映射与函数

内容 :映射、函数、函数的单调性和奇偶性、反函数 .选择题1　下列命题中 ,真命题为 (　　 )(Ａ)若Ａ ={ａ} ,Ｂ≠ ,则Ａ到Ｂ的映射最多能构建 1个 .(Ｂ)若Ａ≠ ,Ｂ ={ｂ} ,则Ａ到Ｂ的映射仅能构建 1个 .(Ｃ)若Ａ =Ｂ≠ ,则Ａ到Ｂ上的一一映射仅能构建 1个 .(Ｄ)若Ａ≠ ,Ｂ≠ ,则Ａ到Ｂ上的一一映射至少能构建 1个 .2　设 ｆ(ｘ)的定义域为Ｒ ,且存在反函数 ｆ- 1(ｘ) ,对任意的ａ∈Ｒ ,则下列说法正确的是 (　　 )(Ａ)方程 ｆ(ｘ) =ａ一定有解 .(Ｂ)方程 ｆ- 1(ｘ) =ａ一定有解 .(Ｃ)方程 ｆ(ｘ) =ａ一定无解 .(Ｄ)方程 ｆ- 1(ｘ) =ａ…     

具有进位吸引子的区间映射

对于一个具有n-进位吸引子的区间连续映射,证明了"n不是2的方幂"是该映射具有正拓扑熵的充分条件但不是必要条件;探讨了函数方程f3(λx)=λf(x)的一类解,并证明这类解中的每一个成员都有3-进位吸引子.     

具有进位吸引子的区间映射

对于一个具有n-进位吸引子的区间连续映射,证明了：“n不是2的方幂”是该映射具有正拓扑熵的充分条件但不是必要条件；探讨了函数方程f~3(λx)=λf(x)的一类解,并证明这类解中的每一个成员都有3-进位吸引子．     

求解混沌映射不变分布的符号计算法

根据Frobenius-Perron方程,可以对混沌映射的不变分布从理论上加以分析,从而对混沌映射不变分布作出大致的估计.由此可以利用符号计算的方法求解得到迭代函数系统不变分布的密度近似函数,从而逼近理论解.用几个计算实例和常见数值解法作了比较,试验结果表明符号计算方法具有一定的优势.     

Ginzburg-Landau旋涡与调和映射的自相似解

得到了带pinning效应的Ginzburg Landau方程组的H1 紧性 ,证明了其二维情况的旋涡被pinning函数的最小点所吸引 ,构造了调和映射发展方程的自相似解     

集值映射最优化问题超有效解集的连通性

本文在局部凸空间中对集值映射最优化问题引入超有效解的概念．首先研究了超 有效点的一些重要特性．其后证明了当目标函数为锥类凸的集值映射时，其目标空间里 的超有效点集是连通的；若目标函数为锥凸的集值映射时，其超有效解集也是连通的．     

一类非线性双曲型发展方程的孤子解

研究了一类非线性发展方程.首先在无扰动情形下,利用待定函数和泛函同伦映射方法得到了非扰动发展方程的孤子精确解和扰动方程的任意次近似行波孤子解.接着引入一个同伦映射,并选取初始近似函数,再用同伦映射理论,依次求出非线性双曲型发展扰动方程孤子解的各次近似解析解.再利用摄动理论举例说明了用该方法得到的近似解析解的有效性和各次近似解的近似度.最后,简述了用同伦映射方法得到的近似解的意义,指出了用上述方法得到的各次近似解具有便于求解、精度高等优点.     

改进集映射下参数广义向量拟平衡问题解映射的Berge下半连续性

该文主要讨论了一类新的参数广义向量拟平衡问题解映射的稳定性.首先,定义了改进集映射,基于改进集映射,将序结构进行推广并应用于拟平衡问题的研究,得到了改进集映射下参数广义向量拟平衡问题(IPGVQEP).然后,给出了一类与改进集映射相关的非线性标量化函数Ψ,利用非线性标量化函数Ψ得到了与原问题(IPGVQEP)对应的标量化问题(IPGVQEP)_Ψ,并获得了原问题与标量化问题解之间的关系.最后,引入了一个关键假设H_Ψ,借助关键假设H_Ψ及原问题与标量化问题间解的关系,获得了IPGVQEP解映射Berge下半连续性的充分必要条件,并举例验证了所得结果.     

二重调和映射在稳态轴对称引力场中的进一步应用

该文运用二重调和映射，将Ｅｒｉｓ引用的复合映射及Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程化为二重形式，从而使Ｅｉｒｓ求解稳态轴对称引力场的方法得到扩展，并具体导出了Ｅｒｎｓｔ解和它的对偶解．最后指出该复合映射的本质及其局限性．     

第二类Feigenbaum函数方程的平顶单谷扩充连续解

本文考虑第二类Ｆｅｉｇｅｎｂａｕｍ函数方程，探讨了该方程的连续解的性态及由某平顶单谷映射扩充所能得到的一切连续解的性态，并给出了构造由某平顶单谷映射扩充所能得到的一切连续解的可行方法。     

Enumeration on Nonseparable Planar Maps

This paper provides some functional equations satisfied by the generating functions for nonseparable rooted planar maps with the valency of root-vertex, the number of edges and the valency of root-faces of the maps as three parameters. But the solutions of these equations can only be obtained indirectly by considering some relations between nonseparable and general rooted planar maps. One of them is an answer to the open problem 6.1 in Liu (1983, Comb. Optim. CORR83-26, University of Waterloo).     

Localized Normal Maps and the Stability of Variational Conditions

This paper introduces a localized version of the normal maps previously employed by the author and others to study solutions of variational inequalities. This localization permits us to deal with variational conditions posed over sets that may not be convex, and to draw conclusions about the existence and continuity of solutions of the variational condition by studying those of the normal map. As an application of these localized normal maps we investigate solutions of variational conditions when the set over which the condition is posed, as well as the function appearing in the condition, may vary continuously. In this situation solutions need be neither locally single-valued nor Lipschitzian in the parameters. Nonetheless, the normal-map approach permits us to draw fairly strong conclusions about the existence and continuity of solutions of perturbed problems if the unperturbed problems satisfy certain regularity conditions and the perturbations satisfy mild continuity requirements.     

上半连续集值优化解在图像逼近意义下的稳定性

给出上半连续集值映射优化问题在图像拓扑逼近意义下的本质弱有效解和本质有效解的概念.利用通有稳定性研究的usco方法, 证明了上半连续集值映射优化问题.在图像拓扑逼近意义下,弱有效解映射在定义域和映射同时扰动下是紧值上半连续的,从而是通有下半连续的,即在Baire纲意义下, 绝大多数上半连续集值映射优化问题, 在图像逼近意义下其弱有效解是稳定的,或者说是本质的. 证明了上半连续集值映射优化问题在图像逼近意义下有效解映射上半连续的一个充要条件,也即是有效解通有稳定的一个重要条件.     

On Optimization Problems with Set-Valued Objective Maps: Existence and Optimality

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets of n-dimensional Euclidean space. Second, by using these quasi orderings, we define the concepts of lower semi-continuity for set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and we give some conditions under which these optimal solutions exist to the problems and give necessary and sufficient conditions for optimality.     

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution

We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map under the Schrödinger maps evolution with respect to non-equivariant perturbations, provided obeys a suitable linearized stability condition. This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper [36]. © 2021 Wiley Periodicals LLC.     

Computing Harmonic Maps and Conformal Maps on Point Clouds

We use a narrow-band approach to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space, using point cloud data only. Given a surface, or a point cloud approximation, we simply use the standard cubic lattice to approximate its $\epsilon$-neighborhood. Then the harmonic map of the surface can be approximated by discrete harmonic maps on lattices. The conformal map, or the surface uniformization, is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant curvature. We propose algorithms and numerical examples for closed surfaces and topological disks. To the best of the authors' knowledge, our approach provides the first meshless method for computing harmonic maps and uniformizations of higher genus surfaces.     

A Remark on the Potentials of Optimal Transport Maps

Optimal maps, solutions to the optimal transportation problems, are completely determined by the corresponding c-convex potential functions. In this paper, we give simple sufficient conditions for a smooth function to be c-convex when the cost is given by minimizing a Lagrangian action.     

关于三商映射

三商映射是完备映射和开映射的共同推广。本文综述三商映射的理论，论术这三商映射，开映射，紧覆盖映射，诱导完备映之间的一些转换关系，提出了几个供进一步研究的问题。     

On a Relation Between Potentials for Pluriharmonic Maps and Para-Pluriharmonic Maps

In this paper, we show that one can interrelate pluriharmonic maps with para-pluriharmonic maps by means of the loop group method. As an appendix, we give examples for the interrelation between pluriharmonic maps and para-pluriharmonic maps. Moreover, we investigate the relation among CMC-surfaces by use of such maps.     

Geometric Transient Solutions of Autonomous Scalar Maps

Smooth autonomous scalar maps with locally asymptotically stable equilibria have families of asymptotically constant solutions which decay geometrically to the equilibria. Locally, all transients converging to the equilibria have this form.