PERIODIC BOUNDARY VALUE PROBLEM FOR SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

This paper investigates the periodic boundary value problems for a class of second order functional differential equations. The monotone iterative technique and the maximum principle are applied to obtain the existence of maximal and minimal solutions.     

Three Types of Expression in Dark-Soliton Perturbation Theory Based on Squared Jost Solutions

Three types of expression in the dark-soliton perturbation theory based on squared Jost solutions are invesgigaged in ghis paper. It is shown that there are three formally different results about the effects of perturbagion on a dark soliton, and it is proved by means of a transformation between two integral variables that they are essentially equivalent.     

Measurement of acoustic particle velocities in enclosed sound field: Assessment of two Laser Doppler Velocimetry measuring systems

The performances of two Laser Doppler Velocimetry (LDV) systems adapted for measuring the acoustic particle velocities are assessed in enclosed sound field. This assessment is performed by comparing the acoustic velocities measured by means of LDV to reference acoustic velocities estimated from sound pressure measurements. The two LDV systems are based on a single optical bench which delivers an optical signal called Doppler signal. The Doppler signal, which is frequency modulated, is analyzed by means of two signal processing systems, the BSA (Burst Spectrum Analyser from Dantec) on the one hand, and a system specifically developed for the estimation of the acoustic velocity on the other hand. Once the experimental setup has been optimized for minimizing the errors made on the reference velocities, the assessment is performed and shows that both systems can measure the acoustic velocity in enclosed field in two the frequency ranges [0-4 kHz] and [0-2 kHz] respectively for acoustic velocity amplitudes of 10 mm/s and 1 mm/s.     

Point-to-Periodic and Periodic-to-Periodic Connections

In this work we consider computing and continuing connecting orbits in parameter dependent dynamical systems. We give details of algorithms for computing connections between equilibria and periodic orbits, and between periodic orbits. The theoretical foundation for these techniques is given by the seminal work of Beyn in 1994, “On well-posed problems for connecting orbits in dynamical systems”, where a numerical technique is also proposed. Our algorithms consist of splitting the computation of the connection from that of the periodic orbit(s). To set up appropriate boundary conditions, we follow the algorithmic approach used by Demmel, Dieci, and Friedman, for the case of connecting orbits between equilibria, and we construct and exploit the smooth block Schur decomposition of the monodromy matrices associated to the periodic orbits. Numerical examples illustrate the performance of the algorithms. This revised version was published online in July 2006 with corrections to the Cover Date.     

Positive periodic solution of two-species ratio-dependent predator–prey system with time delay in two-patch environment

In this paper, we are concerned with the existence of positive periodic solution to a class of two-species ratio-dependent predator–prey diffusion model with time delay. By using the continuation theorem of coincidence degree theory, we transform this problem into a problem of calculating the topological degree of a continuous mapping, and then some sufficient conditions of the existence of positive periodic solution is established for the system.     

Monotone Mappings and Flows of Viscous Media

We establish solvability of the boundary-value problems describing stationary and periodic flows of viscoplastic media. In the case of stationary flows we study the question of convergence of the Galerkin method. For the problem of periodic flows we prove a version of the second Bogolyubov theorem.     

Rigorous results in selection of steady needle crystals

This paper concerns the existence of solutions to a steady needle crystal growth problem in a one-sided model. We rigorously prove that for small nonzero anisotropy γ, analytic symmetric needle crystal solutions exist in the limit of surface tension ε² if only if the stokes constant S for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter β=2^9/7γε^−8/7 and earlier numerical calculations by a number of investigators have shown this to be zero for a discrete set of values of β. It is also proven that for γ=0, there can be no symmetric needle crystal solution in the considered space.The methodology consists of two steps. First, the original problem is reduced to a weak half-strip problem for any γ in a compact set of [0,1) by relaxation of the symmetry condition. The weak problem is shown to have a unique solution in the function space considered for any γ∈[0,γ_m] for some γ_m>0. When a symmetry is invoked, the weak problem is shown equivalent to the original needle crystal problem. Next, by considering the behavior of the solution in neighborhood of an appropriate complex turning point for γ∈(0,γ_m], we extract an exponentially small term in ε as ε→0⁺ that generally violates the symmetric condition. We prove that the symmetry condition is satisfied for small ε when the parameter β is constrained appropriately.     

Sequential Estimation of the Parameters in a Trigonometric Regression Model with the Gaussian Coloured Noise

This paper considers the estimation problem for a trigonometric regression model with the noise specified by the Ornstein–Uhlenbeck process with unknown parameter. We propose a sequential procedure which ensures a prescribed mean square precision uniformly in the nuisance parameter. The asymptotic behaviour of the procedure duration mean has been studied. This revised version was published online in August 2006 with corrections to the Cover Date.     

Galactic Metrics

The paths of stars in galaxies have circular velocity independent of their distance from the centre of the galaxy. Newtonian mechanics with a logarithmic potential has such paths. In relativity these paths can be taken to be geodesics and this requirement places a restricting equation on the spacetime metric. This equation has a non-unique spherically symmetric solution that in the Newtonian limit has a logarithmic potential. It also can be solved in terms of a conformal factor. In particular it has solutions which are conformal to the vacuum-Einstein solutions and such spacetimes are solutions to the vacuum-Bach equations. Therefore it is suggested that the Bach equations describe dynamics on galactic length scales.     

PERIODIC SOLUTIONS OF PLANAR SYSTEM WITH SIX DELAYS

In this paper we obtain some sufficient conditions for the existence of non-constant periodic solutions of the following planar system with six delaysOur approach is based on the continuation theorem of the coincidence degree.