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Low level representations of E10 and E11
 in: Proceedings of the Ramanujan International Symposium on Kac– Moody Algebras and Applications, ISKMAA2002
"... Abstract. We work out the decomposition of the indefinite Kac Moody algebras E10 and E11 w.r.t. their respective subalgebras A9 and A10 at low levels. Tables of the irreducible representations with their outer multiplicities are presented for E10 up to level ℓ = 18 and for E11 up to level ℓ = 10. On ..."
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Cited by 35 (9 self)
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Abstract. We work out the decomposition of the indefinite Kac Moody algebras E10 and E11 w.r.t. their respective subalgebras A9 and A10 at low levels. Tables of the irreducible representations with their outer multiplicities are presented for E10 up to level ℓ = 18 and for E11 up to level ℓ = 10. On the way we confirm and extend existing results for E10 root multiplicities, and for the first time compute nontrivial root multiplicities of E11. 1.
The kissing number in four dimensions
, 2005
"... The kissing number problem asks for the maximal number of equal size nonoverlapping spheres that can touch another sphere of the same size in ndimensional space. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimension ..."
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Cited by 33 (10 self)
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The kissing number problem asks for the maximal number of equal size nonoverlapping spheres that can touch another sphere of the same size in ndimensional space. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schütte and van der Waerden. It was proved that the bounds given by Delsarte’s method are not good enough to solve the problem in 4dimensional space. In this paper we present a solution of the problem in dimension four, based on a modification of Delsarte’s method. Keywords: kissing number, contact number, spherical codes, Delsarte’s method, Gegenbauer (ultraspherical) polynomials
The ClarksonShor Technique Revisited and Extended
 Comb., Prob. & Comput
, 2001
"... We provide an alternative, simpler and more general derivation of the ClarksonShor probabilistic technique [6] and use it to obtain in addition several extensions and new combinatorial bounds. ..."
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Cited by 22 (3 self)
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We provide an alternative, simpler and more general derivation of the ClarksonShor probabilistic technique [6] and use it to obtain in addition several extensions and new combinatorial bounds.
Forest density estimation
 Journal of Machine Learning Research
, 2011
"... We study graph estimation and density estimation in high dimensions, using a family of density estimators based on forest structured undirected graphical models. For density estimation, we do not assume the true distribution corresponds to a forest; rather, we form kernel density estimates of the bi ..."
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Cited by 20 (8 self)
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We study graph estimation and density estimation in high dimensions, using a family of density estimators based on forest structured undirected graphical models. For density estimation, we do not assume the true distribution corresponds to a forest; rather, we form kernel density estimates of the bivariate and univariate marginals, and apply Kruskal’s algorithm to estimate the optimal forest on held out data. We prove an oracle inequality on the excess risk of the resulting estimator relative to the risk of the best forest. For graph estimation, we consider the problem of estimating forests with restricted tree sizes. We prove that finding a maximum weight spanning forest with restricted tree size is NPhard, and develop an approximation algorithm for this problem. Viewing the tree size as a complexity parameter, we then select a forest using data splitting, and prove bounds on excess risk and structure selection consistency of the procedure. Experiments with simulated data and microarray data indicate that the methods are a practical alternative to Gaussian graphical models.
Quantum communication complexity
 Foundations of Physics
"... Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can ..."
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Cited by 19 (6 self)
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Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can be communicated but no more. In apparent contradiction, there are distributed computational tasks for which quantum communication cannot be simulated efficiently by classical means. In some cases, the effect of transmitting quantum bits cannot be achieved classically short of transmitting an exponentially larger number of bits. In a similar vein, can entanglement be used to save on classical communication? It is well known that entanglement on its own is useless for the transmission of information. Yet, there are distributed tasks that cannot be accomplished at all in a classical world when communication is not allowed, but that become possible if the noncommunicating parties share prior entanglement. This leads to the question of how expensive it is, in terms of classical communication, to provide an exact simulation of the spooky power of entanglement. KEY WORDS: Bell’s theorem; communication complexity; distributed computation; entanglement simulation; pseudotelepathy; spooky communication.
Proof Transformation by CERES
 MATHEMATICAL KNOWLEDGE MANAGEMENT (MKM) 2006, VOLUME 4108 OF LECTURE NOTES IN ARTIFICIAL INTELLIGENCE
, 2006
"... Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cutelimination method CERES (cutelimination by resolution) works by constructing a set o ..."
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Cited by 14 (10 self)
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Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cutelimination method CERES (cutelimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of an LKproof with only atomic cuts. In this paper we present an extension of CERES to a calculus LKDe which is stronger than the Gentzen calculus LK (it contains rules for introduction of definitions and equality rules). This extension makes it much easier to formalize mathematical proofs and increases the performance of the cutelimination method. The system CERES already proved efficient in handling very large proofs.
Coloring Hamming graphs, Optimal Binary Codes, and the 0/1Borsuk Problem in Low Dimensions
 IN H. ALT (ED.): COMPUTATIONAL DISCRETE MATHEMATICS, LECTURE NOTES IN COMPUTER SCIENCE 2122
, 2001
"... The 0/1Borsuk problem asks whether every subset of f0; 1g d can be partitioned into at most d + 1 sets of smaller diameter. This is known to be false in high dimensions (in particular for d 561, due to Kahn & Kalai, Nilli, and Raigorodskii), and yields the known counterexamples to Borsuk& ..."
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Cited by 13 (0 self)
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The 0/1Borsuk problem asks whether every subset of f0; 1g d can be partitioned into at most d + 1 sets of smaller diameter. This is known to be false in high dimensions (in particular for d 561, due to Kahn & Kalai, Nilli, and Raigorodskii), and yields the known counterexamples to Borsuk's problem posed in 1933. Here we ask whether there might be counterexamples in low dimension as well. We show that there is no counterexample to the 0/1Borsuk conjecture in dimensions d 9. (In contrast, the general Borsuk conjecture is open even for d = 4.) Our study relates the 0/1case of Borsuk's problem to the coloring problem for the Hamming graphs, to the geometry of a Hamming code, as well as to some upper bounds for the sizes of binary codes.
The kissing problem in three dimensions
 Discrete Comput. Geom
"... The kissing number k(3) is the maximal number of equal size nonoverlapping spheres in three dimensions that can touch another sphere of the same size. This number was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. The first proof that k(3) = 12 was given by Schüt ..."
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Cited by 12 (6 self)
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The kissing number k(3) is the maximal number of equal size nonoverlapping spheres in three dimensions that can touch another sphere of the same size. This number was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. The first proof that k(3) = 12 was given by Schütte and van der Waerden only in 1953. In this paper we present a new solution of the NewtonGregory problem that uses our extension of the Delsarte method. This proof relies on basic calculus and simple spherical geometry. Keywords: Kissing numbers, thirteen spheres problem, NewtonGregory problem, Legendre polynomials, Delsarte’s method