Straight nearness spaces

Straight spaces are spaces for which a continuous map defined on the space which is uniformly continuous on each set of a finite closed cover is then uniformly continuous on the whole space. Previously, straight spaces have been studied in the setting of metric spaces. In this paper, we present a study of straight spaces in the more general setting of nearness spaces. In a subcategory of nearness spaces somewhat more general than uniform spaces, we relate straightness to uniform local connectedness. We investigate category theoretic situations involving straight spaces. We prove that straightness is preserved by final sinks, in particular by sums and by quotients, and also by completions.     

Extension to infinite dimensions of a stochastic second-order model associated with shape splines

Motivated by the development of a probabilistic model for growth of biological shapes in the context of large deformations by diffeomorphisms, we present a stochastic perturbation of the Hamiltonian equations of geodesics on shape spaces. We study the finite-dimensional case of groups of points for which we prove that the strong solutions of the stochastic system exist for all time. We extend the model to the space of parameterized curves and surfaces and we develop a convenient analytical setting to prove a strong convergence result from the finite-dimensional to the infinite-dimensional case. We then present some enhancements of the model.     

Besov type function spaces defined on metric-measure spaces

The purpose of this article is to study the Besov type function spaces for maps which are defined on abstract metric-measure spaces. We extend some of the embedding theorems of the classical Besov spaces to the setting of abstract spaces.     

Hodge Theory on Metric Spaces

Hodge theory is a beautiful synthesis of geometry, topology, and analysis which has been developed in the setting of Riemannian manifolds. However, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step toward understanding the geometry of vision.     

Differential symmetry breaking operators: I. General theory and F-method

We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case, symmetry breaking operators are characterized by differential equations of second order via the F-method.     

Shape fibrations and strong shape theory

The notion of shape fibration was introduced by Marde?i? and Rushing. In this paper we use ‘fibrant space’ techniques in strong shape theory to prove that every shape fibration p:E → B of compact metric spaces is contained in a map of fibrant spaces p′:E′→B′ which enjoys a certain lifting property and whose homotopy properties reflect the strong shape properties of the map p. Standard methods for studying Hurewicz fibrations are readily applied to the map p' and in this way we obtain a number of strong shape generalizations of results of Marde?i? and Rushing. We also prove the following theorem which answers a question of Rushing: A shape fibration of compact metric spaces which is a strong shape equivalence is an hereditary shape equivalence. Since the converse was known, this gives a characterization of hereditary shape equivalences.     

Interpolation between Brezis–Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities

This paper is devoted to investigate an interpolation inequality between the Brezis–Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and only if it is isometric to a Minkowski space. Our arguments explore fine properties of Bessel functions, comparison principles, and anisotropic symmetrization on Minkowski spaces. As an application, we characterize the existence of nonzero solutions for a quasilinear PDE involving the Finsler–Laplace operator and a Hardy-type singularity on Minkowski spaces where the sharp BPV inequality plays a crucial role. The results are also new in the Riemannian/Euclidean setting.     

Riesz–Kantorovich formulas for operators on multi-wedged spaces

We introduce the notions of multi-suprema and multi-infima for vector spaces equipped with a collection of wedges, generalizing the notions of suprema and infima in ordered vector spaces. Multi-lattices are vector spaces that are closed under multi-suprema and multi-infima and are thus an abstraction of vector lattices. The Riesz decomposition property in the multi-wedged setting is also introduced, leading to Riesz–Kantorovich formulas for multi-suprema and multi-infima in certain spaces of operators.     

Merotopological Spaces

In this paper we introduce a new topological-type of structured set called merotopological space. The appropriate morphisms are defined and characterizations of the corresponding initial and final structures are given. The resulting category contains as fully embedded subcategories not only the category of topological spaces and continuous maps but also the category of merotopic spaces and uniformly continuous maps, and, a fortiori, the category of nearness spaces and the category of uniform spaces. A functorial completion is constructed for merotopological spaces using bunches. A problem that has remained long open in the setting of nearness spaces is to find an internal characterization of the epireflective hull of the topological spaces. We solve the analogue of this problem in the setting of merotopological spaces. Applications to the Wyler prime closed filter compactification and to Taimanov's extension theorem are given.     

Variational treatment of electrostatic interaction force in atomic force microscopy

In this paper we introduce the mathematical model for the electrostatic interaction force between an atomic force microscope (AFM) tip and a sample surface. We formulate the electrostatic potential problem in Sobolev spaces and find the corresponding weak solution in terms of the integral potential, which can be approximated numerically by generalized Fourier series and used to find the interaction force between an AFM tip and a sample surface. The formulation of the problem in a weak (Sobolev) space setting allows us to determine the force for AFM tips of arbitrary shape. Efficiency of the method is illustrated using numerical examples for the spherical and tetrahedral AFM tips.     

Tractability of Tensor Product Linear Operators

This paper deals with the worst case setting for approximating multivariate tensor product linear operators defined over Hilbert spaces. Approximations are obtained by using a number of linear functionals from a given class of information. We consider the three classes of information: the class of all linear functionals, the Fourier class of inner products with respect to given orthonormal elements, and the standard class of function values. We wish to determine which problems are tractable and which are strongly tractable. The complete analysis is provided for approximating operators of rank two or more. The problem of approximating linear functionals is fully analyzed in the first two classes of information. For the third class of standard information we show that the possibilities are very rich. We prove that tractability of linear functionals depends on the given space of functions. For some spaces all nontrivial normed linear functionals are intractable, whereas for other spaces all linear functionals are tractable. In “typical” function spaces, some linear functionals are tractable and some others are not.     

Differential Transforms in Weighted Spaces

We extend the results by Jones and Rosenblatt about the series of the differences of differentiation operators along lacunary sequences to BMO and to the setting of weighted L^p spaces. We use a different approach which allows to establish that the one-sided Sawyer A_p weights are the natural ones to study the boundedness and convergence of that series in weighted spaces.     

Spaces of smooth functions and distributions on infinite-dimensional compact groups

Several scales of smooth functions are introduced in the setting of connected infinite-dimensional compact groups. These are spaces of functions on the group with continuous derivatives in certain directions. We study properties of these spaces and of associated distribution spaces. Some of these spaces are intrinsically associated with the infinitesimal generator of a given Gaussian convolution semigroup. One of the reasons for studying these smooth function and distribution spaces is to obtain sharp results concerning the hypoellipticity of the infinitesimal generators of Gaussian convolution semigroups, i.e., invariant sub-Laplacians on compact groups.     

Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals

Though the theory of one-parameter Triebel-Lizorkin and Besov spaces has been very well developed in the past decades, the multi-parameter counterpart of such a theory is still absent. The main purpose of this paper is to develop a theory of multi-parameter Triebel-Lizorkin and Besov spaces using the discrete Littlewood-Paley-Stein analysis in the setting of implicit multi-parameter structure. It is motivated by the recent work of Han and Lu in which they established a satisfactory theory of multi-parameter Littlewood-Paley-Stein analysis and Hardy spaces associated with the flag singular integral operators studied by Muller-Ricci-Stein and Nagel-Ricci-Stein. We also prove the boundedness of flag singular integral operators on Triebel-Lizorkin space and Besov space. Our methods here can be applied to develop easily the theory of multi-parameter Triebel-Lizorkin and Besov spaces in the pure product setting.     

Variational treatment of electrostatic interaction force in atomic force microscopy

In this paper we introduce the mathematical model for the electrostatic interaction force between an atomic force microscope (AFM) tip and a sample surface. We formulate the electrostatic potential problem in Sobolev spaces and find the corresponding weak solution in terms of the integral potential, which can be approximated numerically by generalized Fourier series and used to find the interaction force between an AFM tip and a sample surface. The formulation of the problem in a weak (Sobolev) space setting allows us to determine the force for AFM tips of arbitrary shape. Efficiency of the method is illustrated using numerical examples for the spherical and tetrahedral AFM tips.      

Chebyshev spaces with polynomial blossoms

The use of extended Chebyshev spaces in geometric design is motivated by the interesting shape parameters they provide. Unfortunately the algorithms such spaces lead to are generally complicated because the blossoms themselves are complicated. In order to make up for this inconvenience, we here investigate particular extended Chebyshev spaces, containing the constants and power functions whose exponents are consecutive positive integers. We show that these spaces lead to simple algorithms due to the fact that the blossoms are polynomial functions. Furthermore, we also describe an elegant dimension elevation algorithm which makes it possible to return to polynomial spaces and therefore to use all the classical algorithms for polynomials. This revised version was published online in June 2006 with corrections to the Cover Date.     

Homotopical properties of upper semifinite hyperspaces of compacta

In this paper we study homotopical properties of a special neighborhood system, which is denoted by {Uε}_?>0, for the canonical embedding of a compact metric space in its upper semifinite hyperspace to get results in the shape theory for compacta. We also point out that there are spaces with the shape of finite discrete spaces and having not the homotopy type of any T₁-space     

Approximate extension property of mappings

The extension problem is to determine the extendability of a mapping defined on a closed subset of a space into a nice space such as a CW complex over the whole space. In this paper, we consider the extension problem when the codomains are general spaces. We take a shape theoretic approach to generalize the extension theory so that the codomains are allowed to be general spaces. We extend the notion of extension type which has been defined for the class of CW complexes and introduce the notion of approximate extension type which is defined for general spaces. We define approximate extension dimension analogously to extension dimension, replacing the class of CW complexes by the class of finitistic separable metrizable spaces. For every metrizable space X, we show the existence of approximate extension dimension of X.     

Pointed shape and global attractors for metrizable spaces

In this paper we consider two notions of attractors for semidynamical systems defined in metric spaces. We show that Borsuk's weak and strong shape theories are a convenient framework to study the global properties which the attractor inherits from the phase space.Moreover we obtain pointed equivalences (even in the absence of equilibria) which allow to consider also pointed invariants, like shape groups.     

Gleason’s problem,rational functions and spaces of left-regular functions: The split-quaternion setting

We study Gleason’s problem, rational functions and spaces of regular functions in the setting of split-quaternions. There are two natural symmetries in the algebra of split-quaternions. The first symmetry allows to define positive matrices with split-quaternionic entries, and also reproducing Hilbert spaces of regular functions. The second leads to reproducing kernel Krein spaces.