scholarly journals A Fast Algorithm for MacMahon's Partition Analysis

10.37236/1811 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Guoce Xin
Keyword(s):  
Fast Algorithm ◽  
Special Class ◽  
Laurent Series ◽  
Rational Functions ◽  
Constant Term ◽  
Partial Fraction ◽  
Running Time ◽  
Partition Analysis ◽  
Several Variables ◽  
Partial Fraction Decomposition

This paper deals with evaluating constant terms of a special class of rational functions, the Elliott-rational functions. The constant term of such a function can be read off immediately from its partial fraction decomposition. We combine the theory of iterated Laurent series and a new algorithm for partial fraction decompositions to obtain a fast algorithm for MacMahon's Omega calculus, which (partially) avoids the "run-time explosion" problem when eliminating several variables. We discuss the efficiency of our algorithm by investigating problems studied by Andrews and his coauthors; our running time is much less than that of their Omega package.

scholarly journals New Multiple Harmonic Sum Identities

10.37236/3946 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Roberto Tauraso ◽  
Helmut Prodinger
Keyword(s):  
Infinite Series ◽  
Special Class ◽  
Partial Fraction ◽  
Harmonic Numbers ◽  
Elementary Method ◽  
Partial Fraction Decomposition ◽  
Binomial Sums

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series  and congruences are given.

scholarly journals A Closed Character Formula for Symmetric Powers of Irreducible Representations

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Stavros Kousidis
Keyword(s):  
Irreducible Representation ◽  
Rational Functions ◽  
Character Formula ◽  
Irreducible Representations ◽  
Partition Functions ◽  
Partial Fraction ◽  
Residue Type ◽  
Symmetric Powers ◽  
International Audience ◽  
Partial Fraction Decomposition

International audience We prove a closed character formula for the symmetric powers $S^N V(λ )$ of a fixed irreducible representation $V(λ )$ of a complex semi-simple Lie algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula involves rational functions in rank of $\mathfrak{g}$ many variables which are easier to determine than the weight multiplicities of $S^N V(λ )$ themselves. We compute those rational functions in some interesting cases. Furthermore, we introduce a residue-type generating function for the weight multiplicities of $S^N V(λ )$ and explain the connections between our character formula, vector partition functions and iterated partial fraction decomposition. Nous établissons une formule fermée pour le caractère de la puissance symétrique $S^N V(λ )$ d'une représentation irréductible $V(λ )$ d'une algèbre de Lie semi-simple complexe$\mathfrak{g}$, en utilisant des décompositions en fractions partielles. Cette formule exprime ce caractère en termes de fractions rationnelles en $r$ variables, où $r$ est le rang de $\mathfrak{g}$. Ces fractions sont plus faciles à déterminer que les multiplicités de la décomposition de $S^N V(λ )$ elles-mêmes. Nous calculons ces fonctions rationnelles dans quelques cas intéressants. Nous introduisons par ailleurs une fonction génératrice de type résidu pour les multiplicités de $S^N V(λ )$ et relions notre formule aux fonctions de partitions vectorielles et aux décompositions itérées en fractions partielles.

scholarly journals h-Adic Polynomials and Partial Fraction Decomposition of Proper Rational Functions over R or C

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Kwang Hyun Kim ◽  
Xin Zhang
Keyword(s):  
Commutative Algebra ◽  
Computer Algebra System ◽  
Rational Functions ◽  
Partial Fraction ◽  
Decomposition Technique ◽  
Taylor Polynomial ◽  
Simple Method ◽  
Recursive Computation ◽  
Algebraic Operations ◽  
Partial Fraction Decomposition

The partial fraction decomposition technique is very useful in many areas including mathematics and engineering. In this paper we present a new and simple method on the partial fraction decomposition of proper rational functions which have completely factored denominators over R or C. The method is based on a recursive computation of the h-adic polynomial in commutative algebra which is a generalization of the Taylor polynomial. Since its computation requires only simple algebraic operations, it does not require a computer algebra system to be programmed.

scholarly journals Two-loop leading colour QCD corrections to $$ q\overline{q} $$ → γγg and qg → γγq

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Bakul Agarwal ◽  
Federico Buccioni ◽  
Andreas von Manteuffel ◽  
Lorenzo Tancredi
Keyword(s):  
Numerical Code ◽  
Tree Level ◽  
Partial Fraction ◽  
Integration By Parts ◽  
Analytic Expressions ◽  
Partial Fraction Decomposition ◽  
Loop Amplitudes

Abstract We present the leading colour and light fermionic planar two-loop corrections for the production of two photons and a jet in the quark-antiquark and quark-gluon channels. In particular, we compute the interference of the two-loop amplitudes with the corresponding tree level ones, summed over colours and polarisations. Our calculation uses the latest advancements in the algorithms for integration-by-parts reduction and multivariate partial fraction decomposition to produce compact and easy-to-use results. We have implemented our results in an efficient C++ numerical code. We also provide their analytic expressions in Mathematica format.

A RAPID METHOD FOR FILTER PAPER ELECTROPHORESIS

1954 ◽  
Vol 32 (1) ◽  
pp. 567-570
Author(s):  
S. D. Vesselinovitch ◽  
H. S. Funnell
Keyword(s):  
Rapid Method ◽  
Filter Paper ◽  
Protein Separation ◽  
Paper Electrophoresis ◽  
Running Time ◽  
Electrophoretic Patterns ◽  
Several Variables

A technique for filter paper electrophoresis and strip staining that permits the running time to be reduced to two hours or less is described. Some typical two hour electrophoretic patterns are illustrated and the clinical advantages of the technique mentioned. Details of a simple and inexpensive apparatus for electrophoresis on filter paper are given. Several variables affecting rapid and distinct protein separation, which were modified in the development of this method, and the relationships between these changes and the shortened running time are discussed.

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