Showing papers on "Dirac delta function published in 2008"

Malliavin Calculus for Lévy Processes with Applications to Finance

Abstract: The Continuous Case: Brownian Motion.- The Wiener-Ito Chaos Expansion.- The Skorohod Integral.- Malliavin Derivative via Chaos Expansion.- Integral Representations and the Clark-Ocone formula.- White Noise, the Wick Product, and Stochastic Integration.- The Hida-Malliavin Derivative on the Space ? = S?(?).- The Donsker Delta Function and Applications.- The Forward Integral and Applications.- The Discontinuous Case: Pure Jump Levy Processes.- A Short Introduction to Levy Processes.- The Wiener-Ito Chaos Expansion.- Skorohod Integrals.- The Malliavin Derivative.- Levy White Noise and Stochastic Distributions.- The Donsker Delta Function of a Levy Process and Applications.- The Forward Integral.- Applications to Stochastic Control: Partial and Inside Information.- Regularity of Solutions of SDEs Driven by Levy Processes.- Absolute Continuity of Probability Laws.

Assessment of regularized delta functions and feedback forcing schemes for an immersed boundary method

Abstract: We present an improved immersed boundary method for simulating incompressible viscous flow around an arbitrarily moving body on a fixed computational grid. To achieve a large CFL number and to transfer quantities between Eulerian and Lagrangian domains effectively, we combined the feedback foreing scheme of the virtual boundary method with Peskin’s regularized delta function approach. Stability analysis of the proposed method was carried out for various types of regularized delta function. The stability regime of the 4-point regularized delta function was much wider than that of the 2-point delta function. An optimum regime of the feedback forcing is suggested on the basis of the analysis of stability limits and feedback forcing gains. The proposed method was implemented in a finite difference and fractional step context. The proposed method was tested on several flow problems and the findings were in excellent agreement with previous numerical and experimental results.

Analytical time-domain Green's functions for power-law media.

Abstract: Frequency-dependent loss and dispersion are typically modeled with a power-law attenuation coefficient, where the power-law exponent ranges from 0 to 2. To facilitate analytical solution, a fractional partial differential equation is derived that exactly describes power-law attenuation and the Szabo wave equation [“Time domain wave-equations for lossy media obeying a frequency power-law,” J. Acoust. Soc. Am. 96, 491–500 (1994)] is an approximation to this equation. This paper derives analytical time-domain Green’s functions in power-law media for exponents in this range. To construct solutions, stable law probability distributions are utilized. For exponents equal to 0, 1∕3, 1∕2, 2∕3, 3∕2, and 2, the Green’s function is expressed in terms of Dirac delta, exponential, Airy, hypergeometric, and Gaussian functions. For exponents strictly less than 1, the Green’s functions are expressed as Fox functions and are causal. For exponents greater than or equal than 1, the Green’s functions are expressed as Fox and ...

Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac

Abstract: We consider a stationary nonlinear Schroodinger equation with a repulsive delta-function impurity in one space dimension. This equation admits a unique positive solution and this solution is even. We prove that it is a minimizer of the associated energy on the subspace of even functions of $H^1(\R, \C)$, but not on all $H^1(\R, \C)$, and study its orbital stability.

Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation

Abstract: We consider the interaction between the Hermitian world, represented by a real delta-function potential -{alpha}{delta}(x), and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients i{beta}({delta}(x-L)-{delta}(x+L)). In the context of standard quantum mechanics, the effect of the introduction of the imaginary delta functions on the bound-state energy of the real delta function and its associated wave function is small for L large. However, scattering from the combined potentials does not conserve probability as conventionally defined. Both these problems can be studied instead in the context of quasi-Hermiticity, whereby quantum mechanics is endowed with a new metric {eta}, and consequently a new wave function {psi}(x), defined in terms of the original wave function {psi}(x) by means of {eta}. In this picture, working perturbatively in {beta}, the bound-state wave function is actually unchanged from its unperturbed form for |x| >L, is changed in a significant manner. In particular, there are incoming and outgoing waves on both sides of the potential. One can then no longer talk in terms of reflection and transmission coefficients, but the total right-moving flux is now conserved.

Regularization of fields for self-force problems in curved spacetime: Foundations and a time-domain application

Abstract: We propose an approach for the calculation of self-forces, energy fluxes and waveforms arising from moving point charges in curved spacetimes. As opposed to mode-sum schemes that regularize the self-force derived from the singular retarded field, this approach regularizes the retarded field itself. The singular part of the retarded field is first analytically identified and removed, yielding a finite, differentiable remainder from which the self-force is easily calculated. This regular remainder solves a wave equation which enjoys the benefit of having a nonsingular source. Solving this wave equation for the remainder completely avoids the calculation of the singular retarded field along with the attendant difficulties associated with numerically modeling a delta-function source. From this differentiable remainder one may compute the self-force, the energy flux, and also a waveform which reflects the effects of the self-force. As a test of principle, we implement this method using a 4th-order ($1+1$) code, and calculate the self-force for the simple case of a scalar charge moving in a circular orbit around a Schwarzschild black hole. We achieve agreement with frequency-domain results to $\ensuremath{\sim}0.1%$ or better.

One-particle density matrix and momentum distribution function of one-dimensional anyon gases

Abstract: We present a systematic study of the Green functions of a one-dimensional gas of impenetrable anyons. We show that the one-particle density matrix is the determinant of a Toeplitz matrix whose large N asymptotic is given by the Fisher–Hartwig conjecture. We provide a careful numerical analysis of this determinant for general values of the anyonic parameter, showing in full details the crossover between bosons and fermions and the reorganization of the singularities of the momentum distribution function. We show that the one-particle density matrix satisfies a Painleve VI differential equation that is then used to derive the small distance and large momentum expansions. We find that the first non-vanishing term in this expansion is always k−4, that is proved to be true for all couplings in the Lieb–Liniger anyonic gas and that can be traced back to the presence of a delta function interaction in the Hamiltonian.

Geometric nonlinear analysis of plates and cylindrical shells via a linearly conforming radial point interpolation method

Abstract: In this paper, the linearly conforming radial point interpolation method is extended for geometric nonlinear analysis of plates and cylindrical shells. The Sander’s nonlinear shell theory is utilized and the arc-length technique is implemented in conjunction with the modified Newton–Raphson method to solve the nonlinear equilibrium equations. The radial and polynomial basis functions are employed to construct the shape functions with Delta function property using a set of arbitrarily distributed nodes in local support domains. Besides the conventional nodal integration, a stabilized conforming nodal integration is applied to restore the conformability and to improve the accuracy of solutions. Small rotations and deformations, as well as finite strains, are assumed for the present formulation. Comparisons of present solutions are made with the results reported in the literature and good agreements are obtained. The numerical examples have demonstrated that the present approach, combined with arc-length method, is quite effective in tracing the load-deflection paths of snap-through and snap-back phenomena in shell problems.

Next-to-leading order hard scattering using fully unintegrated parton distribution functions

Abstract: We calculate the next-to-leading order fully unintegrated hard scattering coefficient for unpolarized gluon-induced deep inelastic scattering using the logical framework of parton correlation functions developed in previous work. In our approach, exact four-momentum conservation is maintained throughout the calculation. Hence, all nonperturbative functions, like parton distribution functions, depend on all components of parton four-momentum. In contrast to the usual collinear factorization approach where the hard scattering coefficient involves generalized functions (such as Dirac {delta} functions), the fully unintegrated hard scattering coefficient is an ordinary function. Gluon-induced deep inelastic scattering provides a simple illustration of the application of the fully unintegrated factorization formalism with a nontrivial hard scattering coefficient, applied to a phenomenologically interesting case. Furthermore, the gluon-induced process allows for a parametrization of the fully unintegrated gluon distribution function.

Nonextensive Statistical Mechanics and Nonlinear Dynamics

Abstract: This chapter addresses turbulence and related questions from a statistical– mechanical viewpoint. More precisely, as one of the many existing realizations of nontrivial–nonlinear dynamical phenomena, which stands at the grounding of statistical mechanics itself. Consistently, we shall first review some general aspects of the statistical approach of mechanical phenomena, and only later make a connection with turbulence, in Sect. 4.4. A mechanical foundation of statistical mechanics from first principles should essentially include, in one way or another, the following main steps. (i) Adopt a microscopic dynamics. This dynamics is typically deterministic, i.e. without any phenomenological noise or stochastic ingredient, so that the foundation may be considered as from first principles. This dynamics could be Newtonian, or quantum, or relativistic mechanics (or some other mechanics to be found in future) of a many-body system composed by say N interacting elements or fields. It could also be conservativeor dissipative-coupled maps, or even cellular automata. Consistently, time t could be continuous or discrete. The same is valid for space. The quantity which is defined in space-time could itself be continuous or discrete. For example, in quantum mechanics, the quantity is a complex continuous variable (the wave function) defined in a continuous space-time. On the other extreme, we have cellular automata, for which all three relevant variables—time, space and the quantity therein defined—are discrete. In the case of a Newtonian mechanical system of particles, we may think of N Dirac delta functions localized in continuous spatial positions which depend on a continuous time. Langevin-like equations (and associated Fokker–Planck-like equations) are typically considered not microscopic, but mesoscopic instead. The reason, of course, is the fact that they include at their very formulation, i.e. in an essential manner, some sort of noise. Consequently, they should not be used as a

Generalized Israel Junction Conditions for a Fourth-Order Brane World

Abstract: We discuss a general fourth-order theory of gravity on the brane. In general, the formulation of the junction conditions (except for Euler characteristics such as Gauss-Bonnet term) leads to the higher powers of the delta function and requires regularization. We suggest the way to avoid such a problem by imposing the metric and its first derivative to be regular at the brane, while the second derivative to have a kink, the third derivative of the metric to have a step function discontinuity, and no sooner as the fourth derivative of the metric to give the delta function contribution to the field equations. Alternatively, we discuss the reduction of the fourth-order gravity to the second-order theory by introducing an extra tensor field. We formulate the appropriate junction conditions on the brane. We prove the equivalence of both theories. In particular, we prove the equivalence of the junction conditions with different assumptions related to the continuity of the metric along the brane.

Applications of continuity and discontinuity of a fractional derivative of the wave functions to fractional quantum mechanics

Abstract: The space fractional Schrodinger equation with a finite square potential, periodic potential, and delta-function potential is studied in this paper. We find that the continuity or discontinuity condition of a fractional derivative of the wave functions should be considered to solve the fractional Schrodinger equation in fractional quantum mechanics. More parity states than those given by standard quantum mechanics for the finite square potential well are obtained. The corresponding energy equations are derived and then solved by graphical methods. We show the validity of Bloch’s theorem and reveal the energy band structure for the periodic potential. The jump (discontinuity) condition for the fractional derivative of the wave function of the delta-function potential is given. With the help of the jump condition, we study some delta-function potential fields. For the delta-function potential well, an alternate expression of the wave function (the H function form of it was given by Dong and Xu [J. Math. Phy...

Optimal Image Subtraction Method: Summary Derivations, Applications, and Publicly Shared Application Using IDL

Abstract: To detect objects that vary in brightness or spatial coordinates over time, C. Alard and R. H. Lupton in 1998 proposed an "optimal image subtraction" (OIS) method that constructs a convolution kernel from a set of matching stars distributed across the two images to be subtracted. Using multivariable least squares, the kernel is derived and can be designed to vary by pixel coordinates across the convolved image. Local effects in the optics, including aberrations or other spatially sensitive perturbations to a perfect image, can be mitigated. This paper presents the specific systems of equations that originate from the OIS method. Also included is a complete description of the Gaussian components basis vectors used by Alard & Lupton to construct the convolution kernel. An alternative set of basis vectors, called the delta function basis, is also described. Important issues are addressed, including the selection of the matching stars, differential background correction, constant photometric flux, contaminated pixel masking, and alignment at the subpixel level. Computer algorithms for the OIS method were developed, written using the Interactive Data Language (IDL), and applications demonstrating these algorithms are presented.

Statistical universalities in fragmentation under scaling symmetry with a constant frequency of fragmentation

Abstract: This paper analyses statistical universalities that arise over time during constant frequency fragmentation under scaling symmetry. The explicit expression of particle-size distribution obtained from the evolution kinetic equation shows that, with increasing time, the initial distribution tends to the ultimate steady-state delta function through at least two intermediate universal asymptotics. The earlier asymptotic is the well-known log-normal distribution of Kolmogorov (1941 Dokl. Akad. Nauk. SSSR 31 99–101). This distribution is the first universality and has two parameters: the first and the second logarithmic moments of the fragmentation intensity spectrum. The later asymptotic is a power function (stronger universality) with a single parameter that is given by the ratio of the first two logarithmic moments. At large times, the first universality implies that the evolution equation can be reduced exactly to the Fokker–Planck equation instead of making the widely used but inconsistent assumption about the smallness of higher than second order moments. At even larger times, the second universality shows evolution towards a fractal state with dimension identified as a measure of the fracture resistance of the medium.

One-particle density matrix and momentum distribution function of one-dimensional anyon gases

Abstract: We present a systematic study of the Green functions of a one-dimensional gas of impenetrable anyons. We show that the one-particle density matrix is the determinant of a Toeplitz matrix whose large N asymptotic is given by the Fisher-Hartwig conjecture. We provide a careful numerical analysis of this determinant for general values of the anyonic parameter, showing in full details the crossover between bosons and fermions and the reorganization of the singularities of the momentum distribution function. We show that the one-particle density matrix satisfies a Painleve VI differential equation, that is then used to derive the small distance and large momentum expansions. We find that the first non-vanishing term in this expansion is always k^{-4}, that is proved to be true for all couplings in the Lieb-Liniger anyonic gas and that can be traced back to the presence of a delta function interaction in the Hamiltonian.

High Order Numerical Quadratures to One Dimensional Delta Function Integrals

Abstract: We study high order numerical quadratures to one dimensional delta function integrals in this paper. This is motivated by the fact that traditional numerical quadratures give only first order accuracy in general. We provide criteria on discrete delta functions and support size formulas which ensure any desired accuracy of the numerical quadratures. Discrete delta functions and support size formulas satisfying these criteria are designed. Numerical examples are presented to verify the performed analysis and the high order accuracy of the proposed numerical quadratures.

A fractal quantum mechanical model with Coulomb potential

Abstract: We study the Schrodinger operator $ H = - \Delta + V $ on the product of two copies of an infinite blowup of the Sierpinski gasket, where $ V$ is the analog of a Coulomb potential ($\Delta V$ is a multiple of a delta function). So $H$ is the analog of the standard Hydrogen atom model in nonrelativistic quantum mechanics. Like the classical model, we show that the essential spectrum of $H$ is the same as for $ - \Delta $, and there is a countable discrete spectrum of negative eigenvalues.

Entanglement mechanisms in one-dimensional potential scattering

Abstract: When two non-relativistic particles scatter in one dimension, they can become entangled. This entanglement process is constrained by the symmetries of the scattering system and the boundary conditions on the incoming state. Applying these constraints, three different mechanisms of entanglement can be identified: the superposition of reflected and transmitted modes, momentum correlations of the reflected mode due to inversion of the relative momentum, and momentum correlations in the transmitted and reflected modes due to dependence of the scattering amplitude on the relative momentum. We consider three standard potentials, the hard core, Dirac delta, and double Dirac delta, and show that the relative importance of these mechanisms depends on the interaction and on the properties of the incoming wavefunction. We find that even when the momenta distributions of the incoming articles are sharply peaked, entanglement due to the momentum correlations generated by reflection can be quite large for particles with unequal mass.

Nonstandard Analysis and Jump Conditions for Converging Shock Waves

Abstract: Nonstandard analysis is an area of modern mathematics that studies abstract number systems containing both infinitesimal and infinite numbers. This article applies nonstandard analysis to derive jump conditions for one-dimensional, converging shock waves in a compressible, inviscid, perfect gas. It is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval. Predistributions of the Heaviside function and the Dirac delta measure are introduced to model the flow parameters across a shock wave. The equations of motion expressed in nonconservative form are then applied to derive unambiguous relationships between the jump functions for the flow parameters.

One-dimensional anyons with competing $\delta$-function and derivative $\delta$-function potentials

Abstract: We propose an exactly solvable model of one-dimensional anyons with competing $\delta$-function and derivative $\delta$-function interaction potentials. The Bethe ansatz equations are derived in terms of the $N$-particle sector for the quantum anyonic field model of the generalized derivative nonlinear Schr\"{o}dinger equation. This more general anyon model exhibits richer physics than that of the recently studied one-dimensional model of $\delta$-function interacting anyons. We show that the anyonic signature is inextricably related to the velocities of the colliding particles and the pairwise dynamical interaction between particles.

Fast soliton scattering by attractive delta impurities

Abstract: We study the Gross-Pitaevskii equation with an attractive delta function potential and show that in the high velocity limit an incident soliton is split into reflected and transmitted soliton components plus a small amount of dispersion. We give explicit analytic formulas for the reflected and transmitted portions, while the remainder takes the form of an error. Although the existence of a bound state for this potential introduces difficulties not present in the case of a repulsive potential, we show that the proportion of the soliton which is trapped at the origin vanishes in the limit.

A variational treatment for general elliptic equations of the flame propagation type : regularity of the free boundary

Abstract: We develop a variational theory to study the free boundary regularity problem for elliptic operators: Lu=Dj(aij(x)Diu)+biui+c(x)u=0 in {u>0}, 〈aij(x)∇u,∇u〉=2 on ∂{u>0}. We use a singular perturbation framework to approximate this free boundary problem by regularizing ones of the form: Lue=βe(ue), where βe is a suitable approximation of Dirac delta function δ0. A useful variational characterization to solutions of the above approximating problem is established and used to obtain important geometric properties that enable regularity of the free boundary. This theory has been developed in connection to a very recent line of research as an effort to study existence and regularity theory for free boundary problems with gradient dependence upon the penalization.

One-dimensional anyons with competing δ-function and derivative δ-function potentials

Abstract: We propose an exactly solvable model of one-dimensional anyons with competing δ-function and derivative δ-function interaction potentials. The Bethe ansatz equations are derived in terms of the N-particle sector for the quantum anyonic field model of the generalized derivative nonlinear Schrodinger equation. This more general anyon model exhibits richer physics than that of the recently studied one-dimensional model of δ-function interacting anyons. We show that the anyonic signature is inextricably related to the velocities of the colliding particles and the pairwise dynamical interaction between particles.

Some Applications of Dirac's Delta Function in Statistics for More Than One Random Variable

Abstract: In this paper, we discuss some interesting applications of Dirac's delta function in Statistics We have tried to extend some of the existing results to the more than one variable case While doing that, we particularly concentrate on the bivariate case

Dirac delta methods for Helmholtz transmission problems

Abstract: In this paper we use a boundary integral method with single layer potentials to solve a class of Helmholtz transmission problems in the plane. We propose and analyze a novel and very simple quadrature method to solve numerically the equivalent system of integral equations which provides an approximation of the solution of the original problem with linear convergence (quadratic in some special cases). Furthermore, we also investigate a modified quadrature approximation based on the ideas of qualocation methods. This new scheme is again extremely simple to implement and has order three in weak norms.