BANACH SPACES WITHOUT MINIMAL SUBSPACES - EXAMPLES

We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in [11], by classifying them according to which side of the dichotomies they fall.

How far is C-0(Gamma, X) with Gamma discrete from C-0(K, X) spaces?

For a locally compact Hausdorff space K and a Banach space X we denote by C-0(K, X) the space of X-valued continuous functions on K which vanish at infinity, provided with the supremum norm. Let n be a positive integer, Gamma an infinite set with the discrete topology, and X a Banach space having non-trivial cotype. We first prove that if the nth derived set of K is not empty, then the Banach-Mazur distance between C-0(Gamma, X) and C-0(K, X) is greater than or equal to 2n + 1. We also show that the Banach-Mazur distance between C-0(N, X) and C([1, omega(n)k], X) is exactly 2n + 1, for any positive integers n and k. These results extend and provide a vector-valued version of some 1970 Cambern theorems, concerning the cases where n = 1 and X is the scalar field.

Running coupling and pomeron loop effects on inclusive and diffractive DIS cross sections

Within the framework of a (1 + 1)-dimensional model which mimics high-energy QCD, we study the behavior of the cross sections for inclusive and diffractive deep inelastic gamma*h scattering cross sections. We analyze the cases of both fixed and running coupling within the mean-field approximation, in which the evolution of the scattering amplitude is described by the Balitsky-Kovchegov equation, and also through the pomeron loop equations, which include in the evolution the gluon number fluctuations. In the diffractive case, similarly to the inclusive one, suppression of the diffusive scaling, as a consequence of the inclusion of the running of the coupling, is observed.

Semi-supervised dimensionality reduction based on partial least squares for visual analysis of high dimensional data

Dimensionality reduction is employed for visual data analysis as a way to obtaining reduced spaces for high dimensional data or to mapping data directly into 2D or 3D spaces. Although techniques have evolved to improve data segregation on reduced or visual spaces, they have limited capabilities for adjusting the results according to user's knowledge. In this paper, we propose a novel approach to handling both dimensionality reduction and visualization of high dimensional data, taking into account user's input. It employs Partial Least Squares (PLS), a statistical tool to perform retrieval of latent spaces focusing on the discriminability of the data. The method employs a training set for building a highly precise model that can then be applied to a much larger data set very effectively. The reduced data set can be exhibited using various existing visualization techniques. The training data is important to code user's knowledge into the loop. However, this work also devises a strategy for calculating PLS reduced spaces when no training data is available. The approach produces increasingly precise visual mappings as the user feeds back his or her knowledge and is capable of working with small and unbalanced training sets.

Conformation analysis of a surface loop that controls active site access in the GH11 xylanase A from Bacillus subtilis

Xylanases (EC 3.2.1.8 endo-1,4-glycosyl hydrolase) catalyze the hydrolysis of xylan, an abundant hemicellulose of plant cell walls. Access to the catalytic site of GH11 xylanases is regulated by movement of a short beta-hairpin, the so-called thumb region, which can adopt open or closed conformations. A crystallographic study has shown that the D11F/R122D mutant of the GH11 xylanase A from Bacillus subtilis (BsXA) displays a stable "open" conformation, and here we report a molecular dynamics simulation study comparing this mutant with the native enzyme over a range of temperatures. The mutant open conformation was stable at 300 and 328 K, however it showed a transition to the closed state at 338 K. Analysis of dihedral angles identified thumb region residues Y113 and T123 as key hinge points which determine the open-closed transition at 338 K. Although the D11F/R122D mutations result in a reduction in local inter-intramolecular hydrogen bonding, the global energies of the open and closed conformations in the native enzyme are equivalent, suggesting that the two conformations are equally accessible. These results indicate that the thumb region shows a broader degree of energetically permissible conformations which regulate the access to the active site region. The R122D mutation contributes to the stability of the open conformation, but is not essential for thumb dynamics, i.e., the wild type enzyme can also adapt to the open conformation.

Hypoellipticity in spaces of ultradistributions-Study of a model case

In this work we study C (a)-hypoellipticity in spaces of ultradistributions for analytic linear partial differential operators. Our main tool is a new a-priori inequality, which is stated in terms of the behaviour of holomorphic functions on appropriate wedges. In particular, for sum of squares operators satisfying Hormander's condition, we thus obtain a new method for studying analytic hypoellipticity for such a class. We also show how this method can be explicitly applied by studying a model operator, which is constructed as a perturbation of the so-called Baouendi-Goulaouic operator.

Numerical assessment of stability of interface discontinuous finite element pressure spaces

The stability of two recently developed pressure spaces has been assessed numerically: The space proposed by Ausas et al. [R.F. Ausas, F.S. Sousa, G.C. Buscaglia, An improved finite element space for discontinuous pressures, Comput. Methods Appl. Mech. Engrg. 199 (2010) 1019-1031], which is capable of representing discontinuous pressures, and the space proposed by Coppola-Owen and Codina [A.H. Coppola-Owen, R. Codina, Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions, Int. J. Numer. Methods Fluids, 49 (2005) 1287-1304], which can represent discontinuities in pressure gradients. We assess the stability of these spaces by numerically computing the inf-sup constants of several meshes. The inf-sup constant results as the solution of a generalized eigenvalue problems. Both spaces are in this way confirmed to be stable in their original form. An application of the same numerical assessment tool to the stabilized equal-order P-1/P-1 formulation is then reported. An interesting finding is that the stabilization coefficient can be safely set to zero in an arbitrary band of elements without compromising the formulation's stability. An analogous result is also reported for the mini-element P-1(+)/P-1 when the velocity bubbles are removed in an arbitrary band of elements. (C) 2012 Elsevier B.V. All rights reserved.

Reproducing kernel Hilbert spaces associated with kernels on topological spaces

We analyze reproducing kernel Hilbert spaces of positive definite kernels on a topological space X being either first countable or locally compact. The results include versions of Mercer's theorem and theorems on the embedding of these spaces into spaces of continuous and square integrable functions.

Weighted Fourier–Laplace transforms in reproducing kernel Hilbert spaces on the sphere

We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives.

Pullback exponential attractors for evolution processes in Banach spaces: properties and applications

This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply.