We compute the axial anomaly for the Taub-NUT metric on R. We show that the axial anomaly for the generalized Taub-NUT metrics introduced by Iwai and Katayama is finite, although the Dirac operator is not Fredholm. We show that the essential spectrum of the Dirac operator is the whole real line. Pacs: 04.62.+v

Distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive Hermitian matrices. The inverse Fourier transform of the distribution is shown to be a Fredholm determinant of a certain operator that is an analogue of a Wiener-Hopf operator. The asymptotic formula shows that up to the terms of order o(1), the distributions are Gaussian.

We calculate a correlation function of the Jordan-Wigner operator in a class of free-fermion models formulated on an infinite one-dimensional lattice. We represent this function in terms of the determinant of an integrable Fredholm operator, convenient for analytic and numerical investigations. By using Wick’s theorem, we avoid the form-factor summation customarily used in literature for treati...

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 th...

We compute the axial anomaly for the Taub-NUT metric on R. We show that the axial anomaly for the generalized Taub-NUT metrics introduced by Iwai and Katayama is finite, although the Dirac operator is not Fredholm. We show that the essential spectrum of the Dirac operator is the whole real line. Pacs: 04.62.+v

We apply an analogue of the Zakharov-Shabat dressing method to obtain infinite matrix solutions to the Toda lattice hierarchy. Using an operator transformation we convert some of these into solutions in terms of integral operators and Fredholm determinants. Others are converted into a class of operator solutions to the l-periodic Toda hierarchy.

We explore the relationship between the Evans function, transmission coefficient and Fredholm determinant for systems of first-order linear differential operators on the real line. The applications we have in mind include linear stability problems associated with travelling wave solutions to nonlinear partial differential equations, for example reaction-diffusion or solitary wave equations. The...

We consider methods of reduction of differential operator equations with the Fredholm operator in the main expression to regular problems. Relation between the initial conditions choice problem and the Jordan structure of operator coefficients of equations is shown. The theorem of existence and uniqueness of the initial problem is proved. The method of fundamental operators is used for construc...

Keywords: Approximate solutions to integral equations Radial and kernel-based networks Gaussian kernels Model complexity Analysis of algorithms a b s t r a c t Approximate solutions to inhomogeneous Fredholm integral equations of the second kind by radial and kernel networks are investigated. Upper bounds are derived on errors in approximation of solutions of these equations by networks with in...