Partial Geometric Regularity of Some Optimal Connected Transportation Networks

We consider a continuous optimization model of a one-dimensional connected transportation network under the assumption that the cost of transportation with the use of network is negligible in comparison with the cost of transportation without it. We investigate the connections between this problem and its important special case, the minimization of the average distance functional. For the average distance minimization problem we formulate a number of conditions for the partial geometric regularity of a solution in ℝⁿ with an arbitrary dimension n ⩾ 2. The corresponding results are applied to solutions to the general optimization problem. Bibliography: 26 titles. Illustrations: 1 Figure. __________ Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 129–157.     

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.     

On the Existence of n-Harmonic Spheres

Using the bubbling argument of Sacks and Uhlenbeck, we prove the existence of n-harmonic maps from the n-sphere to Riemannian manifolds. An application is made to a problem concerning manifolds with strongly pth moment stable stochastic dynamical systems.     

Optimal Regularity of Lower-Dimensional Obstacle Problems

In this paper, we prove that solutions to the “boundary obstacle problem” have the optimal regularity, C^1,1/2, in any space dimension. This bound depends only on the local L²-norm of the solution. Main ingredients in the proof are the quasiconvexity of the solution and a monotonicity formula for an appropriate weighted average of the local energy of the normal derivative of the solution. Bibliography: 8 titles. Dedicated to Nina Nikolaevna Uraltseva on the occasion of her 70th birthday __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 49–66.     

常曲率球面到单位球面的等距极小浸入与嵌入

本文主要估计了球面到单位球面的等距极小浸入的覆盖层数，从而导出一些有关嵌入的结果。同时，在本文证明过程中，给出了一个构造指定覆盖层数的等距极小浸入的方法．     

关于运输问题最优解的进一步讨论

首先给出了运输问题最优解的相关概念,将最优解扩展到广义范畴,提出狭义多重最优解和广义多重最优解的概念及其区别.然后给出了惟一最优解、多重最优解、广义有限多重最优解、广义无限多重最优解的判定定理及其证明过程.最后推导出了狭义有限多重最优解个数下限和广义有限多重最优解个数上限的计算公式,并举例验证了结论的正确性.     

Regularity of the Optimal Feedback and the Value Function in Convex Problems of Optimal Control

An optimal feedback mapping, leading to necessary and sufficient conditions for optimality in terms of a closed-loop differential inclusion, is derived in the setting of fully convex generalized problems of Bolza. Results are translated to the format of control problems with linear dynamics and convex costs. Properties of the feedback mapping, with focus on single-valuedness and continuity, are analyzed through those of the value function and of the Hamiltonian. Conditions guaranteeing differentiability of the value function are obtained through the analysis of its subdifferential as a maximal monotone operator and of the generalized Hamiltonian dynamics.     

On the Image Regularity Condition

In this paper, we continue the analysis of the image regularity condition (IRC) as introduced in a previous paper where we have proved that IRC implies the existence of generalized Lagrange-John multipliers with first component equal to 1. The term generalized is connected with the fact that the separation (in the image space) is not necessarily linear (when we have classic Lagrange-John multipliers), but it can be also nonlinear. Here, we prove that the IRC guarantees, also in the nondifferentiable case, the fact that 0 is a solution of the first-order homogeneized (linearized) problem obtained by means of the Dini-Hadamard derivatives.     

On the Regularity of Crossed Products

We study some generalizations of the notion of regular crossed products K * G. For the case when K is an algebraically closed field, we give necessary and sufficient conditions for the twisted group ring K * G to be an n-weakly regular ring, a ξ^* N-ring or a ring without nilpotent elements.     

单位球面间的等距延拓

本文证明了在一定条件下赋范线性空间与其共轭空间的单位球面之间的等距算子可以延拓为全空间的实线性等距算子。进而,刻画了光滑的自反空间的单位球面到其共轭空间的单位球面上的等距算子。     

关于右SF-环的正则性

研究了每一个极大的右理想是拟理想的右SF-环的正则性,得到了右SF-环是正则环的一些新的刻画,推广了一些已知的结论.     

环的弱正则性

本文给出了环的弱正则性的一个充要条件，还讨论了Ｍｕｎｎ环的弱正则性．最后对环的半格和及补半格和，给出了弱正则性的刻画．     

Regularity of Optimal Controls for State Constrained Problems

Conditions are given under which optimal controls are Lipschitz continuous, for dynamic optimization problems with functional inequality constraints. The linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters. Smoothness conditions on the data are also relaxed. A key part of the proof involves an analysis of the implications of first order optimality conditions in the form of a nonsmooth Maximum Principle.     

On the Discrepancy of Point Distributions on Spheres and Hyperbolic Spaces

 Optimal lower bounds are given for the discrepancy of point distributions w.r.t. geodesic balls on spheres and hyperbolic spaces. The mean discrepancy is estimated below by using a non-commutative version of the Fourier transform method developed by Beck for Euclidean spaces.     

On Large Lattice Packings of Spheres

The packing density of large lattice packings of spheres in Euclidean E ^d measured by the parametric density depends on the parameter and on the shape of the convex hull P of the sphere centers; in particular on the isoperimetric coefficient of P and on the second term in the Ehrhart polynomial of the lattice polytope P. We show in E ^d , d 2, that flat or spherelike polytopes generate less dense packings, whereas polytopes with suitably chosen large facets generate dense packings. This indicates that large lattice packings in E ³ of high parametric density may be good models for real crystals.     

On the Discrepancy of Point Distributions on Spheres and Hyperbolic Spaces

 Optimal lower bounds are given for the discrepancy of point distributions w.r.t. geodesic balls on spheres and hyperbolic spaces. The mean discrepancy is estimated below by using a non-commutative version of the Fourier transform method developed by Beck for Euclidean spaces. Received March 2, 2000; in revised form February 28, 2002 Published online August 19, 2002     

Optimal Regularity for the No‐Sign Obstacle Problem

In this paper we prove the optimal $C^{1,1}(B_{1/2})$ ‐regularity for a general obstacle‐type problem under the assumption that $f*N$ is $C^{1,1}(B_1)$ , where N is the Newtonian potential. This is the weakest assumption for which one can hope to get $C^{1,1}$ ‐regularity. As a by‐product of the $C^{1,1}$ ‐regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point $x^0$ , the free boundary is locally a $C^1$ ‐graph close to $x^0$ provided f is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades. © 2012 Wiley Periodicals, Inc.     

On the Stability of the Directional Regularity

In this paper we select two tools of investigation of the classical metric regularity of set-valued mappings, namely the Ioffe criterion and the Ekeland Variational Principle, which we adapt to the study of the directional setting. In this way, we obtain in a unitary manner new necessary and/or sufficient conditions for directional metric regularity. As an application, we establish stability of this property at composition and sum of set-valued mappings. In this process, we introduce directional tangent cones and the associated generalized primal differentiation objects and concepts. Moreover, we underline several links between our main assertions by providing alternative proofs for several results.