell–Chern–Simons O(3) sigma model under the conditions on Chern–Simons coupling parameter and the charge of electron. Besides, we also provide the quantitative analysis of radial solutions of all types for single vortex-point case.

Convexity of spaces plays a very important role in non-linear analysis theory , variational inequality theory , optimal theory and so on .In all these cases of theoretical and applied studies , the dependence on the con-vexity of spaces is presumed .However , a lot of spaces do not possess the convex property from the viewpoint of the usual linear structure .Because of this , it has become recent interest to establish generalized convexity on spaces without linear structure and to generalize the continuity selection theorem , the fixed point theorem and some other important results , to generalized convex spaces .Here we make full use of the H0-condition of T-convex space and classical analysis method , to prove KKM lemma on T-convex space without linear structure .Finally , we study two applications of this lemma on T-convex space .One is the fixed point theorem and the other is the existence theorem for the solution of Ky Fan inequality without quasi-T-concavity on T-convex spaces .