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566
SPECIAL VALUES OF MULTIPLE POLYLOGARITHMS
, 1999
"... Historically, the polylogarithm has attracted specialists and nonspecialists alike withitslovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and highenergy physics. More recently, we ha ..."
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Cited by 94 (25 self)
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Historically, the polylogarithm has attracted specialists and nonspecialists alike withitslovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and highenergy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.
On the Notion of Interestingness in Automated Mathematical Discovery
 International Journal of Human Computer Studies
, 2000
"... We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathema ..."
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Cited by 70 (26 self)
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We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathematical discovery. We detail how empirical evidence is used to give plausibility to conjectures, and the dierent ways in which a result can be thought of as novel. We also look at the ways in which the programs assess how surprising and complex a conjecture statement is, and the dierent ways in which the applicability of a concept or conjecture is used. Finally, we note how a user can set tasks for the program to achieve and how this aects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.
Stability results for random sampling of sparse trigonometric polynomials
, 2006
"... Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of nonzero coefficients, can be reconstructed exactly from a small number of random samples using Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP). In the present article it is shown that ..."
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Cited by 65 (18 self)
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Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of nonzero coefficients, can be reconstructed exactly from a small number of random samples using Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP). In the present article it is shown that recovery both by a BP variant and by OMP is stable under perturbation of the samples values by noise. For BP in addition, the stability result is extended to (nonsparse) trigonometric polynomials that can be wellapproximated by sparse ones. The theoretical findings are illustrated by numerical experiments. Key Words: random sampling, trigonometric polynomials, Orthogonal Matching Pursuit, Basis Pursuit, compressed sensing, stability under noise, fast Fourier transform, nonequispaced
Link spam alliances
 In Proceedings of the 31st International Conference on Very Large Data Bases (VLDB
, 2005
"... Link spam is used to increase the ranking of certain target web pages by misleading the connectivitybased ranking algorithms in search engines. In this paper we study how web pages can be interconnected in a spam farm in order to optimize rankings. We also study alliances, that is, interconnections ..."
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Cited by 64 (1 self)
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Link spam is used to increase the ranking of certain target web pages by misleading the connectivitybased ranking algorithms in search engines. In this paper we study how web pages can be interconnected in a spam farm in order to optimize rankings. We also study alliances, that is, interconnections of spam farms. Our results identify the optimal structures and quantify the potential gains. In particular, we show that alliances can be synergistic and improve the rankings of all participants. We believe that the insights we gain will be useful in identifying and combating link spam. 1
Negative weights make adversaries stronger. To appear in STOC’07
 Algorithmica
, 2002
"... The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations in terms of ..."
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Cited by 63 (7 self)
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The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations in terms of weight schemes, eigenvalues, and Kolmogorov complexity. All these formulations are informationtheoretic and rely on the principle that if an algorithm successfully computes a function then, in particular, it is able to distinguish between inputs which map to different values. We present a stronger version of the adversary method which goes beyond this principle to make explicit use of the existence of a measurement in a successful algorithm which gives the correct answer, with high probability. We show that this new method, which we call ADV ±, has all the advantages of the old: it is a lower bound on boundederror quantum query complexity, its square is a lower bound on formula size, and it behaves well with respect to function composition. Moreover ADV ± is always at least as large as the adversary method ADV, and we show an example of a monotone function for which ADV ± (f) = Ω(ADV(f) 1.098). We also give examples showing that ADV ± does not face limitations of ADV such as the certificate complexity barrier and the property testing barrier. 1
Counting gauge invariants: The Plethystic program
 JHEP 0703 (2007) 090 hepth/0701063
"... We propose a programme for systematically counting the single and multitrace gauge invariant operators of a gauge theory. Key to this is the plethystic function. We expound in detail the power of this plethystic programme for worldvolume quiver gauge theories of Dbranes probing CalabiYau singulari ..."
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Cited by 49 (17 self)
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We propose a programme for systematically counting the single and multitrace gauge invariant operators of a gauge theory. Key to this is the plethystic function. We expound in detail the power of this plethystic programme for worldvolume quiver gauge theories of Dbranes probing CalabiYau singularities, an illustrative case to which the programme is not limited, though in which a full intimate web of relations between the geometry and the gauge theory manifests herself. We can also use generalisations of HardyRamanujan to compute the entropy of gauge theories from the plethystic exponential. In due course, we also touch upon fascinating connections to Young Tableaux, Hilbert schemes and the
A modified splitradix FFT with fewer arithmetic operations
 IEEE TRANS. SIGNAL PROCESSING
, 2006
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The Enumeration of Simple Permutations
 J. Integer Seq
, 2003
"... A simple permutation is one which maps no proper nonsingleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary generating function for simple permutations and that for all ..."
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Cited by 48 (4 self)
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A simple permutation is one which maps no proper nonsingleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary generating function for simple permutations and that for all permutations, that the coe#cients of this series are not P recursive, an asymptotic expansion for these coe# cients, and a number of congruence results.
Automatic Concept Formation in Pure Mathematics
"... The HR program forms concepts and makes conjectures in domains of pure mathematics andusestheoremproverOTTERandmodel generatorMACEtoproveordisprovetheconjectures. HRmeasurespropertiesofconcepts andassessesthetheoremsandproofsinvolving themtoestimatetheinterestingnessofeach concept and employ a best ..."
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Cited by 44 (31 self)
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The HR program forms concepts and makes conjectures in domains of pure mathematics andusestheoremproverOTTERandmodel generatorMACEtoproveordisprovetheconjectures. HRmeasurespropertiesofconcepts andassessesthetheoremsandproofsinvolving themtoestimatetheinterestingnessofeach concept and employ a best first search. This approachhasledHRtothediscoveryofinterestingnewmathematics and enables it to build theories from just the axioms of finite algebras.
Transfer matrices and partitionfunction zeros for antiferromagnetic Potts models. I. General theory and squarelattice chromatic polynomial
 J. Stat. Phys
, 2001
"... We study the chromatic polynomials ( = zerotemperature antiferromagnetic Pottsmodel partition functions) PG(q) for m × n rectangular subsets of the square lattice, with m ≤ 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer ..."
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Cited by 42 (6 self)
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We study the chromatic polynomials ( = zerotemperature antiferromagnetic Pottsmodel partition functions) PG(q) for m × n rectangular subsets of the square lattice, with m ≤ 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the FortuinKasteleyn representation. In particular, we extract the limiting curves of partitionfunction zeros when n → ∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B2,B3,B4,B5 are limiting points of partitionfunction zeros as n → ∞ whenever the strip width m is ≥ 7 (periodic transverse b.c.) or ≥ 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B10) cannot be a chromatic root of any graph. Key Words: Chromatic polynomial; chromatic root; antiferromagnetic Potts model; square lattice; transfer matrix; FortuinKasteleyn representation; TemperleyLieb algebra; Beraha–Kahane–Weiss theorem; Beraha numbers.