Showing papers on "Dirac delta function published in 1987"

On the imaging of reflectors in the earth

Abstract: In this paper, I present a modification of the Beylkin inversion operator. This modification accounts for the band-limited nature of the data and makes the role of discontinuities in the sound speed more precise. The inversion presented here partially dispenses with the small-parameter constraint of the Born approximation. This is shown by applying the proposed inversion operator to upward scattered data represented by the Kirchhoff approximation, using the angularly dependent geometrical-optics reflection coefficient. A fully nonlinear estimate of the jump in sound speed may be extracted from the output of this algorithm interpreted in the context of these Kirchhoff-approximate data for the forward problem. The inversion of these data involves integration over the source-receiver surface, the reflecting surface, and frequency. The spatial integrals are computed by the method of stationary phase. The output is asymptotically a scaled singular function of the reflecting surface. The singular function of a surface is a Dirac delta function whose support is on the surface. Thus, knowledge of the singular functions is equivalent to mathematical imaging of the reflector. The scale factor multiplying the singular function is proportional to the geometrical-optics reflection coefficient. In addition to its dependence on the variations in sound speed, this reflection coefficient depends on an opening angle between rays from a source and receiver pair to the reflector. I show how to determine this unknown angle. With the angle determined, the reflection coefficient contains only the sound speed below the reflector as an unknown, and it can be determined. A recursive application of the inversion formalism is possible. That is, starting from the upper surface, each time a major reflector is imaged, the background sound speed is updated to account for the new information and data are processed deeper into the section until a new major reflector is imaged. Hence, the present inversion formalism lends itself to this type of recursive implementation. The inversion proposed here takes the form of a Kirchhoff migration of filtered data traces, with the space-domain amplitude and frequency-domain filter deduced from the inversion theory. Thus, one could view this type of inversion and parameter estimation as a Kirchhoff migration with careful attention to amplitude.

Analysis of Wiener Functionals (Malliavin Calculus) and its Applications to Heat Kernels

Abstract: An analysis of Wiener functionals is studied as a kind of Schwartz distribution theory on Wiener space. For this, we introduce, besides ordinary $L_p$-spaces of Wiener functionals, Sobolev-type spaces of (generalized) Wiener functionals. Any Schwartz distribution on $\mathbf{R}^d$ is pulled back to a generalized Wiener functional by a $d$-dimensional Wiener map which is smooth and nondegenerate in the sense of Malliavin. As applications, we construct a heat kernel (i.e., the fundamental solution of a heat equation) by a generalized expectation of the Dirac delta function pulled back by an Ito map, i.e., a Wiener map obtained by solving Ito's stochastic differential equations. Short-time asymptotics of heat kernels are studied through the asymptotics, in terms of Sobolev norms, of the generalized Wiener functional under the expectation.

Relativistic quarks in one-dimensional periodic structures

Abstract: We obtain the band condition for relativistic quarks moving in one-dimensional periodic potentials using the transfer matrix method. Using a strong electrostatic type of potential in the Dirac equation does not give confining properties, while Lorentz scalar potentials do. We give an analytic form for the wave function which results when the potential is taken to a delta function limit, and discuss the discrepancy between this result and that obtained by ``solving'' the Dirac equation for a delta function potential.

Proper treatment of the delta function potential in the one‐dimensional Dirac equation

Abstract: It is pointed out that the usual treatment of the delta function potential in the one‐dimensional Dirac equation is incorrect. Steps leading to such incorrect results are identified. The delta function potential is also examined from the limiting case of a nonlocal potential V(x,x’). In this general case, the strength g=∫dx ∫dx’V(x,x’) is no longer a sufficient parameter to characterize a potential.

The nonlinear Schrodinger model as a special continuum limit of the anisotropic Heisenberg model

Abstract: One-dimensional spin chains are investigated by constructing a mapping of the anisotropic Heisenberg model in the limit Delta to 1 onto the non-linear Schrodinger model (a gas of bosons with delta function interactions). Three applications of this mapping are given. First it is shown that the Bethe ansatz solutions of the anisotropic spin-1/2 Heisenberg model go over to the continuous Bethe ansatz solutions of the non-linear Schrodinger model. Then bound-state energies for arbitrary spin are calculated and as a third application a nearly identical mapping of the anisotropic spin-1/2 Heisenberg model in the limit Delta to 0 onto a non-linear Schrodinger model for fermion fields is given. Finally the authors construct an anisotropic SU(N) model. The continuum limit of this model is shown to be the (N-1)-component non-linear Schrodinger model, which can be solved by means of the nested Bethe ansatz.

Analysis of the scattered light component in distorted fluorescence decay profiles using a modified delta function convolution method

Abstract: An interactive reconvolutlon method that permits one to recover the decay parameters from fluorescence decay curves distorted with a scattered ilght component is presented. The aigorlthm is based on the delta function convolutlon method, a technique that corrects for the wavelength dependence of the instrument response function. The expected decay parameters were successfully extracted from slmulated and experimentally measured, single and double exponential fluorescence decay curves polluted with scattered light. I t Is shown that when an extra exponential component In the decay function is used to account for the scattered light component, erroneous values of the decay parameters can be obtained.

K-pulse estimation from the impulse response of a target

Abstract: A simple and general method is used to estimate the K-pulse of a target from its Fourier synthesized impulse response. The approach is based on minimization of energy content outside the K-pulse response duration. The method is simple because it only deals with Dirac delta functions in integrals. It is general because the method only manipulates samples of a continuous function as a computer does. The method does not require a priori information on the target poles. Results obtained from Fourier synthesized data show excellent convergence for the K-pulses. Furthermore, dominant poles of the target can be extracted from the approximated K-pulse. The pole-extraction method is also discussed in association with some results.

General solution of diffusion-induced stresses

Abstract: A method for obtaining the general solution of diffusion-induced stress problems is proposed. The method, which is an extension of the Duhamel method, includes two situations. First, the boundary condition may contain not only spatial derivatives but also time integrals of the surface temperature. Second, the solution of the diffusion equation with time-dependent surface temperature can be expressed in terms of the known solutions with the lime variation of the surface temperature as a Dirac delta function, a constant, or a power function of time. The method is applied to three simple geometries, slab, cylinder, and sphere.

First-order Fermi acceleration in the two-stream limit

Abstract: A study of the first-order Fermi mechanism for accelerating cosmic-rays at relativistic and nonrelativistic shocks is carried out by using the two-stream approximation. Exact steady-state analytic solutions illustrating the shock acceleration process in the test-particle limit in which monoenergetic (relativistic) seed particles enter the shock through an upstream free-escape boundary are obtained. The momentum spectrum of the shock accelerated particles consists of a series of Dirac delta distributions corresponding to particles that have undergone an integral number of acceleration cycles. Since particles in the model have a finite fixed escape probability from the shock and the particle momenta p are equally spaced in log p, the envelope of the delta functions series is a power law in momentum. The solutions are used to discuss time-dependent aspects of the shock acceleration process in terms of the finite cycle time, escape probability, and momentum change per cycle that can be deduced from the steady-state model. The length-scale over which the accelerated particles extend upstream of the shock is shown to depend upon the particle energy, with the higher energy particles extending further upstream. This effect is shown to be intimately related to the kinematic threshold requirement that the particle speed exceed the fluid speed in order for particles to swim upstream of the shock and participate in the shock acceleration process.

Schrödinger‐like equation for relativistic particles

Abstract: The Dirac equation in a spherically symmetric screened Coulomb potential is transformed to a modified Schrodinger equation of the form d2u/dr2+k2(r)u=0. This transformation is induced by expressing the Dirac function as a linear combination of the function u and its derivative du/dr. Various properties of the transformation and of the resulting equations are studied. The close similarity between the modified Schrodinger equation and the Schrodinger equation suggests that methods applied to the Schrodinger equation to derive nonrelativistic relations can be applied to the modified Schrodinger equation to derive the analogous relativistic relations. As an example, this approach is applied to the single channel quantum defect theory to give a new derivation of its relativistic form.

On the limit Gibbs states of the spherical model

Abstract: The effect of a class of Hamiltonian perturbations, vanishing in the thermodynamic limit, on the limit Gibbs states of the spherical and mean spherical models is studied. The perturbation term is taken in the form of interaction energy with uniform magnetic field of strength h0N- alpha , where h0 in r1 and alpha )0 are parameters, and N is the number of particles. For fixed temperatures below the critical temperature, in the absence of constant external magnetic fields and at alpha =1 the authors obtain convex sets of different mixed Gibbs states parametrised by h0. A natural one-parameter generalisation of the Kac-Thompson transformation kernel which relates the states of the mean spherical model to the states of the spherical model is found. When 0( alpha (1 and h0 not=0, or alpha =1 and h0=+or- infinity , this kernel becomes a delta function even below the critical temperature; then the states in both ensembles coincide with each other and with one of the two (depending on the sign of h0) extreme points. The case of alpha >1 is found to lead to the well known results corresponding to the absence of perturbation (h0=0).

Discontinuous and impulsive excitation

Abstract: In this paper, we study the solution of differential equation with Dirac function and Heaviside function, arising from discontinuous and impulsive excitation. Firstly, according to the theory of differential equation, we suggest, then we derive the equation of x 1 (t) and x 2 (t) by terms of property of distribution, and by solving x 1 (t) and x 2 (t) we obtain x(t); finally, we make a thorough investigation about periodic impulsive parametric excitation.

Steady state multiplicity behavior of an adiabatic plug-flow reactor with nonuniformly active catalyst

Abstract: A heterogeneous, adiabatic plug-flow reactor is studied with catalyst pellets having nonuniform distribution of catalytic activity. The activity distribution function used is a Dirac delta function. Because of the specific distribution function used, the solid phase mass and energy balances can be solved analytically, and result in a nonlinear algebraic equation which describes the interactions of the solid and fluid phases. Singularity Theory with a distinguished parameter is used to study the steady-state multiplicity characteristics of this equation, and to find all possible bifurcation diagrams relating the solid and the fluid phase concentrations. Reactions of zeroth order, m-th (m > 0) order, and bimolecular Langmuir-Hinshelwood kinetics are studied. Because of the large number of varieties and boundary sets, the total number of possible bifurcation diagrams is quite large. Two new variety sets are used to further divide the global parameter space into more detailed global bifurcation diagrams. Once...

Deconvolution algorithm for a Fabry-Perot interferometer.

Abstract: Fabry-Perot spectral output is a convolution between the input spectrum and the response function of the Fabry-Perot spectroscopic instrument. We have developed a deconvolution algorithm, where we do not specify the functional forms to the output spectrum and the instrumental response function but use only numerical data as observed. Merits of our algorithm include: the divergence problem is avoided by iterative methods treating data components in the order of their magnitudes; background and delta function type elastic components are treated automatically as part of the built-in signal; and very low level signals can be retrieved from the strong background.

A delta function approach to certain integrals

Abstract: In this paper a simple integral is investigated by first introducing an associated discontinuous function. The evaluation of the integral is then completed by making use of delta functions and Fourier transforms. This technique introduces ideas which, at first sight, might appear to conflict with the traditional mathematical approach to discontinuous functions.