scholarly journals IBP reduction coefficients made simple

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Janko Boehm ◽  
Marcel Wittmann ◽  
Zihao Wu ◽  
Yingxuan Xu ◽  
Yang Zhang
Keyword(s):  
Computer Algebra ◽  
Efficient Method ◽  
Computer Algebra System ◽  
Size Reduction ◽  
Partial Fraction ◽  
Spacetime Dimension ◽  
Integral Basis ◽  
Feynman Integrals ◽  
Analytic Integration ◽  
Fraction Algorithm

Abstract We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas’ multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, we observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension D. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as ∼ 100. We observe that our algorithm also works well for settings without a UT basis.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

The Maple computer algebra system: A review

1995 ◽  
Vol 10 (3) ◽  
pp. 329-337 ◽  
Author(s):  
John Hutton ◽  
James Hutton
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
System A

scholarly journals OPTIMALISASI MOTIVASI DAN PRESTASI BELAJAR MENGGUNAKAN MOODLE BERBANTUAN COMPUTER ALGEBRA SYSTEM (CAS)

2020 ◽  
Vol 9 (1) ◽  
pp. 53
Author(s):  
Kamhar Ngado ◽  
Rosnawati Rosnawati ◽  
Heri Retnawati ◽  
Sri Andayani
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System

scholarly journals Generation of equations for shell models of turbulence in the Maple system

2021 ◽  
Vol 254 ◽  
pp. 02006
Author(s):  
Liubov Feshchenko ◽  
Gleb Vodinchar
Keyword(s):  
Computer Algebra ◽  
Power Law ◽  
Computer Algebra System ◽  
Nonlinear Interactions ◽  
Systems Of Equations ◽  
Shell Models ◽  
Quadratic Invariants ◽  
Automated Compilation ◽  
Shell Models Of Turbulence

The paper describes a technology for the automated compilation of equations for shell models of turbulence in the computer algebra system Maple. A general form of equations for the coefficients of nonlinear interactions is given, which will ensure that the required combination of quadratic invariants and power-law solutions is fulfilled in the model. Described the codes for the Maple system allowing to generate and solve systems of equations for the coefficients. The proposed technology allows you to quickly and accurately generate classes of shell models with the desired properties.

scholarly journals Using Algebraic Computing To Teach General Relativity And Cosmology

2012 ◽  
Vol 56 (1) ◽  
pp. 139-144
Author(s):  
Dumitru N. Vulcanov ◽  
Remus-Ştefan Ş. Boată
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Undergraduate Students ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Einstein Equations ◽  
Algebraic Programming ◽  
Algebraic Computing

AbstractThe article presents some new aspects and experience on the use of computer in teaching general relativity and cosmology for undergraduate students (and not only) with some experience in computer manipulation. Some years ago certain results were reported [1] using old fashioned computer algebra platforms but the growing popularity of graphical platforms as Maple and Mathematica forced us to adapt and reconsider our methods and programs. We will describe some simple algebraic programming procedures (in Maple with GrTensorII package) for obtaining and the study of some exact solutions of the Einstein equations in order to convince a dedicated student in general relativity about the utility of a computer algebra system.

scholarly journals Reduction of Feynman integrals in the parametric representation II: reduction of tensor integrals

2021 ◽  
Vol 81 (3) ◽  
Author(s):  
Wen Chen
Keyword(s):  
Gröbner Basis ◽  
Parametric Representation ◽  
Polynomial Equations ◽  
Grobner Basis ◽  
Spacetime Dimension ◽  
Integration By Parts ◽  
Feynman Integrals ◽  
Parametric Integrals ◽  
Parametric Integration

AbstractIn a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gröbner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.

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