Generalization of the Modified Bessel Function and Its Generating Function

2000 Mathematics Subject Classification: 33C10, 33-02, 60K25

On q-Laplace Transforms of the q-Bessel Functions

Mathematics Subject Classification: 33D15, 44A10, 44A20

Fractional Integration of the Product of Bessel Functions of the First Kind

Dedicated to 75th birthday of Prof. A.M. Mathai, Mathematical Subject Classification 2010:26A33, 33C10, 33C20, 33C50, 33C60, 26A09

Precise Near-Field Focusing exploiting Bessel Beam Launchers for Wireless Power Transfer at millimeter waves

Wireless Power Transfer has become a promising technology to overcome the limits of wired solutions. Within this framework, the objective of this thesis is to study a WPT link at millimeter waves involving a particular type of antenna working in the radiative near-field, known as Bessel Beam (BB) Launcher. This antenna has been chosen for its peculiarity of generating a Bessel Beam which is by nature non-diffractive, showing good focusing and self-healing capabilities. In particular, a Bull-Eye Leaky Wave Antenna is designed and analysed, fed by a loop antenna and resonating at approximately 30 GHz. The structure excites a Hybrid-TE mode showing zeroth-order Bessel function over the z-component of the magnetic field. The same antenna is designed with two different dimensions, showing good wireless power transport properties. The link budgets obtained for different configurations are reported. With the aim of exploiting BB Launchers in wearable applications, a further analysis on the receiving part is conducted. For WPT wearable or implantable devices a reduced dimension of the receiver system must be considered. Therefore, an electrically large loop antenna in planar technology is modified, inserting phase shifters in order to increase the intensity of the magnetic field in its interrogation zone. This is fundamental when a BB Launcher is involved as transmitter. The loop antenna, in reception, shows a further miniaturization level since it is built such that its interrogation zone corresponds to the main beam dimension of transmitting BB Launcher. The link budget is evaluated with the new receiver showing comparable results with respect to previous configurations, showing an efficient WPT link for near-field focusing. Finally, a matching network and a full-wave rectifying circuit are attached to two of the different receiving systems considered. Further analysis will be carried out about the robustness of the square loop over biological tissues.

Bandwidth considerations for multilevel converters

Multilevel converters can achieve an overall effective switch frequency multiplication and consequent ripple reduction through the cancellation of the lowest order switch frequency terms. This paper investigates the harmonic content and the frequency response of these multimodulator converters. It is shown that the transfer function of uniformly sampled modulators is a bessel function associated with the inherent sampling process. Naturally sampled modulators have a flat transfer function, but multiple switchings per switch cycle will occur unless the input is slew-rate limited. Lower sideband harmonics of the effective carrier frequency and, in uniform converters, harmonics of the input signal also limit the useful bandwidth. Observations about the effect of the number of converters, their type (naturally or uniformly sampled), and the ratio of modulating frequency and switch frequency are made.

CR1Dmod: A Matlab program to model 1D complex resistivity effects in electrical and electromagnetic surveys

We have constructed a forward modelling code in Matlab, capable of handling several commonly used electrical and electromagnetic methods in a 1D environment. We review the implemented electromagnetic field equations for grounded wires, frequency and transient soundings and present new solutions in the case of a non-magnetic first layer. The CR1Dmod code evaluates the Hankel transforms occurring in the field equations using either the Fast Hankel Transform based on digital filter theory, or a numerical integration scheme applied between the zeros of the Bessel function. A graphical user interface allows easy construction of 1D models and control of the parameters. Modelling results are in agreement with other authors, but the time of computation is less efficient than other available codes. Nevertheless, the CR1Dmod routine handles complex resistivities and offers solutions based on the full EM-equations as well as the quasi-static approximation. Thus, modelling of effects based on changes in the magnetic permeability and the permittivity is also possible.

Croissance des fonctions propres du laplacien sur un domaine circulaire

Ce mémoire a pour but d'étudier les propriétés des solutions à l'équation aux valeurs propres de l'opérateur de Laplace sur le disque lorsque les valeurs propres tendent vers l'in ni. En particulier, on s'intéresse au taux de croissance des normes ponctuelle et L1. Soit D le disque unitaire et @D sa frontière (le cercle unitaire). On s'inté- resse aux solutions de l'équation aux valeurs propres f = f avec soit des conditions frontières de Dirichlet (fj@D = 0), soit des conditions frontières de Neumann ( @f @nj@D = 0 ; notons que sur le disque, la dérivée normale est simplement la dérivée par rapport à la variable radiale : @ @n = @ @r ). Les fonctions propres correspondantes sont données par : f (r; ) = fn;m(r; ) = Jn(kn;mr)(Acos(n ) + B sin(n )) (Dirichlet) fN (r; ) = fN n;m(r; ) = Jn(k0 n;mr)(Acos(n ) + B sin(n )) (Neumann) où Jn est la fonction de Bessel de premier type d'ordre n, kn;m est son m- ième zéro et k0 n;m est le m-ième zéro de sa dérivée (ici on dénote les fonctions propres pour le problème de Dirichlet par f et celles pour le problème de Neumann par fN). Dans ce cas, on obtient que le spectre SpD( ) du laplacien sur D, c'est-à-dire l'ensemble de ses valeurs propres, est donné par : SpD( ) = f : f = fg = fk2 n;m : n = 0; 1; 2; : : :m = 1; 2; : : :g (Dirichlet) SpN D( ) = f : fN = fNg = fk0 n;m 2 : n = 0; 1; 2; : : :m = 1; 2; : : :g (Neumann) En n, on impose que nos fonctions propres soient normalisées par rapport à la norme L2 sur D, c'est-à-dire : R D F2 da = 1 (à partir de maintenant on utilise F pour noter les fonctions propres normalisées et f pour les fonctions propres quelconques). Sous ces conditions, on s'intéresse à déterminer le taux de croissance de la norme L1 des fonctions propres normalisées, notée jjF jj1, selon . Il est vi important de mentionner que la norme L1 d'une fonction sur un domaine correspond au maximum de sa valeur absolue sur le domaine. Notons que dépend de deux paramètres, m et n et que la dépendance entre et la norme L1 dépendra du rapport entre leurs taux de croissance. L'étude du comportement de la norme L1 est étroitement liée à l'étude de l'ensemble E(D) qui est l'ensemble des points d'accumulation de log(jjF jj1)= log : Notre principal résultat sera de montrer que [7=36; 1=4] E(B2) [1=18; 1=4]: Le mémoire est organisé comme suit. L'introdution et les résultats principaux sont présentés au chapitre 1. Au chapitre 2, on rappelle quelques faits biens connus concernant les fonctions propres du laplacien sur le disque et sur les fonctions de Bessel. Au chapitre 3, on prouve des résultats concernant la croissance de la norme ponctuelle des fonctions propres. On montre notamment que, si m=n ! 0, alors pour tout point donné (r; ) du disque, la valeur de F (r; ) décroit exponentiellement lorsque ! 1. Au chapitre 4, on montre plusieurs résultats sur la croissance de la norme L1. Le probl ème avec conditions frontières de Neumann est discuté au chapitre 5 et on présente quelques résultats numériques au chapitre 6. Une brève discussion et un sommaire de notre travail se trouve au chapitre 7.

Distribución hiperbólica generalizada: una aplicación en la selección de portafolios y en la cuantificación de medidas de riesgo de mercado

En este trabajo se implementa una metodología para incluir momentos de orden superior en la selección de portafolios, haciendo uso de la Distribución Hiperbólica Generalizada, para posteriormente hacer un análisis comparativo frente al modelo de Markowitz.

New fast SCFT algorithm applied to binary diblock copolymer/homopolymer blends

We present an efficient strategy for mapping out the classical phase behavior of block copolymer systems using self-consistent field theory (SCFT). With our new algorithm, the complete solution of a classical block copolymer phase can be evaluated typically in a fraction of a second on a single-processor computer, even for highly segregated melts. This is accomplished by implementing the standard unit-cell approximation (UCA) for the cylindrical and spherical phases, and solving the resulting equations using a Bessel function expansion. Here the method is used to investigate blends of AB diblock copolymer and A homopolymer, concentrating on the situation where the two molecules are of similar size.

Michelson interferometer vibrometer using self-correcting synthetic-heterodyne demodulation

Synthetic-heterodyne demodulation is a useful technique for dynamic displacement and velocity detection in interferometric sensors, as it can provide an output signal that is immune to interferometric drift. With the advent of cost-effective, high-speed real-time signal-processing systems and software, processing of the complex signals encountered in interferometry has become more feasible. In synthetic heterodyne, to obtain the actual dynamic displacement or vibration of the object under test requires knowledge of the interferometer visibility and also the argument of two Bessel functions. In this paper, a method is described for determining the former and setting the Bessel function argument to a set value, which ensures maximum sensitivity. Conventional synthetic-heterodyne demodulation requires the use of two in-phase local oscillators; however, the relative phase of these oscillators relative to the interferometric signal is unknown. It is shown that, by using two additional quadrature local oscillators, a demodulated signal can be obtained that is independent of this phase difference. The experimental interferometer is aMichelson configuration using a visible single-mode laser, whose current is sinusoidally modulated at a frequency of 20 kHz. The detected interferometer output is acquired using a 250 kHz analog-to-digital converter and processed in real time. The system is used to measure the displacement sensitivity frequency response and linearity of a piezoelectric mirror shifter over a range of 500 Hz to 10 kHz. The experimental results show good agreement with two data-obtained independent techniques: the signal coincidence and denominated n-commuted Pernick method.

Modelos de mapas simpléticos para o movimento de deriva elétrica com efeitos de raio de Larmor finito

Mapas simpléticos têm sido amplamente utilizados para modelar o transporte caótico em plasmas e fluidos. Neste trabalho, propomos três tipos de mapas simpléticos que descrevem o movimento de deriva elétrica em plasmas magnetizados. Efeitos de raio de Larmor finito são incluídos em cada um dos mapas. No limite do raio de Larmor tendendo a zero, o mapa com frequência monotônica se reduz ao mapa de Chirikov-Taylor, e, nos casos com frequência não-monotônica, os mapas se reduzem ao mapa padrão não-twist. Mostramos como o raio de Larmor finito pode levar à supressão de caos, modificar a topologia do espaço de fases e a robustez de barreiras de transporte. Um método baseado na contagem dos tempos de recorrência é proposto para analisar a influência do raio de Larmor sobre os parâmetros críticos que definem a quebra de barreiras de transporte. Também estudamos um modelo para um sistema de partículas onde a deriva elétrica é descrita pelo mapa de frequência monotônica, e o raio de Larmor é uma variável aleatória que assume valores específicos para cada partícula do sistema. A função densidade de probabilidade para o raio de Larmor é obtida a partir da distribuição de Maxwell-Boltzmann, que caracteriza plasmas na condição de equilíbrio térmico. Um importante parâmetro neste modelo é a variável aleatória gama, definida pelo valor da função de Bessel de ordem zero avaliada no raio de Larmor da partícula. Resultados analíticos e numéricos descrevendo as principais propriedades estatísticas do parâmetro gama são apresentados. Tais resultados são então aplicados no estudo de duas medidas de transporte: a taxa de escape e a taxa de aprisionamento por ilhas de período um.

Flexible Gaussian process wind field models

This technical report builds on previous reports to derive the likelihood and its derivatives for a Gaussian Process with a modified Bessel function based covariance function. The full derivation is shown. The likelihood (with gradient information) can be used in maximum likelihood procedures (i.e. gradient based optimisation) and in Hybrid Monte Carlo sampling (i.e. within a Bayesian framework).

Hankel Transform in Quantum Calculus and Applications

Mathematics Subject Classification: 44A15, 33D15, 81Q99

On Hankel Transform of Generalized Mathieu Series

Mathematics Subject Classification: Primary 33E20, 44A10; Secondary 33C10, 33C20, 44A20

Gyrokinetic modelling of plasma transport and investigation on test particle dynamics

Turbulent plasmas inside tokamaks are modeled and studied using guiding center theory, applied to charged test particles, in a Hamiltonian framework. The equations of motion for the guiding center dynamics, under the conditions of a constant and uniform magnetic field and turbulent electrostatic field are derived by averaging over the fast gyroangle, for the first and second order in the guiding center potential, using invertible changes of coordinates such as Lie transforms. The equations of motion are then made dimensionless, exploiting temporal and spatial periodicities of the model chosen for the electrostatic potential. They are implemented numerically in Python. Fast Fourier Transform and its inverse are used. Improvements to the original Python scripts are made, notably the introduction of a power-law curve fitting to account for anomalous diffusion, the possibility to integrate the equations in two steps to save computational time by removing trapped trajectories, and the implementation of multicolored stroboscopic plots to distinguish between trapped and untrapped guiding centers. The post-processing of the results is made in MATLAB. The values and ranges of the parameters chosen for the simulations are selected based on numerous simulations used as feedback tools. In particular, a recurring value for the threshold to detect trapped trajectories is evidenced. Effects of the Larmor radius, the amplitude of the guiding center potential and the intensity of its second order term are studied by analyzing their diffusive regimes, their stroboscopic plots and the shape of guiding center potentials. The main result is the identification of cases anomalous diffusion depending on the values of the parameters (mostly the Larmor radius). The transitions between diffusive regimes are identified. The presence of highways for the super-diffusive trajectories are unveiled. The influence of the charge on these transitions from diffusive to ballistic behaviors is analyzed.