Kinematic Self-Similarity

Self-similarity in general relativity is briefly reviewed and the differences between self-similarity of the first kind and generalized self-similarity are discussed. The covariant notion of a kinematic self-similarity in the context of relativistic fluid mechanics is defined. Various mathematical and physical properties of spacetimes admitting a kinematic self-similarity are discussed. The governing equations for perfect fluid cosmological models are introduced and a set of integrability conditions for the existence of a proper kinematic self-similarity in these models is derived. Exact solutions of the irrotational perfect fluid Einstein field equations admitting a kinematic self-similarity are then sought in a number of special cases, and it is found that; (1) in the geodesic case the 3-spaces orthogonal to the fluid velocity vector are necessarily Ricci-flat and (ii) in the further specialisation to dust the differential equation governing the expansion can be completely integrated and the asymptotic properties of these solutions can be determined, (iii) the solutions in the case of zero-expansion consist of a class of shear-free and static models and a class of stiff perfect fluid (and non-static) models, and (iv) solutions in which the kinematic self-similar vector is parallel to the fluid velocity vector are necessarily Friedmann-Robertson-Walker (FRW) models.Comment: 29 pages, AmsTe

Self-Similarity and Localization

The localized eigenstates of the Harper equation exhibit universal self-similar fluctuations once the exponentially decaying part of a wave function is factorized out. For a fixed quantum state, we show that the whole localized phase is characterized by a single strong coupling fixed point of the renormalization equations. This fixed point also describes the generalized Harper model with next nearest neighbor interaction below a certain threshold. Above the threshold, the fluctuations in the generalized Harper model are described by a strange invariant set of the renormalization equations.Comment: 4 pages, RevTeX, 2 figures include

Self-Similarity of Friction Laws

The change of the friction law from a mesoscopic level to a macroscopic level is studied in the spring-block models introduced by Burridge-Knopoff. We find that the Coulomb law is always scale invariant. Other proposed scaling laws are only invariant under certain conditions.}Comment: Plain TEX. Figures not include

Self-similarity in Laplacian Growth

We consider Laplacian Growth of self-similar domains in different geometries. Self-similarity determines the analytic structure of the Schwarz function of the moving boundary. The knowledge of this analytic structure allows us to derive the integral equation for the conformal map. It is shown that solutions to the integral equation obey also a second order differential equation which is the one dimensional Schroedinger equation with the sinh inverse square potential. The solutions, which are expressed through the Gauss hypergeometric function, characterize the geometry of self-similar patterns in a wedge. We also find the potential for the Coulomb gas representation of the self-similar Laplacian growth in a wedge and calculate the corresponding free energy.Comment: 16 pages, 9 figure

Self-similarity of complex networks

Complex networks have been studied extensively due to their relevance to many real systems as diverse as the World-Wide-Web (WWW), the Internet, energy landscapes, biological and social networks \cite{ab-review,mendes,vespignani,newman,amaral}. A large number of real networks are called ``scale-free'' because they show a power-law distribution of the number of links per node \cite{ab-review,barabasi1999,faloutsos}. However, it is widely believed that complex networks are not {\it length-scale} invariant or self-similar. This conclusion originates from the ``small-world'' property of these networks, which implies that the number of nodes increases exponentially with the ``diameter'' of the network \cite{erdos,bollobas,milgram,watts}, rather than the power-law relation expected for a self-similar structure. Nevertheless, here we present a novel approach to the analysis of such networks, revealing that their structure is indeed self-similar. This result is achieved by the application of a renormalization procedure which coarse-grains the system into boxes containing nodes within a given "size". Concurrently, we identify a power-law relation between the number of boxes needed to cover the network and the size of the box defining a finite self-similar exponent. These fundamental properties, which are shown for the WWW, social, cellular and protein-protein interaction networks, help to understand the emergence of the scale-free property in complex networks. They suggest a common self-organization dynamics of diverse networks at different scales into a critical state and in turn bring together previously unrelated fields: the statistical physics of complex networks with renormalization group, fractals and critical phenomena.Comment: 28 pages, 12 figures, more informations at http://www.jamlab.or