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72
On motives associated to graph polynomials
 Commun. Math. Phys
"... Abstract. The appearance of multiple zeta values in anomalous dimensions and βfunctions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a su ..."
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Cited by 50 (12 self)
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Abstract. The appearance of multiple zeta values in anomalous dimensions and βfunctions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions. Calculations of Feynman integrals arising in perturbative quantum field theory [4, 5] reveal interesting patterns of zeta and multiple zeta values. Clearly, these are motivic in origin, arising from the existence of Tate mixed Hodge structures with periods given by Feynman integrals.
Parallel Integer Relation Detection: Techniques and Applications
 Mathematics of Computation
, 2000
"... Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and phy ..."
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Cited by 47 (35 self)
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Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Singleand multilevel implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in mathematical number theory, quantum field theory and chaos theory.
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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Mixed Tate motives and multiple zeta values
 Invent. Math
"... Let l be a positive integer and k1,..., kl be integers such that ki ≥ 1 for i = 1,..., l − 1 and kl ≥ 2. We define the multiple zeta value ζ(k1,..., kl) of index (k1,..., kl) as ..."
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Cited by 37 (0 self)
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Let l be a positive integer and k1,..., kl be integers such that ki ≥ 1 for i = 1,..., l − 1 and kl ≥ 2. We define the multiple zeta value ζ(k1,..., kl) of index (k1,..., kl) as
Combinatorics of (perturbative) quantum field theory
, 2000
"... We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic oper ..."
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Cited by 30 (8 self)
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We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic operation of graph insertions. Particular emphasis is given to the connection with the Riemann–Hilbert problem. Finally, we outline how these structures relate to the numbers which we see in Feynman diagrams.
Structures in Feynman Graphs Hopf Algebras and Symmetries
, 2008
"... We review the combinatorial structure of perturbative quantum field theory with emphasis given to the decomposition of graphs into primitive ones. The consequences in terms of unique factorization of Dyson– Schwinger equations into Euler products are discussed. ..."
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Cited by 24 (10 self)
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We review the combinatorial structure of perturbative quantum field theory with emphasis given to the decomposition of graphs into primitive ones. The consequences in terms of unique factorization of Dyson– Schwinger equations into Euler products are discussed.
Renormalization and motivic Galois theory
 International Math. Research Notices
"... Abstract. We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “ motivic Galois group ” U ∗ , which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identifie ..."
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Cited by 24 (13 self)
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Abstract. We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “ motivic Galois group ” U ∗ , which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup of U ∗. The group U ∗ arises through a Riemann–Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, U ∗ is a semidirect product by the multiplicative group Gm of a prounipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes–Moscovici. When working with formal Laurent series over Q, the data of equisingular flat vector bundles
Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)
, 1998
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Experimental Mathematics: Recent Developments and Future Outlook
 CECM PREPRINT 99:143] FFL J.M. BORWEIN AND P.B. BORWEIN, "CHALLENGES FOR MATHEMATICAL COMPUTING," COMPUTING IN SCIENCE & ENGINEERING, 2001. [CECM PREPRINT 01:160
, 2000
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