Multiplication in Riesz spaces

A.G. Vulih has shown how an essentially unique intrinsic multiplication can be defined in certain types of Riesz spaces (vector lattices) L. In general, the multiplication is not universally defined in L, but L can always be imbedded in a large space L^# in which multiplication is universally defined.

If ф is a normal integral in L, then ф can be extended to a normal integral on a large space L₁(ф) in L^#, and L₁(ф) may be regarded as an abstract integral space. A very general form of the Radon-Nikodym theorem can be proved in L₁(ф), and this can be used to give a relatively simple proof of a theorem of Segal giving a necessary and sufficient condition that the Radon-Nikodym theorem hold in a measure space.

In another application, the multiplication is used to give a representation of certain Riesz spaces as rings of operators on a Hilbert space.

Smooth Banach spaces and approximations

If E and F are real Banach spaces let C^p,q(E, F) O ≤ q ≤ p ≤ ∞, denote those maps from E to F which have p continuous Frechet derivatives of which the first q derivatives are bounded. A Banach space E is defined to be C^p,q smooth if C^p,q(E,R) contains a nonzero function with bounded support. This generalizes the standard C^p smoothness classification.

If an L^p space, p ≥ 1, is C^q smooth then it is also C^q,q smooth so that in particular L^p for p an even integer is C^∞,∞ smooth and L^p for p an odd integer is C^p-1,p-1 smooth. In general, however, a C^p smooth B-space need not be C^p,p smooth. C_o is shown to be a non-C^2,2 smooth B-space although it is known to be C^∞ smooth. It is proved that if E is C^p,1 smooth then C_o(E) is C^p,1 smooth and if E has an equivalent C^p norm then c_o(E) has an equivalent C^p norm.

Various consequences of C^p,q smoothness are studied. If f ϵ C^p,q(E,F), if F is C^p,q smooth and if E is non-C^p,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable then E is C^p,q smooth if and only if E admits C^p,q partitions of unity; E is C^p,psmooth, p ˂∞, if and only if every closed subset of E is the zero set of some C^P function.

f ϵ C^q(E,F), 0 ≤ q ≤ p ≤ ∞, is said to be C_p,q approximable on a subset U of E if for any ϵ ˃ 0 there exists a g ϵ C^p(E,F) satisfying

sup/xϵU, O≤k≤q ‖ D^k f(x) - D^k g(x) ‖ ≤ ϵ.

It is shown that if E is separable and C^p,q smooth and if f ϵ C^q(E,F) is C_p,q approximable on some neighborhood of every point of E, then F is C_p,q approximable on all of E.

In general it is unknown whether an arbitrary function in C¹(l², R) is C_2,1 approximable and an example of a function in C¹(l², R) which may not be C_2,1 approximable is given. A weak form of C_∞,q, q≥1, to functions in C^q(l², R) is proved: Let {U_α} be a locally finite cover of l² and let {T_α} be a corresponding collection of Hilbert-Schmidt operators on l². Then for any f ϵ C^q(l²,F) such that for all α

sup ‖ D^k(f(x)-g(x))[T_αh]‖ ≤ 1.

xϵU_α,‖h‖≤1, 0≤k≤q

Minimum drag profiles in infinite cavity flows

The problem considered is that of minimizing the drag of a symmetric plate in infinite cavity flow under the constraints of fixed arclength and fixed chord. The flow is assumed to be steady, irrotational, and incompressible. The effects of gravity and viscosity are ignored.

Using complex variables, expressions for the drag, arclength, and chord, are derived in terms of two hodograph variables, Γ (the logarithm of the speed) and β (the flow angle), and two real parameters, a magnification factor and a parameter which determines how much of the plate is a free-streamline.

Two methods are employed for optimization:

(1) The parameter method. Γ and β are expanded in finite orthogonal series of N terms. Optimization is performed with respect to the N coefficients in these series and the magnification and free-streamline parameters. This method is carried out for the case N = 1 and minimum drag profiles and drag coefficients are found for all values of the ratio of arclength to chord.

(2) The variational method. A variational calculus method for minimizing integral functionals of a function and its finite Hilbert transform is introduced, This method is applied to functionals of quadratic form and a necessary condition for the existence of a minimum solution is derived. The variational method is applied to the minimum drag problem and a nonlinear integral equation is derived but not solved.

The Riesz space structure of an Abelian W*-algebra

Let M be an Abelian W*-algebra of operators on a Hilbert space H. Let M₀ be the set of all linear, closed, densely defined transformations in H which commute with every unitary operator in the commutant M’ of M. A well known result of R. Pallu de Barriere states that if ɸ is a normal positive linear functional on M, then ɸ is of the form T → (Tx, x) for some x in H, where T is in M. An elementary proof of this result is given, using only those properties which are consequences of the fact that ReM is a Dedekind complete Riesz space with plenty of normal integrals. The techniques used lead to a natural construction of the class M₀, and an elementary proof is given of the fact that a positive self-adjoint transformation in M₀ has a unique positive square root in M₀. It is then shown that when the algebraic operations are suitably defined, then M₀ becomes a commutative algebra. If ReM₀ denotes the set of all self-adjoint elements of M₀, then it is proved that ReM₀ is Dedekind complete, universally complete Riesz spaces which contains ReM as an order dense ideal. A generalization of the result of R. Pallu de la Barriere is obtained for the Riesz space ReM₀ which characterizes the normal integrals on the order dense ideals of ReM₀. It is then shown that ReM₀ may be identified with the extended order dual of ReM, and that ReM₀ is perfect in the extended sense.

Some secondary questions related to the Riesz space ReM are also studied. In particular it is shown that ReM is a perfect Riesz space, and that every integral is normal under the assumption that every decomposition of the identity operator has non-measurable cardinal. The presence of atoms in ReM is examined briefly, and it is shown that ReM is finite dimensional if and only if every order bounded linear functional on ReM is a normal integral.

Embedding quantum simulators

79 p.

Análise da regra de evidência cartesiana

A dissertação examina o critério de evidência cartesiano. O critério de evidência consiste na clareza e distinção das idéias. O objetivo é mostrar que o critério de evidência comporta elementos objetivos de diferenciação entre as idéias claras e distintas e as idéias obscuras e confusas. Primeiramente, demonstra-se que há propriedades intrínsecas constitutivas da clareza e distinção das idéias. Depois, mostra-se que as idéias claras e distintas têm conteúdos proposicionais, e, além disso, estabelece-se que as idéias claras e distintas comportam índices fundadores indubitáveis. Desta forma, perceber clara e distintamente é identificar relações lógicas entre a proposição e o seu índice fundador indubitável. Para cada índice fundador identificado existem várias implicações proposicionais possíveis. Um sujeito finito não pode garantir que apreendeu todas as implicações proposicionais possíveis; no entanto, isso não significa que o sujeito não é capaz de ter uma percepção clara e distinta. Com efeito, pretende-se mostrar que para satisfazer o critério de evidência basta apreender a implicação essencial do índice fundador. Conclui-se, assim, que o critério de evidência comporta elementos objetivos, os quais são fundados nas relações lógicas que existem entre as proposições e seus índices fundadores indubitáveis.

The Forbidden Quantum Adder

Quantum information provides fundamentally different computational resources than classical information. We prove that there is no unitary protocol able to add unknown quantum states belonging to different Hilbert spaces. This is an inherent restriction of quantum physics that is related to the impossibility of copying an arbitrary quantum state, i.e., the no-cloning theorem. Moreover, we demonstrate that a quantum adder, in absence of an ancillary system, is also forbidden for a known orthonormal basis. This allows us to propose an approximate quantum adder that could be implemented in the lab. Finally, we discuss the distinct character of the forbidden quantum adder for quantum states and the allowed quantum adder for density matrices.

Do real da ciência ao real da psicanálise: a função do ato psicanalítico

Esta dissertação versa sobre três aspectos da relação entre psicanálise e ciência tendo como fio condutor o sujeito da psicanálise. Primeiramente a importância da ciência para a delimitação do estatuto do saber psicanalítico, visando, mais do que decidir sobre o estatuto científico da psicanálise, as condições de sua emergência como um campo de saber e como uma práxis a partir da ciência. Neste primeiro momento da dissertação, seguimos as indicações da posição de Freud em A questão de uma Weltanschauung e a problematização levada a cabo por Lacan no seu Seminário - Livro 11 (Os quatro conceitos fundamentais da psicanálise). Num segundo momento, discutimos o lugar e a função da ciência na constituição da psicanálise ela é sua condição discursiva ou obstáculo? a partir do escrito lacaniano A ciência e a verdade e tomando como contraponto alguns autores que se situam como lacanianos mas que sustentam, contra a posição de Lacan, que a ciência se coloca como obstáculo para a psicanálise. No terceiro e último momento da dissertação, discutimos a diferença do estatuto de real na ciência e na psicanálise utilizando como critério demarcador o ato psicanalítico tal como este último é definido por Lacan no Seminário Livro XV O ato psicanalítico.

Rio, cenas decisivas: teatro entre televisão, patrocínio e política

Baseado em estudos de casos e em discussões teóricas, este trabalho analisa as relações entre a estética, a produção e a recepção de uma vertente do teatro produzido no Rio de Janeiro que se propõe manter-se além do entretenimento. O critério para a escolha dos casos estudados é o da inteligibilidade das relações entre estética e contexto sócio-econômico, a partir das próprias referências e vivências do pesquisador enquanto também artista e produtor teatral. A abordagem do teatro como fenômeno estético e cultural que interage com o meio sócio-econômico vale neste trabalho tanto para o teatro da contemporaneidade quanto em suas discussões históricas. A universalização da TV, a falta de público pagante para a arte e a ainda recente resposta do Estado como financiador universal da cultura aparecem aqui intimamente ligadas à proliferação e transformação de poéticas teatrais, principalmente no que concerne ao trabalho do ator em cena, sua formação, e a concepção que ele tem de sociedade e de seu papel dentro dela. Esta mirada a partir do fazer teatral questiona o valor de limites fixos entre arte e indústria cultural, ao sugerir como problemática a definição de arte e cultura na contemporaneidade. As relações de mútua legitimação entre a indústria da televisão e o teatro na cidade são acompanhadas desde a sua descoberta pelo Teatro dos Sete, seu apogeu nos anos 70 e 80, e sua subsequente crise no final da década de 80, quando a simbiose se transformou em competição e a TV passou a ditar as regras do mercado de atores. Os espetáculos escolhidos para análise nesta ótica são justamente O Mambembe de 1959 e A Maldição do Vale Negro, de 1988. Em seguida o texto apresenta um resumo de observações e pesquisas sobre sociologia da cultura, a sócio-economia do teatro e as políticas públicas na França. The Flash and Crash Days, de Gerald Thomas, é o espetáculo escolhido para abordar o início da era dos patrocínios no teatro carioca. A última parte da tese procura juntar os resultados obtidos nos estudos e algumas análises a discussões atuais sobre o teatro no Rio de Janeiro, ao comentar possíveis relações entre a estética e o contexto na cena experimental da zona sul da cidade, e a mudança ideológica e de valores que acompanha a ainda recente mudança de modos de produção. O olhar do teatro como negócio é confrontado por uma alternativa que sugere a abertura de novas perspectivas, através da inclusão de novos sujeitos, como oportunidade de renovação da cena carioca

Boundary integral technique for the numerical modelling of the air flow for the Aaberg exhaust system

A boundary integral technique has been developed for the numerical simulation of the air flow for the Aaberg exhaust system. For the steady, ideal, irrotational air flow induced by a jet, the air velocity is an analytical function. The solution of the problem is formulated in the form of a boundary integral equation by seeking the solution of a mixed boundary-value problem of an analytical function based on the Riemann-Hilbert technique. The boundary integral equation is numerically solved by converting it into a system of linear algebraic equations, which are solved by the process of the Gaussian elimination. The air velocity vector at any point in the solution domain is then computed from the air velocity on the boundary of the solution domains.

Kernelized sorting

Object matching is a fundamental operation in data analysis. It typically requires the definition of a similarity measure between the classes of objects to be matched. Instead, we develop an approach which is able to perform matching by requiring a similarity measure only within each of the classes. This is achieved by maximizing the dependency between matched pairs of observations by means of the Hilbert Schmidt Independence Criterion. This problem can be cast as one of maximizing a quadratic assignment problem with special structure and we present a simple algorithm for finding a locally optimal solution.

Distribution matching for transduction

Many transductive inference algorithms assume that distributions over training and test estimates should be related, e.g. by providing a large margin of separation on both sets. We use this idea to design a transduction algorithm which can be used without modification for classification, regression, and structured estimation. At its heart we exploit the fact that for a good learner the distributions over the outputs on training and test sets should match. This is a classical two-sample problem which can be solved efficiently in its most general form by using distance measures in Hilbert Space. It turns out that a number of existing heuristics can be viewed as special cases of our approach.

Contraction and observer design on cones

We consider the problem of positive observer design for positive systems defined on solid cones in Banach spaces. The design is based on the Hilbert metric and convergence properties are analyzed in the light of the Birkhoff theorem. Two main applications are discussed: positive observers for systems defined in the positive orthant, and positive observers on the cone of positive semi-definite matrices with a view on quantum systems. © 2011 IEEE.

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.