Atsuji completions vis-à-vis hyperspaces

A metric space (X, d) is called an Atsuji space if every real-valued continuous function on (X, d) is uniformly continuous. It is well-known that an Atsuji space must be complete. A metric space (X, d) is said to have an Atsuji completion if its completion (, d) is an Atsuji space. In this paper, we study twelve equivalent (external) characterizations for a metric space to have an Atsuji completion in terms of hyperspace topologies. We also characterize topologically those metrizable spaces whose completions are Atsuji spaces. The first author was supported by the SPM fellowship awarded by the Council of Scientific and Industrial Research, India, during the work of this paper.     

Topologies on function spaces

In the present paper we introduce notions of A-splitting and A-jointly continuous topology on the set C(Y,Z) of all continuous maps of a topological space Y into a topological space Z, where A is any family of spaces. These notions satisfy the basic properties of splitting and jointly continuous topologies on C(Y,Z). In particular, for every A, the greatest A-splitting topology on C(Y,Z) (denoted by τ(A) always exists. We indicate some families A of spaces for which the topology τ(A) coincides with the greatest splitting topology on C(X,Y). We give a notion of equivalent families of spaces and try to find a “simple” family which is equivalent to a given family. In particular, we prove that every family is equivalent to a family consisting of one space, and the family of all spaces is equivalent to a family of all T₁-spaces containing at most one nonisolated point. We compare the topologies τ({X}) for distinct compact metrizable spaces X and give some examples. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 82–97. Translated by A. A. Ivanov.     

Uniformly convex spaces,bead spaces,and equivalence conditions

The notion of a metric bead space was introduced in the preceding paper (L.Pasicki: Bead spaces and fixed point theorems, Topology Appl., vol. 156 (2009), 1811–1816) and it was proved there that every bounded set in such a space (provided the space is complete) has a unique central point. The bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces. It appears that normed bead spaces are identical with uniformly convex spaces. On the other hand the “metric” approach leads to new elementary conditions equivalent to the uniform convexity. The initial part of the paper contains the proof that discus spaces (they seem to have a richer structure) are identical with bead spaces.     

Lipschitzian Properties of Integral Functionals on Lebesgue Spaces Lp, 1 \leqslant p < ￥\bf{L_p, 1 \leqslant {p} < \infty}

We give complete characterizations of integral functionals which are Lipschitzian on a Lebesgue space L _p with p ≠ ∞. When the measure is atomless, we characterize the integral functionals which are locally Lipschitzian on such Lebesgue spaces. In every cases, the Lipchitzian properties of the integral functional can be described by growth conditions on the subdifferentials of the integrand which are equivalent to Lipschitzian properties of the integrand.      

Gabor pairs, and a discrete approach to wave-front sets

We introduce admissible lattices and Gabor pairs to define discrete versions of wave-front sets with respect to Fourier–Lebesgue and modulation spaces. We prove that these wave-front sets agree with each other and with corresponding wave-front sets of “continuous type”. This implies that the coefficients of a Gabor frame expansion of f are parameter dependent, and describe the wave-front set of f.     

The Weizsäcker phenomenon and a canonical definition of Gaussian Lebesgue–Rokhlin measures

The notion of “Gaussian Lebesgue–Rokhlin space” is introduced. The definition is canonical, i.e., is given without use of topological and others irrelevant mathematical structures. The object under discussion completes the category of finite-dimensional Gaussian vector spaces. Some nontrivial examples are considered and historical comments are given. Bibliography: 30 titles.     

Atsuji completions: Equivalent characterisations

A metric space (X,d) is called an Atsuji space if every real-valued continuous function on (X,d) is uniformly continuous. It is well known that an Atsuji space must be complete. A metric space (X,d) is said to have an Atsuji completion if its completion is an Atsuji space. In this paper, we study twenty-nine equivalent characterisations for a metric space to have an Atsuji completion.     

Reverse mathematics of separably closed sets

This paper contains a corrected proof that the statement “every non-empty closed subset of a compact complete separable metric space is separably closed” implies the arithmetical comprehension axiom of reverse mathematics.     

A simple proof of the ergodic theorem using nonstandard analysis

A simple proof of the individual ergodic theorem is given. The essential tool is the nonstandard measure theory developed by P. Loeb. Any dynamical system on an abstract Lebesgue space can be represented as a factor of a “cyclic” system with a hyperfinite cycle. The ergodic theorem for such a “cyclic” system is almost trivial because of its simple structure. The general case follows from this special case.     

Some examples of clans

It is shown that every space obtained by identifying exactly two points of a two-cell supports a clan, including the so-called “French Army Hat”. Examples are given of clans on spaces obtained by sewing together ice-cream cones.     

The geodesic flow of a nonpositively curved graph manifold

We consider discrete cocompact isometric actions where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces — complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov — as Hadamard spaces) and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We identify invariants (“geometric data”) of the action which determine, and are determined by, the equivariant homeomorphism type of the action of G on the ideal boundary of X. Moreover, if are two actions with the same geometric data and is a G-equivariant quasi-isometry, then for every geodesic ray there is a geodesic ray (unique up to equivalence) so that . This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136]. Submitted: May 2001.     

Vitali's Theorem and WWKL

 Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA₀. We show that several well-known measure-theoretic propositions including the Vitali Covering Theorem are equivalent to WWKL over RCA₀. Received: 3 April 1998 / Published online: 3 October 2001     

On Stone's theorem and the Axiom of Choice

It is a well established fact that in Zermelo-Fraenkel set theory, Tychonoff's Theorem, the statement that the product of compact topological spaces is compact, is equivalent to the Axiom of Choice. On the other hand, Urysohn's Metrization Theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. A. H. Stone's Theorem, that every metric space is paracompact, is considered here from this perspective. Stone's Theorem is shown not to be a theorem in ZF by a forcing argument. The construction also shows that Stone's Theorem cannot be proved by additionally assuming the Principle of Dependent Choice.

     

Points de namioka espaces normants applications à la théorie isométrique de la dualité

We study in this work various aspects of the isometric theory of duality. We show that in wide classes of Banach spaces, dual spaces are characterized by the existence of a retraction fromE″ ontoE. The predual of such spaces is then unique. We study the imbedding of regularly normed spaces into dual spaces. We better the known results on loss of regularity of the norm of dual spaces. We characterize the dual norms on an Asplund space in terms of “bad differentiability”.      

Complete observability of differential-algebraic systems with delays

We study the statement and solvability of complete observability problems for linear stationary differential-algebraic dynamical systems with delays (DAD systems. Since in the general case, the state space of such systems is infinite-dimensional and is not necessarily “minimal,” we consider various statements of problems depending on what states are observed. Our attention is focused on the simplest DAD system in symmetric form. We obtain efficient parametric criteria and analyze relationships between various notions of complete observability for DAD systems. In the case of DAD systems with scalar coefficients, we obtain a complete classification of notions of complete observability in the class of continuous initial functions with the continuous matching condition. We analyze the problem of computing the minimum number of outputs of a spectrally observable DAD system.     

Symbolic calculus for boundary value problems on manifolds with edges

Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbolic structure is responsible for ellipticity and for the nature of parametrices within an algebra of “edge-degenerate” pseudo-differential operators. The edge symbolic component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operator-valued Mellin symbols. We establish a calculus in a framework of “twisted homogeneity” that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.     

Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions

We analyze a class of quasilinear elliptic problems involving a p(·)-Laplace-type operator on a bounded domain W ì \mathbb R^N{\Omega\subset{\mathbb R}^N}, N ≥ 2, and we deal with nonlinear conditions on the boundary. Working on the variable exponent Lebesgue–Sobolev spaces, we follow the steps described by the “fountain theorem” and we establish the existence of a sequence of weak solutions.     

Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over Twisted Heisenberg-Virasoro Algebra

We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenber-Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module （and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series）.     

Area Spaces: First Steps

We show that a Riemannian manifold of dimension at least 3 can be recovered from the space of boundaries of rectifiable integral 2-currents (the “lcycles”) equipped with the filling area distance, and discuss possible approaches to “spaces with area structures”.     

Differentiation of Integrals in RΩ

We show that the Lebesgue differentiation theorem does not hold in Ρ with the “product” Lebesgue measure. This revised version was published online in June 2006 with corrections to the Cover Date.