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76
Higherdimensional box integrals
, 2010
"... Herein, with the aid of substantial symbolic computation, we solve previously open problems in the theory of ndimensional box integrals Bn(s): = 〈⃗r  s 〉; ⃗r ∈ [0, 1] n. In particular we resolve an elusive integral called K5 that previously acted as a blockade against closedform evaluation in ..."
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Cited by 7 (7 self)
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Herein, with the aid of substantial symbolic computation, we solve previously open problems in the theory of ndimensional box integrals Bn(s): = 〈⃗r  s 〉; ⃗r ∈ [0, 1] n. In particular we resolve an elusive integral called K5 that previously acted as a blockade against closedform evaluation in n = 5 dimensions. In consequence we now know that Bn (integer) can be given a closed form for n = 1, 2, 3, 4, 5. We also nd the general residue at the pole at s = −n, this leading to new relations and definite integrals for example, we are able to give the first nontrivial closed forms for 6dimensional box integrals and to show hyperclosure of B6(even). The Clausen function and its generalizations play a central role in these higherdimensional evaluations. Our results provide stringent test scenarios for symbolicalgebra simplification methods.
On the link pattern distribution of quarterturn symmetric
 FPL configurations, FPSAC 2008, Valparaiso (Chile), DMTCS proceedings
"... Abstract. We present new conjectures on the distribution of link patterns for fullypacked loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that t ..."
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Cited by 7 (2 self)
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Abstract. We present new conjectures on the distribution of link patterns for fullypacked loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, we get a formula for the enumeration of a new class of quasisymmetry of plane partitions. 1.
Hierarchical Dobińskitype relations via substitution and the moment problem, J.Phys
 A: Math.Gen
, 2004
"... Abstract. We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a, a † ] = 1) monomials of the form exp[λ(a † ) r a], r = 1, 2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffe ..."
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Cited by 5 (3 self)
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Abstract. We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a, a † ] = 1) monomials of the form exp[λ(a † ) r a], r = 1, 2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffertype. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: a) the property of being the solution of the Stieltjes moment problem; and b) the representation of these sequences through infinite series (Dobińskitype relations). We present a number of examples of such composition satisfying properties a) and b). We obtain new Dobińskitype formulas and solve the associated moment problem for several hierarchically defined combinatorial families of sequences. Hierarchical Dobińskitype relations via substitution and the moment problem 2 1.
On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling Number Triangles
, 2006
"... We study a particular number pyramid bn,k,l that relates the binomial, Deleham, Eulerian, MacMahontype and Stirling number triangles. The numbers bn,k,l are generated by a function Bc (x, y, t), c ∈ C, that appears in the calculation of derivatives of a class of functions whose derivatives can be e ..."
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Cited by 5 (0 self)
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We study a particular number pyramid bn,k,l that relates the binomial, Deleham, Eulerian, MacMahontype and Stirling number triangles. The numbers bn,k,l are generated by a function Bc (x, y, t), c ∈ C, that appears in the calculation of derivatives of a class of functions whose derivatives can be expressed as polynomials in the function itself or a related function. Based on the properties of the numbers bn,k,l, we derive several new relations related to these triangles. In particular, we show that the number triangle Tn,k, recently constructed by Deleham (Sloane’s A088874), is generated by the Maclaurin series of sech c t, c ∈ C. We also give explicit expressions and various partial sums for the triangle Tn,k. Further, we find that em 2p, the numbers appearing in the Maclaurin series of cosh m t, for all m ∈ N, equal the number of closed walks, based at a vertex, of length 2p along the edges of an mdimensional cube.
A Partition Formula for Fibonacci Numbers
, 2008
"... We present a partition formula for the even index Fibonacci numbers. The formula is motivated by the appearance of these Fibonacci numbers in the representation theory of the socalled 3Kronecker quiver, i.e., the oriented graph with two vertices and three arrows in the same direction. ..."
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Cited by 4 (1 self)
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We present a partition formula for the even index Fibonacci numbers. The formula is motivated by the appearance of these Fibonacci numbers in the representation theory of the socalled 3Kronecker quiver, i.e., the oriented graph with two vertices and three arrows in the same direction.
THE MODULI SPACE OF n POINTS ON THE LINE IS CUT OUT BY SIMPLE QUADRICS WHEN n IS NOT SIX
"... ABSTRACT. A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory quotients come with a natural ample line bundle, and hence often a natural projective embedding. This question translates to determining the equations of the moduli spac ..."
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ABSTRACT. A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory quotients come with a natural ample line bundle, and hence often a natural projective embedding. This question translates to determining the equations of the moduli space under this embedding. This note deals with one of the most classical quotients, the space of ordered points on the projective line. We show that under any linearization, this quotient is cut out (schemetheoretically) by a particularly simple set of quadric relations, with the single exception of the Segre cubic threefold (the space of six points with equal weight). Unlike many facts in geometric invariant theory, these results (at least for the stable locus) are fieldindependent, and indeed work over the integers. CONTENTS
An efficient algorithm for the computation of Bernoulli numbers
, 2007
"... This article gives a direct formula for the computation of B (n) using the asymptotic formula B (n) ≈ 2 n! π n 2 n where n is even and n ≫ 1. This is simply based on the fact that ζ (n) is very near 1 when n is large and since B (n) = 2 ζ(n)n! πn2n exactly. The formula chosen for the Zeta function ..."
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Cited by 3 (0 self)
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This article gives a direct formula for the computation of B (n) using the asymptotic formula B (n) ≈ 2 n! π n 2 n where n is even and n ≫ 1. This is simply based on the fact that ζ (n) is very near 1 when n is large and since B (n) = 2 ζ(n)n! πn2n exactly. The formula chosen for the Zeta function is the one with prime numbers from the wellknown Euler product for ζ (n). This algorithm is far better than the recurrence formula for the Bernoulli numbers even if each B(n) is computed individually. The author could compute B (750,000) in a few hours. The current record of computation is now (as of Feb. 2007) B (5,000,000) a number of (the numerator) of 27332507 decimal digits is also based on that idea. 1 The need for a single computation This algorithm came once in 1996 when the authors wanted to compute large Bernoulli numbers
On rationally parametrized modular equations
"... Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformati ..."
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Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformation of elliptic curve periods, which are parametrized by a Hauptmodul (function field generator). Since the periods satisfy a Picard–Fuchs equation, which is of hypergeometric, Heun, or more general type, the new equations can be viewed as algebraic transformation formulas for special functions. The ones for N = 4,3, 2 yield parametrized modular transformations of Ramanujan’s elliptic integrals of signatures 2, 3,4. The case of signature 6 will require an extension of the present theory, to one of modular equations for general elliptic surfaces.
Filtering for private collaborative benchmarking
 International Conference on Emergin Trends in Information and Communication Security, LNCS 3995
, 2006
"... Abstract. Collaborative Benchmarking is an important issue for modern enterprises, but the business performance quantities used as input are often highly confidential. Secure MultiParty Computation can offer protocols that can compute benchmarks without leaking the input variables. Benchmarking is ..."
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Cited by 3 (1 self)
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Abstract. Collaborative Benchmarking is an important issue for modern enterprises, but the business performance quantities used as input are often highly confidential. Secure MultiParty Computation can offer protocols that can compute benchmarks without leaking the input variables. Benchmarking is a process of comparing to the “best”, so often it is necessary to only include the kbest enterprises for computing a benchmark to not distort the result with some outlying performances. We present a protocol that can be used as a filter, before running any collaborative benchmarking protocol that restricts the participants to the k best values. Our protocol doesn’t use the general circuit construction technique for SMC aiming to optimize performance. As building blocks we present the fastest implementation of Yao’s millionaires ’ protocol and a protocol that achieves a fair shuffle in O(log n) rounds. 1
WEIGHT MULTIPLICITY POLYNOMIALS OF MULTIVARIABLE WEYL MODULES
, 2009
"... This paper is based on the observation that dimension of weight spaces of multivariable Weyl modules depends polynomially on the highest weight (Conjecture 1). We support this conjecture by various explicit answers for up to three variable cases and discuss the underlying combinatorics. ..."
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This paper is based on the observation that dimension of weight spaces of multivariable Weyl modules depends polynomially on the highest weight (Conjecture 1). We support this conjecture by various explicit answers for up to three variable cases and discuss the underlying combinatorics.