Topological strings, double affine Hecke algebras, and exceptional knot homology

In this thesis, we consider two main subjects: refined, composite invariants and exceptional knot homologies of torus knots. The main technical tools are double affine Hecke algebras ("DAHA") and various insights from topological string theory.

In particular, we define and study the composite DAHA-superpolynomials of torus knots, which depend on pairs of Young diagrams and generalize the composite HOMFLY-PT polynomials from the full HOMFLY-PT skein of the annulus. We also describe a rich structure of differentials that act on homological knot invariants for exceptional groups. These follow from the physics of BPS states and the adjacencies/spectra of singularities associated with Landau-Ginzburg potentials. At the end, we construct two DAHA-hyperpolynomials which are closely related to the Deligne-Gross exceptional series of root systems.

In addition to these main themes, we also provide new results connecting DAHA-Jones polynomials to quantum torus knot invariants for Cartan types A and D, as well as the first appearance of quantum E6 knot invariants in the literature.

像差对星间激光通信相互对准误差的影响

We analyze mutual alignment errors due to wave-front aberrations. To solve the central obscured problem, we introduce modified Zernike polynomials, which are a set of complete orthogonal polynomials. It is found that different aberrations have different effects on mutual alignment errors. Some aberrations influence only the line of sight, while some aberrations influence both the line of sight and the intensity distributions. (c) 2005 Optical Society of America

Estresse no trabalho e pressão arterial: reflexões metodológicas sobre linearidade e operacionalização da exposição

O modelo demanda-controle proposto por Karasek caracteriza as situações de estresse no ambiente laboral que é definido pela alta exigência no trabalho resultante da combinação do efeito da demanda psicológica no trabalho (estressores) e moderadores do estresse, especialmente a capacidade de tomar decisões (controle). Existem diversas formas de operacionalizar a exposição (alta exigência no trabalho) descritas na literatura. No entanto, não há consenso sobre qual a melhor operacionalização. Outra questão negligenciada pela maioria dos autores é a forma funcional da curva exposição-resposta ou não linearidade da associação. Entre os desfechos avaliados em estudos sobre os efeitos do estresse no trabalho, usando o modelo demanda-controle, estão as doenças cardiovasculares, como a hipertensão arterial. Contudo, os resultados relatados parecem conflitantes sobre a associação entre o estresse no trabalho e a hipertensão arterial. Os objetivos deste trabalho são estudar a associação entre o estresse no trabalho e aumento da pressão arterial/hipertensão arterial usando as diferentes formas de operacionalização do estresse no trabalho e avaliar a curva exposição-resposta quanto à linearidade. O trabalho é um estudo seccional desenvolvido na população do Estudo Pró-Saúde que é composta por servidores públicos de uma universidade no Rio de Janeiro. Modelos de regressão usando diferentes distribuições foram usados para estimar a associação entre a alta exigência no trabalho e a pressão arterial/hipertensão arterial. Funções não lineares foram utilizadas para investigar a linearidade da associação entre a exposição e os desfechos. Os resultados sugerem que a relação entre a dimensão controle e a pressão arterial seja não linear. Foram observadas associações significativas entre alta exigência no trabalho e pressão arterial sistólica e diastólica para algumas formas de operacionalização da exposição, quando analisados separadamente. No entanto, quando analisadas de forma conjunta por meio do modelo de resposta bivariada não foi encontrada associação. Também não foi encontrada associação entre a alta exigência no trabalho e hipertensão arterial para nenhuma forma de operacionalização da exposição. Os modelos estatísticos com melhor ajuste foram aqueles em que a exposição foi operacionalizada por meio de quadrantes gerados pela mediana de cada uma das dimensões e de forma dicotômica com termo de interação entre demanda psicológica e controle. Neste estudo estas foram as formas mais adequadas de operacionalização. Os resultados também mostram que a associação entre a dimensão controle e a pressão arterial é não linear. Conclui-se que não há garantia de linearidade da associação entre estresse no trabalho e pressão arterial e que essa associação pode ser influenciada pela forma de operacionalização da exposição e do desfecho ou da estratégia de modelagem.

Symmetric representation of an integral domain over a subdomain

Let F(θ) be a separable extension of degree n of a field F. Let Δ and D be integral domains with quotient fields F(θ) and F respectively. Assume that Δ ᴝ D. A mapping φ of Δ into the n x n D matrices is called a Δ/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dI_n whenever d ϵ D. If the matrices are also symmetric, φ is a Δ/D symrep.

Every Δ/D rep can be extended uniquely to an F(θ)/F rep. This extension is completely determined by the image of θ. Two Δ/D reps are called equivalent if the images of θ differ by a D unimodular similarity. There is a one-to-one correspondence between classes of Δ/D reps and classes of Δ ideals having an n element basis over D.

The condition that a given Δ/D rep class contain a Δ/D symrep can be phrased in various ways. Using these formulations it is possible to (i) bound the number of symreps in a given class, (ii) count the number of symreps if F is finite, (iii) establish the existence of an F(θ)/F symrep when n is odd, F is an algebraic number field, and F(θ) is totally real if F is formally real (for n = 3 see Sapiro, “Characteristic polynomials of symmetric matrices” Sibirsk. Mat. Ž. 3 (1962) pp. 280-291), and (iv) study the case D = Z, the integers (see Taussky, “On matrix classes corresponding to an ideal and its inverse” Illinois J. Math. 1 (1957) pp. 108-113 and Faddeev, “On the characteristic equations of rational symmetric matrices” Dokl. Akad. Nauk SSSR 58 (1947) pp. 753-754).

The case D = Z and n = 2 is studied in detail. Let Δ’ be an integral domain also having quotient field F(θ) and such that Δ’ ᴝ Δ. Let φ be a Δ/Z symrep. A method is given for finding a Δ’/Z symrep ʘ such that the Δ’ ideal class corresponding to the class of ʘ is an extension to Δ’ of the Δ ideal class corresponding to the class of φ. The problem of finding all Δ/Z symreps equivalent to a given one is studied.

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL. Let A_i+1 = A_iB - BA_i, i = 0, 1, 2,…, with A = A_o. Let f_k (A, B; σ) = A_2K+1 - ^σ1^A2K-1 ^{+ σ}2^A2K-3 -… +(-1)^Kσ_KA₁ where σ = (σ₁, σ₂,…, σ_K), σ_i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f_n(A, B; σ) = 0 if σ_i is the i^th elementary symmetric function of (β₄- β_s)², 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β₄ are the characteristic roots of B. In this thesis we discuss relations involving f_k(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f_k(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X₁, X₂…X_r belong to the radical of the algebra generated by A and B over F, where X_i has the form f₂(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists σ so that f_k(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g_k(w; σ) = w^2K+1 - σ₁w^2K-1 + σ₂w^2K-3 - …. +(-1)^K σ_Kw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].

Geometrical effects on the resonance absorption of neutrons

An investigation was conducted to estimate the error when the flat-flux approximation is used to compute the resonance integral for a single absorber element embedded in a neutron source.

The investigation was initiated by assuming a parabolic flux distribution in computing the flux-averaged escape probability which occurs in the collision density equation. Furthermore, also assumed were both wide resonance and narrow resonance expressions for the resonance integral. The fact that this simple model demonstrated a decrease in the resonance integral motivated the more detailed investigation of the thesis.

An integral equation describing the collision density as a function of energy, position and angle is constructed and is subsequently specialized to the case of energy and spatial dependence. This equation is further simplified by expanding the spatial dependence in a series of Legendre polynomials (since a one-dimensional case is considered). In this form, the effects of slowing-down and flux depression may be accounted for to any degree of accuracy desired. The resulting integral equation for the energy dependence is thus solved numerically, considering the slowing down model and the infinite mass model as separate cases.

From the solution obtained by the above method, the error ascribable to the flat-flux approximation is obtained. In addition to this, the error introduced in the resonance integral in assuming no slowing down in the absorber is deduced. Results by Chernick for bismuth rods, and by Corngold for uranium slabs, are compared to the latter case, and these agree to within the approximations made.

Estimation of catalyst activity profiles and deactivation parameters from reactor operating data with applications to naphtha reforming

Techniques are developed for estimating activity profiles in fixed bed reactors and catalyst deactivation parameters from operating reactor data. These techniques are applicable, in general, to most industrial catalytic processes. The catalytic reforming of naphthas is taken as a broad example to illustrate the estimation schemes and to signify the physical meaning of the kinetic parameters of the estimation equations. The work is described in two parts. Part I deals with the modeling of kinetic rate expressions and the derivation of the working equations for estimation. Part II concentrates on developing various estimation techniques.

Part I: The reactions used to describe naphtha reforming are dehydrogenation and dehydroisomerization of cycloparaffins; isomerization, dehydrocyclization and hydrocracking of paraffins; and the catalyst deactivation reactions, namely coking on alumina sites and sintering of platinum crystallites. The rate expressions for the above reactions are formulated, and the effects of transport limitations on the overall reaction rates are discussed in the appendices. Moreover, various types of interaction between the metallic and acidic active centers of reforming catalysts are discussed as characterizing the different types of reforming reactions.

Part II: In catalytic reactor operation, the activity distribution along the reactor determines the kinetics of the main reaction and is needed for predicting the effect of changes in the feed state and the operating conditions on the reactor output. In the case of a monofunctional catalyst and of bifunctional catalysts in limiting conditions, the cumulative activity is sufficient for predicting steady reactor output. The estimation of this cumulative activity can be carried out easily from measurements at the reactor exit. For a general bifunctional catalytic system, the detailed activity distribution is needed for describing the reactor operation, and some approximation must be made to obtain practicable estimation schemes. This is accomplished by parametrization techniques using measurements at a few points along the reactor. Such parametrization techniques are illustrated numerically with a simplified model of naphtha reforming.

To determine long term catalyst utilization and regeneration policies, it is necessary to estimate catalyst deactivation parameters from the the current operating data. For a first order deactivation model with a monofunctional catalyst or with a bifunctional catalyst in special limiting circumstances, analytical techniques are presented to transform the partial differential equations to ordinary differential equations which admit more feasible estimation schemes. Numerical examples include the catalytic oxidation of butene to butadiene and a simplified model of naphtha reforming. For a general bifunctional system or in the case of a monofunctional catalyst subject to general power law deactivation, the estimation can only be accomplished approximately. The basic feature of an appropriate estimation scheme involves approximating the activity profile by certain polynomials and then estimating the deactivation parameters from the integrated form of the deactivation equation by regression techniques. Different bifunctional systems must be treated by different estimation algorithms, which are illustrated by several cases of naphtha reforming with different feed or catalyst composition.

Effects of the frequency chirp on the fields of a chirped Gaussian pulse passing through a hard-edged aperture

Based on the Huygens-Fresnel diffraction integral and Fourier transform, propagation expression of a chirped Gaussian pulse passing through a hard-edged aperture is derived. Intensity distributions of the pulse with different frequency chirp in the near-field and far-field are analyzed in detail by numerical calculations. In the near-field, amplitudes of the intensity peaks generated by the modulation of the hard-edged aperture decrease with increasing the frequency chirp, which results in the improving of the beam uniformity. A physical explanation for the smoothing effect brought by increasing the frequency chirp is given. The smoothing effect is achieved not only in the pulse with Gaussian transverse profile but also in the pulse with Hermite-Gaussian transverse profile when the frequency chirp increases. (C) 2005 Elsevier B.V. All rights reserved.

厄米-高斯光束中带宽对横向强度分布均匀性的影响

使用惠更斯-菲涅耳衍射积分和傅里叶变换，推导了横向模式为厄米高斯的激光束通过硬边光阑的传输表达式。利用所得公式通过数值计算详细分析了光束的频带宽度对光束横向强度分布的影响。分析发现激光束通过硬边光阑后会出现大量的强度尖峰，引起近场的强度分布不均匀。然而，当带宽增加时，强度尖峰的数量减少，幅度减小，光束的均匀性得到改善。在远场，光束的横向强度分布没有强度尖峰出现，但宽度随带宽的增加而减小。因此带宽增加会带来近场强度匀滑化和远场光束宽度变窄的效应。另外，数值结果表明带宽对模指数为1和2的厄米-高斯光束的强度分

O perfil dos diretores e da gestão de escolas públicas urbanas das capitais brasileiras: um estudo a partir do SAEB 2003

O presente trabalho apresentou a evolução do conceito de administração escolar até chegar a gestão, os principais modelos de gestão escolar encontrados na literatura educacional, que podemos assim dizer: gestão participativa, compartilhada e democrática. Buscou compreender o perfil dos diretores e da gestão de escolas públicas e urbanas do Brasil e de que maneira o perfil do diretor e os modelos de gestão apresentados na literatura educacional se relacionam com as características socioeconômicas médias dos alunos das escolas. O estudo fez uso dos dados do Sistema Nacional de Avaliação da Educação Básica SAEB 2003, referentes às séries iniciais do ensino fundamental. A análise baseou-se em estatísticas descritivas uni e bivariadas e os resultados foram analisados à luz da literatura específica. Os dados adquiridos a partir da análise das variáveis do SAEB 2003 foram confrontados com as informações teóricas pesquisadas sobre o tema. Esta pesquisa teve por principal base teórica as idéias de Anísio Teixeira, José Quirino Ribeiro e Vitor Henrique Paro, Lauro C. Wittman, Heloísa Lück, Jaap Scheerens, a respeito da administração escolar até sua evolução a gestão escolar; Pam Sammons juntamente com Fátima C. Alves, Creso Franco, Alícia Bonamino, Francisco C.Gomes e os documentos oficiais que tratam do SAEB contribuíram no estudo para a construção do perfil do diretor brasileiro e de sua gestão juntamente com as variáveis selecionadas do SAEB 2003 e no entendimento do conceito de escolas eficazes. A pesquisa revelou que o desempenho de uma gestão de qualidade pressupõe a conquista de características no sistema educacional que envolve a participação, a autonomia e a descentralização.

Integrated treatment of first episode psychosis with online training (e-learning): study protocol for a randomised controlled trial

Background: The integrated treatment of first episode psychosis has been shown to improve functionality and negative symptoms in previous studies. In this paper, we describe a study of integrated treatment (individual psychoeducation complementary to pharmacotherapy) versus treatment as usual, comparing results at baseline with those at 6-month re-assessment (at the end of the study) for these patients, and online training of professionals to provide this complementary treatment, with the following objectives: 1) to compare the efficacy of individual psychoeducation as add-on treatment versus treatment as usual in improving psychotic and mood symptoms; 2) to compare adherence to medication, functioning, insight, social response, quality of life, and brain-derived neurotrophic factor, between both groups; and 3) to analyse the efficacy of online training of psychotherapists. Methods/design: This is a single-blind randomised clinical trial including patients with first episode psychosis from hospitals across Spain, randomly assigned to either a control group with pharmacotherapy and regular sessions with their psychiatrist (treatment as usual) or an intervention group with integrated care including treatment as usual plus a psychoeducational intervention (14 sessions). Training for professionals involved at each participating centre was provided by the coordinating centre (University Hospital of Alava) through video conferences. Patients are evaluated with an extensive battery of tests assessing clinical and sociodemographic characteristics (Positive and Negative Syndrome Scale, State-Trait Anxiety Inventory, Liebowitz Social Anxiety Scale, Hamilton Rating Scale for Depression, Scale to Assess Unawareness of Mental Disorders, Strauss and Carpenter Prognostic Scale, Global Assessment of Functioning Scale, Morisky Green Adherence Scale, Functioning Assessment Short Test, World Health Organization Quality of Life instrument WHOQOL-BREF (an abbreviated version of the WHOQOL-100), and EuroQoL questionnaire), and brain-derived neurotrophic factor levels are measured in peripheral blood at baseline and at 6 months. The statistical analysis, including bivariate analysis, linear and logistic regression models, will be performed using SPSS. Discussion: This is an innovative study that includes the assessment of an integrated intervention for patients with first episode psychosis provided by professionals who are trained online, potentially making it possible to offer the intervention to more patients.

A formação científica brasileira e o PISA 2006

Este trabalho investigou a formação científica brasileira a partir dos dados do PISA 2006. O PISA - Programa Internacional de Avaliação dos Estudantes - é um programa internacional de avaliação comparada, aplicado a uma amostra de estudantes de 15 anos de idade. O ensino adequado de ciências estimula o raciocínio lógico e a curiosidade, ajuda a formar cidadãos mais aptos a enfrentar os desafios da sociedade contemporânea e fortalece a democracia, dando à população, em geral, melhores condições para participar dos debates cada vez mais comuns sobre temas científicos que afetam nosso cotidiano. Partindo da importância e pertinência de se discutir a formação científica brasileira e fazendo uso dos dados do PISA 2006, buscou-se responder a três questões: (a) qual a percepção dos alunos brasileiros que participaram do PISA 2006 sobre Ciências? (b) como o Brasil se situa no contexto internacional no que se refere às competências científicas priorizadas por esta avaliação? (c) que características dos estudantes estão associadas aos seus resultados no teste de Ciências aplicado pelo PISA 2006? Para a realização do estudo, conduzimos nossas análises baseando-nos em duas lógicas principais: descrição (estatística univariada e bivariada) e explicação (teste de modelos de regressão linear). Os resultados indicam que os alunos brasileiros tem relativo interesse pela Ciência, no entanto apresentam uma visão bastante estereotipada e demonstram não associá-la a si. Aparentemente, a ciência tem relação com fenômenos que estão muito além da compreensão e configura-se em algo abstrato que não está diretamente ligado aos alunos brasileiros. Estes apontam a escola como principal fonte de aprendizagem de tópicos relacionados a Ciência, porém, a escola parece não converter o potencial interesse dos alunos por Ciências em letramento científico. O Brasil ocupa a 52 posição entre os 57 países participantes e está alocado no nível mais baixo de desempenho do PISA 2006. Entre os principais aspectos que impactam significativamente os resultados dos alunos brasileiros no programa estão a Rede de Ensino, o Atraso Escolar, o Nível Socioeconômico e o Nível de Escolaridade dos Pais.

Transverse mode formation in longitudinally pumped miniature slab lasers

The formation of transverse modes in longitudinally pumped miniature slab lasers is investigated theoretically and experimentally. The longitudinally non-uniform gain-guiding is studied by expanding the electric field into the Hermite-Gaussian functions that satisfy boundary conditions of the resonator. Non-Gaussian transversal beam profiles in the near field are found and the beam diameter is reduced when the pump spot becomes smaller. The experimental observation agrees with the theoretical calculation.

Helmholtz Fermi surface harmonics: an efficient approach for treating anisotropic problems involving Fermi surface integrals

We present a new efficient numerical approach for representing anisotropic physical quantities and/or matrix elements defined on the Fermi surface (FS) of metallic materials. The method introduces a set of numerically calculated generalized orthonormal functions which are the solutions of the Helmholtz equation defined on the FS. Noteworthy, many properties of our proposed basis set are also shared by the FS harmonics introduced by Philip B Allen (1976 Phys. Rev. B 13 1416), proposed to be constructed as polynomials of the cartesian components of the electronic velocity. The main motivation of both approaches is identical, to handle anisotropic problems efficiently. However, in our approach the basis set is defined as the eigenfunctions of a differential operator and several desirable properties are introduced by construction. The method is demonstrated to be very robust in handling problems with any crystal structure or topology of the FS, and the periodicity of the reciprocal space is treated as a boundary condition for our Helmholtz equation. We illustrate the method by analysing the free-electron-like lithium (Li), sodium (Na), copper (Cu), lead (Pb), tungsten (W) and magnesium diboride (MgB2)

Robust design optimization of gas turbine compression systems

Gas turbine compression systems are required to perform adequately over a range of operating conditions. Complexity has encouraged the conventional design process for compressors to focus initially on one operating point, usually the most commonor arduous, to draw up an outline design. Generally, only as this initial design is refined is its offdesign performance assessed in detail. Not only does this necessarily introduce a potentially costly and timeconsuming extra loop in the design process, but it also may result in a design whose offdesign behavior is suboptimal. Aversion of nonintrusive polynomial chaos was previously developed in which a set of orthonormal polynomials was generated to facilitate a rapid analysis of robustness in the presence of generic uncertainties with good accuracy. In this paper, this analysis method is incorporated in real time into the design process for the compression system of a three-shaft gas turbine aeroengine. This approach to robust optimization is shown to lead to designs that exhibit consistently improved system performance with reduced sensitivity to offdesign operation.