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Concept

Logarithmic integral function

Summary

In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value x. Integral representation The logarithmic integral has an integral representation defined for all positive real numbers x ≠ 1 by the definite integral : \operatorname{li}(x) = \int_0^x \frac{dt}{\ln t}. Here, ln denotes the natural logarithm. The function 1/(ln t) has a singularity at t = 1, and the integral for x > 1 is interpreted as a Cauchy principal value, : \operatorname{li}(x) = \lim_{\varepsilon \to 0+} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} + \int_{1+\varepsilon}^x \frac{dt}{\ln t} \right). Offset logarithmic integr

Official source

https://en.wikipedia.org/wiki/Logarithmic_integral_function

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A quantitative central limit theorem for the random walk among random conductances

Jean-Christophe Mourrat

We consider the random walk among random conductances on Z(d). We assume that the conductances are independent, identically distributed and uniformly bounded away from 0 and infinity. We obtain a quantitative version of the central limit theorem for this random walk, which takes the form of a Berry-Esseen estimate with speed t(-1/10) for d = 3, up to logarithmic corrections.

Univ Washington, Dept Mathematics2012

The subject of the work presented here is the study of the acoustical ray method, which aims at describing a pressure field by an analogy with light rays. The concept of a wave is therefore replaced by the concept of "acoustical rays", whose paths through the domain under study are described by their reflections off the obstacles they encounter. The postulate of specular reflection is made : only the direct neighborhood of the point where the incoming ray meets the obstacle is taken into account. When absorption is considered at that particular point, it is represented by a localized acoustic impedance. Such a method, also called geometrical acoustics, is widely used for the computation of the acoustical characteristics of a room, for example the reverberation time in a concert hall, etc. In this case, the dimensions of the domain under study are several orders of magnitude larger than the considered wavelengths. Moreover, the information sought resides mostly in the first reflections. In these circumstances, the quality of prediction obtained is satisfactory, and is experimentally verified. Conversely, in geometries of smaller size and when a solution combining a greater number of reflections is required, simulations show a discrepancy between results obtained with the geometrical solution and those obtained via another method, such as finite elements. These differences, mostly in terms of frequency shifts of resonance peaks, are particularly visible in positions near the walls of the domain under study. With the aim of eventually finding ways to improve the prediction quality of the geometrical method, the work proposes a substitute for the intuitive origin of image sources. In fact, a parallel with a form of a solution of the Helmholtz equation by the integral method is shown. With this novel representation, image sources appear to have a more rigorous base than with the optical analogy. Before being able to improve a method, a deep understanding of its foundations is required. Here, the study of two phenomena occurring in reflection problems is proposed, and their influence on the results obtained with the geometrical method is observed. First, the very concept of image sources, which give rise to acoustical rays, is studied. By doing this, an a priori new description of image sources is proposed, which shows a close parallel between the representation of the acoustical field by a sum of rays emitted by image sources and a solution obtained by the integral method. Secondly, the validity of non locally reacting wave reflection, which is implied in geometrical acoustical methods, is studied. This is achieved by replacing the specular (local) coefficient classically used to describe reflection upon an obstacle with a non local coefficient, obtained by identification from an integral representation. Both methods are evaluated in geometries of growing complexity : spaces bounded by one, then two walls, and finally closed by several walls. This enables the study of an increasing number of reflections.

EPFL2006

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Roland Donninger

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as

t^{-2\ell-3}

for

t \to \infty

provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be

t^{-2\ell-4}

. We give a proof of

t^{-2\ell-2}

decay for general data in the form of weighted

L^1

L^\infty

bounds for solutions of the Regge--Wheeler equation. For initially static perturbations we obtain

t^{-2\ell-3}

. The proof is based on an integral representation of the solution which follows from self--adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.

2011