Hypergeometric expression for a three-loop vacuum integral

2020 ◽  
Vol 35 (19) ◽  
pp. 2050089
Author(s):  
Zhi-Hua Gu ◽  
Hai-Bin Zhang ◽  
Tai-Fu Feng
Keyword(s):  
Finite Element ◽  
Hypergeometric Functions ◽  
Dimensional Regularization ◽  
Feynman Integral ◽  
Numerical Continuation ◽  
Residue Theorem ◽  
Linear Partial Differential Equations ◽  
Multidimensional Residue ◽  
Stationary Conditions ◽  
Scalar Integral

Using the corresponding Mellin–Barnes representation, we derive holonomic hypergeometric system of linear partial differential equations (PDEs) satisfied by Feynman integral of a three-loop vacuum with five propagators. Through the multidimensional residue theorem in dimensional regularization, the scalar integral can be written as the summation of multiple hypergeometric functions, whose convergent regions can be obtained by the Horn’s convergent theory. The numerical continuation of the scalar integral from convergent regions to whole kinematic regions can be accomplished with the finite element methods, when the system of PDEs can be treated as the stationary conditions of a functional under the restrictions.

scholarly journals Selberg integrals, super-hypergeometric functions and applications to β-ensembles of random matrices

2015 ◽  
Vol 04 (02) ◽  
pp. 1550007 ◽  
Author(s):  
Patrick Desrosiers ◽  
Dang-Zheng Liu
Keyword(s):  
Hypergeometric Functions ◽  
Matrix Theory ◽  
Direct Application ◽  
Holonomic System ◽  
Expectation Value ◽  
Linear Partial Differential Equations ◽  
Integral Formulas ◽  
Duality Relations ◽  
Selberg Integrals ◽  
Type Integral

We study a new Selberg-type integral with n + m indeterminates, which turns out to be related to the deformed Calogero–Sutherland systems. We show that the integral satisfies a holonomic system of n + m non-symmetric linear partial differential equations. We also prove that a particular hypergeometric function defined in terms of super-Jack polynomials is the unique solution of the system. Some properties such as duality relations, integral formulas, Pfaff–Euler and Kummer transformations are also established. As a direct application, we evaluate the expectation value of ratios of characteristic polynomials in the classical β-ensembles of Random Matrix Theory.

Feynman integrals and hypergeometric functions

2014 ◽  
Vol 05 (supp01) ◽  
pp. 1441001
Author(s):  
Héctor Luna García ◽  
Luz María García
Keyword(s):  
Hypergeometric Functions ◽  
Dimensional Regularization ◽  
Feynman Integrals ◽  
Regularization Scheme

We review Davydychev method for calculating Feynman integrals for massive and no massive propagators, by employing Mellin–Barnes transformation and the dimensional regularization scheme, same that lead to hypergeometric functions. In particular, an example is calculated explicitly from such a method.

scholarly journals Analytical Calculation of Mutual Inductance of Finite-Length Coaxial Helical Filaments and Tape Coils

Energies ◽  
2019 ◽  
Vol 12 (3) ◽  
pp. 566 ◽  
Author(s):  
Xinglong Zhou ◽  
Baichao Chen ◽  
Yao Luo ◽  
Runhang Zhu
Keyword(s):  
Finite Element ◽  
Finite Element Simulation ◽  
Finite Length ◽  
Hypergeometric Functions ◽  
Analytical Calculation ◽  
Mutual Inductance ◽  
General Applicability ◽  
Element Simulation ◽  
Analytical Expressions ◽  
Dimensional Integral

Mutual inductance between finite-length coaxial helical filaments and tape coils are presented analytically. In this paper, a mathematical model for finite-length coaxial helical filaments is established, and subsequently, the mutual inductance of the filaments is derived in a series form, containing a one-dimensional integral. The mutual inductance expression of the filaments is then generalized for a tape conductor. When the tape conductor of each coil is closely wound, then the inverse Mellin transform is further utilized for transforming the generalized integral in the mutual inductance expression into a series involving hypergeometric functions, for increasing the calculation speed. Finally, the obtained expressions are compared numerically with the existing analytical solutions and finite-element simulation in order to verify the correctness and general applicability of the results. In this paper, as all the mutual-inductance analytical expressions are concise with fast convergence, it is easy to obtain the numerical results in software, such as Mathematica. The expressions presented in this paper are applicable to any corresponding geometric parameter, and are thereby more advantageous compared to the existing analytical methods. In addition, evaluation by these expressions is considerably more efficient, as compared to finite element simulation.

EXTRACTION OF INFRARED DIVERGENCES IN THE DIMENSIONAL REGULARIZED TWO-LOOP LADDER GRAPH

1994 ◽  
Vol 09 (20) ◽  
pp. 3535-3553
Author(s):  
M. RAKOWSKI ◽  
F. SAVATIER
Keyword(s):  
Dimensional Regularization ◽  
Finite Part ◽  
Feynman Parameter ◽  
Ladder Graph ◽  
Infrared Divergences ◽  
Scalar Integral ◽  
Transfer Momentum

We consider the evaluation of the fundamental scalar integral in the on-shell two-loop ladder graph with different external masses and arbitrary transfer momentum. A method for cleanly extracting the infrared divergences in the Feynman parameter integrals using dimensional regularization is presented, and we analyze one of the finite part contributions to this integral.

Calculation of the gradient of Tikhonov’s functional in solving coefficient inverse problems for linear partial differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander S. Leonov ◽  
Alexander N. Sharov ◽  
Anatoly G. Yagola
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Partial Differential Equations ◽  
Inverse Problems ◽  
Differential Equations ◽  
Differential Operators ◽  
The Finite Element Method ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Element Method

Abstract A fast algorithm for calculating the gradient of the Tikhonov functional is proposed for solving inverse coefficient problems for linear partial differential equations of a general form by the regularization method. The algorithm is designed for problems with discretized differential operators that linearly depend on the desired coefficients. When discretizing the problem and calculating the gradient, it is possible to use the finite element method. As an illustration, we consider the solution of two inverse problems of elastography using the finite element method: finding the distribution of Young’s modulus in biological tissue from data on its compression and a similar problem of determining the characteristics of local oncological inclusions, which have a special parametric form.

Modal-Dynamic Analysis of the Steering Mechanism on a Vehicle

2016 ◽  
Vol 822 ◽  
pp. 26-35
Author(s):  
Nicolae Dumitru ◽  
Gabriel Marinescu ◽  
Adrian Roşca ◽  
Oana Oţăt
Keyword(s):  
Finite Element ◽  
Dynamic Analysis ◽  
Frequency Variation ◽  
Dynamic Feedback ◽  
The Finite Element Method ◽  
Steering Mechanism ◽  
Stationary Conditions ◽  
Tie Rods ◽  
Phase Angles ◽  
Impulse Force

The present paper aims to identify the dynamic feedback of a steering mechanism in a vehicle, mainly under stationary conditions. The modal – dynamic analysis was established by applying two methods, the experimental method and the finite element modelling. For the experimental results concerning the proper frequencies of the steering box and of the tie rods assembly, a method of excitation with a measured and controlled linear momentum (impulse) force was implemented. The modal-based analysis aimed to measure the proper frequencies and the proper vibration modes. Therefore, the harmonic analysis obtained by using the finite element method is applied in order to determine the frequency variation diagrams of some nodes displacements, from some of the elements of the mechanism. Furthermore, the applied analysis seeks to identify the critical frequencies and the corresponding phase angles by investigating the displacements, the strains and the stresses of the mechanism elements, which correspond to critical frequencies and phase angles.

scholarly journals A Finite Element Simulation of the Active and Passive Controls of the MHD Effect on an Axisymmetric Nanofluid Flow with Thermo-Diffusion over a Radially Stretched Sheet

Processes ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 207 ◽  
Author(s):  
Bagh Ali ◽  
Xiaojun Yu ◽  
Muhammad Tariq Sadiq ◽  
Ateeq Ur Rehman ◽  
Liaqat Ali
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Volume Fraction ◽  
Axisymmetric Flow ◽  
Physical Parameters ◽  
Convective Boundary ◽  
Linear Partial Differential Equations ◽  
Similarity Transformations ◽  
Element Method

The present study investigated the steady magnetohydrodynamics of the axisymmetric flow of a incompressible, viscous, electricity-conducting nanofluid with convective boundary conditions and thermo-diffusion over a radially stretched surface. The nanoparticles’ volume fraction was passively controlled on the boundary, rather than actively controlled. The governing non-linear partial differential equations were transformed into a system of nonlinear, ordinary differential equations with the aid of similarity transformations which were solved numerically, using the very efficient variational finite element method. The coefficient of skin friction and rate of heat transfer, and an exact solution of fluid flow velocity, were contrasted with the numerical solution gotten by FEM. Excellent agreement between the numerical and exact solutions was observed. The influences of various physical parameters on the velocity, temperature, and solutal and nanoparticle concentration profiles are discussed by the aid of graphs and tables. Additionally, authentication of the convergence of the numerical consequences acquired by the finite element method and the computations was acquired by decreasing the mesh level. This exploration is significant for the higher temperature of nanomaterial privileging technology.

Dynamics of Multiple-Drive Belt Conveyors during Starting

2016 ◽  
Vol 842 ◽  
pp. 141-146
Author(s):  
Indraswari Kusumaningtyas ◽  
Ashley J.G. Nuttall ◽  
Gabriel Lodewijks
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Element Model ◽  
Elastic Response ◽  
Belt Conveyor ◽  
Conveyor System ◽  
Start Up ◽  
Drive Belt ◽  
Stationary Conditions ◽  
Belt Conveyors

In this paper, the dynamics of multiple-drive belt conveyors during starting is discussed. The aim of the research is to determine whether the belt sections in a multiple-drive belt conveyor can be viewed as a single-drive belt conveyor, and whether the DIN 22101 standard for the starting of a single-drive belt conveyor can still be used for the starting of a multiple-drive belt conveyor. A finite element model of a belt conveyor system was built in Matlab, consisting of a model of the belt and its support structure, and a model of the drive system. In this work, the simulations were carried out for the starting procedures of empty belt conveyors with varying number of drives. For each simulation case, the linear start-up procedure was tested. The simulations focused on the study of the axial elastic response of the belt. The simulations revealed that, by using more drives, the maximum belt stress during non-stationary as well as stationary conditions decreased. However, when using reduced starting times, negative stresses occur in the system. Overall, it was observed that the behaviour of each section between two drive stations in the multiple-drive belt conveyor differed from those of the single-drive belt conveyor. Therefore, the DIN 22101 guidelines for the start-up of a single-drive belt conveyor cannot be applied directly for the start-up of a multiple-drive belt conveyor.

scholarly journals A novel algorithm for nested summation and hypergeometric expansions

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Andrew J. McLeod ◽  
Henrik Jessen Munch ◽  
Georgios Papathanasiou ◽  
Matt von Hippel
Keyword(s):  
Hypergeometric Functions ◽  
Dimensional Regularization ◽  
Self Energy ◽  
Novel Algorithm

Abstract We consider a class of sums over products of Z-sums whose arguments differ by a symbolic integer. Such sums appear, for instance, in the expansion of Gauss hypergeometric functions around integer indices that depend on a symbolic parameter. We present a telescopic algorithm for efficiently converting these sums into generalized polylogarithms, Z-sums, and cyclotomic harmonic sums for generic values of this parameter. This algorithm is illustrated by computing the double pentaladder integrals through ten loops, and a family of massive self-energy diagrams through $$ \mathcal{O}\left({\epsilon}^6\right) $$ O ϵ 6 in dimensional regularization. We also outline the general telescopic strategy of this algorithm, which we anticipate can be applied to other classes of sums.

An Approximation of Three-Dimensional Semiconductor Devices by Mixed Finite Element Method and Characteristics-Mixed Finite Element Method

2015 ◽  
Vol 8 (3) ◽  
pp. 356-382 ◽  
Author(s):  
Qing Yang ◽  
Yirang Yuan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Electrostatic Potential ◽  
Hole Concentration ◽  
Mixed Finite Element ◽  
Semiconductor Devices ◽  
Three Dimensional ◽  
Mixed Finite Element Method ◽  
Linear Partial Differential Equations ◽  
Element Method

AbstractThe mathematical model for semiconductor devices in three space dimensions are numerically discretized. The system consists of three quasi-linear partial differential equations about three physical variables: the electrostatic potential, the electron concentration and the hole concentration. We use standard mixed finite element method to approximate the elliptic electrostatic potential equation. For the two convection-dominated concentration equations, a characteristics-mixed finite element method is presented. The scheme is locally conservative. The optimalL2-norm error estimates are derived by the aid of a post-processing step. Finally, numerical experiments are presented to validate the theoretical analysis.

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