On the length and area spectrum of analytic convex domains

Area-preserving twist maps have at least two different (p, q)-periodic orbits and every (p, q)-periodic orbit has its (p, q)-periodic action for suitable couples (p, q). We establish an exponentially small upper bound for the differences of (p, q)-periodic actions when the map is analytic on a (m, n)-resonant rotational invariant curve (resonant RIC) and p/q is 'sufficiently close' to m/n. The exponent in this upper bound is closely related to the analyticity strip width of a suitable angular variable. The result is obtained in two steps. First, we prove a Neishtadt-like theorem, in which the n-th power of the twist map is written as an integrable twist map plus an exponentially small remainder on the distance to the RIC. Second, we apply the MacKay-Meiss-Percival action principle. We apply our exponentially small upper bound to several billiard problems. The resonant RIC is a boundary of the phase space in almost all of them. For instance, we show that the lengths (respectively, areas) of all the (1, q)-periodic billiard (respectively, dual billiard) trajectories inside (respectively, outside) analytic strictly convex domains are exponentially close in the period q. This improves some classical results of Marvizi, Melrose, Colin de Verdiere, Tabachnikov, and others about the smooth case.Peer ReviewedPostprint (author's final draft

Exponentially small asymptotic formulas for the length spectrum in some billiard tables

Let $q \ge 3$ be a period. There are at least two $(1,q)$-periodic trajectories inside any smooth strictly convex billiard table, and all of them have the same length when the table is an ellipse or a circle. We quantify the chaotic dynamics of axisymmetric billiard tables close to their borders by studying the asymptotic behavior of the differences of the lengths of their axisymmetric $(1,q)$-periodic trajectories as $q \to +\infty$. Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor $q^{-3} e^{-r q}$ times either a constant or an oscillating function, and the exponent $r$ is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the $(1,q)$-periodic trajectories. Our experiments are restricted to some perturbed ellipses and circles, which allows us to compare the numerical results with some analytical predictions obtained by Melnikov methods and also to detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.Comment: 21 pages, 37 figure

On the length and area spectrum of analytic convex domains

Area-preserving twist maps have at least two different $(p,q)$-periodic orbits and every $(p,q)$-periodic orbit has its $(p,q)$-periodic action for suitable couples $(p,q)$. We establish an exponentially small upper bound for the differences of $(p,q)$-periodic actions when the map is analytic on a $(m,n)$-resonant rotational invariant curve (resonant RIC) and $p/q$ is "sufficiently close" to $m/n$. The exponent in this upper bound is closely related to the analyticity strip width of a suitable angular variable. The result is obtained in two steps. First, we prove a Neishtadt-like theorem, in which the $n$-th power of the twist map is written as an integrable twist map plus an exponentially small remainder on the distance to the RIC. Second, we apply the MacKay-Meiss-Percival action principle. We apply our exponentially small upper bound to several billiard problems. The resonant RIC is a boundary of the phase space in almost all of them. For instance, we show that the lengths (respectively, areas) of all the $(1,q)$-periodic billiard (respectively, dual billiard) trajectories inside (respectively, outside) analytic strictly convex domains are exponentially close in the period $q$. This improves some classical results of Marvizi, Melrose, Colin de Verdi\`ere, Tabachnikov, and others about the smooth case

Tratamiento del neuroma de antepié mediante neurólisis con radiofrecuencia

Treball Final de Grau de Podologia, Facultat de Medicina i Ciències de la Salut, Universitat de Barcelona, curs: 2016-2017, Tutor: Enrique Giralt de VecianaEl neuroma de antepié es un tumor producido por la compresión del espacio intermetatarsal, que tiene el tratamiento conservador e infiltraciones como primeras líneas de actuación. Cuando estas no tienen efecto, consideramos la cirugía. La neurólisis con radiofrecuencia para el tratamiento del neuroma de antepié es una técnica de mínimo abordaje, a través de una aguja, que con corriente de alta frecuencia se calienta e inhibe o disminuye el dolor producido por el nervio afectado. Este es un estudio de revisión bibliográfica que recoge diferentes artículos sobre esta técnica publicados hasta abril de 2017, a partir de bases de datos médicos. Esta investigación tiene como objetivo analizar el procedimiento, valorar los resultados, determinar las indicaciones y contraindicaciones, y relacionar las complicaciones con la técnica, según cada trabajo. Esta técnica tiene variaciones dependiendo del autor: puede ser de diferente tipo de radiofrecuencia (pulsátil o continua), temperaturas y tiempos de aplicación, abordaje, etc. A pesar de los diferentes criterios entre los trabajos, es una cirugía con buenos resultados y pocas recidivas, y es una alternativa a la de campo abierto, cuando el tratamiento conservador o infiltraciones no tienen efecto. A demás, presenta menos complicaciones por su bajo riesgo de tener problemas de infección o de cicatrización

Solving Correspondences for Non-Rigid Deformations

Projecte final de carrera realitzat en col.laboració amb l'IR

Towards energy-autonomous wake-up receiver using visible light communication

The use of Visible Light Communication (VLC) in wake-up communication systems is a potential energy-efficient and low-cost solution for wireless communication of consumer electronics. In this paper, we go one step further and propose the use of visible light both for wake-up communication and energy harvesting purposes, with the final objective of an energy-autonomous wake-up receiver module. We first present the details and the design criteria of this novel system. We then present the results of evaluation of design criteria such as solar panel and capacitor type choices. To evaluate the performance of the developed wake-up system with energy-autonomous receiver system, we perform realistic indoor scenario tests, analyzing the effect of varying distances, angles, and light intensities as well as the effect of presence of interfering lights.Peer ReviewedPostprint (author's final draft

Spare The Rod: Preschoolers’ And Their Mothers’ Perspectives Of Spanking From Within And Outside Conservative Protestantism

A growing body of research suggests that corporal punishment results in an increased risk for behavioral and emotional problems for children who are subjected to it. Corporal punishment is still a widely used form of discipline within the Conservative Protestant religion. This study compares preschoolers’ and their mothers’ views about the acceptability, fairness, and effectiveness of spanking – the most common form of corporal punishment – from within and outside of Conservative Protestantism. Children ages 4-to 5-years-old and their mothers (or primary caregivers) heard nine vignettes about preschool characters who committed common physical (e.g., hitting), social conventional (e.g., using your hands to eat) or prudential (e.g., lighting matches) transgressions and answered several questions about the appropriateness of using spanking as a consequence for each misbehavior. Caregivers also answered a questionnaire about their religious beliefs and parenting practices. Results revealed no difference between conservative Protestant children and their mother’s views but a significant difference between conservative Protestant and non-conservative Protestant mothers. Furthermore, acceptability of spanking varied by transgression, mothers viewed spanking as the most okay for prudential transgressions and children for social transgressions. The current study is a first step in exploring mother-child beliefs about corporal punishment within and outside of Conservative Protestantism

Singular phenomena in the length spectrum of analytic convex curves

Consider the billiard map defined inside an analytic closed strictly convex curve Q. Given q>2 and 0<p<q relatively prime integers, there exist at least two (p,q)-periodic trajectories inside Q. The main goal of this thesis is to study the maximal difference of lengths among (p,q)-periodic trajectories on the billiard, D(p,q). The quantity D(p,q) gives some dynamical and geometrical information. First, it characterizes part of the length spectrum of Q and so it relates to Kac's question, "Can one hear the shape of a drum?''. Second, D(p,q) is an upper bound of Mather's DW(p/q) and so it quantifies the chaotic dynamics of the billiard table. We first focus on the study of the maximal difference of lengths among (1,q)-periodic orbits. These orbits approach the boundary of the billiard table as q tends to infinity. The study of D(1,q) is twofold. On the one hand, we obtain an exponentially small upper bound in the period q for D(1,q). The result is obtained on the general framework of the maximal difference of (p,q)-periodic actions among (p,q)-periodic orbits on analytic exact twist maps. Precisely, we establish an exponentially small upper bound for differences of (p,q)-periodic actions when the map is analytic on a (m,n)-resonant rotational invariant curve and p/q is ``sufficiently close'' to m/n. The exponent in the upper bound is closely related to the analyticity strip width of a suitable angular variable. The result is obtained in two steps. First, we prove a Neishtadt-like theorem. Second, we apply the MacKay-Meiss-Percival action principle. This result implies that the lengths of all the (1,q)-periodic billiard trajectories inside analytic strictly convex domains are exponentially close in the period q, which improves the classical result of Marvizi and Melrose about the smooth case. But it also has several other applications in both classical and dual billiards. For instance, we show that the areas of the (1,q)-periodic dual billiard trajectories outside Q are exponentially close in the period q. This result improves Tabachnikov's classical result about the smooth case. On the other hand, we discuss some exponentially small asymptotic formulas for D(1,q) when the billiard table is a generic axisymmetric analytic strictly convex curve. In this context, we conjecture that the differences behave asymptotically like an exponentially small factor q^(-3)*exp(-rq) times either a constant or an oscillatory function. Also, the exponent r is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the (1,q)-periodic trajectories. This conjecture is strongly supported by numerical experiments. Our computations require a multiple-precision arithmetic and we have used PARI/GP. The experiments are restricted to some perturbed ellipses and circles, which allow us to compare the numerical results with some analytical Melnikov predictions and also to detect some non-generic behaviors due to the presence of extra symmetries. The asymptotic formulas we obtain resemble the ones obtained for the splitting of separatrices on many analytic maps, where the behavior of the splitting size is of order h(-m)*exp(-r/h). In such cases, the parameter h>0 is small and continuous so the formulas are exponentially small in 1/h instead. The exponent r has been proved to be (or is strongly numerically supported, depending on the map studied) 2pi times the distance to the real axis of the set of complex singularities of the homoclinic solution of a limit Hamiltonian flow. We propose and study an equivalent limit problem in the billiard setting. Next, we give some insight on how D(p,q) behaves when (p,q)-periodic orbits do not tend to the boundary of Q but to other regions of the phase space. Namely, we consider the cases of p/q tends to an irrational number or to P/Q. The study of D(p,q) in these cases consists of a phenomenological study based on some numerical results.Considereu l'aplicació billard definida dins d'una corba tancada, analítica i estrictament convexa Q. Per q>2 i 00 és petit i contínu i les fórmules són exponencialment petites en 1/h. S'ha demostrat (o està recolzat fortament per experiments numèriques) que l'exponent r és 2Pi vegades la distància a l'eix real del conjunt de singularitats complexes de la solució homoclínica del flux d'un Hamiltonià límit. Proposem i estudiem un equivalent a problema límit per l'aplicació billar. A continuació, comentem com es comporta D(p,q) per òrbites (p,q)-periòdiques que tendeixen a regions de l'espai de fases diferents de la frontera de Q. En concret, considerem els casos de p/q tendint a un nombre irracional o a P/Q. L'estudi de D(p,q) en aquests casos es basa en un estudi numèric dels fenòmens

Solving Correspondences for Non-Rigid Deformations

Projecte final de carrera realitzat en col.laboració amb l'IR