Convolution Formula Research Articles

ON CONVOLUTION OF SOME TYPE OF THE NUMBERS CONNECTED WITH GENERALIZED REPUNITS

We will concentrate on special types of numbers Jn(k) = n 2 X i=0 � n i � k n 2 i , where k is any nonnegative integer and n is any positive integer greater than 1. These numbers are a generalization of generalized repunits Rn(b). In this paper some results about divisibility of Jn(k) are stated. Further the generating function and a m-fold convolution formula for the numbers Jn(k) is found.

The Finite Difference Methods for Fractional Ordinary Differential Equations

Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.

Some Convolution Identities and an Inverse Relation Involving Partial Bell Polynomials

We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with nested compositions of partial Bell polynomials.

Pion-exchange and Fermi-motion effects on the proton-deuteron Drell-Yan process

Within a nuclear model where the deuteron has $NN$ and $\ensuremath{\pi}NN$ components, we derive a convolution formula for investigating the Drell-Yan process in proton-deuteron ($pd$) reactions. The contribution from the $\ensuremath{\pi}NN$ component is expressed in terms of a pion momentum distribution that depends sensitively on the $\ensuremath{\pi}NN$ form factor. With a $\ensuremath{\pi}NN$ form factor determined by fitting the $\ensuremath{\pi}N$ scattering data up to invariant mass $W=1.3\text{ }\text{ }\mathrm{GeV}$, we find that the pion-exchange and nucleon Fermi-motion effects can change significantly the ratios between the proton-deuteron and proton-proton Drell-Yan cross sections, ${R}_{pd/pp}={\ensuremath{\sigma}}^{pd}/(2{\ensuremath{\sigma}}^{pp})$, in the region where the partons emitted from the target deuteron are in the Bjorken ${x}_{2}\ensuremath{\gtrsim}0.4$ region. The calculated ratios ${R}_{pd/pp}$ at 800 GeV agree with the available data. Predictions at 120 GeV for analyzing the forthcoming data from Fermilab are presented.

A combinatorial proof of Shapiroʼs Catalan convolution

Shapiro proved an elegant convolution formula involving Catalan numbers of even index. This paper gives a combinatorial proof of his formula. In addition, we show that it is equivalent to an alternating convolution formula of central binomial coefficients.

OPE for null Wilson loops and open spin chains

Maximal helicity-violating scattering amplitudes in N=4 supersymmetric Yang–Mills theory are dual to Wilson loops on closed null polygons. We perform their operator product expansion analysis in two-dimensional kinematics in the soft-collinear approximation which corresponds to the case when some light-cone distances vanish. We construct the expansion in terms of multi-particle “heavy”–light operators, where the “heavy” fields are identified with the Wilson lines defining the OPE channel and the light fields emerge from the curvature of the contour. The correlation function of these define the remainder function. We study the dilatation operator for these operators at one-loop order and find that it corresponds to a non-compact open spin chain. This provides an alternative view on elementary excitations propagating on the GKP string at weak coupling, which now correspond to particles traveling along an open spin chain. The factorized structure of the Wilson loop in the soft limit allows one to represent the two-loop correction to the octagon Wilson loop as a convolution formula and find the corresponding remainder function.

A Proof of Catalan's Convolution Formula

Abstract.A new proof is given for the

Fibre de Milnor motivique à l’infini et composition avec un polynôme non dégénéré

Let k be a field of characteristic zero and P be a Laurent polynomial in d variables, with coefficients in k and non degenerate for its Newton polyhedron at infinity. Let (f l ) be d non constant functions with separated variables and defined on smooth varieties. As Guibert, Loeser and Merle in the local case, we compute in this article the motivic Milnor fiber at infinity of P(f) in terms of the Newton polyhedron at infinity of P. For P equal to the sum x 1 +x 2 , we obtained a Thom-Sebastiani formula. Then we can introduce a notion of motivic vanishing cycles of a function g for the infinite value denoted by S g∞,U Φ , and which verified, as in the local case, a convolution formula. In particular if g is the polynomial x 1 +...+x n +1/x 1 ...x n , we show that the spectrum S g,∞,U Φ is 1+t+..+t n which coincides with the spectrum at infinity of g considered by Douai and Sabbah.

Application of the nonlinear antenna theory model to a tall tower struck by lightning for the evaluation of return stroke channel current and radiated electromagnetic fields

[1] In this paper an improved antenna theory (AT) model with nonlinearly varying resistive loading and fixed inductive loading is used to electromagnetically simulate lightning strikes to tall structures. Measurement data captured from Toronto's CN tower are used to verify the validity of the new model. Both the return stroke channel (RSC) and the tower are modeled by straight thin conducting wires. The wire model of the channel is assumed to have distributed nonlinear resistive elements as a function of current and time, adopted from the numerical models of a spark channel and consequent shockwave from a lightning discharge, yielding a varying value of the channel radius from the base to the cloud along the RSC. Such distributed elements are used to take into account the current attenuation while propagating along the channel and varying propagation speeds lower than the speed of light. RSC current distribution and radiated electromagnetic fields in near, intermediate, and far range distances predicted by the proposed model are compared with those obtained from the measurement data and with those of the original AT model and the AT with fixed inductive loading (ATIL-F) model. Current wave propagation speed profile in RSC and tower is investigated as a function of height as well. The effects of applying different tower geometry models are also studied. It is shown that the new model is able to reproduce one of the characteristic features of the electromagnetic fields radiated by lightning, namely, the far-field inversion of polarity with a zero crossing occurring in the tens of microseconds range. We have also investigated the effect of nonlinearity of the channel assumed in the new model. It is shown that among the electromagnetic models, distributed nonlinear resistance along the channel leads to a zero crossing in the tens of microseconds range even for large values of resistance. It is also shown that decreasing the nonlinearity results in the predictions asymptotically converging to those of the ATIL-F model in which a uniformly distributed resistance along the channel is used. Further, the profile of the return stroke current attenuation and speed and removed charge as a function of height are investigated. Finally, the effect of lossy ground is analyzed using Wait-Cooray convolution formula in the time domain to verify the validity of PEC assumption in the proposed model.

Inversion of the generalized Fibonacci matrix by convolution

A convolution formula containing the generalized Fibonacci numbers and applications of this formula are investigated. Starting from the convolution formula, we derive combinatorial identities involving generalized and usual Fibonacci numbers, as well as the Lucas numbers. The inversion of a lower triangular matrix and the generalized inversion of strictly lower triangular Toeplitz matrix whose non-zero elements are generalized Fibonacci numbers are considered.

Five convolution formulae of orthogonal polynomials

By means of the classical Lagrange expansion theorem, five convolution formulae are established for the orthogonal polynomials named after Laguerre, Jacobi, Meixner, Gegenbauer and Pollaczek, that contain the well-known Hagen–Rothe formula for binomial coefficients as common special case.

Convolutions and zeros of orthogonal polynomials

In an attempt to answer a long standing open question of Al-Salam we generate various beautiful formulae for convolutions of orthogonal polynomials similar to U n ( x ) = ∑ k = 0 n P k ( x ) P n − k ( x ) , where U n ( x ) are the Chebyshev polynomials of the second kind and P k ( x ) are the Legendre polynomials. The results are derived both via the generating functions approach and a new convolution formulae for hypergeometric functions. We apply some addition formulae similar to the well-known expansion H n ( x + y ) = 2 − n / 2 ∑ k = 0 n ( n k ) H k ( 2 x ) H n − k ( 2 y ) for the Hermite polynomials, due to Appell and Kampé de Fériet, to obtain new interesting inequalities about the zeros of the corresponding orthogonal polynomials.

Thermal Bending of Circular Plates for Non-Axisymmetrical Problems

Using the Green function, the boundary integral formula and natural boundary integral equation for thermal elastic problems are obtained. Then based on bending solutions to circular plates subjected to the non-axi- symmetrical load, by utilizing the Fourier series and convolution formulae, the bending solutions under non-axisymmetrical thermal conditions have been obtained. The calculating process is simple. Examples show the discussed methods are effective.

Convolution and Reciprocity Formulas for Bernoulli Polynomials

AbstractWe prove a new convolution identity for sums of products of two Bernoulli polynomials. This can be rewritten to obtain a reciprocity relation for a related sum. The proof uses some results on Stirling numbers of both kinds which are of independent interest. In particular, a class of polynomials related to the Stirling numbers of the second kind turns out to be a useful tool.

On another extension of q-Pfaff-Saalschütz formula

In this paper we give an extension of q-Pfaff-Saalschutz formula by means of Andrews-Askey integral. Applications of the extension are also given, which include an extension of q-Chu-Vandermonde convolution formula and some other q-identities.

On Pricing Equity-Linked Investment Products with a Threshold Expense Structure

This paper considers a certain expense structure where a vendor of equity-linked investment product will collect its expenses continuously from the investor's account whenever the investment performance exceeds a certain threshold level. Under the Black-Scholes framework, we derive compact convolution formulas for evaluating the total expenses to be collected during the investment period by using the joint Laplace transform of the Brownian motion and its excursion time. We provide numerical examples for illustration.

Survey of deterministic and stochastic service curve models in the network calculus

In recent years service curves have proven a powerful and versatile model for performance analysis of network elements, such as schedulers, links, and traffic shapers, up to entire computer networks, like the Internet. The elegance of the concept of service curve is due to intuitive convolution formulas that determine the data departures of a system from its arrivals and its service curve. This fundamental relation constitutes the basis of the network calculus and relates it to systems theory, however, under a different, so-called min-plus algebra. As in systems theory, the particular strength of the minplus convolution is the ability to concatenate tandem systems along a network path. This facilitates the notion of network service curve that has the expressiveness to characterize whole networks by a single transfer function. This paper surveys the state-of-the-art of the deterministic and the recent probabilistic network calculus. It discusses the concept of service curves, its use in the network calculus, and the relation to systems theory under the min-plus algebra. Service curve models of common schedulers and different types of networks are reviewed and methods for identification of a system's service curve representation from measurements are discussed. After recapitulating the state of knowledge on time-varying min-plus systems theory, stochastic service curve models are surveyed. These models allow utilizing the statistical multiplexing gain in a network calculus framework that features end-to-end network analysis by convolution of service curves.

ERRATA: "FINITE-WIDTH EFFECTS IN THE MODEL OF UNSTABLE PARTICLES WITH A SMEARED MASS"

The quantum field model of unstable particles with a smeared mass is suggested to describe the finite-width effects. Within the framework of this model, we derive the convolution formula for a width and factorized formula for a cross-section. The phenomenological consequences of the mass smearing is discussed in the various fields of particle physics.

Spatial bandwidth extended reconstruction for digital color Fresnel holograms

This paper presents a reconstruction algorithm based on the convolution formula of diffraction which uses the Fresnel impulse response of free space propagation. The bandwidth of the reconstructing convolution kernel is extended to the one of the object in order to allow the direct reconstruction of objects with size quite larger than the recording area. The spatial bandwidth extension is made possible by the use of a numerical spherical wave as a virtual reconstructing wave, thus modifying the virtual reconstruction distance and increasing the kernel bandwidth. Experimental results confirm the suitability of the proposed method in the case of the simultaneous recording of two-color digital holograms by using a spatial color multiplexing scheme.

FINITE-WIDTH EFFECTS IN THE MODEL OF UNSTABLE PARTICLES WITH A SMEARED MASS

The quantum field model of unstable particles with a smeared mass is suggested to describe the finite-width effects. Within the framework of this model, we derive the convolution formula for a width and factorized formula for a cross-section. The phenomenological consequences of the mass smearing is discussed in the various fields of particle physics.