The Experts below are selected from a list of 1242 Experts worldwide ranked by ideXlab platform
Andrew V. Sills - One of the best experts on this subject based on the ideXlab platform.
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Analysis and combinatorics of partition zeta functions
arXiv: Number Theory, 2019Co-Authors: Robert Schneider, Andrew V. SillsAbstract:We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon's Partial Fraction Decomposition of the generating function for partitions of fixed length.
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Rademacher’s infinite Partial Fraction conjecture is (almost certainly) false
2016Co-Authors: Andrew Sills, Andrew V. Sills, Doron ZeilbergerAbstract:“It is a capital mistake to theorise before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.”—Sherlock Holmes to Dr. Watson [2, p. 63]. Abstract. In his book Topics in Analytic Number Theory, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary Partial Fraction Decomposition of the generating function for parti-tions of integers into at most N parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made toward proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this pa-per, we provide a fast algorithm for calculating the Rademacher coefficients, a large amount of data, direct formulas for certain collections of Rademacher coefficients, and overwhelming evidence against the truth of the conjecture. While the limits of the sequences of Rademacher coefficients do not exist (the sequences oscillate and attain arbitrarily large positive and negative values), the sequences do get very close to Rademacher’s conjectured limits for certai
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Rademacher’s infinite Partial Fraction conjecture is (almost certainly) false
2016Co-Authors: Andrew V. Sills, Doron ZeilbergerAbstract:“It is a capital mistake to theorise before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.”—Sherlock Holmes to Dr. Watson [2, p. 63]. Abstract. In his book Topics in Analytic Number Theory, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary Partial Fraction Decomposition of the generating function for parti-tions of integers into at most N parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made toward proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this pa-per, we provide a fast algorithm for calculating the Rademacher coefficients, a large amount of data, direct formulas for certain collections of Rademacher coefficients, and overwhelming evidence against the truth of the conjecture. While the limits of the sequences of Rademacher coefficients do not exist (the sequences oscillate and attain arbitrarily large positive and negative values), the sequences do get very close to Rademacher’s conjectured limits for certain (predictable) indices in the sequences
Ulf Saalmann - One of the best experts on this subject based on the ideXlab platform.
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Partial Fraction Decomposition of the fermi function
Physical Review B, 2009Co-Authors: Alexander Croy, Ulf SaalmannAbstract:A Partial Fraction Decomposition of the Fermi function resulting in a finite sum over simple poles is proposed. This allows for efficient calculations involving the Fermi function in various contexts of electronic structure or electron transport theories. The proposed Decomposition converges in a well-defined region faster than exponential and is thus superior to the standard Matsubara expansion.
Xin Zhang - One of the best experts on this subject based on the ideXlab platform.
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adic polynomials and Partial Fraction Decomposition of proper rational functions over or
International Journal of Mathematics and Mathematical Sciences, 2018Co-Authors: Kwang Hyun Kim, Xin ZhangAbstract: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 or . The method is based on a recursive computation of the -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.
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h-Adic Polynomials and Partial Fraction Decomposition of Proper Rational Functions over R or C
Hindawi Limited, 2018Co-Authors: Kwang Hyun Kim, Xin ZhangAbstract: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
Page B. - One of the best experts on this subject based on the ideXlab platform.
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Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD
'Springer Science and Business Media LLC', 2019Co-Authors: Abreu S., Dormans J., Cordero F. Febres, Ita H., Page B.Abstract:We present the analytic form of all leading-color two-loop five-parton helicity amplitudes in QCD. The results are analytically reconstructed from exact numerical evaluations over finite fields. Combining a judicious choice of variables with a new approach to the treatment of particle states in $D$ dimensions for the numerical evaluation of amplitudes, we obtain the analytic expressions with a modest computational effort. Their systematic simplification using multivariate Partial-Fraction Decomposition leads to a particularly compact form. Our results provide all two-loop amplitudes required for the calculation of next-to-next-to-leading order QCD corrections to the production of three jets at hadron colliders in the leading-color approximation.Comment: v2: Minor text changes, matches published versio
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Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD
'Springer Science and Business Media LLC', 2019Co-Authors: Abreu S., Dormans J., Ita H., Febres Cordero F., Page B.Abstract:International audienceWe present the analytic form of all leading-color two-loop five-parton helicity amplitudes in QCD. The results are analytically reconstructed from exact numerical evaluations over finite fields. Combining a judicious choice of variables with a new approach to the treatment of particle states in D dimensions for the numerical evaluation of amplitudes, we obtain the analytic expressions with a modest computational effort. Their systematic simplification using multivariate Partial-Fraction Decomposition leads to a particularly compact form. Our results provide all two-loop amplitudes required for the calculation of next-to-next-to-leading order QCD corrections to the production of three jets at hadron colliders in the leading-color approximation
Doron Zeilberger - One of the best experts on this subject based on the ideXlab platform.
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Rademacher’s infinite Partial Fraction conjecture is (almost certainly) false
2016Co-Authors: Andrew Sills, Andrew V. Sills, Doron ZeilbergerAbstract:“It is a capital mistake to theorise before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.”—Sherlock Holmes to Dr. Watson [2, p. 63]. Abstract. In his book Topics in Analytic Number Theory, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary Partial Fraction Decomposition of the generating function for parti-tions of integers into at most N parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made toward proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this pa-per, we provide a fast algorithm for calculating the Rademacher coefficients, a large amount of data, direct formulas for certain collections of Rademacher coefficients, and overwhelming evidence against the truth of the conjecture. While the limits of the sequences of Rademacher coefficients do not exist (the sequences oscillate and attain arbitrarily large positive and negative values), the sequences do get very close to Rademacher’s conjectured limits for certai
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Rademacher’s infinite Partial Fraction conjecture is (almost certainly) false
2016Co-Authors: Andrew V. Sills, Doron ZeilbergerAbstract:“It is a capital mistake to theorise before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.”—Sherlock Holmes to Dr. Watson [2, p. 63]. Abstract. In his book Topics in Analytic Number Theory, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary Partial Fraction Decomposition of the generating function for parti-tions of integers into at most N parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made toward proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this pa-per, we provide a fast algorithm for calculating the Rademacher coefficients, a large amount of data, direct formulas for certain collections of Rademacher coefficients, and overwhelming evidence against the truth of the conjecture. While the limits of the sequences of Rademacher coefficients do not exist (the sequences oscillate and attain arbitrarily large positive and negative values), the sequences do get very close to Rademacher’s conjectured limits for certain (predictable) indices in the sequences
