Tracing KAM tori in presymplectic dynamical systems

We present a KAM theorem for presymplectic dynamical systems. The theorem has a " a posteriori " format. We show that given a Diophantine frequency $\omega$ and a family of presymplectic mappings, if we find an embedded torus which is approximately invariant with rotation $\omega$ such that the torus and the family of mappings satisfy some explicit non-degeneracy condition, then we can find an embedded torus and a value of the parameter close to to the original ones so that the torus is invariant under the map associated to the value of the parameter. Furthermore, we show that the dimension of the parameter space is reduced if we assume that the systems are exact.Comment: 33 pages and one figur

Development and preliminary psychometric characteristics of the PODIUM questionnaire for recreational marathon runners

The purpose of this research was to develop a comprehensive and psychometrically adequate measure of recreational marathon runner’s psychological state during the few days and hours prior to the race. The questionnaire was developed in Spanish. In Study 1, Participants were 1060 recreational runners aged 18-67 years. Exploratory factor analysis revealed five dimensions reflective of motivation, self-confidence, anxiety, perceived physical fitness, and perceived social support. In two subsequent studies, the psychometric properties of a refined version of this measure were examined. In study 2, an independent sample of 801 recreational runners (aged 17-63 years) completed the questionnaire. Confirmatory factor analysis and alternative model testing supported a six-factor model. Internal consistency was .72 to .90. In support of construct validity, the self-confidence scale correlated positively with perceived physical fitness, motivation scalecorrelated positively with social support and self-confidence, and anxiety correlated negatively with motivation and self-confidence factors. In study 3, an independent sample of 22 recreational marathon runners (aged 28-47 years) responded to the PODIUM and MOMS. Additionally, another independent sample of 36 recreational runners (23-57 years) responded the to PODIUM and CSAI-2 scales. In support of concurrent validity of PODIUM, the motivation scale correlated with MOMS, and the anxiety and the self-confidence scales correlated with CSAI-2

On the continuum limit for discrete NLS with long-range lattice interactions

We consider a general class of discrete nonlinear Schroedinger equations (DNLS) on the lattice $h \mathbb{Z}$ with mesh size $h>0$. In the continuum limit when $h \to 0$, we prove that the limiting dynamics are given by a nonlinear Schroedinger equation (NLS) on $\mathbb{R}$ with the fractional Laplacian $(-\Delta)^\alpha$ as dispersive symbol. In particular, we obtain that fractional powers $1/2 < \alpha < 1$ arise from long-range lattice interactions when passing to the continuum limit, whereas NLS with the non-fractional Laplacian $-\Delta$ describes the dispersion in the continuum limit for short-range lattice interactions (e.g., nearest-neighbor interactions). Our results rigorously justify certain NLS model equations with fractional Laplacians proposed in the physics literature. Moreover, the arguments given in our paper can be also applied to discuss the continuum limit for other lattice systems with long-range interactions.Comment: 26 pages; no figures. Some minor revisions. To appear in Comm. Math. Phy

Canonical Melnikov theory for diffeomorphisms

We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection. We show how our definition reproduces the classical methods of Poincar\'{e} and Melnikov and specializes to methods previously used for exact symplectic and volume-preserving maps. We use the method to detect the transverse intersection of stable and unstable manifolds and relate this intersection to the set of zeros of the Melnikov displacement.Comment: laTeX, 31 pages, 3 figure

Boundaries of Siegel Disks: Numerical Studies of their Dynamics and Regularity

Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there exists a natural parametrization which transforms the dynamics on the boundary into a rotation. We compute numerically this parameterization and use methods of harmonic analysis to compute the global Holder regularity of the parametrization for different maps and rotation numbers. We obtain that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory (i.e., they do not depend on the map when the map ranges in an open set), and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. We also discuss some possible relations between the regularity of the parametrization of the boundaries and the corresponding scaling exponents. (C) 2008 American Institute of Physics.NSFMathematic

Regularity of critical invariant circles of the standard nontwist map

We study critical invariant circles of several noble rotation numbers at the edge of break-up for an area-preserving map of the cylinder, which violates the twist condition.These circles admit essentially unique parametrizations by rotational coordinates. We present a high accuracy computation of about 107 Fourier coefficients. This allows us to compute the regularity of the conjugating maps and to show that, to the extent of numerical precision, it only depends on the tail of the continued fraction expansion.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49075/2/non5_3_013.pd

Entropy production in Gaussian thermostats

We show that an arbitrary Anosov Gaussian thermostat on a surface is dissipative unless the external field has a global potential

Do Short-Term Effects Predict Long-Term Improvements in Women Who Receive Manual Therapy or Surgery for Carpal Tunnel Syndrome? A Bayesian Network Analysis of a Randomized Clinical Trial.

The purpose of this study was to develop a data-driven Bayesian network approach to understand the potential multivariate pathways of the effect of manual physical therapy in women with carpal tunnel syndrome (CTS). Data from a randomized clinical trial (n = 104) were analyzed comparing manual therapy including desensitization maneuvers of the central nervous system versus surgery in women with CTS. All variables included in the original trial were included in a Bayesian network to explore its multivariate relationship. The model was used to quantify the direct and indirect pathways of the effect of physical therapy and surgery on short-term, mid-term, and long-term changes in the clinical variables of pain, related function, and symptom severity. Manual physical therapy improved function in women with CTS (between-groups difference: 0.09; 95% CI = 0.07 to 0.11). The Bayesian network showed that early improvements (at 1 month) in function and symptom severity led to long-term (at 12 months) changes in related disability both directly and via complex pathways involving baseline pain intensity and depression levels. Additionally, women with moderate CTS had 0.14-point (95% CI = 0.11 to 0.17 point) poorer function at 12 months than those with mild CTS and 0.12-point (95% CI = 0.09 to 0.15 point) poorer function at 12 months than those with severe CTS. Current findings suggest that short-term benefits in function and symptom severity observed after manual therapy/surgery were associated with long-term improvements in function, but mechanisms driving these effects interact with depression levels and severity as assessed using electromyography. Nevertheless, it should be noted that between-group differences depending on severity determined using electromyography were small, and the clinical relevance is elusive. Further data-driven analyses involving a broad range of biopsychosocial variables are recommended to fully understand the pathways underpinning CTS treatment effects. Short-term effects of physical manual therapy seem to be clinically relevant for obtaining long-term effects in women with CTS

Families of Canonical Transformations by Hamilton-Jacobi-Poincar\'e equation. Application to Rotational and Orbital Motion

The Hamilton-Jacobi equation in the sense of Poincar\'e, i.e. formulated in the extended phase space and including regularization, is revisited building canonical transformations with the purpose of Hamiltonian reduction. We illustrate our approach dealing with orbital and attitude dynamics. Based on the use of Whittaker and Andoyer symplectic charts, for which all but one coordinates are cyclic in the Hamilton-Jacobi equation, we provide whole families of canonical transformations, among which one recognizes the familiar ones used in orbital and attitude dynamics. In addition, new canonical transformations are demonstrated.Comment: 21 page