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. / The goal of this master's thesis is to explore the properties of the solutions of the eigenvalue problem for the Laplace operator on a disk as the eigenvalues go to in nity. More speci cally, we study the growth rate of the pointwise and the L1 norms of the eigenfunctions. Let D be the unit disk and @D be its boundary (the unit circle). We study the solutions of the eigenvalue problem f = f with either Dirichlet boundary condition (fj@D = 0) or Neumann boundary condition ( @f @nj@D = 0; note that for the disk the normal derivative is simply the derivative with respect to the radial variable: @ @n = @ @r ). The corresponding eigenfunctions are given by: 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) where Jn is the nth order Bessel function of the rst type, kn;m is its mth zero and k0 n;m is the mth zero of its derivative (here we denote the eigenfunctions for the Dirichlet problem by f and those for the Neumann problem by fN). The spectrum of the Laplacian on D, SpD( ), that is the set of its eigenvalues, is given by: 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) Finally, we normalize the L2 norm of the eigenfunctions on D, namely: R D F2 da = 1 (here and further on we use the notation F for the normalized eigenfunctions and f for arbitrary eigenfunctions). Under these conditions, we study the growth rate of the L1 norm of the normalized eigenfunctions, jjF jj1, in relation to . It is important to mention that the L1 norm of a function on a given domain corresponds to the iv maximum of its absolute value on the domain. Note that depends on two parameters, m and n, and the relation between and the L1 norm depends on the regime at which m and n change as goes to in nity. Studying the behavior of the L1 norm is linked to the study of the set E(D) which is the set of accumulation points of log(jjF jj1)= log : One of our main results is that [7=36; 1=4] E(B2) [1=18; 1=4]: The thesis is organized as follows. Introduction and main results are presented in chapter 1. In chapter 2 we review some well-known facts regarding the eigenfunctions of the Laplacian on the disk and the properties of the Bessel functions. In chapter 3 we prove results on pointwise growth of eigenfunctions. In particular, we show that, if m=n ! 0, then, for any xed point (r; ) on D, the value of F (r; ) decreases exponentially as ! 1. In chapter 4 we study the growth of the L1 norm. Eigenfunctions of the Neumann problem are discussed in chapter 5. Some numerical results are presented in chapter 6. A discussion and a summary of our work could be found in chapter 7.

Στη παρούσα εργασία ασχοληθήκαμε με ιδιότητες μονοτονίας των Τροποποιημένων συναρτήσεων Bessel 1ου και 2ου είδους. Συγκεκριμένα ομαδοποιήσαμε ήδη υπάρχοντα φράγματα για τα κλάσματα των συναρτήσεων αυτών. Η εύρεση φραγμάτων για τα κλάσματα των Τροποποιημένων Συναρτήσεων Bessel είναι σημαντική, λόγω της χρησιμότητάς τους σε διάφορους κλάδους των Μαθηματικών και όχι μόνο, όπως ενδεικτικά, στην Πεπερασμένη Ελαστικότητα, στην Στατιστική και στις Πιθανότητες, στην Ειδική Θεωρία Σχετικότητας, στην Μηχανική των Ρευστών, στην Ηλεκτρομηχανική, στη Βιοφυσική, στη Μαθηματική Φυσική και αλλού. Αρχικά, στο Κεφάλαιο 1, παρατέθηκαν κάποια βασικά στοιχεία, όπως ορισμοί των συναρτήσεων Bessel 1ου και 2ου είδους (Τροποποιημένων και μη) και αναδρομικές σχέσεις που ικανοποιούν. Στο Κεφάλαιο 2, γίνεται η καταγραφή και σύγκριση άνω και κάτω φραγμάτων για τα διάφορα κλάσματα των Τροποποιημένων συναρτήσεων Bessel 1ου είδους, καθώς και αναφορά σε ανισότητες τύπου Turán για τις συναρτήσεις αυτές. Επίσης, αναφέρεται η μεθοδολογία στην οποία στηρίχθηκε ο κάθε ερευνητής για να πάρει τα αντίστοιχα αποτελέσματα. Στο Κεφάλαιο 3, γίνεται η αντίστοιχη διαδικασία για τα κλάσματα και εκ νέου αναφορά σε ανισότητες τύπου Turán για αυτές τις συναρτήσεις. / In this project we described properties of Modified Bessel functions of the 1st and 2nd kind. Specifically we have grouped existing bounds for the quotients of these functions. These bounds of the Modified Bessel functions is very importand and could be found in different branches of Mathematics and other sciences, such as in Finite Elasticity, in Statistics and Probability Theory, in Relativity Theory, in Fluid Mechanics, in Engineering, in Biophysics, in Mathematical Physics and so on. Firsty, in Chapter 1, we cited some basic data, such as definitions of definitions of Bessel fynctions of the 1st and 2nd kind (both simple and Modified) and recurrence relations that they satisfy. In Chapter 2, we describe upper and lower bounds of different quotients of Modified Bessel functions of the 1st kind and reference to Turán type Inequalities of those functions. Moreover, we refer to the method that each recearcher based on in order to prove the required results. In Chapter 3, we have the same process but for Modified Bessel functons of the 2nd kind as well as reference to Turán type Inequalities for the corresponding functions.

Orientador: Edmundo Capelas de Oliveira / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-11T05:13:17Z (GMT). No. of bitstreams: 1 Coelho_JoaoBosco_M.pdf: 1003588 bytes, checksum: c8b5b0bbc0f7fe49adbeacc39f398bcf (MD5) Previous issue date: 2008 / Resumo: Efetua-se um estudo sistemático das equações diferenciais parciais, lineares, de segunda ordem e do tipo hiperbólico, isto é, aquelas equações que estão associadas com o problema envolvendo a propagação de ondas. Como uma aplicação, discute-se o problema de ondas de corrente e ondas de tensão, através da chamada equação do telégrafo, também conhecida como equação dos telegrafistas. Casos particulares são discutidos tanto do ponto de vista matemático quanto do ponto de vista físico. Apresenta-se o método de Riemann como ferramenta para discutir a solução geral / Abstract: We perform a systematic way to study the linear, second order partial differential equation of the hyperbolic type, that is, those equations which are associated with the problem involving wave propagation. As an application, we discuss the problem associated with the current waves and tension waves by means of the so-called telegraph equation, also known as telephone equation. Particular cases are discussed in both sense, Mathematic and Physical point of view. We also present the Riemann¿s method as a powerful tool to discuss the general solution / Mestrado / Mestre em Matemática

Orientador: Edmundo Capelas de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T23:00:41Z (GMT). No. of bitstreams: 1 Rodrigues_FabioGrangeiro_D.pdf: 1185818 bytes, checksum: 96f82c6ff4622e4ecdd3ccae79803dae (MD5) Previous issue date: 2015 / Resumo: Neste trabalho é apresentado um modo de se obter soluções de um caso particular da equação hipergeométrica confluente, a equação de Bessel de ordem p, utilizando-se da teoria do cálculo de ordem arbitrária, também conhecido popularmente por cálculo fracionário. Em particular, discutimos alguns equívocos identificados na literatura e levantamos questionamentos sobre algumas interpretações a respeito dos operadores formulados segundo Riemann-Liouville quando aplicados a certos tipos de funções. Para tanto, apresentamos inicialmente os operadores de integração e diferenciação fracionárias segundo as formulações mais clássicas (Riemann-Liouville, Caputo e Grünwald-Letnikov) e, em seguida, apresentamos o operador de integrodiferenciação fracionária que é a tentativa de unificar as operações de integração e diferenciação sob um único operador. Ao longo do texto indicamos as principais propriedades destes operadores e citamos algumas das suas aplicações comumente encontrados na Matemática, Física e Engenharias / Abstract: In this thesis we discuss the solvability of the Bessel's differential equation of order p, which is a particular case of the confluent hypergeometric equation, from the perspective of the theory of calculus of arbitrary order, also commonly known as fractional calculus. In particular, we expose some misconceptions encountered in the literature and we raise some questions about interpretations of the Riemann-Liouville operators when acting on certain types of functions. In order to do so, we present the main fractional operators (Riemann-Liouville, Caputo and Grünwald-Letnikov) as well as the fractional integrodifferential operator, which is an unified view of both integration and differentiation under a single operator. We also show the main properties of these operators and mention some of its applications in Mathematics, Physics and Engeneering / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

International Telemetering Conference Proceedings / October 20-23, 2003 / Riviera Hotel and Convention Center, Las Vegas, Nevada / The drive towards higher accuracy and sampling rates has raised the bar for modern FTI signal conditioning. This paper focuses on the issue of anti-alias filtering. Today's 16-bit (and greater resolution) ADC’s, coupled with the drive for optimum sampling rates, means that filters have to be more accurate and yet more flexible than ever before. However, in order to take full advantage of these advances, it is important to understand the trade-offs involved and to correctly specify the system filtering requirements. Trade-offs focus on: • Analog vs. Digital signal conditioning • FIR vs. IIR Digital Filters • Signal bandwidth vs. Sampling rate • Coherency issues such as filter phase distortion vs. delay This paper will discuss each of these aspects. In particular, it will focus on some of the advantages of digital filtering various analog filter techniques. This paper will also look at some ideas for specifying filter cut-off and characteristics.

We present Bessel function and Gaussian beam models for a study of microwave power deposition in bounded and inhomogeneous lossy media. The aim is to develop methods that can accurately simulate practical results commonly found in electromagnetic hyperthermic treatment, which is a noninvasive method. The Bessel function method has a closed form solution and can be used to compute accurate results of electromagnetic fields emanating from applicators with cosinusoidal aperture fields. On the other hand, the Gaussian beam method is approximate but has the capability to simplify boundary value problems and to compute fields in three-dimensions with extremely low CPU time (less than 30 sec). Although the Gaussian beam method is derived from geometrical optics theory, it performs very well in domains outside the realm of geometrical optics which stipulates that aperture dimension/λ ≥ 5 in the design of microwave systems. This condition has no relevance to the Gaussian beam method since the method shows that a limit of aperture dimension/ λ ≥ 0.9 is possible, which is a very important achievement in the design and application of microwave systems. Experimental verifications of the two theoretical models are integral parts of the presentation and show the viability of the methods.

acase@tulane.edu / Modular-type transformation formulas are the identities that are invariant under the transformation α → 1/α, and they can be represented as F (α) = F (β) where α β = 1. We derive a new transformation formula of the form F (α, z, w) = F (β, z, iw) that is a one-variable generalization of the well-known Ramanujan-Guinand identity of the form F (α, z) = F (β, z) and a two-variable generalization of Koshliakov’s formula of the form F (α) = F (β) where α β = 1. The formula is generated by first finding an integral J that is comprised of an invariance function Z and evaluating the integral to give F (α, z, w) mentioned above. The modified Bessel function K z (x) appearing in Ramanujan-Guinand identity is generalized to a new function, denoted as K z,w (x), that yields a pair of functions reciprocal in the Koshliakov kernel, which in turn yields the invariance function Z and hence the integral J and the new formula. The special function K z,w (x), first defined as the inverse Mellin transform of a product of two gamma functions and two confluent hypergeometric functions, is shown to exhibit a rich theory as evidenced by a number of integral and series representations as well as a differential-difference equation. The second topic of the thesis is 2-adic valuations of integer sequences associated with quadratic polynomials of the form x 2 +a. The sequence {n 2 +a : n ∈ Z} contains numbers divisible by any power of 2 if and only if a is of the form 4 m (8l+7). Applying this result to the sequences derived from the sums of four or fewer squares when one or more of the squares are kept constant leads to interesting results, that also points to an inherent connection with the functions r k (n) that count the number of ways to represent n as sums of k integer squares. Another class of sequences studied is the shifted sequences of the polygonal numbers given by the quadratic formula, for which the most common examples are the triangular numbers and the squares. / 1 / Aashita Kesarwani

This thesis presents the first-known exact solutions for vibration of stepped circular Mindlin plates. The considered circular plate is of several step-wise variation in thickness in the radial direction. The Mindlin first order shear deformable plate theory is employed to derive the governing differential equations for the annular and circular segments. The exact solutions to these differential equations may be expressed in terms of the Bessel functions of the first and second kinds and the modified Bessel functions of the first and second kinds. The governing homogenous system of equations is assembled by implementing the essential and natural boundary conditions and the segment interface conditions. Vibration solutions are presented for circular Mindlin plates of different edge support conditions and various combinations of step-wise thickness variations. These exact vibration results may serve as important benchmark values for researchers to validate their numerical methods for such circular plate problems / Master of Engineering (Civil)

We present the results of a numerical simulation tool for modeling the second-harmonic generation (SHG) interaction experienced by a diffracting beam. This code is used to study the simultaneous frequency and spatial profile conversion of a truncated Bessel beam that closely resembles a higher-order mode (HOM) of an optical fiber. SHG with Bessel beams has been investigated in the past and was determined have limited value because it is less efficient than SHG with a Gaussian beam in the undepleted pump regime. This thesis considers, for the first time to the best of our knowledge, whether most of the power from a Bessel-like beam could be converted into a second-harmonic beam (full depletion), as is the case with a Gaussian beam. We study this problem because using HOMs for fiber lasers and amplifiers allows reduced optical intensities, which mitigates nonlinearities, and is one possible way to increase the available output powers of fiber laser systems. The chief disadvantage of using HOM fiber amplifiers is the spatial profile of the output, but this can be transformed as part of the SHG interaction, most notably to a quasi-Gaussian profile when the phase mismatch meets the noncollinear criteria. We predict, based on numerical simulation, that noncollinear SHG (NC-SHG) can simultaneously perform highly efficient (90%) wavelength conversion from 1064 nm to 532 nm, as well as concurrent mode transformation from a truncated Bessel beam to a Gaussian-like beam (94% overlap with a Gaussian) at modest input powers (250 W, peak power or continuous-wave operation). These simulated results reveal two attractive features – the feasibility of efficiently converting HOMs of fibers into Gaussian-like beams, and the ability to simultaneously perform frequency conversion. Combining the high powers that are possible with HOM fiber amplifiers with access to non-traditional wavelengths may offer significant advantages over the state of the art for many important applications, including underwater communications, laser guide stars, and theater projectors.

Magíster en Ciencias de la Ingeniería, Mención Matemáticas Aplicadas. Ingeniero Civil Matemático / En la presente tesis se estudian las distribuciones cuasi-estacionarias para el proceso de Bessel en el intervalo (0,1]. Este proceso corresponde a una difusión uni-dimensional con coeficiente de drift singular en 0, la cual se extingue al llegar a 1. Debido a la naturaleza del problema, se hace un estudio sobre difusiones uni-dimensionales, tocando temas tales como condiciones de explosión, existencia y unicidad. Posteriormente se trata el problema en cuestión. La principal herramienta consiste en una representación espectral adecuada para el núcleo de transición del proceso de Bessel, obtenido a partir del Movimiento Browniano en la bola unitaria que se extingue al llegar a la frontera. Se demuestra que existe una única distribución cuasi-estacionaria para el proceso, que además resulta ser su límite de Yaglom. Se tocan algunos tópicos adicionales sobre el proceso de Bessel tales como su tipo de frontera y operadores diferenciales asociados. Esto dará orientación a una posible generalización de estos resultados a difusiones más generales.