借助几何图形理解高等数学中的抽象概念和结论

高等数学的很多内容比较抽象,学生不易理解.通过几个例子说明如何将抽象的数学概念和结论与几何图形有机的结合起来,加深对这些概念结论的理解,激发学生的学习兴趣.     

Toll number of the Cartesian and the lexicographic product of graphs

Toll convexity is a variation of the so-called interval convexity. A tolled walk $T$ between two non-adjacent vertices $u$ and $v$ in a graph $G$ is a walk, in which $u$ is adjacent only to the second vertex of $T$ and $v$ is adjacent only to the second-to-last vertex of $T$. A toll interval between $u,v\in V\left(G\right)$ is a set $T_G(u,v)=\{x\in V\left(G\right):x\text{ lies on a tolled walk between }u\text{ and }v\}$. A set $S?V\left(G\right)$ is toll convex, if $T_G(u,v)?S$ for all $u,v\in S$. A toll closure of a set $S?V\left(G\right)$ is the union of toll intervals between all pairs of vertices from $S$. The size of a smallest set $S$ whose toll closure is the whole vertex set is called a toll number of a graph $G$, $tn\left(G\right)$. The first part of the paper reinvestigates the characterization of convex sets in the Cartesian product of two graphs. It is proved that the toll number of the Cartesian product of two graphs equals 2. In the second part, the toll number of the lexicographic product of two graphs is studied. It is shown that if $H$ is not isomorphic to a complete graph, $tn(G°H)\le 3?tn\left(G\right)$. We give some necessary and sufficient conditions for $tn(G°H)=3?tn\left(G\right)$. Moreover, if $G$ has at least two extreme vertices, a complete characterization is given. Furthermore, graphs with $tn(G°H)=2$ are characterized. Finally, the formula for $tn(G°H)$ is given — it is described in terms of the so-called toll-dominating triples or, if $H$ is complete, toll-dominating pairs.     

Hölder Continuity of the Saddle Point Set for Real-Valued Functions

This paper is concerned with Hölder continuity of the solution to a saddle point problem. Some new su?cient conditions for the uniqueness and Hölder continuity of the solution for a perturbed saddle point problem are established. Applications of the result on Hölder continuity of the solution for perturbed constrained optimization problems are presented under mild conditions. Examples are given to illustrate the obtained results.     

局部凸空间P-自反下的中点局部k-一致凸性(光滑性)

在局部凸空间已有的中点局部kk-一致凸性和中点局部k-一致光滑性这一对对偶概念的基础上,证明了中点局部kk-一致凸性与中点局部(k+1)-一致凸性的关系,给出了在P-自反的条件下它们之间的等价对偶定理.     

Nonlinear Stability of a One-Dimensional Boussinesq Equation

We study nonlinear orbital stability and instability of the set of ground state solitary wave solutions of a one-dimensional Boussinesq equation or one-dimensional Benney–Luke equation. It is shown that a solitary wave (traveling wave with finite energy) may be orbitally stable or unstable depending on the range of the wave's speed of propagation.     

CHARACTERISTICS OF SUBDIFFERENTIALS OF FUNCTIONS

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New geometric properties of Banach spaces

In this paper, we introduce new geometric properties as generalizations of p‐uniform smoothness and q‐uniform convexity of Banach spaces. Furthermore, using generalized Beckner's inequality, we characterize the properties in terms of norm inequalities. As an application, we consider the duality relation.     

ω-非常凸空间与ω-非常光滑空间

本文研究了ω-非常凸空间和ω-非常光滑空间的问题.利用局部自反原理和切片证明了ω-非常凸空间和ω-非常光滑空间的对偶关系,讨论了ω-非常凸空间和ω-非常光滑空间与其它凸性和光滑性的关系,给出了ω-非常凸空间与ω-非常光滑空间的若干特征刻画,所得结果完善了关于Banach空间凸性与光滑性理论的研究.     

A Minty variational principle for set optimization

Extremal problems are studied involving an objective function with values in (order) complete lattices of sets generated by so-called set relations. Contrary to the popular paradigm in vector optimization, the solution concept for such problems, introduced by F. Heyde and A. Löhne, comprises the attainment of the infimum as well as a minimality property. The main result is a Minty type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions. The variational inequality is based on a new Dini directional derivative for set-valued functions which is defined in terms of a “lattice difference quotient.” A residual operation in a lattice of sets replaces the inverse addition in linear spaces. Relationships to families of scalar problems are pointed out and used for proofs. The appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as Diewert's theorem to improper functions.