Contractive Hilbert modules and their dilations

In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z,w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S (-1)(z, w)k(z, w) is a positive kernel function, where S(z,w) is the Szego kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale [8] and Ambrozie, Englis and Muller [2]. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in a'', (m) . Some consequences of this more general result are then explored in the case of several natural function algebras.

Unitary invariants for Hilbert modules of finite rank

We associate a sheaf model to a class of Hilbert modules satisfying a natural finiteness condition. It is obtained as the dual to a linear system of Hermitian vector spaces (in the sense of Grothendieck). A refined notion of curvature is derived from this construction leading to a new unitary invariant for the Hilbert module. A division problem with bounds, originating in Douady's privilege, is related to this framework. A series of concrete computations illustrate the abstract concepts of the paper.

Hilbert W*-modules and coherent states

Hilbert C*-module valued coherent states was introduced earlier by Ali, Bhattacharyya and Shyam Roy. We consider the case when the underlying C*-algebra is a W*-algebra. The construction is similar with a substantial gain. The associated reproducing kernel is now algebra valued, rather than taking values in the space of bounded linear operators between two C*-algebras.

Resolution of Singularities for a Class of Hilbert Modules

Let M be the completion of the polynomial ring C(z) under bar] with respect to some inner product, and for any ideal I subset of C (z) under bar], let I] be the closure of I in M. For a homogeneous ideal I, the joint kernel of the submodule I] subset of M is shown, after imposing some mild conditions on M, to be the linear span of the set of vectors {p(i)(partial derivative/partial derivative(w) over bar (1),...,partial derivative/partial derivative(w) over bar (m)) K-I] (., w)vertical bar(w=0), 1 <= i <= t}, where K-I] is the reproducing kernel for the submodule 2] and p(1),..., p(t) is some minimal ``canonical set of generators'' for the ideal I. The proof includes an algorithm for constructing this canonical set of generators, which is determined uniquely modulo linear relations, for homogeneous ideals. A short proof of the ``Rigidity Theorem'' using the sheaf model for Hilbert modules over polynomial rings is given. We describe, via the monoidal transformation, the construction of a Hermitian holomorphic line bundle for a large class of Hilbert modules of the form I]. We show that the curvature, or even its restriction to the exceptional set, of this line bundle is an invariant for the unitary equivalence class of I]. Several examples are given to illustrate the explicit computation of these invariants.

Fractional Hilbert transform extensions and associated analytic signal construction

The analytic signal (AS) was proposed by Gabor as a complex signal corresponding to a given real signal. The AS has a one-sided spectrum and gives rise to meaningful spectral averages. The Hilbert transform (HT) is a key component in Gabor's AS construction. We generalize the construction methodology by employing the fractional Hilbert transform (FrHT), without going through the standard fractional Fourier transform (FrFT) route. We discuss some properties of the fractional Hilbert operator and show how decomposition of the operator in terms of the identity and the standard Hilbert operators enables the construction of a family of analytic signals. We show that these analytic signals also satisfy Bedrosian-type properties and that their time-frequency localization properties are unaltered. We also propose a generalized-phase AS (GPAS) using a generalized-phase Hilbert transform (GPHT). We show that the GPHT shares many properties of the FrHT, in particular, selective highlighting of singularities, and a connection with Lie groups. We also investigate the duality between analyticity and causality concepts to arrive at a representation of causal signals in terms of the FrHT and GPHT. On the application front, we develop a secure multi-key single-sideband (SSB) modulation scheme and analyze its performance in noise and sensitivity to security key perturbations. (C) 2013 Elsevier B.V. All rights reserved.

Sign changes of Fourier coefficients of Hilbert modular forms

Sign changes of Fourier coefficients of various modular forms have been studied. In this paper, we analyze some sign change properties of Fourier coefficients of Hilbert modular forms, under the assumption that all the coefficients are real. The quantitative results on the number of sign changes in short intervals are also discussed. (C) 2014 Elsevier Inc. All rights reserved.

ON THE ROLE OF THE HILBERT TRANSFORM IN BOOSTING THE PERFORMANCE OF THE ANNIHILATING FILTER

We consider the problem of parameter estimation from real-valued multi-tone signals. Such problems arise frequently in spectral estimation. More recently, they have gained new importance in finite-rate-of-innovation signal sampling and reconstruction. The annihilating filter is a key tool for parameter estimation in these problems. The standard annihilating filter design has to be modified to result in accurate estimation when dealing with real sinusoids, particularly because the real-valued nature of the sinusoids must be factored into the annihilating filter design. We show that the constraint on the annihilating filter can be relaxed by making use of the Hilbert transform. We refer to this approach as the Hilbert annihilating filter approach. We show that accurate parameter estimation is possible by this approach. In the single-tone case, the mean-square error performance increases by 6 dB for signal-to-noise ratio (SNR) greater than 0 dB. We also present experimental results in the multi-tone case, which show that a significant improvement (about 6 dB) is obtained when the parameters are close to 0 or pi. In the mid-frequency range, the improvement is about 2 to 3 dB.

On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.

Paranormal operator on a Hilbert space

In this thesis an extensive study is made of the set P of all paranormal operators in B(H), the set of all bounded endomorphisms on the complex Hilbert space H. T ϵ B(H) is paranormal if for each z contained in the resolvent set of T, d(z, σ(T))//(T-zI)^-1 = 1 where d(z, σ(T)) is the distance from z to σ(T), the spectrum of T. P contains the set N of normal operators and P contains the set of hyponormal operators. However, P is contained in L, the set of all T ϵ B(H) such that the convex hull of the spectrum of T is equal to the closure of the numerical range of T. Thus, N≤P≤L.

If the uniform operator (norm) topology is placed on B(H), then the relative topological properties of N, P, L can be discussed. In Section IV, it is shown that: 1) N P and L are arc-wise connected and closed, 2) N, P, and L are nowhere dense subsets of B(H) when dim H ≥ 2, 3) N = P when dimH ˂ ∞ , 4) N is a nowhere dense subset of P when dimH ˂ ∞ , 5) P is not a nowhere dense subset of L when dimH ˂ ∞ , and 6) it is not known if P is a nowhere dense subset of L when dimH ˂ ∞.

The spectral properties of paranormal operators are of current interest in the literature. Putnam [22, 23] has shown that certain points on the boundary of the spectrum of a paranormal operator are either normal eigenvalues or normal approximate eigenvalues. Stampfli [26] has shown that a hyponormal operator with countable spectrum is normal. However, in Theorem 3.3, it is shown that a paranormal operator T with countable spectrum can be written as the direct sum, N ⊕ A, of a normal operator N with σ(N) = σ(T) and of an operator A with σ(A) a subset of the derived set of σ(T). It is then shown that A need not be normal. If we restrict the countable spectrum of T ϵ P to lie on a C²-smooth rectifiable Jordan curve G_o, then T must be normal [see Theorem 3.5 and its Corollary]. If T is a scalar paranormal operator with countable spectrum, then in order to conclude that T is normal the condition of σ(T) ≤ G_o can be relaxed [see Theorem 3.6]. In Theorem 3.7 it is then shown that the above result is not true when T is not assumed to be scalar. It was then conjectured that if T ϵ P with σ(T) ≤ G_o, then T is normal. The proof of Theorem 3.5 relies heavily on the assumption that T has countable spectrum and cannot be generalized. However, the corollary to Theorem 3.9 states that if T ϵ P with σ(T) ≤ G_o, then T has a non-trivial lattice of invariant subspaces. After the completion of most of the work on this thesis, Stampfli [30, 31] published a proof that a paranormal operator T with σ(T) ≤ G_o is normal. His proof uses some rather deep results concerning numerical ranges whereas the proof of Theorem 3.5 uses relatively elementary methods.

Intuicionismo y formalismo en la filosofía de la matemática de I. Kant y H. Hilbert. Sobre función y significado de la intuición matemática

514 p.

Estudo do efeito magnetocalórico nos compostos da série Gd1-xDyxAl2

Nesta dissertação, foram investigadas as propriedades magnéticas e magnetocalóricas nos compostos intermetálicos de terras-raras Gd1-xDyxAl2 (x = 0, 0.25, 0.50, 0.75 e 1.00) usando abordagens teórica e experimental. Do ponto de vista teórico, a série Gd1-xDyxAl2 foi descrita através de um modelo para o hamiltoniano magnético, incluindo o efeito Zeeman, interação de troca e a anisotropia de campo elétrico cristalino. As entropias da rede e eletrônica foram consideradas nas aproximações de Debye e de gás de elétrons livres, respectivamente. A parte experimental inclui a preparação do material, sua caracterização e medidas das quantidades magnéticas e magnetocalóricas. Os resultados experimentais e os cálculos teóricos da variação adiabática da temperatura (ΔTad) e da variação isotérmica da entropia (ΔS T), sob variações de campo magnético ao longo da direção de fácil magnetização, estão de bom acordo. O efeito da aplicação do campo magnético ao longo de uma direção de difícil magnetização foi estudado e as componentes da magnetização em função da temperatura foram investigadas. Também foi observado que a temperatura de reorientação de spin, TR, diminui quando a intensidade do campo magnético aumenta. Além disso, as concentrações molares ótimas de um material híbrido formado pelos compostos Gd1-xDyxAl2 (x = 0, 0.25, 0.50, 0.75 e 1.00) foram simuladas usando um método numérico de matriz proposto por Smaili e Chahine. O compósito apresenta um bom intervalo de temperatura para um refrigerador magnético de 60 até 170 K.

Do fracasso escolar à distorção idade-série: caminhos percorridos pelas classes de aceleração do Projeto Acelerar para Vencer

Este estudo parte das observações e análises realizadas em classes de aceleração do Projeto Acelerar para Vencer (PAV- 2009/2012), desenvolvido pela Secretaria de Educação de Minas Gerais, tendo a discussão sobre o fracasso escolar e a distorção idade-série como centrais dentro das políticas adotadas. A pesquisa busca ampliar nosso entendimento quanto à relação entre escola e expectativas individuais, levando-nos a refletir para além do direito à educação. Dentre as reflexões, destacamos: desigualdade de oportunidades, relações de poder dentro de um sistema que, de forma hegemônica, se mantém estável, mas desestabiliza vidas ao negar às camadas desprivilegiadas direitos básicos: acesso à alfabetização na idade certa, à leitura, ao conhecimento escolar e a uma educação atraente e de qualidade que atenda às necessidades dos sujeitos de acordo com as realidades em que estão inseridos. Certos de que tais problemas perpassam questões políticas, econômicas e sociais, pretendemos ater-nos às diferenças existentes dentro do espaço escolar, o que nos leva a tentar desvendar, - no sentido de não apenas repetir, mas também compreender -, as causas que levam à distorção idade-série e à sua inserção, ou disfarce, no processo de universalização do ensino, chegando à forma como a escola e seus agentes percebem as diferenças e lidam com ela. Para o desenvolvimento do estudo, recorremos à pesquisa qualitativa de cunho etnográfico, buscando subsídios em autores que discutem fracasso escolar, distorção idade-série, teoria curricular, mecanismos de exclusão social e respeito às diferenças culturais. Nesse sentido, concluímos que compreender as funcionalidades sociais da escola implica arregimentar, ou fazer coexistir, em um mesmo viés de observação, elementos interdependentes: política, escola, demandas sociais e cultura. Reconhecemos que tais elementos são imprescindíveis para pensarmos os sujeitos e suas distinções, a cultura e suas representações, o poder e as hegemonias presentes em todas as instâncias da vida escolar, transitando por uma via de mão dupla que envolve a enunciação das diferenças e seus atores: Secretaria de Educação, instituição pesquisada, gestão escolar, professores, alunos e responsáveis

Ecuaciones integrales en el contexto de espacios de Hilbert

Las ideas básicas de la teoría de los espacios de Hilbert tienen como origen diversos problemas del análisis funcional, entre los cuales podemos citar los relativos a ciertas ecuaciones integrales lineales. Concretamente, un precedente de los métodos de la teoría espectral de operadores fue precisamente el enfoque de I. Fredholm de resolución de ciertas ecuaciones integrales mediante la teoría de matrices y determinantes infinitos utilizando el método de coeficientes indeterminados. Imitando la técnica de von Koch para desarrollar determinantes infinitos, Fredholm desarrolló su famoso teorema de alternativa en la resolución de las ecuaciones que llevan su nombre. Algunos tipos de ecuaciones integrales lineales están relacionados con operadores acotados completamente continuos y la teoría espectral para esta clase de operadores se podrá aplicar en la resolución de estas ecuaciones. En esta memoria se estudian distintos aspectos de estas y otras ecuaciones integrales. En el capítulo 1 se definen los conceptos básicos necesarios para el seguimiento de la misma, como es la de operador lineal y sus propiedades. Se distingue una clase importante de operadores, los compactos. Y se demuestra que todo operador integral pertenece a esta clase de operadores. En los capítulos 2 y 3 se introduce el concepto de ecuación integral, diferenciando las de Fredholm de las de Volterra, y se estudian diferentes técnicas de resolución de dichas ecuaciones, como son el teorema de alternativa, el teorema espectral para operadores compactos y autoadjuntos, ecuaciones integrales con núcleos degenerados y resolución por el método de aproximaciones sucesivas. Para finalizar, en el apéndice se resuelven algunos ejercicios utilizando los diferentes métodos estudiados.