度量空间中有限点集的费马点

本文首先对紧致的度量拓扑空间证明了有限点集的费马点是存在的，其次，运用度量几何的经典方法考察了度量空间（包括双曲空间和Ｂａｎａｃｈ空间）中有限点集的费马点的唯一性，此外，还对ｎ维欧氏空间Ｅ＾ｎ中有限点集的费马作了进一步研究。     

二次曲线与其任意两条切线所围面积为定常值的问题研究

本文得到了二次曲线的任意两条相交切线与曲线本身围成的面积如果为定常值，则切线交点的轨迹仍为同类型二次曲线．又若给定两条同类的二次曲线，由其中一条上的每一点向另一条引出两条切线，则这两条切线与另一条曲线围成的面积为定常值．     

Über Böschungslinien im dreidimensionalen elliptischen Raum

In accordance withH. R. Müller [3] we understand under a curve of constant slope in the elliptic 3-space an isogonal trajectory of the generators of an arbitrary Clifford cylinder. Using linegeometric methods in a special projective model, we study in particular those curves of constant slope, whose tangents form also a constant angle with a fixed plane. Thereby we meet with well-known classes of curves in the Euclidean space, such as spherical involutoids and tractrices of circles and loxodromes on a torus.     

Singularities of Solution Surfaces for Quasilinear First-Order Partial Differential Equations

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.     

On positively differentiable curves with values in normed linear spaces

We prove a theorem that characterizes continuous normed linear space-valued curves allowing differentiable parameterizations with non-zero derivatives as those curves, all the points of which are regular (in Choquet's sense). We also state an equivalent geometric condition not involving any homeomorphisms. This extends a theorem due to Choquet, who proved a similar result for curves with values in Euclidean spaces.     

Finding curves on general spaces through quantitative topology,with applications to Sobolev and Poincaré inequalities

In many metric spaces one can connect an arbitrary pair of points with a curve of finite length, but in Euclidean spaces one can connect a pair of points with a lot of rectifiable curves, curves that are well distributed across a region. In the present paper we give geometric criteria on a metric space under which we can find similar families of curves. We shall find these curves by first solving a dual problem of building Lipschitz maps from our metric space into a sphere with good topological properties. These families of curves can be used to control the values of a function in terms of its gradient (suitably interpreted on a general metric space), and to derive Sobolev and Poincaré inequalities.The author is supported by the U.S. National Science Foundation and grateful to IHES for its hospitality.     

Orthogonal sequences and regularity of embeddings into finite-dimensional spaces

This paper focuses on the regularity of linear embeddings of finite-dimensional subsets of Hilbert and Banach spaces into Euclidean spaces. We study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c₀ of null sequences. The examples studied show that the results due to Hunt and Kaloshin (Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity 12 (1999) 1263-1275) and Robinson (Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity 22 (2009) 711-728) for subsets of Hilbert and Banach spaces with finite box-counting dimension are asymptotically sharp. An analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by Olson and Robinson (Almost bi-Lipschitz embeddings and almost homogeneous sets, Trans. Amer. Math. Soc. 362 (1) (2010) 145-168) for subsets X of Hilbert spaces when X−X has finite Assouad dimension.     

Locally flat Banach spaces

The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher order smoothness is involved. In this note we investigate the structural properties of Banach spaces admitting (arbitrary) bump functions depending locally on finitely many coordinates. This work was supported by the research project MSM 0021620839 and by the grant of the Grant Agency of the Czech Republic No. 201/05/P582 and No. 201/06/0018.     

Minimal vectors in arbitrary Banach spaces

We extend the method of minimal vectors to arbitrary Banach spaces. It is proved, by a variant of the method, that certain quasinilpotent operators on arbitrary Banach spaces have hyperinvariant subspaces.

     

Geometrically induced discrete spectrum in curved tubes

The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating w.r.t. the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincides with the spectrum of the straight tube of the same cross-section and that the discrete spectrum is not empty.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

A strong open mapping theorem for surjections from cones onto Banach spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.     

Flat Chains in Banach Spaces

We extend the notion of a flat chain with coefficients in a normed abelian group from Euclidean space to an arbitrary Banach space and prove a compactness result. We also remove the condition that a flat chain with arbitrary coefficients have finite mass in order for its support to exist. This research was part of the author’s Ph. D. dissertation at Stanford University.     

具随机性误差隐迭代程序的收敛性

在任意Banauch空间中,证明了有限族渐近半压缩映象具随机性误差的隐迭代程序逼近其公共不动点的强收敛性定理.所得结论推广和改进了引文中的相应结果.     

Banach Space-Valued Ornstein–Uhlenbeck Processes with General Drift Coefficients

We construct Ornstein–Uhlenbeck processes with values in Banach space and with continuous paths. The drift coefficient must only generate a strongly continuous semigroup on the Hilbert space which determines the Brownian motion. We admit arbitrary starting points and consider also invariant measures for the process, generalizing earlier work in many directions. A price for the generality is that sometimes one has to enlarge the phase space but most previously known results are covered.The constructions are based on abstract Wiener space methods, more precisely on images of abstract Wiener spaces under suitable linear transformations of the Cameron–Martin space. The image abstract Wiener measures are then given by stochastic extensions. We present the basic spaces and operators and the most important results on image spaces and stochastic extensions in some detail.     

Compact spaces of Szlenk index ω

The Szlenk index has found many applications in the isomorphic theory of Banach spaces. Its definition is based in some kind of interplay between a weak topology and the norm metric with not much care on the linear structure. There is no obstacle to consider the notion of Szlenk index in more general settings. In this paper we study the compact spaces of Szlenk index ω at most with respect to an associated metric. We include new applications to Banach spaces of the these methods, where the estimations of the growth speed of the finite Szlenk indices play a fundamental role.     

From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics

In this paper, we first gather existence results for linear and for pseudo-monotone variational inequalities in reflexive Banach spaces. We discuss the necessity of the involved coerciveness conditions and their relationship. Then, we combine Mosco convergence of convex closed sets with an approximation of pseudo-monotone bifunctions and provide a convergent approximation procedure for pseudo-monotone variational inequalities in reflexive Banach spaces. Since hemivariational inequalities in linear elasticity are pseudo-monotone, our approximation method applies to nonmonotone contact problems. We sketch how regularization of the involved nonsmooth functionals together with finite element approximation lead to an efficient numerical solution method for these nonconvex nondifferentiable optimization problems. To illustrate our theory, we give a numerical example of a 2D linear elastic block under a given nonmonotone contact law.     

Sull'estensione dell'integrale debole alla Burkill-Cesari ad una misura

We extend the weak Burkill-Cesari integral of a vector valued set function, with values in a Banach space, to a measure, following an idea developed by Cesari ([4], [5]) in Euclidean spaces.     

On a class of locally convex closed curves

Summary For a certain type of locally convex plane curves some relations are shown between the numbers of double points and double tangents and the order and class of the curve. It is proved that the curve has an equal number of double points and double tangents and at least v-1 where 2v denotes the order (and class) of the curve. At last it is shown how a curve with more that v-1 double points (tangents) may be deformed into a curve with the minimum number. To Enrico Bompiani on his scientific Jubilee     

A generalization of the set averaging theorem

We consider the possibility of generalizing the averaging theorem from the case of sets from n-dimensional Euclidean space to the case of sets from Banach spaces. The result is a cornerstone for constructing the theory of the Riemann integral for non-convex-valued multivalued mappings and for proving the convexity of this multivalued integral. We obtain a generalization of the averaging theorem to the case of sets from uniformly smooth Banach spaces as well as some corollaries.