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 hardy space H¹ on products of half-spaces

We show that the Hardy space H¹ anal (R2+ x R2+) can be identified with the class of functions f such that f and all its double and partial Hubert transforms Hk f belong to L¹ (R2). A basic tool used in the proof is the bisubharmonicity of |F|q, where F is a vector field that satisfies a generalized conjugate system of Cauchy-Riemann type.

Mesoproterozoic rapakivi granites of the Rondonia Tin Province, southwestern border of the Amazonian craton, Brazil - I. Reconnaissance U-Pb geochronology and regional implications

Rapakivi granites and associated mafic and ultramafic rocks in the Rondonia Tin Province, southwestern Amazonian craton, Brazil were emplaced during six discrete episodes of magmatism between ca 1600 and 970 Ma. The seven rapakivi granite suites emplaced at this time were the Serra da Providencia Intrusive Suite (U-Pb ages between 1606 and 1532 Ma); Santo Antonio Intrusive Suite(U-Pb age 1406 Ma), Teotonio Intrusive Suite (U-Pb age 1387 Ma); Alto Candeias Intrusive Suite (U-Pb ages between 1346 and 1338 Ma); Sao Lourenco-Caripunas Intrusive Suite (U-Pb ages between 1314 and 1309 Ma); Santa Clara Intrusive Suite (U-Pb ages between 1082 and 1074 Ma); and Younger Granites of Rondonia (U-Pb ages between 998 and 974 Ma). The Serra da Providencia Intrusive Suite intruded the Paleoproterozoic (1.80 to 1.70 Ga) Rio Negro-Juruena crust whereas the other suites were emplaced into the 1.50 to 1.30 Ga Rondonia-San Ignacio crust. Their intrusion was contemporaneous with orogenic activity in other parts of the southwestern Amazonian craton, except for the oldest, Serra da Providencia Intrusive Suite. Orogenic events coeval with emplacement of the Serra da Providencia Intrusive Suite are not clearly recognized in the region. The Santo Antonio, Teotonio, Alto Candeias and Sao Lourenco-Caripunas Intrusive Suites are interpreted to represent extensional anorogenic magmatism associated with the terminal stages of the Rondonian-San Ignacio orogeny. At least the Sao Lourenco-Caripunas rapakivi granites and coeval intra-continental rift sedimentary rocks may, in contrast, represent the products of extensional tectonics and rifting preceding the Sunsas/Aguapei orogeny (1.25 to 1.0 Ga). The two youngest rapakivi suites, the Santa Clara Intrusive Suite and Younger Granites of Rondonia, seemingly represent inboard magmatism in the Rondonian-San Ignacio Province during a younger episode of reworking in the Rio Negro-Juruena Province during the waning stages of the collisional 1.1 to 1.0 Ga Sunsas/Aguapei orogeny. The six intra-plate rapakivi granite episodes in the southwestern part of the Amazonian craton form three broad periods of anorogenic magmatism that have age-correlative events composed of similar rocks and geologic environments in eastern Laurentia and Baltica, although the exact timing of magmatism appears slightly different. Recognition of lithologic and chronological correlations between various cratons provide important constraints to models explaining the interplay between rapakivi granite magmatism and deep crustal evolution of an early Mesoproterozoic supercontinent. They are, furthermore, important to plate tectonic models for the assembly, dispersal and reassembly of Amazonia, Laurentia and Baltica in the Mesoproterozoic and Neoproterozoic. (C) 1999 Elsevier B.V. B.V. All rights reserved.

Discrete coherent states and probability distributions in finite-dimensional spaces

Operator bases are discussed in connection with the construction of phase space representatives of operators in finite-dimensional spaces, and their properties are presented. It is also shown how these operator bases allow for the construction of a finite harmonic oscillator-like coherent state. Creation and annihilation operators for the Fock finite-dimensional space are discussed and their expressions in terms of the operator bases are explicitly written. The relevant finite-dimensional probability distributions are obtained and their limiting behavior for an infinite-dimensional space are calculated which agree with the well known results. (C) 1996 Academic Press, Inc.

Dynamics in discrete phase spaces and time interval operators

The von Neumann-Liouville time evolution equation is represented in a discrete quantum phase space. The mapped Liouville operator and the corresponding Wigner function are explicitly written for the problem of a magnetic moment interacting with a magnetic field and the precessing solution is found. The propagator is also discussed and a time interval operator, associated to a unitary operator which shifts the energy levels in the Zeeman spectrum, is introduced. This operator is associated to the particular dynamical process and is not the continuous parameter describing the time evolution. The pair of unitary operators which shifts the time and energy is shown to obey the Weyl-Schwinger algebra. (C) 1999 Elsevier B.V. B.V. All rights reserved.

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.

A stratigraphic chart of the Late Carboniferous/Permian succession of the eastern border of the Parana Basin, Brazil, South America

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

An extended Weyl-Wigner transformation for special finite spaces

We extend the Weyl-Wigner transformation to those particular degrees of freedom described by a finite number of states using a technique of constructing operator bases developed by Schwinger. Discrete transformation kernels are presented instead of continuous coordinate-momentum pair system and systems such as the one-dimensional canonical continuous coordinate-momentum pair system and the two-dimensional rotation system are described by special limits. Expressions are explicitly given for the spin one-half case. © 1988.

Symmetries and time evolution in discrete phase spaces: a soluble model calculation

Using the flexibility and constructive definition of the Schwinger bases, we developed different mapping procedures to enhance different aspects of the dynamics and of the symmetries of an extended version of the two-level Lipkin model. The classical limits of the dynamics are discussed in connection with the different mappings. Discrete Wigner functions are also calculated. © 1995.

Optimality conditions for pareto nonsmooth nonconvex programming in banach spaces

In this paper, we consider a vector optimization problem where all functions involved are defined on Banach spaces. We obtain necessary and sufficient criteria for optimality in the form of Karush-Kuhn-Tucker conditions. We also introduce a nonsmooth dual problem and provide duality theorems.

Discrete generator coordinate: Symmetry restoration in finite-dimensional spaces

Group theoretical-based techniques and fundamental results from number theory are used in order to allow for the construction of exact projectors in finite-dimensional spaces. These operators are shown to make use only of discrete variables, which play the role of discrete generator coordinates, and their application in the number symmetry restoration is carried out in a nuclear BCS wave function which explicitly violates that symmetry. © 1999 Published by Elsevier Science B.V. All rights reserved.

Border detection in digital images: An approach by fuzzy numbers

The purpose of this paper is to introduce a new approach for edge detection in gray shaded images. The proposed approach is based on the fuzzy number theory. The idea is to deal with the uncertainties concerning the gray shades making up the image, and thus calculate the appropriateness of the pixels in relation to an homogeneous region around them. The pixels not belonging to the region are then classified as border pixels. The results have shown that the technique is simple, computationally efficient and with good results when compared with both the traditional border detectors and the fuzzy edge detectors. © 2007 IEEE.

Open urban spaces quality: A study in a historical square in Bath - UK

The quality of open urban spaces is very important for urban vitality. Nowadays urban designers have to face the great challenge of designing urban spaces able to respond to people's need for liveable spaces. The success of these spaces depends on various aspects and the microclimatic condition has been recognized as one of the most influential. However, studies on thermal comfort in open space have shown that the user's thermal sensation does not depend only on microclimate parameters but also on other local qualitative aspects. Thus, environmental quality evaluation of successful public spaces can contribute to understand this issue. This paper focuses on a case study regarding Queen Square's environmental quality, a public space of historical importance in Bath-UK. The first stage of the research, a study on local characteristics and people observations, allowed a preliminary evaluation of the space performance, their social aspects, while it characterized and quantified the hourly variation of the space use in different days and seasons. In the second stage, short microclimatic surveys were carried out simultaneously with a perception survey through a questionnaire. The results show the strong vitality of the square and socioenvironmental significance, not only for its location in the urban context, but also for its historical value. The environmental quality of the square contributes to the users' sensation of comfort even in adverse climatic conditions. This research is part of a project that aims to investigate the impact of the environmental stimuli in the use of open spaces and intend to develop design strategies that aim to maximise the use of open spaces in different weather conditions.

Composite pontics for orthodontic patients with extraction spaces.

Esthetic orthodontic appliances continue to appeal to more patients, which results in objections to extraction spaces that remain for several months during orthodontic therapy. This has led orthodontists to design temporary pontics that fill extraction sites and that can be reduced as the spaces close. This report describes a simple, efficient, and expeditious technique for making such pontics. © 2010 Quintessence Publishing Co, Inc.