Showing papers on "Dirac delta function published in 1985"

Delta function convolution method (DFCM) for fluorescence decay experiments

Abstract: A rigorous and convenient method of correcting for the wavelength variation of the instrument response function in time correlated photon counting fluorescence decay measurements is described. The method involves convolution of a modified functional form Fs of the physical model with a reference data set measured under identical conditions as the measurement of the sample. The method is completely general in that an appropriate functional form may be found for any physical model of the excited state decay process. The modified function includes a term which is a Dirac delta function and terms which give the correct decay times and preexponential values in which one is interested. None of the data is altered in any way, permitting correct statistical analysis of the fitting. The method is readily adaptable to standard deconvolution procedures. The paper describes the theory and application of the method together with fluorescence decay results obtained from measurements of a number of different samples including diphenylhexatriene, myoglobin, hemoglobin, 4’, 6‐diamidine–2‐phenylindole (DAPI), and lysine–trytophan–lysine.

Radiation from a point source and scattering theory in a fluid-saturated porous solid

Abstract: The time harmonic Green function for a point load in an unbounded fluid‐saturated porous solid is derived in the context of Biot’s theory. The solution contains the two compressional waves and one transverse wave that are predicted by the theory and have been observed in experiments. At low frequency, the slow compressional wave is diffusive and only the fast compressional and transverse waves radiate energy. At high frequency, the slow wave radiates, but with a decay radius which is on the order of cm in rocks. The general problem of scattering by an obstacle is considered. The point load solution may be used to obtain scattered fields in terms of the fields on the obstacle. Explicit expressions are presented for the scattering amplitudes of the three waves. Simple reciprocity relations between the scattering amplitudes for plane‐wave incidence are also given. These hold under the interchange of incident and observation directions and are completely general results. Finally, the point source solution is Fourier transformed to get the solution for a load which is a delta function in time as well as space. We obtain a closed form expression when there is no damping. The three waves radiate from the source as distinct delta function pulses. With damping present, asymptotic approximations show the slow wave to be purely diffusive. The fast and transverse waves propagate as pulses. The pulses are Gaussian‐shaped, which broaden with increasing time or radial distance.

Dirac-delta function approximations to the scattering phase function

Abstract: Dirac-delta function approximations are used to represent the single scattering phase function of large spherical particles or voids. The phase function for a spherical particle or void can be represented by a series of Legendre polynomials; however, as the diameter is increased, forward scattering becomes dominant and the number of terms in the series becomes very large. A Dirac-delta function approximation consists of a Dirac-delta function in the forward direction plus a finite series of Legendre polynomials. The Dirac-delta function accounts for strong forward scattering. Particular attention is given to large ice spheres and spherical voids in ice. The Dirac-delta function is shown effective in reducing the number of terms needed to describe the phase function.

An extension of the born inversion method to a depth dependent reference profile

Abstract: An extension of the multidimensional Born inversion technique for acoustic waves is described. In earlier work, a perturbation in reference sound velocity was determined by assuming that the reference velocity was constant. In this extension, we allow the reference velocity to be a function of the depth variable z. The output of this method is a high-frequency bandlimited reflectivity function of the subsurface. The reflectivity function is an array of bandlimited singular functions scaled by the normal reflection strength. Each singular function is a Dirac delta function of a scalar argument which measures distance normal to a reflecting interface. Thus, the reflectivity function is an indicator map of subsurface reflectors equivalent to the map produced by migration. In addition to the assumption of small perturbation, the method requires that the reflection data reside in the high frequency regime in a well-defined sense. The method is based on the derivation of an integral equation for the perturbation in sound velocity from a known reference velocity. When the reference velocity is constant, the integral equation admits an analytic solution as a multifold integral of the reflection data. Further high frequency asymptotic analysis simplifies this integral considerably and leads to an extremely efficient numerical algorithm for computing the reflectivity function. The development of a computer code to implement this constant-reference-velocity solution is published elsewhere. For a reference velocity c(z) we can no longer invert the integral equation exactly. However, we can write down an asymptotic high-frequency approximation for the kernal of the integral equation and an asymptotic solution for the perturbation. The computer implementation of this result is designed along the same lines as the code for constant background velocity. In tests the total processing time for this algorithm with depth-dependent background velocity is usually considerably less than that required by a standard Kirchhoff migration algorithm. The method is implemented as a migration technique and compared with alternative migration algorithms on the flanks of the salt dome.

Derivative dependent surface energy terms in nematic liquid crystals

Abstract: The boundary energy terms dependent on the first derivative of the director are analysed theoretically. Although these terms necessarily arise from nonlocal interactions, they are generally written as purely surface terms. We show that, in this case, the stable director configurations are described by functions with discontinuous derivatives at the boundaries. All the solutions given in literature, which are found by explicitly assuming the continuity of the derivatives, must, therefore, be revised. In a more correct formulation of the problem, the ranges of the interaction forces should be taken into account and continuous solutions are then obtained, which are generally well approximated by the discontinuous ones. The approximation, instead, is not good in the case of the energy term proposed by Dubois-Violette and Parodi, where the discontinuity has the form of a Dirac function. In this case a physically meaningful solution can be found only on the basis of a more suitable expression of the boundary energy.

The half-space problem for the boltzmann equation at zero temperature

Abstract: At zero temperature the Maxwellian distribution is a delta function of velocity. In this paper the Boltzmann equation is linearized around a delta function and then analyzed by a comparison method. Using these results and similar bounds for the nonlinear collision operator, a nonlinear boundary value problem at zero temperature is solved. The results are applied to the asymptotic description at the cold end of the shock profile at infinite Mach number. All solutions F are assumed to have the form F(x, ξ) = (1 - a(x))δ(ξ) + f(x, ξ) in which a and f are regular functions.

Relativistic scattering states for a finite chain with delta-function potentials of arbitrary position and strength

Abstract: With the use of a Green's-function method, the scattering states of the one-dimensional Dirac equation are obtained. The potential is assumed to be a sum of $\ensuremath{\delta}$ functions of arbitrary strengths and positions on a chain of finite length.

Analytical Treatment of a Periodic $\delta $-Function Potential in the Path-Integral Formalism

Abstract: A method for evaluating the density matrix, written as a path-integral, is proposed and applied to a periodic delta function potential. It is the first time that a model which gives rise to a band structure is analytically treated in the framework of the Feynman path-integral formalism. It is shown that the results for the energy spectrum and the wave functions completely coincide with those deduced in the Schrodinger formalism.

Taylor series for the Dirac function on perturbed surfaces with applications to mechanics

Abstract: Let H0 be the Hamiltonian of an unperturbed oscillating system and let V be a small perturbation which is analytic in a neighborhood of the surface H0 = E, E > 0. If H0 = H + λ V is the total Hamiltonian, we prove that there is a Taylor expansion of the Dirac function δ for small |λ| When applied to test functions which are analytic near H0 = E. The connection of this result with the classical Lagrange-Burmann theorem and some applications to the mechanics of non-linear systems are discussed.

Effect of continuum-continuum interaction on the ionisation rate of a model atom

Abstract: A one-dimensional model atom consisting of an attractive delta function at the origin is subjected to a train of delta kicks modelling microwave (laser) ionisation of a real atom. The continuum is taken into account in various approximations which are finally compared with the full (numerical) solution of the model. It is shown that neglecting continuum-continuum interactions is an excellent approximation in the case of kicks of alternating sign.

Calculation of positron annihilation in helium using a global operator

Abstract: The authors applied the global operator of Drachman (1981) to the case of positron scattering from helium atoms. The global operator is identical to the usual delta function operator for exact wavefunctions and is expected to yield superior results in the case of approximate wavefunctions. This expectation is borne out in the present work.

Demonstration of a Jacobi Polynomial Representation of a Kronecker Delta Function.

Abstract: : A formula derived b Noble has been generalized to obtain an expansion of the Kronecker delta function as and infinite series involving the products of two Jacobi polynomials. For the; inviscid, incompressible flow of rotating fluid shells confined between concentric, spherical, rigid, co-rotating boundaries one needs this proof. Keywords: Kronecker delta function; Jacobi polynomials; Fluid shells.

Shear waves from an external acoustic Wave Generator acting on a crack in a rectangular strip

Abstract: An analytical solution for the wave equation in a rectangular elastic strip and in the presence of an external harmonic wave generator acting on a crack is derived This source is spatially distributed in a generalized mathematical form near the vicinity of the tip of the crack and acts normal to its faces Two limiting cases (1) a uniform distribution and (2) Dirac's delta function distribution are solved Examples of the relation between the displacement at the epicenter and the crack length for different body force distributions are shown

Functional Integrals In Stellar Dynamics

Abstract: In the late 70s, the functional integral formulation equivalent to the Fokker-Planck equation was worked out (Graham). We apply this functional integral formulation to gravitationally interacting systems, whose dynamics may be analyzed by separating the forces operating on a particle into a mean field force and fluctuations due to random collisions at intermediate range (scattering at small angles). In this poster the formalism is presented for short periods (in the stochastical meaning) to systems with isotropic distribution background in velocity space (different spatial densities are possible). Later the functional integral for the local change for the distribution function is evaluated in the steepest descent approximation. In the end we point out the applicability of the method to slowly evolving globular star clusters near thermal equilibrium. In conditions of slow evolution we can express the evolution of the orbits in terms of local deviation from equilibrium.

Statistics of photocounts of a field formed by superposition of a coherent signal and Brownian noise

Abstract: The authors consider the photocount statistics of a field formed by superposition of a coherent signal of an arbitrary form, and the Brownian noise (Ornstein-Uhlenbeck process). An analytical expression is obtained for the generating function of photocounts. In the case of a harmonic coherent signal, the authors analyze the count statistics of the superposed field as a function of the shift of the carrier frequency of the harmonic signal relative to the maximum of the Lorentz spectral contour, as a function of the partial intensity components of the superposed radiation, and also as a function of the product of the duration of photodetection and the spectral line width of the noise.

On a relationship between the delta function and the two-dimensional Fourier transform

Abstract: (1985). On a relationship between the delta function and the two-dimensional Fourier transform. International Journal of Electronics: Vol. 59, No. 5, pp. 653-654.