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29
On the spheredecoding algorithm I. Expected complexity
 IEEE Trans. Sig. Proc
, 2005
"... Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The ..."
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Cited by 76 (5 self)
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Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NPhard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worstcase complexity of the integer leastsquares problem, we study its expected complexity, averaged over the noise and over the lattice. For the “sphere decoding” algorithm of Fincke and Pohst, we find a closedform expression for the expected complexity, both for the infinite and finite lattice.
Advanced determinant calculus: a complement
 Linear Algebra Appl
"... Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particu ..."
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Cited by 49 (6 self)
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Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441– 456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems. 1.
Advanced Determinant Calculus
, 1999
"... The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have ..."
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Cited by 37 (0 self)
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The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.
ELLIPTIC HYPERGEOMETRIC SERIES ON ROOT SYSTEMS
, 2002
"... We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems An, Cn and Dn. In the special cases of classical and qseries, our approach leads to new elementary proofs of the corresponding identities. ..."
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Cited by 27 (8 self)
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We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems An, Cn and Dn. In the special cases of classical and qseries, our approach leads to new elementary proofs of the corresponding identities.
Sums of triangular numbers from the Frobenius determinant
"... Abstract. We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitra ..."
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Cited by 7 (5 self)
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Abstract. We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitrary number as a sum of 4m 2 /d triangles, whenever d  2m, and 4m(m + 1)/d triangles, when d  2m or d  2m + 2. This extends recent results of Getz and Mahlburg, Milne, and Zagier. 1.
Qpolynomials, multiple hypergeometric series and enumeration of marked shifted tableaux
 J. Combin. Theory Ser. A
"... Abstract. We study Schur Qpolynomials evaluated on a geometric progression, or equivalently qenumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q = 1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians ..."
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Cited by 5 (3 self)
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Abstract. We study Schur Qpolynomials evaluated on a geometric progression, or equivalently qenumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q = 1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of qultraspherical polynomials. As special cases, we obtain simple closed formulas for staircasetype partitions. 1.
Ramanujan’s contributions to Eisenstein series, especially in his lost notebook
 in Number Theoretic Methods  Future Trends
"... In contemporary notation, the Eisenstein series G2j(τ) and E2j(τ) of weight 2j on the full modular group Γ(1), where j is a positive integer exceeding one, are defined for Im τ> 0 by ..."
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Cited by 3 (3 self)
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In contemporary notation, the Eisenstein series G2j(τ) and E2j(τ) of weight 2j on the full modular group Γ(1), where j is a positive integer exceeding one, are defined for Im τ> 0 by
Ramanujan’s “Lost Notebook ” and the Virasoro Algebra
, 2003
"... In this note, by using the theory of vertex operator algebras, we gave a new proof of a famous Ramanujan’s (modulus 5) modular equation from his ”Lost Notebook ” (p.139 [R]). Moreover, we obtained an infinite list of q–identities for all the odd moduli; thus we generalized the result of Ramanujan. 1 ..."
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Cited by 3 (2 self)
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In this note, by using the theory of vertex operator algebras, we gave a new proof of a famous Ramanujan’s (modulus 5) modular equation from his ”Lost Notebook ” (p.139 [R]). Moreover, we obtained an infinite list of q–identities for all the odd moduli; thus we generalized the result of Ramanujan. 1
Sums of squares from elliptic pfaffians
"... Abstract. We give a new proof of Milne’s formulas for the number of representations of an integer as a sum of 4m 2 and 4m(m + 1) squares. The proof is based on explicit evaluation of pfaffians with elliptic function entries, and relates Milne’s formulas to Schur Qpolynomials and to correlation func ..."
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Cited by 2 (2 self)
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Abstract. We give a new proof of Milne’s formulas for the number of representations of an integer as a sum of 4m 2 and 4m(m + 1) squares. The proof is based on explicit evaluation of pfaffians with elliptic function entries, and relates Milne’s formulas to Schur Qpolynomials and to correlation functions for continuous dual Hahn polynomials. We also state a new formula for 2m 2 squares. 1.
HANKEL DETERMINANTS OF EISENSTEIN SERIES
, 2000
"... Abstract. In this paper we prove Garvan’s conjectured formula for the square of the modular discriminant ∆ as a 3 by 3 Hankel determinant of classical Eisenstein series E2n. We then obtain similar formulas involving minors of Hankel determinants for E2r ∆ m, for m = 1,2,3 and r = 2,3, 4,5,7, and E14 ..."
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Cited by 2 (1 self)
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Abstract. In this paper we prove Garvan’s conjectured formula for the square of the modular discriminant ∆ as a 3 by 3 Hankel determinant of classical Eisenstein series E2n. We then obtain similar formulas involving minors of Hankel determinants for E2r ∆ m, for m = 1,2,3 and r = 2,3, 4,5,7, and E14 ∆ 4. We next use mathematica to discover, and then the standard structure theory of the ring of modular forms, to derive the general form of our infinite family of formulas extending the classical formula for ∆ and Garvan’s formula for ∆2. This general formula expresses the n ×n Hankel determinant det(E2(i+j)(q))1≤i,j≤n as the product of ∆n−1 (τ), a homogeneous polynomial in E3 4 and E2 6, and if needed, E4. We also include a simple verification proof of the classical 2 by 2 Hankel determinant formula for ∆. This proof depends upon polynomial properties of elliptic function parameters from Jacobi’s Fundamenta Nova. The modular forms approach provides a convenient explaination for the determinant identities in this paper. 1.