Propagation Theorem Research Articles

H-measures applied to symmetric systems

H-measures were recently introduced by Luc Tartar as a tool which might provide better understanding of propagating oscillations. Independently, Patrick Gerard introduced the same objects under the name of microlocal defect measures. Partial differential equations of mathematical physics can often be written in the form of a symmetric system:where Ak and B are matrix functions, while u is an unknown vector function, and f a known vector function. In this work we prove a general propagation theorem for H-measures associated to symmetric systems. This result, combined with the localisation property, is then used to obtain more precise results on the behaviour of H-measures associated to the wave equation, Maxwell's and Dirac's systems, and second-order equations in two variables.

A robust continuous model for evidential reasoning

A 2-D model for evidential reasoning is proposed, in which the belief function of evidence is represented as a belief density function which can be in a continuous or discrete form. A vector form of mutual dependency relationship of the evidence is considered and a dependency propagation theorem is proved. This robust method can resolve the conflicts resulting from either the mutual dependency among evidences or the structural dependency in an inference network due to the evidence combination order. Belief conjunction, belief combination, belief propagation procedures, and AND/OR operations of an inference network based on the proposed 2-D model are all presented, followed by some examples demonstrating the advantages of this method over the conventional methods.

Three-dimensional phase contrast imaging by an annular illumination microscope

A method of observing 3-D phase structures through a microscope incorporating computer reconstruction is discussed. This microscope is equipped with an annular pupil in illumination optics, but no phase shifter is included in the imaging optics. The sample stage is longitudinally (z-axial) scanned to collect a focus image series. The 3-D phase transfer function is derived and computer-plotted based on Streibl's 3-D image transfer theory under the first-order Born approximation and the mutual intensity propagation theorem. Experimental results of 3-D phase reconstruction are shown with cultured tobacco cells by Helstrom's inverse filtering of the transfer function to a series of focused images.

A new proof of the propagation theorem forN-body quantum systems

A new proof of I. Sigal's and A. Soffer's propagation theorem is given. This theorem describes a large class of operators which are Kato-smooth with respect to anN-body Schrodinger operator.

Second Microlocalization and Propagation Theorems for the Wave Front Sets

There is showed how to define the second analytic wave front set in order to obtain propagation properties for differentiable, Gevrey, or analytic singularities of distributions without second analytic support.

Systems of microdifferential equations with involutory double characteristics propagation theorem for sheaves in the frameworkof microlocal study of sheaves

Systems of microdifferential equations with involutory double characteristics propagation theorem for sheaves in the frameworkof microlocal study of sheaves

Wave propagation and Thom's theorem

Scalar waves propagated from a finite emitter in an inhomogeneous medium form caustics whose geometry and location are functions of both the boundary conditions on the emitter and the inhomogeneity of the medium. The wave amplitude satisfies a Helmholtz equation. For the class of problems considered, a phase is derived from a characterisation of the medium and the boundary conditions on the emitter. The location and geometry of the caustic in configuration space is determined from phase space considerations through the Lagrange manifold. An oscillatory integral whose asymptotic expansion is that of the amplitude, is determined by a sharpening of Maslov's method (1972) of characteristics. Thom's theorem prescribes a normal form for the phase function in the integral. Transformations carrying the phase function to the canonical form are determined; then, following Duistermaat (1973), the complete asymptotic series of the integral, and hence of the amplitude, is obtained. The entire algorithm is illustrated with a specific example.

Wave propagation and uniqueness theorems for elastic materials with internal state variables

In this paper we consider the general, one-dimensional theory of thermoelastic materials with internal state variables and show that such materials have the wave propagation property, i.e. all smooth structured waves propagate with bounded velocity. We further show that this boundedness property ensures the uniqueness of solutions of unbounded domains.

Numerical Methods for Nonlinear Volterra Integro-Differential Equations

Let $u'(x) = f(x,u(x),Iu(x))$ and $u(a) = \mu $, the initial value, where I is the Volterra functional $Iu(x) = \int_a^x {k(x,s,u(s))ds} $. Call this the general nonlinear Volterra integro-differential equation (VIDE). Approximations are given for the functional I and its derivatives. These approximations are applied to create one-step algorithms for the constructive solution of the VIDE problem. Let $x \in [a,b]$, integer $N \geqq 1$, and ${{h = (b - a)} / N}$. Define $x_n = a - nh$ for $n = 0(1)N$. A class of algorithms (each depending upon an order parameter $p \geqq 1$ and upon the step size h) are defined. For each fixed p and h, each algorithm recursively generates a sequence $\{ u_n \} $ which approximates $\{ u(x_n) \} $. Two error propagation theorems are presented. THEOREM 1. Under suitable smoothness conditions, $u_n = u(x_n ) + O(h^p )$, where$p \geqq 1$is the order parameter. The convergence is uniform on closed intervals. THEOREM 2. Under further smoothness conditions, $u_n = u(x_n ) + h^p e(x_n ) + O(h^{p + 1} )$, where the principal error function$e(x)$is defined as the solution of another Volterra integro-differential equation. The results of various numerical experiments, including 4th order Runge–Kutta type algorithms, are presented. These experiments serve to confirm the validity of our results.