Continued fractions

[ES]Las fracciones continuas han tenido un papel fundamental en el desarrollo de numerosas teorías matemáticas y actualmente siguen siendo un tema de investigación muy activo. El estudio de las fracciones continuas con coeficientes en los complejos se asienta en el análisis de las regiones donde la fracción continua converge. Esta teoría de convergencia nos permite definir funciones meromorfas como fracciones continuas a partir de su serie de potencias mediante una sucesión de aproximaciones racionales conocidas como aproximaciones de Padé. Además, se puede establecer una equivalencia entre los números reales y las fracciones continuas simples (un tipo especial de fracciones continuas con coeficientes enteros) y, mediante esta equivalencia, también se pueden estudiar numerosos problemas de teoría de números como la aproximación de números irracionales por aproximaciones racionales o la resolución de la ecuación de Pell, una ecuación diofántica[EN]Continued fractions have played a central role in the development of manu mathematical theories and, even today, they are still a very active line of research. The study of continued fractions with complex coefficients relies o the analysis fo the regions where the continued fraction converges. This convergence theory allows us to define meromorphic functions as continued fractions fron their formal power series with the help of a sequence of rational approximations known as Padé approximants. Furthermore, there is an equivalence between real numbers and simple cotinued fractions (a special case of continued fractions with integer coefficients) and, based on this equivalence, one can study problems of number theory such as how well irrational numbers can be approximated by rational numbers or how to solve Pell's equation, a kind of Diophantine equatio

Forms of the symmetry energy relevant to neutron stars

The symmetry energy is an invaluable tool for studying dense nuclearmatter. Unfortunately, its definition is somewhat implicit, and therefore, phenomenologicalmethods are necessary to describe experimental facts. This paper discusses the differences arising from the use of Taylor series expansion and Padé approximation to determine theoretically the symmetry energy and the possible consequences for neutron stars. For this purpose, a form of the nuclear matter equation of state that explicitly depends on the symmetry energy is used. The obtained results point out that the applied approximations lead to modifications of the equilibrium proton fractions and equation of state, especially in their high-density limit. However, this effect is small near the saturation density n0

Coarse-graining Kohn-Sham Density Functional Theory

We present a real-space formulation for coarse-graining Kohn-Sham Density Functional Theory that significantly speeds up the analysis of material defects without appreciable loss of accuracy. The approximation scheme consists of two steps. First, we develop a linear-scaling method that enables the direct evaluation of the electron density without the need to evaluate individual orbitals. We achieve this by performing Gauss quadrature over the spectrum of the linearized Hamiltonian operator appearing in each iteration of the self-consistent field method. Building on the linear-scaling method, we introduce a spatial approximation scheme resulting in a coarse-grained Density Functional Theory. The spatial approximation is adapted so as to furnish fine resolution where necessary and to coarsen elsewhere. This coarse-graining step enables the analysis of defects at a fraction of the original computational cost, without any significant loss of accuracy. Furthermore, we show that the coarse-grained solutions are convergent with respect to the spatial approximation. We illustrate the scope, versatility, efficiency and accuracy of the scheme by means of selected examples

Towards efficient three-dimensional wide-angle beam propagation methods and theoretical study of nanostructures for enhanced performance of photonic devices

In this dissertation, we have proposed a novel class of approximants, the so-called modified Padé approximant operators for the wide-angle beam propagation method (WA-BPM). Such new operators not only allow a more accurate approximation to the true Helmholtz equation than the conventional operators, but also give evanescent modes the desired damping. We have also demonstrated the usefulness of these new operators for the solution of time-domain beam propagation problems. We have shown this both for a wideband method, which can take reflections into account, and for a split-step method for the modeling of ultrashort unidirectional pulses. The resulting approaches achieve high-order accuracy not only in space but also in time. In addition, we have proposed an adaptation of the recently introduced complex Jacobi iterative (CJI) method for the solution of wide-angle beam propagation problems. The resulting CJI-WA-BPM is very competitive for demanding problems. For large 3D waveguide problems with refractive index profiles varying in the propagation direction, the CJI method can speed-up beam propagation up to 4 times compared to other state-of-the-art methods. For practical problems, the CJI-WA-BPM is found to be very useful to simulate a big component such as an arrayed waveguide grating (AWG) in the silicon-on-insulator platform, which our group is looking at. Apart from WA beam propagation problems for uniform waveguide structures, we have developed novel Padé approximate solutions for wave propagation in graded-index metamaterials. The resulting method offers a very promising tool for such demanding problems. On the other hand, we have carried out the study of improved performance of optical devices such as label-free optical biosensors, light-emitting diodes and solar cells by means of numerical and analytical methods. We have proposed a solution for enhanced sensitivity of a silicon-on-insulator surface plasmon interference biosensor which had been previously proposed in our group. The resulting sensitivity has been enhanced up to 5 times. Furthermore, we have developed an improved model to investigate the influence of isolated metallic nanoparticles on light emission properties of light-emitting diodes. The resulting model compares very well to experimental results. Finally, we have proposed the usefulness of core-shell nanostructures as nanoantennas to enhance light absorption of thin-film amorphous silicon solar cells. An increased absorption up to 33 % has theoretically been demonstrated

Métodos numérico-simbólicos para calcular soluciones liouvillianas de ecuaciones diferenciales lineales

El objetivo de esta tesis es dar un algoritmo para decidir si un sistema explicitable de ecuaciones diferenciales kJiferenciales de orden superior sobre las funciones racionales complejas, dado simbólicamente,admite !Soluciones liouvillianas no nulas, calculando una (de laforma dada por un teorema de Singer) en caso !afirmativo. mediante métodos numérico-simbólicos del tipo Introducido por van der Hoeven.donde el uso de álculo numérico no compromete la corrección simbólica. Para ello se Introduce untipo de grupos algebraicos lineales, los grupos euriméricos, y se calcula el cierre eurimérico del grupo de Galois diferencial,mediante una modificación del algoritmo de Derksen y van der Hoeven, dado por los generadores de Ramis.Departamento de Algebra, Análisis Matemático, Geometría y Topologí

A simple algorithm for expanding a power series as a continued fraction

I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980)

Novel Statistical Methods in Conjoint Analysis and Padé Approximation of the Profile Likelihood

This thesis addresses two topics in statistics that rely on the likelihood function for statistical inference. These topics include novel statistical research in conjoint analysis and a method to approximate the profile likelihood and create approximate profile likelihood confidence intervals. After introducing choice-based conjoint analysis, we introduce a method to empirically assess and compare different conjoint analysis design strategies. A key feature of this method of comparison is that it makes very few assumptions about the data-generating process (essentially just that the respondents answer the surveys independently of one another) while remaining statistically valid; in particular, the respondents are not assumed to use the multinomial logit model. We then turn to the statistical analysis of conjoint analysis survey results. We introduce a method to plot in two-dimensional space the heterogeneity in preferences across the respondents as inferred from the conjoint analysis survey results that can be used for visualization as well as mixture model assessment. We also introduce a novel method to accurately infer the number of natural clusters in preferences across the respondents. This method is shown in simulation studies to give more accurate results than latent class segmentation with AIC and BIC. We additionally suggest regressing estimated respondent preferences on their demographic variables as a strategy for hypothesis generation and model building. We show that the profile likelihood can be well approximated near the maximum likelihood estimate by the [2,2] Padé approximant. This approximation is shown to be better than that provided by a second-order Taylor series approximation. However, like a second-order Taylor series approximation, it can be used to construct a confidence interval with endpoints found using the quadratic formula. The resulting confidence interval is similar to a profile likelihood interval but with computation time similar to that of a Wald interval

Multigrid Preconditioners for the Discontinuous Galerkin Spectral Element Method : Construction and Analysis

Discontinuous Galerkin (DG) methods offer a great potential for simulations of turbulent and wall bounded flows with complex geometries since these high-order schemes offer a great potential in handling eddies. Recently, space-time DG methods have become more popular. These discretizations result in implicit schemes of high order in both spatial and temporal directions. In particular, we consider a specific DG variant, the DG Spectral Element Method (DG-SEM), which is suitable to construct entropy stable solvers for conservation laws. Since the size of the corresponding nonlinear systems is dependent on the order of the method in all dimensions, the problem arises to efficiently solve these huge nonlinear systems with regards to CPU time as well as memory consumption.Currently, there is a lack of good solvers for three-dimensional DG applications, which is one of the major obstacles why these high order methods are not used in e.g. industry. We suggest to use Jacobian-free Newton- Krylov (JFNK) solvers, which are advantageous in memory minimization. In order to improve the convergence speed of these solvers, an efficient preconditioner needs to be constructed for the Krylov sub-solver. However, if the preconditioner requires the storage of the DG system Jacobian, the favorable memory consumption of the JFNK approach is obsolete.We therefore present a multigrid based preconditioner for the Krylov sub-method which retains the low mem- ory consumption, i.e. a Jacobian-free preconditioner. To achieve this, we make use of an auxiliary first order finite volume replacement operator. With this idea, the original DG mesh is refined but can still be implemented algebraically. As smoother, we consider the pseudo time iteration W3 with a symmetric Gauss-Seidel type approx- imation of the Jacobian. Numerical results are presented demonstrating the potential of the new approach.In order to analyze multigrid preconditioners, a common tool is the Local Fourier Analysis (LFA). For a space- time model problem we present this analysis and its benefits for calculating smoothing and two-grid convergence factors, which give more insight into the efficiency of the multigrid method

Row Reduction Applied to Decoding of Rank Metric and Subspace Codes

We show that decoding of $\ell$-Interleaved Gabidulin codes, as well as list-$\ell$ decoding of Mahdavifar--Vardy codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of \F[x] matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form. We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding $\ell$-Interleaved Gabidulin codes. We obtain an algorithm with complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem and is proportional to the code length $n$ in the case of decoding. Further, we show how to perform the interpolation step of list-$\ell$-decoding Mahdavifar--Vardy codes in complexity $O(\ell n^2)$, where $n$ is the number of interpolation constraints.Comment: Accepted for Designs, Codes and Cryptograph

Renormalization Group Summation at High Orders and Implications to the Determination of Some Standard Model Parameters

We use renormalization group summed perturbation theory (RGSPT) to improve perturbation series in quantum chromodynamics in the determination of some of the standard model parameters.Comment: PhD Thesi