Nonlinear Propagation of Light in One Dimensional Periodic Structures

We consider the nonlinear propagation of light in an optical fiber waveguide as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is assumed to have an index of refraction which varies periodically along its length. The wavelength of light is selected to be in resonance with the periodic structure (Bragg resonance). The AMLE system considered incorporates the effects non-instantaneous response of the medium to the electromagnetic field (chromatic or material dispersion), the periodic structure (photonic band dispersion) and nonlinearity. We present a detailed discussion of the role of these effects individually and in concert. We derive the nonlinear coupled mode equations (NLCME) which govern the envelope of the coupled backward and forward components of the electromagnetic field. We prove the validity of the NLCME description and give explicit estimates for the deviation of the approximation given by NLCME from the {\it exact} dynamics, governed by AMLE. NLCME is known to have gap soliton states. A consequence of our results is the existence of very long-lived {\it gap soliton} states of AMLE. We present numerical simulations which validate as well as illustrate the limits of the theory. Finally, we verify that the assumptions of our model apply to the parameter regimes explored in recent physical experiments in which gap solitons were observed.Comment: To appear in The Journal of Nonlinear Science; 55 pages, 13 figure

Theory and Application of Roscoe Pound\u27s Sociological Jurisprudence: Crime Prevention or Control?

The current interest in reforming the administration of justice has been triggered by a number of factors including the 1967 report of the President\u27s Commission on Law Enforcement and the Administration of Justice and the treatment afforded arrestees during the civil disorders of the past few years. The nation is alarmed at the reported annual increases in crime, and this alarm was manifested in the 1968 presidential election when law and order became a major issue. Superficially the answer may seem clear: more effective enforcement of the law and, when necessary, more stringent laws. The critical issue, however, is a jurisprudential-philosophical one: ought the proper approach to crime essentially be its prevention through methods such as the rehabilitation of criminal offenders, or its control through efficient administrative procedures? This is not a new question in jurisprudence, but it remains an important and unresolved one. This article will examine an analytical approach to this problem which was developed and applied by Roscoe Pound, one of America\u27s most eminent jurists. After describing and interpreting Pound\u27s concept of sociological jurisprudence, we will relate it generally to the reform of criminal justice administration and analyze Pound\u27s attempt to apply his theory as -Director of the Cleveland Crime Survey of 1921. Finally, we will compare the recommendations of that Cleveland study and the recent report of the President\u27s Commission in a modest effort to assess their impact on the administration of criminal justice and to draw some lessons for future reform endeavors

Wave dynamics in optically modulated waveguide arrays

A model describing wave propagation in optically modulated waveguide arrays is proposed. In the weakly guided regime, a two-dimensional semidiscrete nonlinear Schrodinger equation with the addition of a bulk diffraction term and an external optical trap is derived from first principles, i.e., Maxwell equations. When the nonlinearity is of the defocusing type, a family of unstaggered localized modes are numerically constructed. It is shown that the equation with an induced potential is well-posed and gives rise to localized dynamically stable nonlinear modes. The derived model is of the Gross-Pitaevskii type, a nonlinear Schrodinger equation with a linear optical potential, which also models Bose-Einstein condensates in a magnetic trap

Wave operator bounds for 1-dimensional Schr\"odinger operators with singular potentials and applications

Boundedness of wave operators for Schr\"odinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive estimates and commutator bounds.Comment: 16 pages, 0 figure

Boundary Avoidance Tracking for Instigating Pilot Induced Oscillations

In order to advance research in the area of pilot induced oscillations, a reliable method to create PIOs in a simulated environment is necessary. Using a boundary avoidance tracking task, researchers performing an evaluation of control systems were able to create PIO events in 42% of cases using a nominal aircraft, and 91% of cases using an aircraft with reduced actuator rate limits. The simulator evaluation took place in the NASA Ames Vertical Motion Simulator, a high-fidelity motion-based simulation facility

Palmoplantar keratoderma along with neuromuscular and metabolic phenotypes in Slurp1-deficient mice.

Mutations in SLURP1 cause mal de Meleda, a rare palmoplantar keratoderma (PPK). SLURP1 is a secreted protein that is expressed highly in keratinocytes but has also been identified elsewhere (e.g., spinal cord neurons). Here, we examined Slurp1-deficient mice (Slurp1(-/-)) created by replacing exon 2 with β-gal and neo cassettes. Slurp1(-/-) mice developed severe PPK characterized by increased keratinocyte proliferation, an accumulation of lipid droplets in the stratum corneum, and a water barrier defect. In addition, Slurp1(-/-) mice exhibited reduced adiposity, protection from obesity on a high-fat diet, low plasma lipid levels, and a neuromuscular abnormality (hind-limb clasping). Initially, it was unclear whether the metabolic and neuromuscular phenotypes were due to Slurp1 deficiency, because we found that the targeted Slurp1 mutation reduced the expression of several neighboring genes (e.g., Slurp2, Lypd2). We therefore created a new line of knockout mice (Slurp1X(-/-) mice) with a simple nonsense mutation in exon 2. The Slurp1X mutation did not reduce the expression of adjacent genes, but Slurp1X(-/-) mice exhibited all of the phenotypes observed in the original line of knockout mice. Thus, Slurp1 deficiency in mice elicits metabolic and neuromuscular abnormalities in addition to PPK

Motion-Based Piloted Simulation Evaluation of a Control Allocation Technique to Recover from Pilot Induced Oscillations

This paper describes the maturation of a control allocation technique designed to assist pilots in the recovery from pilot induced oscillations (PIOs). The Control Allocation technique to recover from Pilot Induced Oscillations (CAPIO) is designed to enable next generation high efficiency aircraft designs. Energy efficient next generation aircraft require feedback control strategies that will enable lowering the actuator rate limit requirements for optimal airframe design. One of the common issues flying with actuator rate limits is PIOs caused by the phase lag between the pilot inputs and control surface response. CAPIO utilizes real-time optimization for control allocation to eliminate phase lag in the system caused by control surface rate limiting. System impacts of the control allocator were assessed through a piloted simulation evaluation of a non-linear aircraft simulation in the NASA Ames Vertical Motion Simulator. Results indicate that CAPIO helps reduce oscillatory behavior, including the severity and duration of PIOs, introduced by control surface rate limiting

Algebraic Torsion in Contact Manifolds

We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order zero if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order one (though the converse is not true). We also construct examples for each nonnegative k of contact 3-manifolds that have algebraic torsion of order k but not k - 1, and derive consequences for contact surgeries on such manifolds. The appendix by Michael Hutchings gives an alternative proof of our cobordism obstructions in dimension three using a refinement of the contact invariant in Embedded Contact Homology.Comment: 53 pages, 4 figures, with an appendix by Michael Hutchings; v.3 is a final update to agree with the published paper, and also corrects a minor error that appeared in the published version of the appendi

Dimension dependent energy thresholds for discrete breathers

Discrete breathers are time-periodic, spatially localized solutions of the equations of motion for a system of classical degrees of freedom interacting on a lattice. We study the existence of energy thresholds for discrete breathers, i.e., the question whether, in a certain system, discrete breathers of arbitrarily low energy exist, or a threshold has to be overcome in order to excite a discrete breather. Breather energies are found to have a positive lower bound if the lattice dimension d is greater than or equal to a certain critical value d_c, whereas no energy threshold is observed for d<d_c. The critical dimension d_c is system dependent and can be computed explicitly, taking on values between zero and infinity. Three classes of Hamiltonian systems are distinguished, being characterized by different mechanisms effecting the existence (or non-existence) of an energy threshold.Comment: 20 pages, 5 figure