Abstract not found

Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random

Abstract: Using the result of a matrix model computation of the exact glueball superpotential, we investigate the relevant mass perturbations of the Leigh-Strassler marginal “q” deformation of N = 4 supersymmetric gauge theory. We recall a conjecture for the elliptic superpotential that describes the theory compactified on a circle and identify this superpotential as one of the Hamiltonians of the elliptic Ruijsenaars-Schneider integrable system. In the limit that the Leigh-Strassler deformation is turned off, the integrable system reduces to the elliptic Calogero-Moser system which describes the N = 1 ∗ theory. Based on these results, we identify the Coulomb branch of the partially mass-deformed Leigh-Strassler theory as the spectral curve of the Ruijsenaars-Schneider system. We also show how the Leigh-Strassler deformation may be obtained by suitably modifying Witten’s M theory brane construction of N = 2 theories

The goal of the paper is to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operator d(z) = Pk di i=1 Qi(z) dzi with polynomial coefficients set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) deg Qk(z) = k + r we call it a non-degenerate higher Lamé operator. Following the classical approach of E. Heine and T. Stieltjes, see [18], [41] we study the multiparameter spectral problem of finding all polynomials V (z) of degree at most r such that the equation: d(z)S(z) + V (z)S(z) = 0 has for a given positive integer n a polynomial solution S(z) of degree n. We show that under some mild non-degeneracy assumptions there exist exactly n+r ´ such polynomials Vn,i(z) whose corresponding eigenpolynomials Sn,i(z)

Abstract: We show how the Dijkgraaf-Vafa matrix model proposal can be extended to describe five-dimensional gauge theories compactified on a circle to four dimensions. This involves solving a certain quantum mechanical matrix model. We do this for the lift of the N = 1 ∗ theory to five dimensions. We show that the resulting expression for the superpotential in the confining vacuum is identical with the elliptic superpotential approach based on Nekrasov’s five-dimensional generalization of Seiberg-Witten theory involving the relativistic elliptic Calogero-Moser, or Ruijsenaars-Schneider, integrable system.

In this paper we give analogues of the Ramanujan functions and nonlinear dierential equations for them. Investigating a modular structure of solutions for nonlinear dierential systems, we deduce new identities between the Ramanujan and hypergeometric functions. Another result of this paper is a solution of transcendence problems concerning nonlinear systems. In 1916 S. Ramanujan has proved [12] that the functions P (q) = 1 24 1 X n=1 1 (n)q n ; Q(q) = 1 + 240 1 X n=1 3 (n)q n ; R(q) = 1 504 1 X n=1 5 (n)q n ; (1) where k (n) = P djn d k , satisfy the system of nonlinear dierential equations q dP dq = 1 12 (P 2 Q); q dQ dq = 1 3 (PQ R); q dR dq = 1 2 (PR Q 2 ) (2) (see also [5, Chapter X, Sect. 5]). Note that Q and R are modular as functions of = 1 2i log q.

Abstract. We provide certain unusual generalizations of Clausen’s and Orr’s theorems for solutions of generalized hypergeometric equations of order 4 and 5. As application, we present several examples of algebraic transformations of Calabi– Yau differential equations.

Abstract: In this note we explore the stringy interpretation of non-perturbative effects in N = 1 ∗ deformations of the Ak−1 quiver models. For certain types of deformations we argue that the massive vacua are described by Nk fractional D3-branes at the orbifold polarizing into k concentric 5-brane spheres each carrying fractional brane charge. The polarization of the D3-branes induces a polarization of D-instantons into string world-sheets wrapped on the Myers spheres. We show that the superpotentials in these models are indeed generated by these world-sheet instantons. We point out that for certain parameter values the condensates yield the exact superpotential for a relevant deformation of the Klebanov-Witten conifold theory. Contents

In this paper we will use the residue theorem of elliptic functions to prove some theta function identities of Ramanujan. We also derive some new identities by this method. Key words: residue theorem, elliptic functions, theta function identities 2000 Mathematics Subject Classification: Primary---11F11, 11E25, 11F27, 33E05 1.

In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equations u tt = (u + flu 2 ) xx + uu xxxx + ¯u xxtt + ffu x u xxx + fiu 2 xx ; (1) where ff; fi; fl; and ¯ are arbitrary constants. This equation may be thought of as a fourth order analogue of a generalization of the Camassa-Holm equation, about which there has been considerable recent interest. Further equation (1) is a "Boussinesq-type" equation which arises as a model of vibrations of an anharmonic mass-spring chain and admits both "compacton" and conventional solitons. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole. In particular we obtain several reductions using the nonclassical method which are not obtainable through the classical method.