12,249 research outputs found
Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds
We reconsider the problem of calculating a general spectral correlation
function containing an arbitrary number of products and ratios of
characteristic polynomials for a N x N random matrix taken from the Gaussian
Unitary Ensemble (GUE).
Deviating from the standard "supersymmetry" approach, we integrate out
Grassmann variables at the early stage and circumvent the use of the
Hubbard-Stratonovich transformation in the "bosonic" sector. The method,
suggested recently by one of us, is shown to be capable of calculation when
reinforced with a generalization of the Itzykson-Zuber integral to a
non-compact integration manifold. We arrive to such a generalisation by
discussing the Duistermaat-Heckman localization principle for integrals over
non-compact homogeneous Kaehler manifolds.
In the limit of large the asymptotic expression for the correlation
function reproduces the result outlined earlier by Andreev and Simons.Comment: 34 page, no figures. In this version we added a few references and
modified the introduction accordingly. We also included a new Appendix on
deriving our Itzykson-Zuber type integral following the diffusion equation
metho
Motivic integration and the Grothendieck group of pseudo-finite fields
We survey our recent work on an extension of the theory of motivic
integration, called arithmetic motivic integration. We developed this theory to
understand how p-adic integrals of a very general type depend on p.Comment: 11 page
Mellin Transforms of the Generalized Fractional Integrals and Derivatives
We obtain the Mellin transforms of the generalized fractional integrals and
derivatives that generalize the Riemann-Liouville and the Hadamard fractional
integrals and derivatives. We also obtain interesting results, which combine
generalized operators with generalized Stirling numbers and Lah
numbers. For example, we show that corresponds to the Stirling
numbers of the kind and corresponds to the unsigned Lah
numbers. Further, we show that the two operators and
, , generate the same sequence given by the
recurrence relation
with and for and or
. Finally, we define a new class of sequences for and in turn show
that corresponds to the generalized Laguerre
polynomials.Comment: 17 pages, 1 figure, 9 tables, Accepted for publication in Applied
Mathematics and Computatio
Estimation with Numerical Integration on Sparse Grids
For the estimation of many econometric models, integrals without analytical solutions have to be evaluated. Examples include limited dependent variables and nonlinear panel data models. In the case of one-dimensional integrals, Gaussian quadrature is known to work efficiently for a large class of problems. In higher dimensions, similar approaches discussed in the literature are either very specific and hard to implement or suffer from exponentially rising computational costs in the number of dimensions - a problem known as the "curse of dimensionality" of numerical integration. We propose a strategy that shares the advantages of Gaussian quadrature methods, is very general and easily implemented, and does not suffer from the curse of dimensionality. Monte Carlo experiments for the random parameters logit model indicate the superior performance of the proposed method over simulation techniques
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