Generic Emergence of Power Law Distributions and L\'evy-Stable Intermittent Fluctuations in Discrete Logistic Systems

The dynamics of generic stochastic Lotka-Volterra (discrete logistic) systems of the form \cite{Solomon96a} $w_i (t+1) = \lambda(t) w_i (t) + a {\bar w (t)} - b w_i (t) {\bar w(t)}$ is studied by computer simulations. The variables $w_i$, $i=1,...N$, are the individual system components and ${\bar w (t)} = {1\over N} \sum_i w_i (t)$ is their average. The parameters $a$ and $b$ are constants, while $\lambda(t)$ is randomly chosen at each time step from a given distribution. Models of this type describe the temporal evolution of a large variety of systems such as stock markets and city populations. These systems are characterized by a large number of interacting objects and the dynamics is dominated by multiplicative processes. The instantaneous probability distribution $P(w,t)$ of the system components $w_i$, turns out to fulfill a (truncated) Pareto power-law $P(w,t) \sim w^{-1-\alpha}$. The time evolution of ${\bar w (t)}$ presents intermittent fluctuations parametrized by a truncated L\'evy distribution of index $\alpha$, showing a connection between the distribution of the $w_i$'s at a given time and the temporal fluctuations of their average.Comment: 18 pages and 5 figures (in one zipped file); [email protected], [email protected], [email protected], [email protected], http://shum.huji.ac.il/~sori

Cosmological Tests from the New Surveys

We review cosmological inference from galaxy surveys at low and high redshifts, with emphasis on new Southern sky surveys. We focus on several issues: (i) The importance of understanding selection effects in catalogues and matching Northern and Southern surveys; (ii) The 2dF galaxy redshift survey of 250,000 galaxies (iii) The proposed 6dF redshift and peculiar velocity survey of near-infrared galaxies (iv) Radio sources and the X-Ray Background as useful probes of the density fluctuations on large scales, and (v) How to combine large scale structure and Cosmic Microwave Background measurements to estimate cosmological parameters.Comment: Invited talk, to appear in the Proceedings of the ESO/ATNF Workshop "Looking Deep in the Southern Sky", 10-12 December 1997, Sydney, Australia, Eds. R. Morganti and W. Couch; Springer; 9 pages, 2 ps figures (Latex, epsf.sty, lamuphys.sty

Testing Homogeneity on Large Scales

We review observational tests for the homogeneity of the Universe on large scales. Redshift and peculiar velocity surveys, radio sources, the X-Ray Background, the Lyman-alpha forest and the Cosmic Microwave Background are used to set constraints on inhomogeneous models and in particular on fractal-like models. Assuming the Cosmological Principle and the FRW metric, we estimate cosmological parameters by joint analysis of peculiar velocities, the CMB, cluster abundance, IRAS and Supernovae. Under certain assumptions the best fit density parameter is Omega_m = 1 - lambda = 0.3-0.5.Comment: Review talk, to appear in the proceedings of the Cosmic Flows Workshop, Victoria, Canada, July 1999, ed. S. Courteau, M. Strauss & J. Willick, ASP series 13 pages Latex, with 5 embedded figures. Uses paspconf.st

Deciding Disjunctive Linear Arithmetic with SAT

Disjunctive Linear Arithmetic (DLA) is a major decidable theory that is supported by almost all existing theorem provers. The theory consists of Boolean combinations of predicates of the form $\Sigma_{j=1}^{n}a_j\cdot x_j \le b$, where the coefficients $a_j$, the bound $b$ and the variables $x_1 >... x_n$ are of type Real ($\mathbb{R}$). We show a reduction to propositional logic from disjunctive linear arithmetic based on Fourier-Motzkin elimination. While the complexity of this procedure is not better than competing techniques, it has practical advantages in solving verification problems. It also promotes the option of deciding a combination of theories by reducing them to this logic. Results from experiments show that this method has a strong advantage over existing techniques when there are many disjunctions in the formula

Continuous time random walk as a random walk in a random environment

We show that for a weakly dense subset of the domain of attraction of a positive stable random variable of index $0<\alpha<1$($DOA\left(\alpha\right))$ the functional stable convergence is a time-changed renewal convergence of distribution of finite mean. Applied to Continuous Time Random Walk(CTRW) \'a la Montroll and Wiess we show that CTRW with renewal times in a weakly dense set of $DOA\left(\alpha\right)$ can be realized as random walk in a random environment. We find the quenched limit and give a bound on the error of the approximation

Theoretical Analysis and Simulations of the Generalized Lotka-Volterra Model

The dynamics of generalized Lotka-Volterra systems is studied by theoretical techniques and computer simulations. These systems describe the time evolution of the wealth distribution of individuals in a society, as well as of the market values of firms in the stock market. The individual wealths or market values are given by a set of time dependent variables $w_i$, $i=1,...N$. The equations include a stochastic autocatalytic term (representing investments), a drift term (representing social security payments) and a time dependent saturation term (due to the finite size of the economy). The $w_i$'s turn out to exhibit a power-law distribution of the form $P(w) \sim w^{-1-\alpha}$. It is shown analytically that the exponent $\alpha$ can be expressed as a function of one parameter, which is the ratio between the constant drift component (social security) and the fluctuating component (investments). This result provides a link between the lower and upper cutoffs of this distribution, namely between the resources available to the poorest and those available to the richest in a given society. The value of $\alpha$ %as well as the position of the lower cutoff is found to be insensitive to variations in the saturation term, that represent the expansion or contraction of the economy. The results are of much relevance to empirical studies that show that the distribution of the individual wealth in different countries during different periods in the 20th century has followed a power-law distribution with $1 < \alpha < 2$

Observational Tests for the Cosmological Principle and World Models

We review observational tests for the homogeneity of the Universe on large scales. Redshift and peculiar velocity surveys, radio sources, the X-Ray Background, the Lyman-$\alpha$ forest and the Cosmic Microwave Background are used to set constraints on inhomogeneous models and in particular on fractal-like models. Assuming the Cosmological Principle and the FRW metric, we estimate cosmological parameters by joint analysis of peculiar velocities, the CMB, cluster abundance, IRAS and Supernovae. Under certain assumptions the best fit density parameter is Omega_m = 1 - lambda \approx 0.4 . We present a new method for joint estimation by combining different data sets in a Bayesian way, and utilising `Hyper-Parameters'.Comment: Review talk, to appear in the proceedings of the NATO ASI `Structure Formation in the Universe', Isaac Newton Institute, Cambridge, July 1999, ed. R. Crritenden & N, Turok. Kluwer; 12 pages Latex, with 3 embedded figures. Uses crckapb.st

Finite Dimensional Fokker-Planck Equations for Continuous Time Random Walks

Continuous Time Random Walk(CTRW) is a model where particle's jumps in space are coupled with waiting times before each jump. A Continuous Time Random Walk Limit(CTRWL) is obtained by a limit procedure on a CTRW and can be used to model anomalous diffusion. The distribution $p\left(dx,t\right)$ of a CTRWL $X_{t}$ satisfies a Fractional Fokker-Planck Equation(FFPE). Since CTRWLs are usually not Markovian, their one dimensional FFPE is not enough to completely define them. In this paper we find the FFPEs of the distribution of $X_{t}$ at multiple times , i.e. the distribution of the random vector $\left(X_{t_{1}},...,X_{t_{n}}\right)$ for $t_{1}<...<t_{n}$ for a large class of CTRWLs. This allows us to define CTRWLs by their finite dimensional FFPEs

Moriond Conference Summary: The Cosmological Model(s)

The XXXVIIth Rencontres de Moriond on "The Cosmological Model" is briefly summarized. Almost none of the current observations argues against the popular Cold Dark Matter + Lambda concordance model. However, it remains to be tested how astrophysical uncertainties involved in the interpretation of the different data sets affect the derived cosmological parameters. Independent tests are still required to establish if the Cold Dark Matter and Dark Energy components are `real', or just `epicycles' that happen to fit the current data sets well.Comment: 4 pages; to appear in the proceedings of the XXXVIIth Moriond Astrophysics Meeting "The Cosmological Model", Les Arcs, France, March 200

Random walks in random environments

Random walks in random environments (RWRE's) have been a source of surprising phenomena and challenging problems since they began to be studied in the 70's. Hitting times and, more recently, certain regeneration structures, have played a major role in our understanding of RWRE's. We review these and provide some hints on current research directions and challenges