Basic Sequences in the Dual of a Fréchet Space

In this paper we study some properties of basic sequences in the dual of a Fréechet space. As a consequence we obtain that if E is a Fréechet space with the property that for each closed subspace F of E and each bounded subset B of E/F there is a bounded subset A of E with φ(A) = B, where φ denotes the canonical surjection of E onto E/F, then one of the following conditions is at least satisfied: 1. E is a Banach space, 2. E is a Schwartz space, 3. E is the product of a Banach space by ω. Finally, we also obtain some results concerning totally reflexive spaces.     

Some Applications of Critical Point Theory of Distance Functions on Riemannian Manifolds

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature K _M 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than /2 and contains a geodesic loop of length 2. We also prove that if M is an n(3)-dimensional complete connected Riemannian manifold with K _M 1 and radius bigger than /2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.     

Bootstrap estimation of the asymptotic variances of statistical functionals

A modified bootstrap estimator of the asymptotic variance of a statistical functional is studied. The modified bootstrap variance estimator circumvents the problem of the original bootstrap when the population distribution has heavy tails, and requires less stringent conditions for its consistency than the ordinary bootstrap variance estimator. The consistency of the modified bootstrap variance estimator is established for differentiable statistical functionals.     

On the global theory of some classes of mappings

This paper is devoted to the study of the global theory of certain mappings between Riemannian manifolds. We generalize results by Vilms, Yano and Ishihara, and study in detail projective, umbilical and harmonic maps.The work of the author was supported by RBRF, grant 94-01-01595 (Russia).     

On Ricci eigenvalues of locally homogeneous Riemannian 3-manifolds

We find the necessary and sufficient conditions for three constants ₁, ₂, ₃ ³ to be the principal Ricci curvatures of some 3-dimensional locally homogeneous Riemannian space.The first author was supported by the grant GAR 201/93/0469; the second author was supported by the grant SFS, Project #0401.     

The Fréchet distance between multivariate normal distributions

The Fréchet distance between two multivariate normal distributions having means μ_X, μ_Y and covariance matrices Σ_X, Σ_Y is shown to be given by $\text{d}{}^2\text{= |μ}{}_{\mathrm{X}}\text{? μ}{}_{\mathrm{Y}}\text{|}{}^2\text{+}\text{tr}\text{(Σ}{}_{\mathrm{X}}\text{+ Σ}{}_{\mathrm{Y}}\text{? 2(Σ}{}_{\mathrm{X}}\text{Σ}{}_{\mathrm{Y}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$. The quantity d₀ given by $\text{d}{}_0{}^2\text{=}\text{tr}\text{(Σ}{}_{\mathrm{X}}\text{+ Σ}{}_{\mathrm{Y}}\text{? 2(Σ}{}_{\mathrm{X}}\text{Σ}{}_{\mathrm{Y}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is a natural metric on the space of real covariance matrices of given order.     

Geometric properties of Fréchet spaces and selection of basis sequences

For an arbitrary nuclear Fréchet spaceE possessing the well-known geometric propertiesD ₁ (DN) and Ω, certain sequences of functionals and elements with power-type estimates of the norms are selected. With the help of these objects, two isomorphic spacesK ₁ andK ₂ of power series of infinite type and continuous mapsJ ₂: K₂→E andJ ₁: E→K₁ are defined. Under some auxiliary conditions, it is proved that the_i are isomorphisms and the spaceE possesses a basis. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 102–111, July, 1999.     

On the Subdifferentiability of the Difference of Two Functions and Local Minimization

We set up a formula for the Fréchet and ε-Fréchet subdifferentials of the difference of two convex functions. We even extend it to the difference of two approximately starshaped functions. As a consequence of this formula, we give necessary and sufficient conditions for local optimality in nonconvex optimization. Our analysis relies on the notion of gap continuity of multivalued maps and involves concepts of independent interest such as the notions of blunt and sharp minimizers and the notion of equi-subdifferentiability.      

Subdifferentiability and superdifferentiability of distance functions

We obtain necessary and sufficient conditions for the subdifferentiability and superdifferentiability (in the Dem'yanov-Rubinov sense) of the distance in an arbitrary norm from a point to a set for the finitedimensional case. The geometric structure of the subdifferential and the superdifferential is described. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 530–542, April, 1997. Translated by N. K. Kulman     

一类完备Riemann流形上的有界调和函数

本文我们将对一类完备Ｒｉｅｍａｎｎ流形上的有界调和函数所组成的线性空间的维数的上界进行估计,同时给出了一个关于测地球体积的Ｂｉｓｈｏｐ－Ｇｒｏｍｏｖ型体积比较定理。     

Fuzzy and Exact Optimality Conditions for a Bilevel Set-Valued Problem via Extremal Principles

In this article, we are concerned with a sequence of two set valued optimization problems in which the feasible region of the first one (the upper-level problem) is determined implicitly by the solution set of the second (the lower-level problem). Since bilevel programming problems are in general nonconvex problems even if the problem data are convex, we use the exact extremal principle and the approximate extremal principle introduced by Mordukhovich [14 B.S. Mordukhovich ( 2001 ). The extremal principle and its applications to optimization and economics . In: Optimization and Related Topics ( A. Rubinov and B. Glover , eds.). Applied Optimization Volumes 47 , Kluwer , Dordrecht , The Netherlands , pp. 343 – 369 . [Google Scholar], 15 B.S. Mordukhovich ( 2006 ). Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), 330 and 331, Springer, Berlin . [Google Scholar]] in order to get optimality conditions for this bilevel problem.