Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse

We study billiard dynamics inside an ellipse for which the axes lengths are changed periodically in time and an $O(\delta)$-small quartic polynomial deformation is added to the boundary. In this situation the energy of the particle in the billiard is no longer conserved. We show a Fermi acceleration in such system: there exists a billiard trajectory on which the energy tends to infinity. The construction is based on the analysis of dynamics in the phase space near a homoclinic intersection of the stable and unstable manifolds of the normally hyperbolic invariant cylinder $\Lambda$, parameterised by the energy and time, that corresponds to the motion along the major axis of the ellipse. The proof depends on the reduction of the billiard map near the homoclinic channel to an iterated function system comprised by the shifts along two Hamiltonian flows defined on $\Lambda$. The two flows approximate the so-called inner and scattering maps, which are basic tools that arise in the studies of the Arnold diffusion; the scattering maps defined by the projection along the strong stable and strong unstable foliations $W^{ss,uu}$ of the stable and unstable invariant manifolds $W^{s,u}(\Lambda)$ at the homoclinic points. Melnikov type calculations imply that the behaviour of the scattering map in this problem is quite unusual: it is only defined on a small subset of $\Lambda$ that shrinks, in the large energy limit, to a set of parallel lines $t=const$ as $\delta\to 0$.Comment: 25 page

What\u27s in a Name? The Worrisome Interchange of Juvenile Adjudications with Criminal Convictions

Juvenile delinquency adjudications are increasingly considered to be criminal convictions for purposes of sentencing enhancement in subsequent adult proceedings, leading to a renewed call for extension of the right to jury trial in the juvenile court so as to legitimize the use of adjudications. Such an extension is troubling, however, because regardless of the factfinder, it is improper to equate juvenile delinquency adjudications with criminal convictions for several reasons. First, the juvenile court remains distinct from the criminal court in its purpose and procedure. Second, the prevalence of juvenile pleas raises questions about whether juveniles defend against delinquency charges with the same vigor as they would against criminal charges. Finally, the necessity for a system of transfer into adult criminal court is questionable if adjudications can ex post facto be considered convictions. Although fairness dictates that juvenile adjudications should not be considered convictions, if they are consistently used as such, the infancy defense should be available in the juvenile court

Linear and nonlinear stability of periodic orbits in annular billiards

An annular billiard is a dynamical system in which a particle moves freely in a disk except for elastic collisions with the boundary, and also a circular scatterer in the interior of the disk. We investigate stability properties of some periodic orbits in annular billiards in which the scatterer is touching or close to the boundary. We analytically show that there exist linearly stable periodic orbits of arbitrary period for scatterers with decreasing radii that are located near the boundary of the disk. As the position of the scatterer moves away from a symmetry line of a periodic orbit, the stability of periodic orbits changes from elliptic to hyperbolic, corresponding to a saddle-center bifurcation. When the scatterer is tangent to the boundary, the periodic orbit is parabolic. We prove that slightly changing the reflection angle of the orbit in the tangential situation leads to the existence of KAM islands. Thus we show that there exists a decreasing to zero sequence of open intervals of scatterer radii, along which the billiard table is not ergodic.Comment: 15 pages, 7 figure

Oxygenated pitch and processing same

A method is provided which includes infusing oxygen into pitch material without stabilizing the oxygen-infused pitch material. In addition, the invention includes further processing steps (including heat stabilization in either an inert atmosphere or an oxygen-containing atmosphere, deformation, pyrolysis, and/or composite formation) performed after or in conjunction with the oxygenation process. Moreover, the invention includes the composition of matter (in any of a number of different physical forms such as powder, fiber, shaped article, composites) resulting from the practice of this oxygenation process, either alone or in conjunction with the further processing steps. The composition has a homogeneous distribution of oxygen and can be heat stabilized in an inert atmosphere

Oxygenated pitch and processing same

A method is provided which includes infusing oxygen into pitch material without stabilizing the oxygen-infused pitch material. In addition, the invention includes further processing steps (including heat stabilization in either an inert atmosphere or an oxygen-containing atmosphere, deformation, pyrolysis, and/or composite formation) performed after or in conjunction with the oxygenation process. Moreover, the invention includes the composition of matter (in any of a number of different physical forms such as powder, fiber, shaped article, composites) resulting from the practice of this oxygenation process, either alone or in conjunction with the further processing steps. The composition has a homogeneous distribution of oxygen and can be heat stabilized in an inert atmosphere

Setting the absolute threshold of vision

The performance of sensory systems in many cases is limited by the physical nature of the stimulus. For vision, the quantal nature of light limits detection by dark-adapted observers; only now are we beginning to be aware of the subtleties in the biophysical mechanisms underlying this exquisite sensitivity

Fluorescence from a few electrons

Systems containing few Fermions (e.g., electrons) are of great current interest. Fluorescence occurs when electrons drop from one level to another without changing spin. Only electron gases in a state of equilibrium are considered. When the system may exchange electrons with a large reservoir, the electron-gas fluorescence is easily obtained from the well-known Fermi-Dirac distribution. But this is not so when the number of electrons in the system is prevented from varying, as is the case for isolated systems and for systems that are in thermal contact with electrical insulators such as diamond. Our accurate expressions rest on the assumption that single-electron energy levels are evenly spaced, and that energy coupling and spin coupling between electrons are small. These assumptions are shown to be realistic for many systems. Fluorescence from short, nearly isolated, quantum wires is predicted to drop abruptly in the visible, a result not predicted by the Fermi-Dirac distribution. Our exact formulas are based on restricted and unrestricted partitions of integers. The method is considerably simpler than the ones proposed earlier, which are based on second quantization and contour integration.Comment: 10 pages, 3 figures, RevTe

Constraints, Histones, and the 30 Nanometer Spiral

We investigate the mechanical stability of a segment of DNA wrapped around a histone in the nucleosome configuration. The assumption underlying this investigation is that the proper model for this packaging arrangement is that of an elastic rod that is free to twist and that writhes subject to mechanical constraints. We find that the number of constraints required to stabilize the nuclesome configuration is determined by the length of the segment, the number of times the DNA wraps around the histone spool, and the specific constraints utilized. While it can be shown that four constraints suffice, in principle, to insure stability of the nucleosome, a proper choice must be made to guarantee the effectiveness of this minimal number. The optimal choice of constraints appears to bear a relation to the existence of a spiral ridge on the surface of the histone octamer. The particular configuration that we investigate is related to the 30 nanometer spiral, a higher-order organization of DNA in chromatin.Comment: ReVTeX, 15 pages, 18 figure

Funnels in Energy Landscapes

Local minima and the saddle points separating them in the energy landscape are known to dominate the dynamics of biopolymer folding. Here we introduce a notion of a "folding funnel" that is concisely defined in terms of energy minima and saddle points, while at the same time conforming to a notion of a "folding funnel" as it is discussed in the protein folding literature.Comment: 6 pages, 3 figures, submitted to European Conference on Complex Systems 200