By Andrea Gabrielli, F. Sylos Labini, Michael Joyce, Luciano Pietronero

ISBN-10: 3540269991

ISBN-13: 9783540269991

ISBN-10: 3540407456

ISBN-13: 9783540407454

The physics of scale-invariant and intricate platforms is a singular interdisciplinary box. Its principles let us examine normal phenomena in a extensively new and unique means, finally resulting in unifying strategies self sustaining of the unique constitution of the platforms. the target is the examine of advanced, scale-invariant, and extra common stochastic buildings that seem either in area and time in an unlimited number of common phenomena, which convey new varieties of collective behaviors, and the fostering in their knowing. This publication has been conceived as a methodological monograph within which the most tools of recent statistical physics for cosmological constructions and density fields (galaxies, Cosmic Microwave heritage Radiation, etc.) are awarded intimately. the most function is to give sincerely, to a attainable point, those equipment, with a definite mathematical accuracy, offering additionally a few paradigmatic examples of purposes. this could bring about a brand new and extra basic framework for the statistical research of the numerous new info about the diverse cosmic constructions which signify the big scale Universe and for his or her theoretical interpretation and modeling.

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9 for details). In order to explain the origin of the second (and principal) point of controversy, it is useful to clarify what we mean by the term “robust”, as behind it lies our determination of the scale up to which one can obtain constraints from data sets which sample ﬁnite regions around us. In order to clarify it, one must analyze the role of ﬁnite size eﬀects which enter when one estimates the conditional density in such a ﬁnite sample6 . e. of translational and rotational invariance of the statistical properties) about the nature of the underlying distribution at the scales of the order of the sample size.

See [97] for a clear discussion of this point. A practical working deﬁnition of the homogeneity scale applicable in the analysis of galaxy surveys, and a discussion of the current status of this scale, is given in [140] and in Chap. 8). 30). 31) d3 rξ(r) where the integral is extended to all the space. 31) is a much stronger special condition which holds for certain mass density ﬁelds – the ones to which we will ascribe the name “super-homogeneous” [91] (in Chap. 3). 29). Such an approximation is not always valid even for large R, and the convergence properties of the double integral need to be examined carefully to establish it.

Let p(x) be the joint probability density function of all the components of X. 12) where k ≡ {k1 , k2 , . . , kN }. The fact that p(x) is a probability density function, and the general theorems of the Fourier transform (FT) imply, among others, the following simple properties of the characteristic function ˜(k): 1) ˜(0) = 1. 2) |˜ p(k)| ≤ 1. 3) ˜(k) is a uniformly continuous function of its argument for all real k. N i=1 4) If the moment Xili exists, it is given by: Xili = n n i=1 i i=1 ∂ ∂ki li p˜(k) .

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