Convex functions, subdifferentiability and renormings

This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces. Supported by NSFC     

New smooth C-functions for symmetric cone complementarity problems

In this paper, with the help of the Jordan-algebraic technique we introduce two new complementarity functions (C-functions) for symmetric cone complementary problems, and show that they are continuously differentiable and strongly semismooth everywhere. The work was partly supported by the National Natural Science Foundation of China (10671010, 70471002).     

一阶线性模糊微分方程的解

利用文[8]中的扩展原理求解了一阶线性模糊微分方程,给出了解可表示的条件,讨论了同其他求解方法之间的关系,最后给出了具体算例。     

Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Let G be a Lie group which is the union of an ascending sequence G₁⊆G₂⊆? of Lie groups (all of which may be infinite-dimensional). We study the question when in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C^∞-diffeomorphisms of a σ-compact smooth manifold M; and for test function groups of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.     

Elliptic equations on convex domains with nonhomogeneous Dirichlet boundary conditions

In this paper, we will study the differentiability on the boundary of solutions of elliptic non-divergence differential equations on convex domains. The results are divided into two cases: (i) at the boundary points where the blow-up of the domain is not the half-space, if the boundary function is differentiable then the solution is differentiable; (ii) at the boundary points where the blow-up of the domain is the half-space, the differentiability of the solution needs an extra Dini condition for the boundary function. Counterexample is given to show that our results are optimal.     

A STRONG OPTIMIZATION THEOREM IN LOCALLY CONVEX SPACES

This paper presents a geometric characterization of convex sets in locally convex spaces on which a strong optimization theorem of the Stegall-type holds, and gives Collier's theorem of w Asplund spaces a localized setting.     

An envelope theorem and some applications to discounted Markov decision processes

In this paper, an Envelope Theorem (ET) will be established for optimization problems on Euclidean spaces. In general, the Envelope Theorems permit analyzing an optimization problem and giving the solution by means of differentiability techniques. The ET will be presented in two versions. One of them uses concavity assumptions, whereas the other one does not require such kind of assumptions. Thereafter, the ET established will be applied to the Markov Decision Processes (MDPs) on Euclidean spaces, discounted and with infinite horizon. As the first application, several examples (including some economic models) of discounted MDPs for which the et allows to determine the value iteration functions will be presented. This will permit to obtain the corresponding optimal value functions and the optimal policies. As the second application of the ET, it will be proved that under differentiability conditions in the transition law, in the reward function, and the noise of the system, the value function and the optimal policy of the problem are differentiable with respect to the state of the system. Besides, various examples to illustrate these differentiability conditions will be provided. This work was partially supported by Benemérita Universidad Aut ónoma de Puebla (BUAP) under grant VIEP-BUAP 38/EXC/06-G, by Consejo Nacional de Ciencia y Tecnología (CONACYT), and by Evaluation-orientation de la COopération Scientifique (ECOS) under grant CONACyT-ECOS M06-M01.     

A Note on Differentiability in a Markov Chain Market Using Stochastic Flows

Using stochastic flows of diffeomorphisms relating to a Markov chain together with the Itô's differentiation rule, the differentiability of the price of a European-style contingent claim with respect to the underlying state variables is proved in a continuous-time Markov chain market. The differentiability results are also used to calculate the Greeks for hedging.     

Differentiability and its asymptotic analysis for nonlinear singularly perturbed boundary value problem

In this paper, we show differentiability of solutions with respect to the given boundary value data for nonlinear singularly perturbed boundary value problems and its corresponding asymptotic expansion of small parameter. This result fills the gap caused by the solvability condition in Esipova’s result so as to lay a rigorous foundation for the theory of boundary function method on which a guideline is provided as to how to apply this theory to the other forms of singularly perturbed nonlinear boundary value problems and enlarge considerably the scope of applicability and validity of the boundary function method. A third-order singularly perturbed boundary value problem arising in the theory of thin film flows is revisited to illustrate the theory of this paper. Compared to the original result, the imposed potential condition is completely removed by the boundary function method to obtain a better result. Moreover, an improper assumption on the reduced problem has been corrected.     

w-Frchet可微性质和Radon-Nikod(?)m性质以及w-Asplund空间

我们称定义在一个Banach空间的对偶空间上的广义实值w*-下半连续凸函数f具有w*-Frechet可微性质(w*-FDP),如果对于该对偶空间上的每个w*-下半连续的广义实值凸函数g,只要g≤f,就有g在intdom g的某个稠密的Gδ-子集上处处Frechet可微.本文用集合的Radon-Nikodym性质刻划了该种函数的特征.作为它的一个直接推论,给出了局部化的Collier定理.