In this thesis, we study the contractivity preserving, high order, time discretization methods for solving non-stiff ordinary differential equations. We construct a class of one-step, explicit, contractivity preserving, multi-stage, multi-derivative, Hermite-Birkhoff-Taylor methods of order p=5,6, ..., 15, that we denote by CPHBT, with nonnegative coefficients by casting s-stage Runge-Kutta methods of order 4 and 5 with Taylor methods of order p-3 and p-4, respectively. The constructed CPHBT methods are implemented using an efficient variable step algorithm and are compared to other well-known methods on a variety of initial value problems. The results show that CPHBT methods have larger regions of absolute stability, require less function evaluations and hence they require less CPU time to achieve the same accuracy requirements as other methods in the literature. Also, we show that the contractivity preserving property of CPHBT is very efficient in suppressing the effect of the propagation of discretization errors when a long-term integration of a standard N-body problem is considered. The formulae of 49 CPHBT methods of various orders are provided in Butcher form.

In this report we will plunge down in the fascinating world of the generating functions. Generating functions showcase the "power of power series", giving more depth to the word "power" in power series. We start off small to get a good understanding of the generating function and what it does. Also, off course, explaining why it works and why we can do some of the things we do with them. We will see alot of examples throughout the text that helps the reader to grasp the mathematical object that is the generating function. We will look at several kinds of generating functions, the main focus when we establish our understanding of these will be the "ordinary power series" generating function ("ops") that we discuss before moving on to the "exponential generating function" ("egf"). During our discussion on ops we will see a "first time in literature" derivation of the generating function for a recurrence relation regarding "branched coverings". After finishing the discussion regarding egf we move on the Hermite polynomials and show how we derive their generating function. Which is a generating function that generates functions. Lastly we will have a quick look at the "moment generating function".

In this thesis, we consider resource limitations on machine learning algorithms in a variety of settings. In the first two chapters, we study how to learn nonlinear model classes (monomials and neural nets) which are structured in various ways -- we consider sparse monomials and deep neural nets whose weight-matrices are low-rank respectively. These kinds of restrictions on the model class lead to gains in resource efficiency -- sparse and low-rank models are computationally easier to deploy and train. We prove that sparse nonlinear monomials are easier to learn (smaller sample complexity) while still remaining computationally efficient to both estimate and deploy, and we give both theoretical and empirical evidence for the benefit of novel nonlinear initialization schemes for low-rank deep networks. In both cases, we showcase a blessing of nonlinearity -- sparse monomials are in some sense easier to learn compared to a linear class, and the prior state-of-the-art linear low-rank initialization methods for deep networks are inferior to our proposed nonlinear method for initialization. To achieve our theoretical results, we often make use of the theory of Hermite polynomials -- an orthogonal function basis over the Gaussian measure. In the last chapter, we consider resource limitations in an online streaming setting. In particular, we consider how many data points from an oblivious adversarial stream we must store from one pass over the stream to output an additive approximation to the Support Vector Machine (SVM) objective, and prove stronger lower bounds on the memory complexity.

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Generalized linear mixed models (GLMMs) have become extremely popular in recent years. The main computational problem in parameter estimation for GLMMs is that, in contrast to linear mixed models, closed analytical expressions for the likelihood are not available. To overcome this problem, several approaches have been proposed in the literature. For this study we have used one quasi-likelihood approach, penalized quasi-likelihood (PQL), and two integral approaches: Laplace and adaptive Gauss-Hermite quadrature (AGHQ) approximation. Our primary objective was to measure the performances of each estimation method. AGHQ approximation is more accurate than Laplace approximation, but slower. So the question is when Laplace approximation is adequate, versus when AGHQ approximation provides a significantly more accurate result. We have run two simulations using PQL, Laplace and AGHQ approximations with different quadrature points for varying random effect standard deviation (Ɵ) and number of replications per cluster. The performances of these three methods were measured base on the root mean square error (RMSE) and bias. Based on the simulated data, we have found that for both smaller values of Ɵ and small number of replications and for larger values of and for larger values of Ɵ and lager number of replications, the RMSE of PQL method is much higher than Laplace and AGHQ approximations. However, for intermediate values of Ɵ (random effect standard deviation) ranging from 0.63 to 3.98, regardless of number of replications per cluster, both Laplace and AGHQ approximations gave similar estimates. But when both number of replications and Ɵ became small, increasing quadrature points increases RMSE values indicating that Laplace approximation perform better than the AGHQ method. When random effect standard deviation is large, e.g. Ɵ=10, and number of replications is small the Laplace RMSE value is larger than that of AGHQ approximation. Increasing quadrature points decreases the RMSE values. This indicates that AGHQ performs better in this situation. The difference in RMSE between PQL vs Laplace and AGHQ vs Laplace is approximately 12% and 10% respectively. In addition, we have tested the relative performance and the accuracy between two different packages of R (lme4, glmmML) and SAS (PROC GLIMMIX) based on real data. Our results suggested that all of them perform well in terms of accuracy, precision and convergence rates. In most cases, glmmML was found to be much faster than lme4 package and SAS. The only difference was found in the Contraception data where the required computational time for both R packages was exactly the same. The difference in required computational times for these two platforms decreases as the number of quadrature points increases. / Thesis / Master of Science (MSc)

<p>L^p is supposed to be L with a superscript lower case 'p.'</p> / <p>This thesis investigates a 1-dimensional Gross-Pitaevskii (GP) equation from the viewpoint of a system of Hamiltonian partial differential equations (PDEs). A theorem on Birkhoff normal forms is a particularly important goal of this study. The resulting system is a perturbed system of a completely resonant system, which we analyze, using several forms of perturbation theory.</p> <p>In chapter two, we study estimates 011 integrals of products of four Hermite functions, which represent coefficients of mode coupling, and play an important role in the proof of the Birkhoff normal form theorem. This is a basic problem, which has a close relationship with a problem of Besicovitch, namely the behavior of the L^p norms of L² -normalized Hermite functions.</p> <p>In chapter three we carefully reconsider the linear Schrodinger equation with a harmonic potential, and we introduce a family of Hilbert spaces for studying the GP equation, which generalize the traditional energy spaces in which one works. One unexpected fact is that these function spaces have a close relationship with the former works for the tempered distributions, in particular the N-representation theory due to B. Simon, and V. Bargmann's theory, which uncovers relationship between the tempered distributions and his function spaces through the so-called Segal-Bargmann transformation. In addition, our function spaces have a nice relationship with the Sobolev spaces. In this chapter, a few other questions regarding these function spaces are discussed.</p> <p>In chapter four the proof of the Birkhoff normal form theorem on spaces we have introduced are provided. The analysis is divided into two cases according to the regularity of the related function space. After proving the Birkhoff normal form theorem, we made an analysis of the impact of the perturbation on the main part of the GP system, which we remark is completel:y resonant.</p> / Doctor of Philosophy (PhD)

A new N-dimensional digital modulation technique is proposed as a bandwidth efficient method for the transmission of digital data. The technique uses an antipodal scheme in which parallel binary data signs baseband orthogonal waveforms derived from Hermite polynomials. Orthogonality guarantees recoverability of the data from N simultaneously transmitted Hermite waveforms. The signed Hermite waveform is transmitted over a radio link using either amplitude or frequency modulation. The bandwidth efficiency of the amplitude Hermite method is found to be as good as filtered BPSK in practice, while the bit error rates for both modulations are identical. Hermite Keying (HK), the FM modulation version of the N-dimensional Hermite transmission, outperforms constant envelope FSK in terms of spectrum efficiency. With a simple FM detector, the bit error rate of HK is as good as that of non-coherent FSK. In a frequency selective fading channel, the simulation results suggest that specific data bits of HK are relatively secure from errors, which is beneficial in some applications. Symbol synchronization is critical to the system. An optimal synchronization method for the N-dimensional antipodal scheme in additive white Gaussian noise channel is derived. Simulation results confirm that the synchronizer can operate successfully at E/No of 3 dB. / Ph. D.

Em modelos de regressão em que uma covariável é medida com erro, é comum o uso de estruturas que relacionam a covariável observada com a verdadeira covariável não observada. Essas estruturas são usualmente aditivas ou multiplicativas. Na literatura existem diversos trabalhos interessantes que tratam de modelos de regressão com erro de medida aditivo, muitos dos quais são modelos lineares com covariáveis e erro de medida normalmente distribuídos. Para modelos em que o erro de medida é multipicativo, não se encontra na literatura o mesmo desenvolvimento teórico encontrado para modelos em que o erro de medida é aditivo. O mesmo vale para situações em que as suposições de normalidade para as covariáveis e erro de medida não se aplicam. Este trabalho propõe a construção, definição, métodos de estimação e análise de diagnóstico para modelos de regressão com erro de medida multiplicativo em uma das covariáveis. Para esses modelos, consideramos que a variável resposta possa pertencer ou à classe de modelos de regressão série de potências modificadas ou à família exponencial. O rol de distribuições pertencentes à família série de potências modificada é bem abrangente, portanto, neste trabalho, desenvolvemos a teoria de estimação e validação do modelo primeiramente de forma geral e, para exemplificar, apresentamos o modelo de regressão binomial negativa com erro de medida. para o caso em que a variável resposta pertença à família exponencial. apresentamos o modelo de regressão beta com erro de medida multiplicativo. Todos os modelos propostos foram analisados através de estados de simulação e aplicados a conjuntos de dados reais. / In regression models in which a covariate is measured with erros, it is common to use structures that correlate the observed covariate with the true non-observed covariate. Such structures are usually additive or multiplicative. In the literatue there are several interesting works that deal with regression models having an additive measuremsnt error, many of which are linear models with covariate and measurement error normally distributed. For models having a multiplicative measurement error, one does not find in the literature the same theoretical amount of works as one finds for models in which the measurement error is additive. The same happens in situations where the supositions of normality for the covariates and the measurement errors do not apply. The presente work proposes the construction,definition, estimation methods, and diagnostic analysis for the regression models with a multiplicative measurement error in one of the covariates. For these models it is considered that the response variable may belong either to the class of modified power series regression models or to the exponential family. The list of distributions belonging to the family modified power series is rather comprehensive; for this reason this work develops, firstly and in a general way, the models estimation and validation theory, and, as an example, presents the model of negative binomial regression with measurement error. In the case where response variable belongs to the exponential family, the model of beta regression with multiplicative measurement error is presented. All proposed models were analysed through simulationb studies and applied to real data sets.

Neste trabalho são apresentadas algumas extensões dos modelos potência-alfa assumindo o contexto em que as observações estão censuradas ou limitadas. Inicialmente propomos um novo modelo assimétrico que estende os modelos t-assimétrico (Azzalini e Capitanio, 2003) e t-potência (Zhao e Kim, 2016) e inclui a distribuição t de Student como caso particular. Este novo modelo é capaz de ajustar dados com alto grau de assimetria e curtose, ainda maior do que os modelos t-assimétrico e t-potência. Em seguida estendemos o modelo t-potência às situações em que os dados apresentam censura, com alto grau de assimetria e caudas pesadas. Este modelo generaliza o modelo de regressão linear t de Student para dados censurados por Arellano-Valle et al. (2012). O trabalho também introduz o modelo linear misto normal-potência para dados assimétricos. Aqui a inferência estatística é realizada desde uma perspectiva clássica usando o método de máxima verossimilhança junto com o método de integração numérica de Gauss-Hermite para aproximar as integrais envolvidas na função de verossimilhança. Mais tarde, o modelo linear com interceptos aleatórios para dados duplamente censurados é estudado. Este modelo é desenvolvido sob a suposição de que os erros e os efeitos aleatórios seguem distribuições normal-potência e normal- assimétrica. Para todos os modelos estudados foram realizados estudos de simulação a fim de estudar as suas bondades de ajuste e limitações. Finalmente, ilustram-se todos os métodos propostos com dados reais. / In this work some extensions of the alpha-power models are presented, assuming the context in which the observations are censored or limited. Initially we propose a new asymmetric model that extends the skew-t (Azzalini e Capitanio, 2003) and power-t (Zhao e Kim, 2016) models and includes the Students t-distribution as a particular case. This new model is able to adjust data with a high degree of asymmetry and cursose, even higher than the skew-t and power-t models. Then we extend the power-t model to situations in which the data present censorship, with a high degree of asymmetry and heavy tails. This model generalizes the Students t linear censored regression model t by Arellano-Valle et al. (2012) The work also introduces the power-normal linear mixed model for asymmetric data. Here statistical inference is performed from a classical perspective using the maximum likelihood method together with the Gauss-Hermite numerical integration method to approximate the integrals involved in the likelihood function. Later, the linear model with random intercepts for doubly censored data is studied. This model is developed under the assumption that errors and random effects follow power-normal and skew-normal distributions. For all the models studied, simulation studies were carried out to study their benefits and limitations. Finally, all proposed methods with real data are illustrated.