The paper deals with the calculation of the fractional part of the η-invariant for elliptic self-adjoint operators in topological terms. The method used to obtain the corresponding formula is based on the index theorem for elliptic operators in subspaces obtained in [1], [2]. It also utilizes K-theory with coefficients Zsub(n). In particular, it is shown that the group K(T*M,Zsub(n)) is realized by elliptic operators (symbols) acting in appropriate subspaces.

An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric n-tuple V. The structure of a free semigroup algebra contains a great deal of information about V. Thus it is natural to study this algebra in order to study V. A free semigroup algebra is said to be analytic if it is isomorphic to the noncommutative analytic Toeplitz algebra, which is a higher-dimensional generalization of the classical algebra of bounded analytic functions on the complex unit disk. This notion of analyticity is of central importance in the general theory of free semigroup algebras. A vector x in H is said to be wandering for an isometric n-tuple V if the set of words in the entries of V map x to an orthonormal set. As in the classical case, the analytic structure of the noncommutative analytic Toeplitz algebra is determined by the existence of wandering vectors for the generators of the algebra. In the first part of this thesis, we prove the following dichotomy: either an isometric n-tuple V has a wandering vector, or the free semigroup algebra it generates is a von Neumann algebra. This implies the existence of wandering vectors for every analytic free semigroup algebra. As a consequence, it follows that every free semigroup algebra is reflexive, in the sense that it is completely determined by its invariant subspace lattice. In the second part of this thesis we prove a decomposition for an isometric tuple of operators which generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. The key result is an operator-algebraic characterization of an absolutely continuous isometric tuple in terms of analyticity. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.

Cancer can develop through a series of genetic events in combination with external influential factors that alter the progression of the disease. Gene expression studies are designed to provide an enhanced understanding of the progression of cancer and to develop clinically relevant biomarkers of disease, prognosis and response to treatment. One of the main aims of microarray gene expression analyses is to develop signatures that are highly predictive of specific biological states, such as the molecular stage of cancer. This dissertation analyzes the classification complexity inherent in gene expression studies, proposing both techniques for measuring complexity and algorithms for reducing this complexity. Classifier algorithms that generate predictive signatures of cancer models must generalize to independent datasets for successful translation to clinical practice. The predictive performance of classifier models is shown to be dependent on the inherent complexity of the gene expression data. Three specific quantitative measures of classification complexity are proposed and one measure ( f) is shown to correlate highly (R 2=0.82) with classifier accuracy in experimental data. Three quantization methods are proposed to enhance contrast in gene expression data and reduce classification complexity. The accuracy for cancer prognosis prediction is shown to improve using quantization in two datasets studied: from 67% to 90% in lung cancer and from 56% to 68% in colorectal cancer. A corresponding reduction in classification complexity is also observed. A random subspace based multivariable feature selection approach using costsensitive analysis is proposed to model the underlying heterogeneous cancer biology and address complexity due to multiple molecular pathways and unbalanced distribution of samples into classes. The technique is shown to be more accurate than the univariate ttest method. The classifier accuracy improves from 56% to 68% for colorectal cancer prognosis prediction.  A published gene expression signature to predict radiosensitivity of tumor cells is augmented with clinical indicators to enhance modeling of the data and represent the underlying biology more closely. Statistical tests and experiments indicate that the improvement in the model fit is a result of modeling the underlying biology rather than statistical over-fitting of the data, thereby accommodating classification complexity through the use of additional variables.

An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric n-tuple V. The structure of a free semigroup algebra contains a great deal of information about V. Thus it is natural to study this algebra in order to study V. A free semigroup algebra is said to be analytic if it is isomorphic to the noncommutative analytic Toeplitz algebra, which is a higher-dimensional generalization of the classical algebra of bounded analytic functions on the complex unit disk. This notion of analyticity is of central importance in the general theory of free semigroup algebras. A vector x in H is said to be wandering for an isometric n-tuple V if the set of words in the entries of V map x to an orthonormal set. As in the classical case, the analytic structure of the noncommutative analytic Toeplitz algebra is determined by the existence of wandering vectors for the generators of the algebra. In the first part of this thesis, we prove the following dichotomy: either an isometric n-tuple V has a wandering vector, or the free semigroup algebra it generates is a von Neumann algebra. This implies the existence of wandering vectors for every analytic free semigroup algebra. As a consequence, it follows that every free semigroup algebra is reflexive, in the sense that it is completely determined by its invariant subspace lattice. In the second part of this thesis we prove a decomposition for an isometric tuple of operators which generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. The key result is an operator-algebraic characterization of an absolutely continuous isometric tuple in terms of analyticity. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.

Made available in DSpace on 2015-05-14T12:17:07Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 7728873 bytes, checksum: 53e38033eca02db2fa8a5a48e8af7bcf (MD5) Previous issue date: 2013-07-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The research aimed at analyzing the city-countryside relationship and had as an objective of study the city of Campina Grande-PB. With the technological modernization and the expansion of urban spaces, the city-countryside theme is standing out at present in the geographic science, from the productions which aim at showing the new relationships and the new objects which are being introduced into the countryside. However, contrary to this perspective, the objective consisted in understanding the existence and the (re)production of the rural subspaces in the interior of the urban net of the city analyzed. Thus, the city was investigated regarding it as a space produced by means of differentiated logics, in which stand out not only the spaces conceived by the dominant classes and the governments, but also the inhabited and appropriated spaces, built by the population daily. Among such spaces, rural subspaces kept and recreated in the city were examined. However, due to the great number of mixed-farming establishments spread out in the city, an area was chosen situated under a high tension net as a central objective for the investigation. Throughout the research we sought to talk with the authors who might give an aid to the discussion of the city-countryside relationship, both in past periods and in the present context; field work was carried out with the aim of observing, registering and investigating the areas with mixed-farming establishments in the city and the data and information about mixed-farming activities in the governmental institutions were consulted, such as the State Office of Development of Mixed-farming and Fishing - SEDAP. After the survey in field and the analysis in the office, we sought to answer the questions made in the beginning of the research. Accordingly, it is a study of qualitative nature, in which the observations, the interviews, the descriptions and the theoretical foundation were fundamental for the accomplishment of the task. At the end of the investigation, it was verified that, even with the intensification of the process of urbanization in Campina Grande, as well as in other Brazilian cities, the customs and the rural activities remain constantly being adapted to the city reality, due to the people s wish to reproduce a way of life similar to that experienced in the countryside. These people cattle breeders and farmers take possession of the uninhabited areas every day (like the ones under the high tension net) and create rural spaces, that is, inhabited spaces which are opposed to the dominant logic of production of the urban space. / A pesquisa visou analisar a relação cidade-campo e teve como objeto de estudo a cidade de Campina Grande-PB. Com a modernização tecnológica e a expansão dos espaços urbanos, a temática cidade-campo, atualmente, vem se destacando na ciência geográfica, a partir de produções que buscam evidenciar as novas relações e os novos objetos que estão sendo inseridos no campo. Todavia, ao contrário dessa perspectiva, aqui, o objetivo consistiu em compreender a existência e a (re)produção dos subespaços rurais no interior da malha urbana da cidade analisada. Assim, investigou-se a cidade considerando-a como espaço produzido por meio de lógicas diferenciadas, em que se destacam não apenas os espaços concebidos pelas classes dominantes e pelos governantes, mas também os espaços vividos e apropriados, construídos cotidianamente pela população. Dentre tais espaços, examinaram-se os subespaços rurais mantidos e recriados na cidade. Contudo, devido ao grande número de estabelecimentos agropecuários espalhados pela cidade, elegeu-se a área localizada sob a rede alta tensão como objeto central para a investigação. Ao longo da pesquisa, procurou-se dialogar com autores que dessem subsídio às discussões acerca da relação cidade-campo, tanto em períodos passados quanto no contexto atual; realizaram-se trabalhos de campo, com a intenção de observar, registrar e investigar as áreas com estabelecimentos agropecuários na cidade e foram consultados os dados e as informações sobre atividades agropecuárias nas instituições governamentais, como a Secretaria de Estado do Desenvolvimento da Agropecuária e da Pesca SEDAP. Após o levantamento em campo e a análise em gabinete, buscou-se responder aos questionamentos elaborados no início da pesquisa. Nesse sentido, trata-se de um estudo de natureza qualitativa, em que as observações, as entrevistas, as descrições e a fundamentação teórica foram fundamentais para a realização do trabalho. Ao término da investigação, verificou-se que, mesmo com a intensificação do processo de urbanização, em Campina Grande, assim como em outras cidades brasileiras, os costumes e as atividades rurais permanecem sendo constantemente adaptadas à realidade citadina, devido ao desejo das pessoas de reproduzirem um modo de vida semelhante ao que vivenciaram no campo. Essas pessoas criadores de gado e agricultores apropriam-se cotidianamente de áreas não edificadas da cidade (como a rede de alta tensão) e criam subespaços rurais, ou seja, espaços vividos que se contrapõem à lógica dominante de produção do espaço urbano.

A crescente aplicação da estatística às mais diversas áreas de pesquisa, tem definido delineamentos complexos, dificultando assim seu planejamento e análise. O diagrama de Hasse é uma ferramenta gráfica, que tem como objetivo facilitar a compreensão da estrutura presente entre os fatores experimentais. Além de uma melhor visualização do experimento o mesmo fornece, através de regras propostas na literatura, os números de graus de liberdade de cada fator. Sob a condição de ortogonalidade do delineamento, podem-se obter também as matrizes núcleo das formas quadráticas para as somas de quadrados e as esperanças dos quadrados médios, propiciando a razão adequada para a aplicação do teste F. O presente trabalho trata-se de uma revisão, fundamentada na álgebra linear, dos conceitos presentes na estrutura do diagrama. Com base nos mesmos, demonstrou-se o desdobramento do espaço vetorial do experimento em subespaços gerados por seus respectivos fatores, de tal modo que fossem ortogonais entre si. E, a fim de exemplificar as regras e o emprego desta ferramenta, utilizaram-se dois conjuntos de dados, o primeiro de um experimento realizados com cabras Saanen e segundo com capim Marandu, detalhando-se a estrutura experimental, demonstrando-se a ortogonalidade entre os fatores e indicando-se o esquema da análise da vari^ancia. Cabe salientar que o diagrama não substitui o uso de softwares, mas tem grande importância quando o interesse está em se comparar resultados e principalmente verificar o quociente adequado para o teste F. / The increase of statistics applications on the most diverse research areas has defined complex statistics designs turn its planning and analysis really hard. The Hasse diagram is a graphic tool that has as objective turn the comprehension of the present structure among the experimental factors easiest. More than a better experiment overview, by the rules proposed on the literature, this diagram gives the degrees of freedom for each factor. By the condition of design orthogonality, the nucleus matrix of quadratic form for the sum of squares and the expected values for the mean squares can also be obtained, given the proper ratio for F test application. The present work is a review, with its foundations on linear algebra, of the present\'s concepts on the diagram structure. With this basis were demonstrated the development of the vectorial space of the experiment in subspaces generated by its own factors, in a way that it was orthogonal within themselves. And, to give examples about the rules and the application of this tool, experimental data of Saanen goats and other set of data of Marandu grazing were used, with a detailed experiment structure, showing the orthogonality within the factors and with an indication of the analysis of variance model. Has to be emphasized that the diagram do not substitute the usage of software but has a great meaning when the interest is about results comparisons and most of all to check the proper quotient for the F test.

[pt] Este trabalho aborda o problema da estabilidade de sistemas lineares, invariantes no tempo, a tempo discreto, com o espaço de estado sendo um espaço de Hilbert complexo e separável de dimensão infinita. São investigadas condições necessárias e/ou suficientes para quatro conceitos diferentes de estabilidade: estabilidade assintótica uniforme e estabilidade assintótica forte, estabilidade assintótica fraca e estabilidade limitada. Identifica-se e analisa-se as conexões entre os problemas de estabilidade e dois problemas em aberto da teoria de operadores em espaços de Hilbert: o problema do subespaço invariante e o problemas da similaridade e contração. Diversos resultados, oriundos de tentativas de solução para os dois problemas acima, ou motivados por aquelas tentativas, são utilizadas para fornecer caracterizações adicionais (principalmente caracterizações espectrais) para os quatro conceitos de estabilidade em questão. / [en] This work deals with the stability problem for time- invariant discrete linear systems evolving in a separable infinite-dimensional Hilbert space. Necessary and/or sufficient conditions for uniform, strong and weak asymptotic stability, as well as to bounded stability problems to two open problems in operator theory, namely, the invariant subspace and the similarity to contractions, are identified and analysed in detail. Several results from the many attempts, of solving the above mentioned open problems, or motivated by those attempts, are used to supply additional characterizations (mainly spectral characterization) for the four stabilty concepts under consideration.

A perturbation analysis shows that if a numerically stable procedure is used to compute the matrix sign function, then it is competitive with conventional methods for computing invariant subspaces. Stability analysis of the Newton iteration improves an earlier result of Byers and confirms that ill-conditioned iterates may cause numerical instability. Numerical examples demonstrate the theoretical results.

Advances in modern technologies have led to an abundance of high-dimensional time series data in many fields, including finance, economics, health, engineering, and meteorology, among others. This causes the “curse of dimensionality” problem in both univariate and multivariate time series data. The main objective of time series analysis is to make inferences about the conditional distributions. There are some methods in the literature to estimate the conditional mean and conditional variance functions in time series. However, most of those are inefficient, computationally intensive, or suffer from the overparameterization. We propose some dimension reduction techniques to address the curse of dimensionality in high-dimensional time series dataFor high-dimensional matrix-valued time series data, there are a limited number of methods in the literature that can preserve the matrix structure and reduce the number of parameters significantly (Samadi, 2014, Chen et al., 2021). However, those models cannot distinguish between relevant and irrelevant information and yet suffer from the overparameterization. We propose a novel dimension reduction technique for matrix-variate time series data called the "envelope matrix autoregressive model" (EMAR), which offers substantial dimension reduction and links the mean function and the covariance matrix of the model by using the minimal reducing subspace of the covariance matrix. The proposed model can identify and remove irrelevant information and can achieve substantial efficiency gains by significantly reducing the total number of parameters. We derive the asymptotic properties of the proposed maximum likelihood estimators of the EMAR model. Extensive simulation studies and a real data analysis are conducted to corroborate our theoretical results and to illustrate the finite sample performance of the proposed EMAR model.For univariate time series, we propose sufficient dimension reduction (SDR) methods based on some integral transformation approaches that can preserve sufficient information about the response. In particular, we use the Fourier and Convolution transformation methods (FM and CM) to perform sufficient dimension reduction in univariate time series and estimate the time series central subspace (TS-CS), the time series mean subspace (TS-CMS), and the time series variance subspace (TS-CVS). Using FM and CM procedures and with some distributional assumptions, we derive candidate matrices that can fully recover the TS-CS, TS-CMS, and TS-CVS, and propose an explicit estimate of the candidate matrices. The asymptotic properties of the proposed estimators are established under both normality and non-normality assumptions. Moreover, we develop some data-drive methods to estimate the dimension of the time series central subspaces as well as the lag order. Our simulation results and real data analyses reveal that the proposed methods are not only significantly more efficient and accurate but also offer substantial computational efficiency compared to the existing methods in the literature. Moreover, we develop an R package entitled “sdrt” to easily perform our program code in FM and CM procedures to estimate suffices dimension reduction subspaces in univariate time series.

We consider low-rank solution methods for certain classes of large-scale linear matrix equations. Our aim is to adapt existing low-rank solution methods based on standard, extended and rational Krylov subspaces to solve equations which may viewed as extensions of the classical Lyapunov and Sylvester equations. The first class of matrix equations that we consider are constrained Sylvester equations, which essentially consist of Sylvester's equation along with a constraint on the solution matrix. These therefore constitute a system of matrix equations. The second are generalized Lyapunov equations, which are Lyapunov equations with additional terms. Such equations arise as computational bottlenecks in model order reduction. / Mathematics