The classical algebra of symmetric functions has a remarkable deformation which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions s , where ranges over the set of all partitions. The main significance of the shifted Schur functions is that they determine a natural basis in Z(gl(n)), the center of the universal enveloping algebra U(gl(n)), n = 1; 2; : : : .

We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call "value recycling".

Arithmetic functions related to number representation systems exhibit various periodicity phenomena. For instance, a well known theorem of Delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function. We show that such periodicity phenomena can be analyzed rather systematically using classical tools from analytic number theory, namely the Mellin-Perron formulae. This approach yields naturally the Fourier series involved in the expansions of a variety of digital sums related to number representation systems.

It is shown that if the automorphism group of a binary self-dual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4. In particular, no cyclic binary self-dual code can have all its weights divisible by 4. The number of cyclic binary self-dual codes of length n is determined, and the shortest nontrivial code in this class is shown to have length 14. 1.

Dedicated to the memory of Gian-Carlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the η-function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing

Abstract. We give an integral representation of solutions of the elliptic Knizhnik– Zamolodchikov–Bernard equations for arbitrary simple Lie algebras. If the level is a positive integer, we obtain formulas for conformal blocks of the WZW model on a torus. The asymptotics of our solutions at critical level gives eigenfunctions of Euler–Calogero–Moser integrable N-body systems. As a by-product, we obtain some remarkable integral identities involving classical theta functions. To appear in International Mathematics Research Notices 1.

This paper discusses quasi--interpolation and interpolation with Gaussians from a new point of view concerning accuracy in numerical computations. Estimates are obtained showing a high order approximation up to some saturation error negligible in numerical applications. The construction of local high order quasi--interpolation formulas is given.

We consider a class of random walks on a lattice, introduced by Gessel and Zeilberger, for which the reflection principle can be used to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We prove three independent results about such "reflectable walks": first, a classification of all such walks; second, many determinant formulas for walk-numbers and their generating functions; third, an equality between the walk-numbers and the multiplicities of irreducibles in the kth tensor power of certain Lie group representations associated to the walk-types. Our results apply to the defining representations of the classical groups, as well as some spin representations of the orthogonal groups.

Abstract not found

: We present the detailed derivation of the charge one periodic instantons - or calorons - with non-trivial holonomy for SU(2). We use a suitable combination of the Nahm transformation and ADHM techniques. Our results rely on our ability to compute explicitly the relevant Green's function in terms of which the solution can be conveniently expressed. We also discuss the properties of the moduli space, IR 3 \Theta S 1 \Theta Taub-NUT/Z 2 and its metric, relating the holonomy to the Taub-NUT mass parameter. We comment on the monopole constituent description of these calorons, how to retrieve topological charge in the context of abelian projection and possible applications to QCD. 1 Introduction Instantons [1] and Bogomolny-Prasad-Sommerfield (BPS) monopoles [2] possess remarkable properties. They exist as exact solutions with arbitrary charges and with an action or energy, proportional to their integer charge. Therefore the multi-charge solutions can be seen as built from constituen...