Resolvents of cone pseudodifferential operators,asymptotic expansions and applications

We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula.      

Scaling spectra and return times of dynamical systems

The grand canonical version of the spectrum of singularities formalism is presented, relying naturally upon certain Markov transition graphs. The structure of a graph is simply determined by the close return times of the dynamical system described. Thus, an intimate connection exists between the shape of the singularity curve and a small but interesting set of dynamical properties.     

Sharp van der Corput estimates and minimal divided differences

We find the sharp constant in a sublevel set estimate which arises in connection with van der Corput's lemma. In order to do this, we find the nodes that minimise divided differences. We go on to find the sharp constant in the first instance of the van der Corput lemma. With these bounds we improve the constant in the general van der Corput lemma, so that it is asymptotically sharp.

     

Bifurcation problems in nonlinear parametric programming

The nonlinear parametric programming problem is reformulated as a closed system of nonlinear equations so that numerical continuation and bifurcation techniques can be used to investigate the dependence of the optimal solution on the system parameters. This system, which is motivated by the Fritz John first-order necessary conditions, contains all Fritz John and all Karush-Kuhn-Tucker points as well as local minima and maxima, saddle points, feasible and nonfeasible critical points. Necessary and sufficient conditions for a singularity to occur in this system are characterized in terms of the loss of a complementarity condition, the linear dependence of the gradients of the active constraints, and the singularity of the Hessian of the Lagrangian on a tangent space. Any singularity can be placed in one of seven distinct classes depending upon which subset of these three conditions hold true at a solution. For problems with one parameter, we analyze simple and multiple bifurcation of critical points from a singularity arising from the loss of the complementarity condition, and then develop a set of conditions which guarantees the unique persistence of a minimum through this singularity. The research of this author was supported by National Science Foundation through NSF Grant DMS-85-10201 and by the Air Force Office of Scientific Research through instrument number AFOSR-ISSA-85-00079.     

Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control

The stability and bifurcation of a van der Pol-Duffing oscillator with the delay feedback are investigated, in which the strength of feedback control is a nonlinear function of delay. A geometrical method in conjunction with an analytical method is developed to identify the critical values for stability switches and Hopf bifurcations. The Hopf bifurcation curves and multi-stable regions are obtained as two parameters vary. Some weak resonant and non-resonant double Hopf bifurcation phenomena are observed due to the vanishing of the real parts of two pairs of characteristic roots on the margins of the “death island” regions simultaneously. By applying the center manifold theory, the normal forms near the double Hopf bifurcation points, as well as classifications of local dynamics are analyzed. Furthermore, some quasi-periodic and chaotic motions are verified in both theoretical and numerical ways.     

Asymptotic regularity of solutions to Hartree–Fock equations with Coulomb potential

We study the asymptotic regularity of solutions to Hartree–Fock (HF) equations for Coulomb systems. To deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo‐differential operators on conical manifolds. First, the non‐self‐consistent‐field case is considered, which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator, which guarantee ellipticity for the local part of the Fock operator on the open stretched cone ?₊ × S². This proves the existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed‐point approach based on Cancès and Le Bris analysis of the level‐shifting algorithm, we show via another bootstrap argument that the corresponding self‐consistent‐field solutions to the HF equation have the same type of asymptotic regularity. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations

In this paper, homotopy perturbation method (HPM) and variational iteration method (VIM) are applied to solve nonlinear oscillator differential equations. Illustrative examples reveal that these methods are very effective and convenient for solving nonlinear differential equations. Moreover, the methods do not require linearization or small perturbation. Comparisons are also made between the exact solutions and the results of the homotopy perturbation method and variational iteration method in order to prove the precision of the results obtained from both methods mentioned.     

Polarization singularities in Young's two-slit experiment

Young's interference experiment is regarded as a two-slit diffraction phenomenon, the polarization singularities in Young's two-slit configuration illuminated with two linearly, orthogonally polarized Gaussian vortex beams are studied. It is shown that generally, there exist L-lines (linearly polarization) and polarization singularities including C-points (circular polarization), S₂₃ and S₃₁ singularities even though the parameters of two beams are the same. The pair creation-annihilation and motion of polarization singularities take place upon propagation, or by varying a control parameter, such as the amplitude ratio of two beams or obscure ratio of slits etc. For a special case of the illumination with two linearly, orthogonally polarized Gaussian vortex-free beams, polarization singularities, in particular, C-points may occur if a parameter of two beams is not equal.     

Models of thermodiffusion in 1D

The goal of the paper is to study the Cauchy problem for 1D models of thermodiffusion. We explain qualitative properties of solutions. In particular, we show which part of the model has a dominant influence on well‐posedness, propagation of singularities, L^p ? L^q decay estimates on the conjugate line, and on the diffusion phenomenon. Copyright © 2013 John Wiley & Sons, Ltd.