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

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 Verdiere, Tabachnikov, and others about the smooth case.Peer ReviewedPostprint (author's final draft

The billiard inside an ellipse deformed by the curvature flow

The billiard dynamics inside an ellipse is integrable. It has zero topological entropy, four separatrices in the phase space, and a continuous family of convex caustics: the confocal ellipses. We prove that the curvature flow destroys the integrability, increases the topological entropy, splits the separatrices in a transverse way, and breaks all resonant convex caustics.Comment: 13 pages, 1 figur

Arterial mechanical motion estimation based on a semi-rigid body deformation approach

Arterial motion estimation in ultrasound (US) sequences is a hard task due to noise and discontinuities in the signal derived from US artifacts. Characterizing the mechanical properties of the artery is a promising novel imaging technique to diagnose various cardiovascular pathologies and a new way of obtaining relevant clinical information, such as determining the absence of dicrotic peak, estimating the Augmentation Index (AIx), the arterial pressure or the arterial stiffness. One of the advantages of using US imaging is the non-invasive nature of the technique unlike Intra Vascular Ultra Sound (IVUS) or angiography invasive techniques, plus the relative low cost of the US units. In this paper, we propose a semi rigid deformable method based on Soft Bodies dynamics realized by a hybrid motion approach based on cross-correlation and optical flow methods to quantify the elasticity of the artery. We evaluate and compare different techniques (for instance optical flow methods) on which our approach is based. The goal of this comparative study is to identify the best model to be used and the impact of the accuracy of these different stages in the proposed method. To this end, an exhaustive assessment has been conducted in order to decide which model is the most appropriate for registering the variation of the arterial diameter over time. Our experiments involved a total of 1620 evaluations within nine simulated sequences of 84 frames each and the estimation of four error metrics. We conclude that our proposed approach obtains approximately 2.5 times higher accuracy than conventional state-of-the-art techniques.The authors thank Ana Palomares for revising their English text. This work has been supported by the National Grant (AP2007-00275), the projects ARC-VISION (TEC2010-15396), ITREBA (TIC-5060), and the EU project TOMSY (FP7-270436)

Origin of the negative differential resistance in the output characteristics of a picene-based thin-film transistor

© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.In this work, we have fabricated and studied p-type picene thin-film transistors. Although the devices exhibited good electrical performance with high field-effect mobility (up to 1.3 cm2/V¿s) and on/off ratios above 105, the output electric characteristics of the devices exhibited a Negative Differential Resistance for higher drain-source voltage. Finally, a possible explanation for this phenomenon is developed.Peer ReviewedPostprint (author's final draft

Nonpersistence of resonant caustics in perturbed elliptic billiards

Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics ---the ones whose tangent trajectories are closed polygons--- are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.Comment: 14 pages, 3 figure

Singular separatrix splitting and the Poincare-Melnikov method for area preserving maps

The splitting of separatrices of area preserving maps close to the identity is one of the most paradigmatic examples of an exponentially small or singular phenomenon. The intrinsic small parameter is the characteristic exponent h > 0 of the saddle fixed point. A standard technique to measure the splitting of separatrices is the so-called Poincaré-Melnikov method, which has several specic features in the case of analytic planar maps. The aim of this talk is to compare the predictions for the splitting of separatrices provided by the Poincaré-Melnikov method, with the analytic and numerical results in a simple example where computations in multiple-precision arithmetic are performed

On Birkhoff's conjecture about convex billiards

Birkhoff conjectured that the elliptic billiard was the only integrable convex billiard. Here we prove a local version of this conjecture: any non-trivial symmetric entire perturbation of an elliptic billiard is non-integrable