Boundary case of equality in optimal Loewner-type inequalities

We prove certain optimal systolic inequalities for a closed Riemannian manifold , depending on a pair of parameters, and . Here  is the dimension of , while is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from to its Jacobi torus  , which are area-decreasing (on -dimensional areas), with respect to suitable norms. These norms are the stable norm of , the conformally invariant norm, as well as other -norms. Here we exploit -minimizing differential 1-forms in cohomology classes. We characterize the case of equality in our optimal inequalities, in terms of the criticality of the lattice of deck transformations of  , while the Abel-Jacobi map is a harmonic Riemannian submersion. That the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of Federer and Fleming, under the assumption of the nonvanishing of the homology class of the lift of the typical fiber of the Abel-Jacobi map to the maximal free abelian cover.

     

Musielak–Orlicz BMO-type Spaces Associated with Generalized Approximations to the Identity

Let X be a space of homogenous type and ψ: X × [0,∞) → [0,∞) be a growth function such that ψ(·,t) is a Muckenhoupt weight uniformly in t and ψ(x,·) an Orlicz function of uniformly upper type 1 and lower type p ∈(0,1].In this article,the authors introduce a new Musielak–Orlicz BMO-type space BMOψA(X) associated with the generalized approximation to the identity,give out its basic properties and establish its two equivalent characterizations,respectively,in terms of the spaces BMOψA,max(X) and BMOψA(X).Moreover,two variants of the John–Nirenberg inequality on BMOψA(X) are obtained.As an application,the authors further prove that the space BMOψΔ1/2(Rn),associated with the Poisson semigroup of the Laplace operator Δ on Rn,coincides with the space BMOψ(Rn) introduced by Ky.     

Linear Connectivity,Schwarz–Pick Lemma and Univalency Criteria for Planar Harmonic Mapping

In this paper, we first establish a Schwarz–Pick lemma for higher-order derivatives of planar harmonic mappings, and apply it to obtain univalency criteria. Then we discuss distortion theorems,Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.     

11th International Conference on Stochastic Programming

In this article we establish necessary conditions for local Pareto and weak minima of multiobjective programming problems involving inequality, equality and set constraints in Banach spaces in terms of convexificators.     

REMARKS ON JOHN DISKS

Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,then D is an inner linearly locally connected domain;(3)A homeomorphism f:R 2 →R 2 is a quasiconformal mapping if and only if f(D)is a John disk for any John disk D■R 2 ;and(4)If D is a bounded quasidisk,then D is a John disk,and there exists an unbounded quasidisk which is not a John disk.     

Stability of mappings with bounded distortion in the Sobolev norm on the John domains of Heisenberg groups

This article completes the authors’s series on stability in the Liouville theorem on the Heisenberg group. We show that every mapping with bounded distortion on a John domain of the Heisenberg group is approximated by a conformal mapping with order of closeness √K ? 1 in the uniform norm and with order of closeness K ? 1 in the Sobolev L _p ¹ -norm for all p < C/K?1. We construct two examples, demonstrating the asymptotic sharpness of our results.     

Extremal problems related to Gauss-John position

In this paper,we consider the extremal problem of the lp-norm:min {lp(TK),o∈TK■L,T∈GL(n)},where K,L are two convex bodies in Rn.Using the optimization theorem of John,we give necessary conditions for K to be in extremal position in terms of a decomposition of the identity.Furthermore,the weaker optimization problem,min{(lp(TK))p:TK■B2n,TK∩Sn-1≠,T∈GL(n)},is also considered.As an application,the geometric distance between the unit ball B2n and a centrally symmetric convex body K is obtained.     

John Bardeen and the Theory of Superconductivity: A Late Revision of a Homework Assignment for J. M. Luttinger

This account of the history of the Bardeen–Cooper–Schrieffer theory of superconductivity has its roots in an assignment that Quin Luttinger designed for me in 1963 when I was his graduate student. It improves on my student work by tracing the story from historical source materials as well as published papers, thus reflecting the contingencies and human elements that shape all research in physics. I also made an attempt to portray general features of how John Bardeen approached solving problems in physics.     

Benson真有效意义下向量集值优化的广义Fritz John条件

引入了一种有关集值映射的切导数和强、弱*伪凸的概念。借助凸集分离定理及锥分离定理建立了Benson真有效意义下向量集值优化导数型的FritzJohn最优性条件,并对条件的充分性进行了讨论。当特殊到单值映射时这些最优性条件与经典的结果完全吻合。