20,942 research outputs found
Truncated generalized averaged Gauss quadrature rules
Generalized averaged Gaussian quadrature formulas may yield higher accuracy than Gauss quadrature formulas that use the same moment information. This makes them attractive to use when moments or modified moments are cumbersome to evaluate. However, generalized averaged Gaussian quadrature formulas may have nodes outside the convex hull of the support of the measure defining the associated Gauss rules. It may therefore not be possible to use generalized averaged Gaussian quadrature formulas with integrands that only are defined on the convex hull of the support of the measure. Generalized averaged Gaussian quadrature formulas are determined by symmetric tridiagonal matrices. This paper investigates whether removing some of the last rows and columns of these matrices gives quadrature rules whose nodes live in the convex hull of the support of the measure
Truncated generalized averaged Gauss quadrature rules
Generalized averaged Gaussian quadrature formulas may yield higher accuracy than Gauss quadrature formulas that use the same moment information. This makes them attractive to use when moments or modified moments are cumbersome to evaluate. However, generalized averaged Gaussian quadrature formulas may have nodes outside the convex hull of the support of the measure defining the associated Gauss rules. It may therefore not be possible to use generalized averaged Gaussian quadrature formulas with integrands that only are defined on the convex hull of the support of the measure. Generalized averaged Gaussian quadrature formulas are determined by symmetric tridiagonal matrices. This paper investigates whether removing some of the last rows and columns of these matrices gives quadrature rules whose nodes live in the convex hull of the support of the measure
A note on generalized averaged Gaussian formulas
We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule
Generalized averaged gaussian formulas for certain weight functions
In this paper we analyze the generalized averaged Gaussian quadrature formulas and the simplest truncated variant for one of them for some weight functions on the interval [0, 1] considered by Milovanovic in [10]. We shall investigate internality of these formulas for the equivalents of the Jacobi polynomials on this interval and, in some special cases, show the existence of the Gauss-Kronrod quadrature formula. We also include some examples showing the corresponding error estimates for some non-classical orthogonal polynomials
The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source
In classical random matrix theory the Gaussian and chiral Gaussian random
matrix models with a source are realized as shifted mean Gaussian, and chiral
Gaussian, random matrices with real , complex ( and
real quaternion ) elements. We use the Dyson Brownian motion model
to give a meaning for general . In the Gaussian case a further
construction valid for is given, as the eigenvalue PDF of a
recursively defined random matrix ensemble. In the case of real or complex
elements, a combinatorial argument is used to compute the averaged
characteristic polynomial. The resulting functional forms are shown to be a
special cases of duality formulas due to Desrosiers. New derivations of the
general case of Desrosiers' dualities are given. A soft edge scaling limit of
the averaged characteristic polynomial is identified, and an explicit
evaluation in terms of so-called incomplete Airy functions is obtained.Comment: 21 page
Random matrices, log-gases and Holder regularity
The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue
statistics of large real and complex Hermitian matrices with independent,
identically distributed entries are universal in a sense that they depend only
on the symmetry class of the matrix and otherwise are independent of the
details of the distribution. We present the recent solution to this
half-century old conjecture. We explain how stochastic tools, such as the Dyson
Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory,
were combined in the solution.
We also show related results for log-gases that represent a universal model
for strongly correlated systems. Finally, in the spirit of Wigner's original
vision, we discuss the extensions of these universality results to more
realistic physical systems such as random band matrices.Comment: Proceedings of ICM 201
Time-averaged MSD of Brownian motion
We study the statistical properties of the time-averaged mean-square
displacements (TAMSD). This is a standard non-local quadratic functional for
inferring the diffusion coefficient from an individual random trajectory of a
diffusing tracer in single-particle tracking experiments. For Brownian motion,
we derive an exact formula for the Laplace transform of the probability density
of the TAMSD by mapping the original problem onto chains of coupled harmonic
oscillators. From this formula, we deduce the first four cumulant moments of
the TAMSD, the asymptotic behavior of the probability density and its accurate
approximation by a generalized Gamma distribution
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