This dissertation develops and analyzes differential equation-based mathematical models and efficient numerical methods and algorithms for genetic regulatory network identification. The primary objectives of the dissertation are to design, analyze, and test a general variational framework and numerical methods for seeking its approximate solutions for reverse engineering genetic regulatory networks from microarray datasets using the approach based on differential equation modeling. In the proposed variational framework, no structure assumption on the genetic network is presumed, instead, the network is solely determined by the microarray profile of the network components and is identified through a well chosen variational principle which minimizes a biological energy functional. The variational principle serves not only as a selection criterion to pick up the right biological solution of the underlying differential equation model but also provide an effective mathematical characterization of the small-world property of genetic regulatory networks which has been observed in lab experiments. Five specific models within the variational framework and efficient numerical methods and algorithms for computing their solutions are proposed and analyzed in the dissertation. Model validations using both synthetic network datasets and real world subnetwork datasets of Saccharomyces cerevisiae (yeast) and E. Coli are done on all five proposed variational models and a performance comparison vs some existing genetic regulatory network identification methods is also provided. As microarray data is typically noisy, in order to take into account the noise effect in the mathematical models, we propose a new approach based on stochastic differential equation modeling and generalize the deterministic variational framework to a stochastic variational framework which relies on stochastic optimization. Numerical algorithms are also proposed for computing solutions of the stochastic variational models. To address the important issue of post-processing computed networks to reflect the small-world property of underlying genetic regulatory networks, a novel threshholding technique based on the Random Matrix Theory is proposed and tested on various synthetic network datasets.

In this paper I investigate properties of square complex matrices of the form Ak, where A is also a complex matrix, and k is a nonnegative integer. I look at several ways of representing Ak. In particular, I present an identity expressing the kth power of the Schur form T of A in terms of the elements of T, which can be used together with the Schur decomposition to provide an expression of Ak. I also explain bounds on the norm of Ak, including some based on the element-based expression of Tk. Finally, I provide a detailed exposition of the most current form of the Kreiss Matrix Theorem.

O objetivo deste trabalho é analisar a regra-matriz constitucional do ICMS mercantil e dos princípios que regem esse imposto, a fim de verificar se os Estados e o Distrito Federal detêm competência para a instituição do ICMS na extração do petróleo, a exemplo do que fez o Estado do Rio de Janeiro por meio da Lei nº 4.117/03, conhecida como Lei Noel. Para tal desiderato, será investigado, inicialmente, o tratamento constitucional atribuído ao petróleo, e serão abordadas questões atinentes às atividades econômicas do monopólio da União sobre esse bem público, prescritas no artigo 177 da Carta Maior, e a sua flexibilização com o advento da Emenda Constitucional nº 9/95. Com supedâneo nas conclusões obtidas a partir da regra-matriz de incidência do ICMS mercantil e dos enunciados constitucionais relativos ao petróleo, serão analisados os os critérios componentes da regra-matriz de incidência construída a partir dos enunciados veiculados pela Lei nº 4.117/03 para que seja possível, dessa forma, verificar se o legislador ordinário estadual observou os limites para o exercício da competência ao instituir o ICMS na extração do petróleo. / The purpose of this study is to analyze the ICMS constitutional matriz-rule of tax incidence and principles that govern such tax, in order to verify the capability of States to create this tax on oil extraction, as it did Rio de Janeiro through Law No. 4.117/03, known as Lei Noel. This study will initially analyze constitutional rules regarding oil as well as aspects related to the economic activities of monopoly, prescribed in Article 177 of Federal Constitution and its flexibility with the advent of Constitutional Amendment No. 9, occurred in 1995. Based on the conclusions obtained from the analysis of ICMS matriz-rule of tax incidence and constitutional rules concerning oil, we will investigate Law No. 4.117/03 in order to check if the Rio de Janeiro State respected the limits imposed by Federal Constitution to create ICMS on oil extraction.

O objetivo deste trabalho é analisar a regra-matriz constitucional do ICMS mercantil e dos princípios que regem esse imposto, a fim de verificar se os Estados e o Distrito Federal detêm competência para a instituição do ICMS na extração do petróleo, a exemplo do que fez o Estado do Rio de Janeiro por meio da Lei nº 4.117/03, conhecida como Lei Noel. Para tal desiderato, será investigado, inicialmente, o tratamento constitucional atribuído ao petróleo, e serão abordadas questões atinentes às atividades econômicas do monopólio da União sobre esse bem público, prescritas no artigo 177 da Carta Maior, e a sua flexibilização com o advento da Emenda Constitucional nº 9/95. Com supedâneo nas conclusões obtidas a partir da regra-matriz de incidência do ICMS mercantil e dos enunciados constitucionais relativos ao petróleo, serão analisados os os critérios componentes da regra-matriz de incidência construída a partir dos enunciados veiculados pela Lei nº 4.117/03 para que seja possível, dessa forma, verificar se o legislador ordinário estadual observou os limites para o exercício da competência ao instituir o ICMS na extração do petróleo. / The purpose of this study is to analyze the ICMS constitutional matriz-rule of tax incidence and principles that govern such tax, in order to verify the capability of States to create this tax on oil extraction, as it did Rio de Janeiro through Law No. 4.117/03, known as Lei Noel. This study will initially analyze constitutional rules regarding oil as well as aspects related to the economic activities of monopoly, prescribed in Article 177 of Federal Constitution and its flexibility with the advent of Constitutional Amendment No. 9, occurred in 1995. Based on the conclusions obtained from the analysis of ICMS matriz-rule of tax incidence and constitutional rules concerning oil, we will investigate Law No. 4.117/03 in order to check if the Rio de Janeiro State respected the limits imposed by Federal Constitution to create ICMS on oil extraction.

New algorithms for the matrix exponential and its Fréchet derivative are presented. First, we derive a new scaling and squaring algorithm (denoted expm[new]) for computing eA, where A is any square matrix, that mitigates the overscaling problem. The algorithm is built on the algorithm of Higham [SIAM J.Matrix Anal. Appl., 26(4): 1179-1193, 2005] but improves on it by two key features. The first, specific to triangular matrices, is to compute the diagonal elements in the squaring phase as exponentials instead of powering them. The second is to base the backward error analysis that underlies the algorithm on members of the sequence {||Ak||1/k} instead of ||A||. The terms ||Ak||1/k are estimated without computing powers of A by using a matrix 1-norm estimator. Second, a new algorithm is developed for computing the action of the matrix exponential on a matrix, etAB, where A is an n x n matrix and B is n x n₀ with n₀ << n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n x n₀ matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix etAB or a sequence etkAB on an equally spaced grid of points tk. It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree using the strategy of expm[new].Preprocessing steps are used to reduce the cost of the algorithm. An important application of the algorithm is to exponential integrators for ordinary differential equations. It is shown that the sums of the form $\sum_{k=0}^p\varphi_k(A)u_k$ that arise in exponential integrators, where the $\varphi_k$ are related to the exponential function, can be expressed in terms of a single exponential of a matrix of dimension $n+p$ built by augmenting $A$ with additional rows and columns. Third, a general framework for simultaneously computing a matrix function, $f(A)$, and its Fréchet derivative in the direction $E$, $L_f(A,E)$, is established for a wide range of matrix functions. In particular, we extend the algorithm of Higham and $\mathrm{expm_{new}}$ to two algorithms that intertwine the evaluation of both $e^A$ and $L(A,E)$ at a cost about three times that for computing $e^A$ alone. These two extended algorithms are then adapted to algorithms that simultaneously calculate $e^A$ together with an estimate of its condition number. Finally, we show that $L_f(A,E)$, where $f$ is a real-valued matrix function and $A$ and $E$ are real matrices, can be approximated by $\Im f(A+ihE)/h$ for some suitably small $h$. This approximation generalizes the complex step approximation known in the scalar case, and is proved to be of second order in $h$ for analytic functions $f$ and also for the matrix sign function. It is shown that it does not suffer the inherent cancellation that limits the accuracy of finite difference approximations in floating point arithmetic. However, cancellation does nevertheless vitiate the approximation when the underlying method for evaluating $f$ employs complex arithmetic. The complex step approximation is attractive when specialized methods for evaluating the Fréchet derivative are not available.

This thesis concerns itself with word classes and their application to language modelling.Considering a purely statistical Markov model trained on sequences of word classes in theSwedish language different problems in language engineering are examined. Problemsconsidered are part-of-speech tagging, evaluating text modifiers such as translators withthe help of probability measurements and matrix norms, and lastly detecting differenttypes of text using the Fourier transform of cross entropy sequences of word classes.The results show that the word class language model is quite weak by itself but that itis able to improve part-of-speech tagging for 1 and 2 letter models. There are indicationsthat a stronger word class model could aid 3-letter and potentially even stronger models.For evaluating modifiers the model is often able to distinguish between shuffled andsometimes translated text as well as to assign a score as to how much a text has beenmodified. Future work on this should however take better care to ensure large enoughtest data. The results from the Fourier approach indicate that a Fourier analysis of thecross entropy sequence between word classes may allow the model to distinguish betweenA.I. generated text as well as translated text from human written text. Future work onmachine learning word class models could be carried out to get further insights into therole of word class models in modern applications. The results could also give interestinginsights in linguistic research regarding word classes.

Made available in DSpace on 2016-04-26T20:21:11Z (GMT). No. of bitstreams: 1 Lucas Galvao de Britto.pdf: 1356483 bytes, checksum: da73d05015996a2d841a7d4e9a63f490 (MD5) Previous issue date: 2012-10-05 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / This study's purpose is the spatial criterion of the matrix-norm of tax incidence. Investigates it from an hermeneutic-analytical point of view, departing from the understanding of how spatial categories help humans in understanding the objects of experience, in order to isolate and transpose these categories to the study of law, testing them with the established theory of place of crime and conducting the analysis results for the formulation of a theory of the place of tribute / Esta pesquisa tem por objeto o critério espacial da regra-matriz de incidência tributária. Investiga-o desde um ponto de vista analítico- hemenêutico para, a partir da compreensão de como as categorias espaciais ajudam o ser humano na compreensão dos objetos da experiência,transpor as categorias isolada são direito,testando-as com a já estabelecida teoria do lugar do crime e, conduzindo os resultados da análise para a formação de uma teoria do lugar do tributo

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