We give congruences modulo powers of 2 for the Fourier coefficients of certain level 2 modular functions with poles only at 0, answering a question posed by Andersen and Jenkins. The congruences involve a modulus that depends on the binary expansion of the modular form's order of vanishing at infinity. We also demonstrate congruences for Fourier coefficients of some level 4 modular functions.

Lehmer's conjecture asserts that $\tau(p) \neq 0$, where $\tau$ is the Ramanujan $\tau$-function. This is equivalent to the assertion that $\tau(n) \neq 0$ for any $n$. A related problem is to find the distribution of primes $p$ for which $\tau(p) \equiv 0 \text{ } (\text{mod } p)$. These are open problems. However, the variant of estimating the number of integers $n$ for which $n$ and $\tau(n)$ do not have a non-trivial common factor is more amenable to study. More generally, let $f$ be a normalized eigenform for the Hecke operators of weight $k \geq 2$ and having rational integer Fourier coefficients $\{a(n)\}$. It is interesting to study the quantity $(n,a(n))$. It was proved by S. Gun and V. K. Murty (2009) that for Hecke eigenforms $f$ of weight $2$ with CM and integer coefficients $a(n)$ \begin{equation} \{ n \leq x \text { } | \text{ } (n,a(n))=1\} = \displaystyle\frac{(1+o(1)) U_f x}{\sqrt{\log x \log \log \log x}} \end{equation} for some constant $U_f$. We extend this result to higher weight forms. \\ We also show that \begin{equation} \{ n \leq x \ | (n,a(n)) \text{ \emph{is a prime}}\} \ll \displaystyle\frac{ x \log \log \log \log x}{\sqrt{\log x \log \log \log x}}. \end{equation}

Lehmer's conjecture asserts that $\tau(p) \neq 0$, where $\tau$ is the Ramanujan $\tau$-function. This is equivalent to the assertion that $\tau(n) \neq 0$ for any $n$. A related problem is to find the distribution of primes $p$ for which $\tau(p) \equiv 0 \text{ } (\text{mod } p)$. These are open problems. However, the variant of estimating the number of integers $n$ for which $n$ and $\tau(n)$ do not have a non-trivial common factor is more amenable to study. More generally, let $f$ be a normalized eigenform for the Hecke operators of weight $k \geq 2$ and having rational integer Fourier coefficients $\{a(n)\}$. It is interesting to study the quantity $(n,a(n))$. It was proved by S. Gun and V. K. Murty (2009) that for Hecke eigenforms $f$ of weight $2$ with CM and integer coefficients $a(n)$ \begin{equation} \{ n \leq x \text { } | \text{ } (n,a(n))=1\} = \displaystyle\frac{(1+o(1)) U_f x}{\sqrt{\log x \log \log \log x}} \end{equation} for some constant $U_f$. We extend this result to higher weight forms. \\ We also show that \begin{equation} \{ n \leq x \ | (n,a(n)) \text{ \emph{is a prime}}\} \ll \displaystyle\frac{ x \log \log \log \log x}{\sqrt{\log x \log \log \log x}}. \end{equation}

We give congruences modulo powers of p in {3, 5, 7} for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and Jenkins and continuing work done by the Jenkins, the author, and Moss. The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at infinity.

Faraday modulation spectroscopy (FAMOS) is a laser-based spectroscopic dispersion technique for detection of paramagnetic molecules in gas phase. This thesis presents both a new theoretical description of FAMOS and experimental results from the ultra-violet (UV) as well as the mid-infrared (MIR) regions. The theoretical description, which is given in terms of the integrated linestrength and Fourier coefficients of modulated dispersion and absorption lineshape functions, facilitates the description and the use of the technique considerably. It serves as an extension to the existing FAMOS model that thereby incorporates also the effects of lineshape asymmetries primarily originating from polarization imperfections. It is shown how the Fourier coefficients of modulated Lorentzian lineshape functions, applicable to the case with fully collisionally broadened transitions, can be expressed in terms of analytical functions. For the cases where also Doppler broadening needs to be included, resulting in lineshapes of Voigt type, the lineshape functions can be swiftly evaluated (orders of magnitude faster than previous procedures) by a newly developed method for rapid calculation of modulated Voigt lineshapes (the WWA-method). All this makes real-time curve fitting to FAMOS spectra feasible. Two experimental configurations for sensitive detection of nitric oxide (NO) by the FAMOS technique are considered and their optimum conditions are determined. The two configurations target transitions originating from the overlapping Q22(21=2) and QR12(21=2) transitions in the ultra-violet (UV) region (227nm) and the Q3=2(3=2)-transition in the fundamental rotational-vibrational band in the mid-infrared (MIR) region (5.33 µm). It is shown that the implementations of FAMOS in the UV- and MIR-region can provide detection limits in the low ppb range, which opens up the possibility for applications where high detection sensitivities of NO is required.

It is important to study gene expression and change patterns over a time period because biologically related gene groups are likely to share similar patterns. In this study, similar gene expression and change patterns are found via model-based clustering method. Fourier and wavelet coefficients of gene expression data are used as the clustering variables. A two-stage model-based method is proposed for stepwise clustering of expression and change patterns. Simulation study is performed to investigate the effectiveness of the proposed methodology. Yeast cell cycle data are analyzed.

We study some topological properties of dynamical systems. In particular the rela- tionship between spatio-temporal chaotic and Li-Yorke sensitive dynamical systems establishing that for minimal dynamical systems those properties are equivalent. In the same direction we show that being a Li-Yorke sensitive dynamical system implies that the system is also Li-Yorke chaotic. On the other hand we survey the possibility of lifting some topological properties from a given dynamical system (Y, S) to an- other (X, T). After studying some basic facts about topological dynamical systems, we move to the particular case of interval maps. We know that through the knowl- edge of interval maps, f : I → I, precious information about the chaotic behavior of general nonlinear dynamical systems can be obtained. It is also well known that the analysis of the spectrum of time series encloses important material related to the signal itself. In this work we look for possible connections between chaotic dynamical systems and the behavior of its Fourier coefficients. We have found that a natural bridge between these two concepts is given by the total variation of a function and its connection with the topological entropy associated to the n-th iteration, fn(x), of the map. Working in a natural way using the Sobolev spaces Wp,q(I) we show how the Fourier coefficients are related to the chaoticity of interval maps.

In this dissertation we present four papers as chapters. In Chapter 2, we extended the techniques used for the Klein-Gordon Chain by Iooss, Kirchgässner, James, and Sire, to chains with non-nearest neighbor interactions. We look for travelling waves by reducing the Klein-Gordon chain with second nearest neighbor interaction to an advance-delay equation. Then we reduce the equation to a finite dimensional center manifold for some parameter regimes. By using the normal form expansion on the center manifold we were able to prove the existence of three different types of travelling solutions for the Klein Gordon Chain: periodic, quasi-periodic and homoclinic to periodic orbits with exponentially small amplitude. In Chapter 3 we include numerical methods for computing quasi-periodic solutions. We developed very efficient algorithms to compute smooth quasiperiodic equilibrium states of models in 1-D statistical mechanics models allowing non-nearest neighbor interactions. If we discretize a hull function using N Fourier coefficients, the algorithms require O(N) storage and a Newton step for the equilibrium equation requires only O(N log(N)) arithmetic operations. This numerical methods give rise to a criterion for the breakdown of quasi-periodic solutions. This criterion is presented in Chapter 4. In Chapter 5, we justify rigorously the criterion in Chapter 4. The justification of the criterion uses both Numerical KAM algorithms and rigorous results. The hypotheses of the theorem concern bounds on the Sobolev norms of a hull function and can be verified rigorously by the computer. The argument works with small modifications in all cases where there is an a posteriori KAM theorem. / text

Boundary conditions for the cable equation - such as voltage-clamped or sealed cable ends, branchpoints, somatic shunts, and current clamps - result in multi-exponential series representations of the voltage or current. Each term in the series expansion is characterized by a decay rate (eigenvalue) and an initial amplitude (Fourier coefficient). The eigenvalues are determined numerically and the Fourier coefficients are subsequently given by the residues at the eigenvalues of the Laplace transform of the solution. In this paper, we introduce an alternative method for estimating the Fourier coefficients which works for all types of boundary conditions and is practical even when analytic expressions for the Fourier coefficients become intractable. It is shown that terms in the analytic expressions for the Fourier coefficients result from derivatives of the equation for the eigenvalues, and that simple numerical estimates for the amplitude coefficients are easily derived by replacing analytical derivatives by numerical eigenslope. The physical quantity represented by the slope is identified as effective neuron capacitance.

Reconhecimento de objetos é uma aplicação básica da área de processamento de imagens e visão computacional. O procedimento comum do reconhecimento consiste em achar ocorrências de uma imagem modelo numa outra imagem a ser analisada. Consequentemente, se as imagens apresentarem mudanças no ponto de vista da câmera o algoritmo normalmente falha. A invariância a pontos de vista é uma qualidade que permite reconhecer um objeto, mesmo que este apresente distorções resultantes de uma transformação em perspectiva causada pela mudança do ponto de vista. Uma abordagem baseada na simulação de pontos de vista, chamada ASIFT, tem sido recentemente proposta no entorno desta problemática. O ASIFT é invariante a pontos de vista, no entanto falha na presença de padrões repetitivos e baixo contraste. O objetivo de nosso trabalho é utilizar uma variante da técnica de simulação de pontos de vista em combinação com a técnica de extração dos coeficientes de Fourier de projeções radiais e circulares (FORAPRO), para propor um algoritmo invariante a pontos de vista, e robusto a padrões repetitivos e baixo contraste. De maneira geral, a nossa proposta resume-se nas seguintes fases: (a) Distorcemos a imagem, variando os parâmetros de inclinação e rotação da câmera, para gerar alguns modelos e conseguir a invariância a deformações em perspectiva, (b) utilizamos cada como modelo a ser procurado na imagem, para escolher o que melhor case, (c) realizamos o casamento de padrões. As duas últimas fases do processo baseiam-se em características invariantes por rotação, escala, brilho e contraste extraídas pelos coeficientes de Fourier. Nossa proposta, que chamamos AFORAPRO, foi testada com 350 imagens que continham diversidade nos requerimentos, e demonstrou ser invariante a pontos de vista e ter ótimo desempenho na presença de padrões repetitivos e baixo contraste. / Object recognition is a basic application from the domain of image processing and computer vision. The common process recognition consists of finding occurrences of an image query in another image to be analyzed A. Consequently, if the images changes viewpoint in the camera it will normally result in the algorithm failure. The invariance viewpoints are qualities that permit recognition of an object, even if this present distortion resultant of a transformation of perspective is caused by the change in viewpoint. An approach based on viewpoint simulation, called ASIFT, has recently been proposed surrounding this issue. The ASIFT algorithm is invariant viewpoints; however there are flaws in the presence of repetitive patterns and low contrast. The objective of our work is to use a variant of this technique of viewpoint simulating, in combination with the technique of extraction of the Coefficients of Fourier Projections Radials and Circulars (FORAPRO), and to propose an algorithm of invariant viewpoints and robust repetitive patterns and low contrast. In general, our proposal summarizes the following stages: (a) We distort the image, varying the parameters of inclination and rotation of the camera, to produce some models and achieve perspective invariance deformation, (b) use as the model to be search in the image, to choose the that match best, (c) realize the template matching. The two last stages of process are based on invariant features by images rotation, scale, brightness and contrast extracted by Fourier coefficients. Our approach, that we call AFORAPRO, was tested with 350 images that contained diversity in applications, and demonstrated to have invariant viewpoints, and to have excellent performance in the presence of patterns repetitive and low contrast.