Terms Of Delta Functions Research Articles

Distributional Representation of a Special Fox–Wright Function with an Application

A review of the literature demonstrates that the Fox–Wright function is not only a mathematical puzzle, but its role is naturally to represent basic physical phenomena. Motivated by this fact, we studied a new representation of this function in terms of complex delta functions. This representation was useful to compute its Laplace transform with respect to the third parameter γ for which it also generalizes the one and two-parameter Mittag-Leffler functions. New identities involving the Fox–Wright function were discussed and used to simplify the results. Different fractional transforms were evaluated and the solution of a fractional kinetic equation was obtained by using its new representation. Several new properties of this function were discussed as a distribution.

Компактность линейных операторов на квазибанаховых пространствах голоморфных функций

We state conditions under which some classical operators acting from abstract quasi-Banach spaces of functions holomorphic in a plain domain into a weighted space of the same functions with sup-norm are compact. It is obtained abstract criteria for the compactness of a linear operator on an arbitrary quasi-Banach space which are stated in terms of delta-functions and formulate their realizations for both classical and generalized Fock spaces. The above results are applied to the weighted composition operator. It is established some conditions for the compactness of this operator which are given in terms of norms of delta-functions in the corresponding dual spaces. These results are essential generalizations of the known Zorboska’s ones. Namely, we significantly extended the class of weighted spaces of holomorphic functions with uniform norms for which one can state some conditions for the compactness of an arbitrary linear operator or the weighted composition operator.

A regularization approach for solving the super-Gaussian Poisson-Boltzmann model with heterogeneous dielectric functions

A regularization method is introduced for the first time in the literature for treating charge singularities in the heterogeneous dielectric Poisson-Boltzmann (PB) model. In the existing Gaussian and super-Gaussian electrostatic analysis, the singular sources in terms of Dirac delta functions are approximated by trilinear interpolation, which introduces a large error in potential solutions, and has to rely on an error cancellation for delivering acceptable estimates of the electrostatic free energy. To overcome this difficulty, a dual decomposition of potential and dielectric function is carried out in the proposed regularization, so that the charge singularities can be analytically captured by the Coulomb potential, while the reaction field potential satisfies a regularized PB equation with a new source term. A rigorous analysis has been conducted to show that a super-Gaussian density, instead of a Gaussian one, is required to guarantee a small neighborhood around each atom center with a nearly homogeneous dielectric medium, so that the source term can be well defined in the sense of distribution. Moreover, the well-posedness of the regularized formulation has been proved, and the regularity of the weak solution has been clarified. In the numerical implementation, special considerations are given in calculating the source term, by using analytical differentiations, instead of numerical approximations, whenever possible. The accuracy, convergence, efficiency, and robustness of the proposed regularization are numerically verified via benchmark studies. It is found that the regularization is more accurate than the trilinear method, as well as produces a faster convergence in energy estimation. Moreover, the grid artifact or artificial grid energy is completely eliminated in the present finite difference PB model.

Comments on Computation of Free-Free Transitions in Atomic Physics

The amplitude T for ‘free-free’ processes such as bremsstrahlung or photoabsorption by an electron in the continuum in the presence of an external field, is usually written as the matrix element of the radiation operator taken between two continuum states. However, unlike the case when at least one of the states is bound, as in radiative transitions, electron capture or the photo-effect, this expression contains unphysical term, proportional to a delta function, and is not really the physical amplitude Tphys. We first give an a priori definition of Tphys in terms of the scattering parts of the continuum functions, which does not have this delta function term and has an obvious interpretation in terms of time-ordered diagrams. We then show that when the formal amplitude T is modified by a long-distance cutoff, the modified form Tα approaches Tphys as the cutoff is removed. The modified form may be used as the basis for calculation and approximations without the need to introduce further cutoffs at a later stage.

Fractional Schrödinger equation and anomalous relaxation: Nonlocal terms and delta potentials

We review and extend some results for the fractional Schrödinger equation by considering nonlocal terms or potential given in terms of delta functions. For each case, we have obtained the solution in terms of the Green function approach.

Regularization methods for the Poisson-Boltzmann equation: Comparison and accuracy recovery

A significant challenge in numerical solution to the Poisson-Boltzmann equation is due to singular charge sources in terms of Dirac delta functions. To overcome this difficulty, several regularization methods have been developed, in which the potential function is decomposed into two or three parts so that the singular component can be analytically solved using the Green's function, while other components become bounded. However, it was observed in the literature that some regularization methods are significantly less accurate than the others for unclear reasons, even though they are analytically equivalent. To understand this discrepancy, the numerical performance of four popular regularization methods is investigated in this work by implementing them with the Matched Interface and Boundary (MIB) approach, which is a sophisticated finite difference method for treating elliptic interface problems with discontinuous coefficients. With all four methods showing second order convergence, accuracy reduction is numerically observed in two schemes. This paper provides numerical analysis and experiment to trace the source of such reduction, and links the error to the fact that the Laplacian of Green's function is dropped outside the protein domain. While this term is analytically vanishing, its numerical negligence introduces a discretization error. Formulating via a proper elliptic interface problem, an effective accuracy recovery technique is proposed so that all four methods yield the same high precision. With this study, all involved regularization schemes are better understood and well connected into a unified framework.

Singularities in twist-3 quark distributions

We find in one-loop calculations and spectator models that twist-3 GPDs exhibit discontinuities. In the forward limit these discontinuities grow into Dirac delta functions which are essential to satisfy the sum rules involving twist-3 PDFs. We calculate twist-3 quasi-PDFs as a function of longitudinal momentum and identify the Dirac delta function terms with momentum components in the nucleon state that do not scale as the nucleon is boosted to the infinite momentum frame.

A regularization approach for solving Poisson’s equation with singular charge sources and diffuse interfaces

Singular charge sources in terms of Dirac delta functions present a well-known numerical challenge for solving Poisson’s equation. For a sharp interface between inhomogeneous media, singular charges could be analytically treated by fundamental solutions or regularization methods. However, no analytical treatment is known in the literature in case of a diffuse interface of complex shape. This letter reports the first such regularization method that represents the Coulomb potential component analytically by Green’s functions to account for singular charges. The other component, i.e., the reaction field potential, then satisfies a regularized Poisson equation with a smooth source and the original elliptic operator. The regularized equation can then be simply solved by any numerical method. By considering two benchmark problems, the proposed regularization method is numerically validated and compared with a semi-analytical quasi-harmonic method.

A New Representation of the k-Gamma Functions

The products of the form z ( z + l ) ( z + 2 l ) … ( z + ( k − 1 ) l ) are of interest for a wide-ranging audience. In particular, they frequently arise in diverse situations, such as computation of Feynman integrals, combinatory of creation, annihilation operators and in fractional calculus. These expressions can be successfully applied for stated applications by using a mathematical notion of k-gamma functions. In this paper, we develop a new series representation of k-gamma functions in terms of delta functions. It led to a novel extension of the applicability of k-gamma functions that introduced them as distributions defined for a specific set of functions.

Statistical Analysis of the Reaction Progress Variable and Mixture Fraction Gradients in Flames Propagating into Droplet Mist: A Direct Numerical Simulation Analysis

ABSTRACTThe statistics of reaction progress variable, , and mixture fraction, , and their gradients (i.e., and ) in flames propagating in droplet mist, where the fuel was supplied in the form of monodisperse droplets, have been analyzed for different values of turbulent velocity fluctuations (), droplet equivalence ratios (, and droplet diameters ( based on three-dimensional direct numerical simulations (DNS) in a canonical configuration under decaying turbulence. The combustion process in the gaseous phase has been found to take place predominantly in fuel-lean mode, even for . The probability of finding fuel-lean mixture increases with increasing initial droplet diameter due to slower evaporation of larger droplets. It has been shown that the joint probability density function (i.e., joint PDF) of and (i.e., ), cannot be approximated in terms of discrete delta functions throughout the flame brush for the cases considered here. Furthermore, the magnitude of cannot be adequately approximated by the product of marginal PDFs of , and variable, (i.e., ). The statistical properties of the Favre probability density functions (Favre-PDFs) of the mixture fraction, , and oxidizer-based reaction progress variable, , have been analyzed at several locations across the flame brush and a -function distribution has been found to capture the Favre-PDFs of and obtained from the DNS data. Furthermore, a log-normal distribution has been shown to capture the qualitative behaviors of the PDFs of the gradient of the mixture fraction and the gradient of the reaction progress variable, and , respectively, but discrepancies between the log-normal distribution and the DNS data were observed at the tails of PDFs. In addition, the interrelation between and was examined in terms of the PDFs of the cosine of the angle between them (i.e., and it was observed that most droplet cases exhibited much greater likelihood of positive values of than negative values. Finally, the joint PDF of and , , has been compared with that of P( (i.e., assuming statistical independence of and ) and a good level of agreement has been obtained. The bivariate log-normal distribution has been considered both assuming correlation between and and assuming no correlation for the purpose of modeling , and the variant with no correlation has been found to be more successful in capturing qualitative behavior of although quantitative discrepancies have been observed due to inaccuracies involved in parameterizing P( and P(by log-normal distributions.

Consistency and derangements in brane tilings

Brane tilings describe Lagrangians (vector multiplets, chiral multiplets, and the superpotential) of four-dimensional supersymmetric gauge theories. These theories, written in terms of a bipartite graph on a torus, correspond to worldvolume theories on N D3-branes probing a toric Calabi–Yau threefold singularity. A pair of permutations compactly encapsulates the data necessary to specify a brane tiling. We show that geometric consistency for brane tilings, which ensures that the corresponding quantum field theories are well behaved, imposes constraints on the pair of permutations, restricting certain products constructed from the pair to have no one-cycles. Permutations without one-cycles are known as derangements. We illustrate this formulation of consistency with known brane tilings. Counting formulas for consistent brane tilings with an arbitrary number of chiral bifundamental fields are written down in terms of delta functions over symmetric groups.

Scalar scattering via conformal higher spin exchange

Theories containing infinite number of higher spin fields require a particular definition of summation over spins consistent with their underlying symmetries. We consider a model of massless scalars interacting (via bilinear conserved currents) with conformal higher spin fields in flat space. We compute the tree-level four-scalar scattering amplitude using a natural prescription for summation over an infinite set of conformal higher spin exchanges and find that it vanishes (modulo delta-function terms having support on measure-zero domain in phase space). Independently, we show that this vanishing of the scalar scattering amplitude is, in fact, implied by the global conformal higher spin symmetry of this model. We also discuss one-loop corrections to the four-scalar scattering amplitude.

Metric perturbations from eccentric orbits on a Schwarzschild black hole. I. Odd-parity Regge-Wheeler to Lorenz gauge transformation and two new methods to circumvent the Gibbs phenomenon

We calculate the odd-parity, radiative ($\ensuremath{\ell}\ensuremath{\ge}2$) parts of the metric perturbation in Lorenz gauge caused by a small compact object in eccentric orbit about a Schwarzschild black hole. The Lorenz gauge solution is found via gauge transformation from a corresponding one in Regge-Wheeler gauge. Like the Regge-Wheeler gauge solution itself, the gauge generator is computed in the frequency domain and transferred to the time domain. The wave equation for the gauge generator has a source with a compact, moving delta-function term and a discontinuous noncompact term. The former term allows the method of extended homogeneous solutions to be applied (which circumvents the Gibbs phenomenon). The latter has required the development of new means to use frequency domain methods and yet be able to transfer to the time domain while avoiding Gibbs problems. Two new methods are developed to achieve this: a partial annihilator method and a method of extended particular solutions. We detail these methods and show their application in calculating the odd-parity gauge generator and Lorenz gauge metric perturbations. A subsequent paper will apply these methods to the harder task of computing the even-parity parts of the gauge generator.

Maximally helicity-violating diagrams in twistor space and the twistor action

MHV diagrams give an efficient Feynman diagram-like formalism for calculating gauge theory scattering amplitudes on momentum space. Although they arise as the Feynman diagrams from an action on twistor space in an axial gauge, the main ingredients were previously expressed only in momentum space and momentum twistor space. Here we show how the formalism can be elegantly derived and expressed entirely in twistor space. This brings out the underlying superconformal invariance of the framework (up to the choice of a reference twistor used to define the axial gauge) and makes the twistor support transparent. Our treatment is largely independent of signature, although we focus on Lorentz signature. Starting from the N=4 super-Yang-Mills twistor action, we obtain the propagator for the anti-holomorphic Dolbeault-operator as a delta function imposing collinear support with the reference twistor defining the axial gauge. The MHV vertices are also expressed in terms of similar delta functions. We obtain concrete formulae for tree-level N^{k}MHV diagrams as a product of MHV amplitudes with an R-invariant for each propagator; here the R-invariant manifests superconformal as opposed to dual-superconformal invariance. This gives the expected explicit support on k+1 lines linked by k further lines associated to the propagators. The R-invariants arising correspond to those obtained in the dual conformal invariant momentum twistor version of the formalism, but differences arise in the specification of the boundary terms. Surprisingly, in this framework, some finite loop integrals can be performed as simply as those for tree diagrams.

Soft-gluon resummation for boosted top-quark production at hadron colliders

We investigate the production of highly energetic top-quark pairs at hadron colliders, focusing on the case where the invariant mass of the pair is much larger than the mass of the top quark. In particular, we set up a factorization formalism appropriate for describing the differential partonic cross section in the double soft and small-mass limit, and explain how to resum simultaneously logarithmic corrections arising from soft gluon emission and from the ratio of the pair-invariant mass to that of the top quark to next-to-next-to-leading logarithmic accuracy. We explore the implications of our results on approximate next-to-next-to-leading order formulas for the differential cross section in the soft limit, pointing out that they offer a simplified calculational procedure for determining the currently unknown delta-function terms in the limit of high invariant mass.

Ordinary differential equations with delta function terms

This article is devoted to nonlinear ordinary differential equations with additive or multiplicative terms consisting of Dirac delta functions or derivatives thereof. Regularizing the delta function terms produces a family of smooth solutions. Conditions on the nonlinear terms, relating to the order of the derivatives of the delta function part, are established so that the regularized solutions converge to a limiting distribution.

Hyperfine splittings in the system

Recent measurements of the $\eta_b(1S)$, the ground state of the $b\bar{b}$ system, show the splitting between it and the $\Up(1S)$ to be 69.5$\pm$3.2 MeV, considerably larger than lattice QCD and potential model predictions, including recent calculations published by us. The models are unable to incorporate such a large hyperfine splitting within the context of a consistent description of the energy spectrum and decays. We demonstrate that in our model, which incorporates a relativistic kinetic energy term, a linear confining term including its scalar-exchange relativistic corrections, and the complete one-loop QCD short distance potential, such a consistent description, including the measured hyperfine splitting, can be obtained by not softening the delta function terms in the hyperfine potential. We calculate the hyperfine splitting to be 67.5 MeV.

What is the boundary condition for the radial wave function of the Schrödinger equation?

There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.

Double excitation effect in non-adiabatic time-dependent density functional theory with an analytic construction of the exchange–correlation kernel in the common energy denominator approximation

Time-dependent density functional (response) theory (TDDF(R)T) is applied almost exclusively in its adiabatic approximation (ATDDFT), which is restricted to predominantly single electronic excitations and neglects additional roots of the TDDFT eigenvalue problem stemming from the interaction between single and double excitations. We incorporate the effect of the latter interaction into a non-adiabatic frequency-dependent and spatially non-local Hartree-exchange-correlation (Hxc) kernel fCEDAHxc (r1, r2, omega), the explicit analytical expression of which is derived for interacting single and double excitations well separated from the other excitations, within the common energy denominator approximation (CEDA) for the Kohn-Sham (KS) and interacting density response functions, chis and chi, respectively. The kernel fCEDAHxc (r1, r2, omega) obtained from the direct analytical inverse of chiCEDAs and chiCEDA is a sum of the delta-function and non-local orbital-dependent spatial terms with frequency-dependent factors, with which fCEDAHxc acquires a modulated quadratic dependence on omega. The effective incorporation in fCEDAHxc of the complete manifold of excited states (through the delta function term) represents an extension of the kernel reported by Maitra, Zhang, Cave, and Burke [J. Chem. Phys., 2004, 120, 5932]. In the TDDFT eigenvalue equations considered in the diagonal approximation, fCEDAHxc generates two excitation energies omegaq and omegaq+1, which both correspond to the same single KS excitation omegasq, thus producing the effect of the single-double excitation interaction.

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

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 Wright functions and are noncausal. However, numerical computations demonstrate that for observation points only one wavelength from the radiating source, the Green's function is effectively causal for power-law exponents greater than or equal to 1. The analytical time-domain Green's function is numerically verified against the material impulse response function, and the results demonstrate excellent agreement.