The Omega Dependence of the Evolution of xi(r)

The evolution of the two-point correlation function, xi(r,z), and the pairwise velocity dispersion, sigma(r,z), for both the matter and halo population, in three different cosmological models: (Omega_M,Omega_Lambda)=(1,0), (0.2,0) and (0.2,0.8) are described. If the evolution of xi is parameterized by xi(r,z)=(1+z)^{-(3+eps)}xi(r,0), where xi(r,0)=(r/r_0)^{-gamma}, then eps(mass) ranges from 1.04 +/- 0.09 for (1,0) to 0.18 +/- 0.12 for (0.2,0), as measured by the evolution of at 1 Mpc (from z ~ 5 to the present epoch). For halos, eps depends on their mean overdensity. Halos with a mean overdensity of about 2000 were used to compute the halo two-point correlation function tested with two different group finding algorithms: the friends of friends and the spherical overdensity algorithm. It is certainly believed that the rate of growth of this xihh will give a good estimate of the evolution of the galaxy two-point correlation function, at least from z ~ 1 to the present epoch. The values we get for eps(halos) range from 1.54 for (1,0) to -0.36 for (0.2,0), as measured by the evolution of xi(halos) from z ~ 1.0 to the present epoch. These values could be used to constrain the cosmological scenario. The evolution of the pairwise velocity dispersion for the mass and halo distribution is measured and compared with the evolution predicted by the Cosmic Virial Theorem (CVT). According to the CVT, sigma(r,z)^2 ~ G Q rho(z) r^2 xi(r,z) or sigma proportional to (1+z)^{-eps/2}. The values of eps measured from our simulated velocities differ from those given by the evolution of xi and the CVT, keeping gamma and Q constant: eps(CVT) = 1.78 +/- 0.13 for (1,0) or 1.40 +/- 0.28 for (0.2,0).Comment: Accepted for publication in the ApJ. Also available at http://manaslu.astro.utoronto.ca/~carlberg/cnoc/xiev/xi_evo.ps.g

Bogoliubov Hamiltonian as Derivative of Dirac Hamiltonian via Braid Relation

In this paper we discuss a new type of 4-dimensional representation of the braid group. The matrices of braid operations are constructed by q-deformation of Hamiltonians. One is the Dirac Hamiltonian for free electron with mass m, the other, which we find, is related to the Bogoliubov Hamiltonian for quasiparticles in $^3$He-B with the same free energy and mass being m/2. In the process, we choose the free q-deformation parameter as a special value in order to be consistent with the anyon description for fractional quantum Hall effect with $\nu = 1/2$.Comment: 3 pages, 5 figure

A solvable model of the evolutionary loop

A model for the evolution of a finite population in a rugged fitness landscape is introduced and solved. The population is trapped in an evolutionary loop, alternating periods of stasis to periods in which it performs adaptive walks. The dependence of the average rarity of the population (a quantity related to the fitness of the most adapted individual) and of the duration of stases on population size and mutation rate is calculated.Comment: 6 pages, EuroLaTeX, 1 figur

Non-abelian statistics of half-quantum vortices in p-wave superconductors

Excitation spectrum of a half-quantum vortex in a p-wave superconductor contains a zero-energy Majorana fermion. This results in a degeneracy of the ground state of the system of several vortices. From the properties of the solutions to Bogoliubov-de-Gennes equations in the vortex core we derive the non-abelian statistics of vortices identical to that for the Moore-Read (Pfaffian) quantum Hall state.Comment: 5 pages, 3 figures, REVTeX, epsf. Reference adde

Teleportation, Braid Group and Temperley--Lieb Algebra

We explore algebraic and topological structures underlying the quantum teleportation phenomena by applying the braid group and Temperley--Lieb algebra. We realize the braid teleportation configuration, teleportation swapping and virtual braid representation in the standard description of the teleportation. We devise diagrammatic rules for quantum circuits involving maximally entangled states and apply them to three sorts of descriptions of the teleportation: the transfer operator, quantum measurements and characteristic equations, and further propose the Temperley--Lieb algebra under local unitary transformations to be a mathematical structure underlying the teleportation. We compare our diagrammatical approach with two known recipes to the quantum information flow: the teleportation topology and strongly compact closed category, in order to explain our diagrammatic rules to be a natural diagrammatic language for the teleportation.Comment: 33 pages, 19 figures, latex. The present article is a short version of the preprint, quant-ph/0601050, which includes details of calculation, more topics such as topological diagrammatical operations and entanglement swapping, and calls the Temperley--Lieb category for the collection of all the Temperley--Lieb algebra with physical operations like local unitary transformation

Entwined Paths, Difference Equations and the Dirac Equation

Entwined space-time paths are bound pairs of trajectories which are traversed in opposite directions with respect to macroscopic time. In this paper we show that ensembles of entwined paths on a discrete space-time lattice are simply described by coupled difference equations which are discrete versions of the Dirac equation. There is no analytic continuation, explicit or forced, involved in this description. The entwined paths are `self-quantizing'. We also show that simple classical stochastic processes that generate the difference equations as ensemble averages are stable numerically and converge at a rate governed by the details of the stochastic process. This result establishes the Dirac equation in one dimension as a phenomenological equation describing an underlying classical stochastic process in the same sense that the Diffusion and Telegraph equations are phenomenological descriptions of stochastic processes.Comment: 15 pages, 5 figures Replacement 11/02 contains minor editorial change

Relaxation, closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks

Attractors in asymmetric neural networks with deterministic parallel dynamics were shown to present a "chaotic" regime at symmetry eta < 0.5, where the average length of the cycles increases exponentially with system size, and an oscillatory regime at high symmetry, where the typical length of the cycles is 2. We show, both with analytic arguments and numerically, that there is a sharp transition, at a critical symmetry \e_c=0.33, between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size, and the weights of their attraction basins are distributed as in a Random Map with reversal symmetry. The time-scale after which cycles are reached grows exponentially with system size $N$, and the exponent vanishes in the symmetric limit, where $T\propto N^{2/3}$. The transition can be related to the dynamics of the infinite system (where cycles are never reached), using the closing probabilities as a tool. We also study the relaxation of the function $E(t)=-1/N\sum_i |h_i(t)|$, where $h_i$ is the local field experienced by the neuron $i$. In the symmetric system, it plays the role of a Ljapunov function which drives the system towards its minima through steepest descent. This interpretation survives, even if only on the average, also for small asymmetry. This acts like an effective temperature: the larger is the asymmetry, the faster is the relaxation of $E$, and the higher is the asymptotic value reached. $E$ reachs very deep minima in the fixed points of the dynamics, which are reached with vanishing probability, and attains a larger value on the typical attractors, which are cycles of length 2.Comment: 24 pages, 9 figures, accepted on Journal of Physics A: Math. Ge

The Asymptotic Number of Attractors in the Random Map Model

The random map model is a deterministic dynamical system in a finite phase space with n points. The map that establishes the dynamics of the system is constructed by randomly choosing, for every point, another one as being its image. We derive here explicit formulas for the statistical distribution of the number of attractors in the system. As in related results, the number of operations involved by our formulas increases exponentially with n; therefore, they are not directly applicable to study the behavior of systems where n is large. However, our formulas lend themselves to derive useful asymptotic expressions, as we show.Comment: 16 pages, 1 figure. Minor changes. To be published in Journal of Physics A: Mathematical and Genera

Competition in Social Networks: Emergence of a Scale-free Leadership Structure and Collective Efficiency

Using the minority game as a model for competition dynamics, we investigate the effects of inter-agent communications on the global evolution of the dynamics of a society characterized by competition for limited resources. The agents communicate across a social network with small-world character that forms the static substrate of a second network, the influence network, which is dynamically coupled to the evolution of the game. The influence network is a directed network, defined by the inter-agent communication links on the substrate along which communicated information is acted upon. We show that the influence network spontaneously develops hubs with a broad distribution of in-degrees, defining a robust leadership structure that is scale-free. Furthermore, in realistic parameter ranges, facilitated by information exchange on the network, agents can generate a high degree of cooperation making the collective almost maximally efficient.Comment: 4 pages, 2 postscript figures include

Genetic algorithm dynamics on a rugged landscape

The genetic algorithm is an optimization procedure motivated by biological evolution and is successfully applied to optimization problems in different areas. A statistical mechanics model for its dynamics is proposed based on the parent-child fitness correlation of the genetic operators, making it applicable to general fitness landscapes. It is compared to a recent model based on a maximum entropy ansatz. Finally it is applied to modeling the dynamics of a genetic algorithm on the rugged fitness landscape of the NK model.Comment: 10 pages RevTeX, 4 figures PostScrip