Geometry in preduals of spaces of 2-homogeneous polynomials on Hilbert spaces

Let H be a (real or complex) Hilbert space. Using spectral theory and properties of the Schatten–Von Neumann operators, we prove that every symmetric tensor of unit norm in HoH is an infinite absolute convex combination of points of the form xox with x in the unit sphere of the Hilbert space. We use this to obtain explicit characterizations of the smooth points of the unit ball of HoH .

Lower bounds on blow up solutions of the three-dimensional Navier-Stokes equations in homogeneous Sobolev spaces

Suppose that u(t) is a solution of the three-dimensional Navier-Stokes equations, either on the whole space or with periodic boundary conditions, that has a singularity at time T. In this paper we show that the norm of u(T - t) in the homogeneous Sobolev space (H)over dot(s) must be bounded below by c(s)t(-(2s-1)/4) for 1/2 < s < 5/2 (s not equal 3/2), where c(s) is an absolute constant depending only on s; and by c(s)parallel to u(0)parallel to((5-2s)/5)(L2)t(-2s/5) for s > 5/2. (The result for 1/2 < s < 3/2 follows from well-known lower bounds on blowup in Lp spaces.) We show in particular that the local existence time in (H)over dot(s)(R-3) depends only on the (H)over dot(s)-norm for 1/2 < s < 5/2, s not equal 3/2. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4762841]

On the approximation by convolution operators in homogeneous Banach spaces on R^d

AMS Subject Classification 2010: 41A25, 41A35, 41A40, 41A63, 41A65, 42A38, 42A85, 42B10, 42B20

On the Approximation by Convolution Operators in Homogeneous Banach Spaces of Periodic Functions

AMS Subject Classification 2010: 41A25, 41A27, 41A35, 41A36, 41A40, 42Al6, 42A85.

A critical regionalist approach to housing design in Vietnam : socio-environmental organisation of living spaces in pre- and post-reform houses

Vietnam has a unique culture which is revealed in the way that people have built and designed their traditional housing. Vietnamese dwellings reflect occupants’ activities in their everyday lives, while adapting to tropical climatic conditions impacted by seasoning monsoons. It is said that these characteristics of Vietnamese dwellings have remained unchanged until the economic reform in 1986, when Vietnam experienced an accelerated development based on the market-oriented economy. New housing types, including modern shop-houses, detached houses, and apartments, have been designed in many places, especially satisfying dwellers’ new lifestyles in Vietnamese cities. The contemporary housing, which has been mostly designed by architects, has reflected rules of spatial organisation so that occupants’ social activities are carried out. However, contemporary housing spaces seem unsustainable in relation to socio-cultural values because they has been influenced by globalism that advocates the use of homogeneous spatial patterns, modern technologies, materials and construction methods. This study investigates the rules of spaces in Vietnamese houses that were built before and after the reform to define the socio-cultural implications in Vietnamese housing design. Firstly, it describes occupants’ views of their current dwellings in terms of indoor comfort conditions and social activities in spaces. Then, it examines the use of spaces in pre-reform Vietnamese housing through occupants’ activities and material applications. Finally, it discusses the organisation of spaces in both pre- and post-reform housing to understand how Vietnamese housing has been designed for occupants to live, act, work, and conduct traditional activities. Understanding spatial organisation is a way to identify characteristics of the lived spaces of the occupants created from the conceived space, which is designed by designers. The characteristics of the housing spaces will inform the designers the way to design future Vietnamese housing in response to cultural contexts. The study applied an abductive approach for the investigation of housing spaces. It used a conceptual framework in relation to Henri Lefebvre’s (1991) theory to understand space as the main factor constituting the language of design, and the principles of semiotics to examine spatial structure in housing as a language used in the everyday life. The study involved a door-knocking survey to 350 households in four regional cities of Vietnam for interpretation of occupancy conditions and levels of occupants’ comfort. A statistical analysis was applied to interpret the survey data. The study also required a process of data selection and collection of fourteen cases of housing in three main climatic regions of the country for analysing spatial organisation and housing characteristics. The study found that there has been a shift in the relationship of spaces from the pre- to post-reform Vietnamese housing. It also indentified that the space for guest welcoming and family activity has been the central space of the Vietnamese housing. Based on the relationships of the central space with the others, theoretical models were proposed for three types of contemporary Vietnamese housing. The models will be significant in adapting to Vietnamese conditions to achieve socioenvironmental characteristics for housing design because it was developed from the occupants’ requirements for their social activities. Another contribution of the study is the use of methodological concepts to understand the language of living spaces. Further work will be needed to test future Vietnamese housing designs from the applications of the models.

Homogeneous model theory of metric structures

This thesis studies homogeneous classes of complete metric spaces. Over the past few decades model theory has been extended to cover a variety of nonelementary frameworks. Shelah introduced the abstact elementary classes (AEC) in the 1980s as a common framework for the study of nonelementary classes. Another direction of extension has been the development of model theory for metric structures. This thesis takes a step in the direction of combining these two by introducing an AEC-like setting for studying metric structures. To find balance between generality and the possibility to develop stability theoretic tools, we work in a homogeneous context, thus extending the usual compact approach. The homogeneous context enables the application of stability theoretic tools developed in discrete homogeneous model theory. Using these we prove categoricity transfer theorems for homogeneous metric structures with respect to isometric isomorphisms. We also show how generalized isomorphisms can be added to the class, giving a model theoretic approach to, e.g., Banach space isomorphisms or operator approximations. The novelty is the built-in treatment of these generalized isomorphisms making, e.g., stability up to perturbation the natural stability notion. With respect to these generalized isomorphisms we develop a notion of independence. It behaves well already for structures which are omega-stable up to perturbation and coincides with the one from classical homogeneous model theory over saturated enough models. We also introduce a notion of isolation and prove dominance for it.

Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite-dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We write down an equivariant constant coefficient differential operator that intertwines the bundle with the direct sum of its irreducible factors. As an application, we show that in the case of the closed unit ball in C-n all homogeneous n-tuples of Cowen-Douglas operators are similar to direct sums of certain basic n-tuples. (c) 2015 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Consensus in non-commutative spaces

Convergence analysis of consensus algorithms is revisited in the light of the Hilbert distance. The Lyapunov function used in the early analysis by Tsitsiklis is shown to be the Hilbert distance to consensus in log coordinates. Birkhoff theorem, which proves contraction of the Hilbert metric for any positive homogeneous monotone map, provides an early yet general convergence result for consensus algorithms. Because Birkhoff theorem holds in arbitrary cones, we extend consensus algorithms to the cone of positive definite matrices. The proposed generalization finds applications in the convergence analysis of quantum stochastic maps, which are a generalization of stochastic maps to non-commutative probability spaces. ©2010 IEEE.

Polynomials on Banach spaces with unconditional bases

We study the classes of homogeneous polynomials on a Banach space with unconditional Schauder basis that have unconditionally convergent monomial expansions relative to this basis. We extend some results of Matos, and we show that the homogeneous polynomials with unconditionally convergent expansions coincide with the polynomials that are regular with respect to the Banach lattices structure of the domain.

Extensions of homogeneous polynomials ON c(0)(l(2)(i))

We show that a 2-homogeneous polynomial on the complex Banach space c(0)(l(2)(i)) is norm attaining if and only if it is finite (i.e, depends only on finite coordinates). As the consequence, we show that there exists a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on c(0)(l(2)(i)).

UNIQUENESS OF THE EXTENSION OF 2-HOMOGENEOUS POLYNOMIALS

Homogeneous polynomials of degree 2 on the complex Banach space c(0)(l(n)(2)) are shown to have unique norm-preserving extension to the bidual space. This is done by using M-projections and extends the analogous result for c(0) proved by P.-K. Lin.

CLOSED SPACES IN COSMOLOGY

This paper deals with two aspects of relativistic cosmologies with closed spatial sections. These spacetimes are based on the theory of general relativity, and admit a foliation into space sections S(t), which are spacelike hypersurfaces satisfying the postulate of the closure of space: each S(t) is a three-dimensional closed Riemannian manifold. The topics discussed are: (i) a comparison, previously obtained, between Thurston geometries and Bianchi-Kantowski-Sachs metrics for such three-manifolds is here clarified and developed; and (ii) the implications of global inhomogeneity for locally homogeneous three-spaces of constant curvature are analyzed from an observational viewpoint.

Hardy Inequality in Variable Exponent Lebesgue Spaces

Mathematics Subject Classification: 26D10, 46E30, 47B38

Equivalent norms in polynomial spaces and applications.

In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy Littlewood constants for 2-homogeneous polynomials on l(p)(2) spaces, 2 < p <= infinity. We also provide lower estimates for the Hardy-Littlewood constants for polynomials of higher degrees.