Existence of equilibria in countable games: an algebraic approach

Although mixed extensions of finite games always admit equilibria, this is not the case for countable games, the best-known example being Wald's pick-the-larger-integer game. Several authors have provided conditions for the existence of equilibria in infinite games. These conditions are typically of topological nature and are rarely applicable to countable games. Here we establish an existence result for the equilibrium of countable games when the strategy sets are a countable group and the payoffs are functions of the group operation. In order to obtain the existence of equilibria, finitely additive mixed strategies have to be allowed. This creates a problem of selection of a product measure of mixed strategies. We propose a family of such selections and prove existence of an equilibrium that does not depend on the selection. As a byproduct we show that if finitely additive mixed strategies are allowed, then Wald's game admits an equilibrium. We also prove existence of equilibria for nontrivial extensions of matching-pennies and rock-scissors-paper. Finally we extend the main results to uncountable games

General Reaction-Diffusion Processes With Separable Equations for Correlation Functions

We consider general multi-species models of reaction diffusion processes and obtain a set of constraints on the rates which give rise to closed systems of equations for correlation functions. Our results are valid in any dimension and on any type of lattice. We also show that under these conditions the evolution equations for two point functions at different times are also closed. As an example we introduce a class of two species models which may be useful for the description of voting processes or the spreading of epidemics.Comment: 17 pages, Latex, No figure

Plankton patchiness investigated using simultaneous nitrate and chlorophyll observations

The complex patterns observed in marine phytoplankton distributions arise from the interplay of biological and physical processes, but the nature of the balance remains uncertain centuries after the first observations. Previous observations have shown a consistent trend of decreasing variability with decreasing length-scale. Influenced by similar scaling found for the properties of the water that the phytoplankton inhabit, ‘universal' theories have been proposed that simultaneously explain the variability seen from meters to hundreds of kilometers. However, data on the distribution of phytoplankton alone has proved insufficient to differentiate between the many causal mechanisms that have been suggested. Here we present novel observations from a cruise in the North Atlantic in which fluorescence (proxy for phytoplankton), nitrate and temperature were measured simultaneously at scales from 10 m to 100 km for the first time in the open ocean. These show a change in spectra between the small scale (10–100 m) and the mesoscale (10–100 km) which is different for the three tracers. We discuss these observations in relation to the current theories for phytoplankton patchiness

Plankton patchiness investigated using simultaneous nitrate and chlorophyll observations

The complex patterns observed in marine phytoplankton distributions arise from the interplay of biological and physical processes, but the nature of the balance remains uncertain centuries after the first observations. Previous observations have shown a consistent trend of decreasing variability with decreasing length-scale. Influenced by similar scaling found for the properties of the water that the phytoplankton inhabit, ‘universal' theories have been proposed that simultaneously explain the variability seen from meters to hundreds of kilometers. However, data on the distribution of phytoplankton alone has proved insufficient to differentiate between the many causal mechanisms that have been suggested. Here we present novel observations from a cruise in the North Atlantic in which fluorescence (proxy for phytoplankton), nitrate and temperature were measured simultaneously at scales from 10 m to 100 km for the first time in the open ocean. These show a change in spectra between the small scale (10–100 m) and the mesoscale (10–100 km) which is different for the three tracers. We discuss these observations in relation to the current theories for phytoplankton patchiness

Hierarchical Structure of Azbel-Hofstader Problem: Strings and loose ends of Bethe Ansatz

We present numerical evidence that solutions of the Bethe Ansatz equations for a Bloch particle in an incommensurate magnetic field (Azbel-Hofstadter or AH model), consist of complexes-"strings". String solutions are well-known from integrable field theories. They become asymptotically exact in the thermodynamic limit. The string solutions for the AH model are exact in the incommensurate limit, where the flux through the unit cell is an irrational number in units of the elementary flux quantum. We introduce the notion of the integral spectral flow and conjecture a hierarchical tree for the problem. The hierarchical tree describes the topology of the singular continuous spectrum of the problem. We show that the string content of a state is determined uniquely by the rate of the spectral flow (Hall conductance) along the tree. We identify the Hall conductances with the set of Takahashi-Suzuki numbers (the set of dimensions of the irreducible representations of $U_q(sl_2)$ with definite parity). In this paper we consider the approximation of noninteracting strings. It provides the gap distribution function, the mean scaling dimension for the bandwidths and gives a very good approximation for some wave functions which even captures their multifractal properties. However, it misses the multifractal character of the spectrum.Comment: revtex, 30 pages, 6 Figs, 8 postscript files are enclosed, important references are adde

Phase transition in an asymmetric generalization of the zero-temperature Glauber model

An asymmetric generalization of the zero-temperature Glauber model on a lattice is introduced. The dynamics of the particle-density and specially the large-time behavior of the system is studied. It is shown that the system exhibits two kinds of phase transition, a static one and a dynamic one.Comment: LaTeX, 9 pages, to appear in Phys. Rev. E (2001

Phase transition in an asymmetric generalization of the zero-temperature q-state Potts model

An asymmetric generalization of the zero-temperature q-state Potts model on a one dimensional lattice, with and without boundaries, has been studied. The dynamics of the particle number, and specially the large time behavior of the system has been analyzed. In the thermodynamic limit, the system exhibits two kinds of phase transitions, a static and a dynamic phase transition.Comment: 11 pages, LaTeX2

Reaction-Diffusion Processes from Equivalent Integrable Quantum Chains

One-dimensional reaction-diffusion systems are mapped through a similarity transformation onto integrable (and a priori non-stochastic) quantum chains. Time-dependent properties of these chemical models can then be found exactly. The reaction-diffusion processes related to free fermion systems with site-independent interactions are classified. The time-dependence of the mean particle density is calculated. Furthermore new integrable stochastic processes related to the Heisenberg XXZ chain are identified and the relaxation times for the particle density and density correlation for these systems are found.Comment: 67 pages, Latex, 3 eps figures. (final version, typos corrected

Autonomous multispecies reaction-diffusion systems with more-than-two-site interactions

Autonomous multispecies systems with more-than-two-neighbor interactions are studied. Conditions necessary and sufficient for closedness of the evolution equations of the $n$-point functions are obtained. The average number of the particles at each site for one species and three-site interactions, and its generalization to the more-than-three-site interactions is explicitly obtained. Generalizations of the Glauber model in different directions, using generalized rates, generalized number of states at each site, and generalized number of interacting sites, are also investigated.Comment: 9 pages, LaTeX2