Characterisation of compactness through convergence in Tychonoff spaces

In this article we first give a characterisation of compact spaces among spaces by improving a theorem of J. Ewert. Then, with the aid of a new type of convergence, we give a characterisation of the pseudocompact and of the Lindelöf spaces.

     

On the Asymptotic Geometry of Nonpositively Curved Manifolds

In this paper we derive new asymptotic properties of all Hadamard manifolds admitting compact quotients. We study the growth function of the volume of geodesic spheres, generalizing the work of Margulis in the case of negative curvature. We show that the growth is of purely exponential type if and only if the Hadamard manifold is of rank 1. In general, there is a polynomial deviation from purely exponential behavior, depending in an unexpected way on the rank of the manifold. Furthermore, we obtain new results on the growth rate of closed geodesics on compact rank 1 spaces. Submitted: August 1996, final version: April 1997     

A note on an approximate mean value theorem for Fréchet subgradients

In this note, we derive a new characterization of Asplund spaces and give a clarification of the proof of the approximate mean value theorem in Mordukhovich (2006) [5] and Mordukhovich and Shao (1996) [1].     

Gross spaces

A Gross space is a vector space of infinite dimension over some field , which is endowed with a symmetric bilinear form and has the property that every infinite dimensional subspace satisfies dim dim. Gross spaces over uncountable fields exist (in certain dimensions). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariant b equals , a Gross space in dimension exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.

     

Extremal characterizations of asplund spaces

We prove new characterizations of Asplund spaces through certain extremal principles in nonsmooth analysis and optimization. The latter principles provide necessary conditions for extremal points of set systems in terms of Fréchet normals and -normals.

     

Characterizations of the Gelfand-Shilov spaces via Fourier transforms

We give symmetric characterizations, with respect to the Fourier transformation, of the Gelfand-Shilov spaces of (generalized) type and type . These results explain more clearly the invariance of these spaces under the Fourier transformations.

     

On Feller processes with sample paths in Besov spaces

Under mild regularity assumptions on its domain the infinitesimal generator of a Feller process is known to be a pseudo-differential operator. We give a simple condition on the symbol of the generator in order to characterize the smoothness of the sample paths of real-valued Feller processes in terms of Besov spaces . Our result extends previous papers on the paths of Gaussian, symmetric -stable [6], [20], and Lévy processes [11]. Received: 31 May 1996 / Revised version: 10 December 1996     

Equivariant Reidemeister torsion on symmetric spaces

We calculate explicitly the equivariant Ray-Singer torsion for all symmetric spaces of compact type with respect to the action of . We show that it equals zero except for the odd-dimensional Gra?mannians and the space . As a corollary, we classify up to diffeomorphism all isometries of these spaces which are homotopic to the identity; also, we classify their quotients by finite group actions up to homeomorphism. Received: 31 May 1995 / In revised form: 9 January 1996     

The Kakutani fixed point theorem for Roberts spaces

Roberts spaces were the first examples of compact convex subsets of Hausdorff topological vector spaces (HTVS) where the Krein–Milman theorem fails. Because of this exotic quality they were candidates for a counterexample to Schauder's conjecture: any compact convex subset of a HTVS has the fixed point property. However, extending the notion of admissible subsets in HTVS of Klee [Math. Ann. 141 (1960) 286–296], Ngu [Topology Appl. 68 (1996) 1–12] showed the fixed point property for a class of spaces, including the Roberts spaces, he called weakly admissible spaces. We prove the Kakutani fixed point theorem for this class and apply it to show the non-linear alternative for weakly admissible spaces.     

Regularity results for the quasi–static Bingham variational inequality in dimensions two and three

We consider the variational inequality describing the stationary flow of a Bingham type fluid in bounded domains. Differentiability properties of weak solutions in suitable energy spaces providing existence theorems are studied. We suppose that the volume forces belong to classes of Morrey type and generalize our previous regularity results concerning slow, steady–state flow of Bingham fluids. Received: 12 February 1996; in final form 16 July 1996     

Recombination induced hypergraphs: A new approach to mutation-recombination isomorphism

Natural selection acts on genetic variation that comes from two principal sources: mutation and recombination. Because of the inherent differences between mutation and recombination, it is often assumed that they are qualitatively different ways to explore the genotype space. In this paper a new way of constructing recombination spaces is introduced and the topological features of the resulting hypergraphs are analyzed. It is shown that types which are neighbors in the point mutation space are also neighbors in the recombination space, i.e., mutation and recombination spaces are homomorphic. This implies that the shapes of the fitness functions explored by mutation and recombination are similar. However, the potential of one- and two-point recombination operators to explore the fitness landscape may differ dramatically from uniform recombination operators or mutation operators because of the limited number of recombinant types they can produce. © 1996 John Wiley & Sons, Inc.     

Hyperbolic Wavelet Approximation

We study the multivariate approximation by certain partial sums (hyperbolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets. We characterize spaces of functions which have a prescribed approximation error by hyperbolic wavelet sums in terms of a K -functional and interpolation spaces. The results parallel those for hyperbolic trigonometric cross approximation of periodic functions [DPT]. October 16, 1995. Date revised: August 28, 1996.     

Lindelöf powers and products of function spaces

We give criteria for finite and countable powers of a space similar to the Michael line being Lindelöf. As applications, we give examples related to Lindelöf property in products of spaces of Michael line type and in products of spaces of continuous functions on separable -compact spaces.

     

Complex convexity in Lebesgue-Bochner Function Spaces

Complex geometric properties of continuously quasi-normed
spaces are introduced and their relationship to their analogues in real Banach spaces is discussed. It is shown that these properties lift from a continuously quasi-normed space to , for . Local versions of these properties and results are also considered.

     

Some applications of the KKM-mapping principle in hyperconvex metric spaces

In 1996 Khamsi proved a result in hyperconvex metric spaces which is an analogue of the famous KKM-maps principle proved by Ky Fan in 1972 for topological vector spaces.In this note, using the above result due to Khamsi, we establish two results in hyperconvex metric spaces which are analogues of the famous Ky Fan inequality and Sion minimax theorem proved for topological vector spaces in 1972 and 1958, respectively.     

Extreme values of operator norms in spaces with equivalent norms

The paper is concerned with various characteristics of spaces (L, ‖·‖) with equivalent norms and of operators on these spaces. In particular, the notion of distance between equivalent norms in the space L is introduced and the relationship between norms of the space and norms of operators in L is studied. Bibliography: 4 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 25–32.     

Classifying spaces and homotopy sets of axes of pairings

We consider the maps between classifying spaces of the form . The main theorem shows that if the restriction map on is a weak epimorphism, then the restriction on should factor through the classifying spaces of the center of the compact Lie group . An application implies that is an H-space (Hopf space) if and only if is abelian.

     

Some New Homogeneous Einstein Metrics on Symmetric Spaces

We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup of Isom acting transitively on we find all -homogeneous (non-symmetric) Einstein metrics on .

     

Products of images

Let be the \u{C}ech-Stone remainder . We show that there exists a large class of images of such that whenever is a subset of of cardinality at most the continuum, then is again an image of . The class contains all separable compact spaces, all compact spaces of weight at most and all perfectly normal compact spaces.