Abstract—The problem of finding the least-squares 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 NP-hard. 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 worst-case complexity of the integer least-squares 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 closed-form expression for the expected complexity, both for the infinite and finite lattice.

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.

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 q-series, our approach leads to new elementary proofs of the corresponding identities.

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.

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.

Abstract. We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration 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 q-ultraspherical polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions. 1.

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

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

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 Q-polynomials and to correlation functions for continuous dual Hahn polynomials. We also state a new formula for 2m 2 squares. 1.

Abstract. We obtain expressions for the Christoffel–Darboux kernel of antisymmetric multivariable orthogonal polynomials as determinants and pfaffians. These kernels include correlation functions of orthogonal polynomial ensembles (with β = 2). In subsequent work, our results are applied in combinatorics (enumeration of marked shifted tableaux) and number theory (representation of integers as sums of squares). 1.