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31
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... 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 ..."
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Cited by 10 (5 self)
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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
ALGEBROGEOMETRIC ASPECTS OF HEINESTIELTJES THEORY
, 2008
"... The goal of the paper is to develop a HeineStieltjes 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 co ..."
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Cited by 9 (2 self)
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The goal of the paper is to develop a HeineStieltjes 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 nondegenerate 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 nondegeneracy assumptions there exist exactly n+r ´ such polynomials Vn,i(z) whose corresponding eigenpolynomials Sn,i(z)
New Results from Glueball Superpotentials and Matrix Models: the LeighStrassler Deformation
, 2002
"... Abstract: Using the result of a matrix model computation of the exact glueball superpotential, we investigate the relevant mass perturbations of the LeighStrassler marginal “q” deformation of N = 4 supersymmetric gauge theory. We recall a conjecture for the elliptic superpotential that describes th ..."
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Abstract: Using the result of a matrix model computation of the exact glueball superpotential, we investigate the relevant mass perturbations of the LeighStrassler 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 RuijsenaarsSchneider integrable system. In the limit that the LeighStrassler deformation is turned off, the integrable system reduces to the elliptic CalogeroMoser system which describes the N = 1 ∗ theory. Based on these results, we identify the Coulomb branch of the partially massdeformed LeighStrassler theory as the spectral curve of the RuijsenaarsSchneider system. We also show how the LeighStrassler deformation may be obtained by suitably modifying Witten’s M theory brane construction of N = 2 theories
Fivedimensional gauge theories and quantum mechanical matrix models
 JHEP 0303
, 2003
"... Abstract: We show how the DijkgraafVafa matrix model proposal can be extended to describe fivedimensional 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. W ..."
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Abstract: We show how the DijkgraafVafa matrix model proposal can be extended to describe fivedimensional 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 fivedimensional generalization of SeibergWitten theory involving the relativistic elliptic CalogeroMoser, or RuijsenaarsSchneider, integrable system.
Generalizations of Clausen’s formula and algebraic transformations of CalabiYau differential equations
, 2009
"... 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. ..."
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Cited by 4 (0 self)
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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.
Hypergeometric Equation And Ramanujan Functions
, 2001
"... 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 solu ..."
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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.
World sheet instantons via the Myers effect and N = 1* quiver superpotentials,” arXiv:hepth/0206051
"... Abstract: In this note we explore the stringy interpretation of nonperturbative 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 D3branes at the orbifold polarizing into k concentric 5brane ..."
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Abstract: In this note we explore the stringy interpretation of nonperturbative 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 D3branes at the orbifold polarizing into k concentric 5brane spheres each carrying fractional brane charge. The polarization of the D3branes induces a polarization of Dinstantons into string worldsheets wrapped on the Myers spheres. We show that the superpotentials in these models are indeed generated by these worldsheet instantons. We point out that for certain parameter values the condensates yield the exact superpotential for a relevant deformation of the KlebanovWitten conifold theory. Contents
Residue Theorem and Theta Function Identities
, 2001
"... 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: Primary11 ..."
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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: Primary11F11, 11E25, 11F27, 33E05 1.
Integral representations of qanalogues of the Hurwitz zeta function
, 2008
"... Two integral representations of qanalogues of the Hurwitz zeta function are established. Each integral representation allows us to obtain an analytic continuation including also a full description of poles and special values at nonpositive integers of the qanalogue of the Hurwitz zeta function, a ..."
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Two integral representations of qanalogues of the Hurwitz zeta function are established. Each integral representation allows us to obtain an analytic continuation including also a full description of poles and special values at nonpositive integers of the qanalogue of the Hurwitz zeta function, and to study the classical limit of this qanalogue. All the discussion developed here is entirely different from the previous work in [4].