Dynamics of Three Agent Games

We study the dynamics and resulting score distribution of three-agent games where after each competition a single agent wins and scores a point. A single competition is described by a triplet of numbers $p$, $t$ and $q$ denoting the probabilities that the team with the highest, middle or lowest accumulated score wins. We study the full family of solutions in the regime, where the number of agents and competitions is large, which can be regarded as a hydrodynamic limit. Depending on the parameter values $(p,q,t)$, we find six qualitatively different asymptotic score distributions and we also provide a qualitative understanding of these results. We checked our analytical results against numerical simulations of the microscopic model and find these to be in excellent agreement. The three agent game can be regarded as a social model where a player can be favored or disfavored for advancement, based on his/her accumulated score. It is also possible to decide the outcome of a three agent game through a mini tournament of two-a gent competitions among the participating players and it turns out that the resulting possible score distributions are a subset of those obtained for the general three agent-games. We discuss how one can add a steady and democratic decline rate to the model and present a simple geometric construction that allows one to write down the corresponding score evolution equations for $n$-agent games

Discrete Analog of the Burgers Equation

We propose the set of coupled ordinary differential equations dn_j/dt=(n_{j-1})^2-(n_j)^2 as a discrete analog of the classic Burgers equation. We focus on traveling waves and triangular waves, and find that these special solutions of the discrete system capture major features of their continuous counterpart. In particular, the propagation velocity of a traveling wave and the shape of a triangular wave match the continuous behavior. However, there are some subtle differences. For traveling waves, the propagating front can be extremely sharp as it exhibits double exponential decay. For triangular waves, there is an unexpected logarithmic shift in the location of the front. We establish these results using asymptotic analysis, heuristic arguments, and direct numerical integration.Comment: 6 pages, 5 figure

Statistics of Partial Minima

Motivated by multi-objective optimization, we study extrema of a set of N points independently distributed inside the d-dimensional hypercube. A point in this set is k-dominated by another point when at least k of its coordinates are larger, and is a k-minimum if it is not k-dominated by any other point. We obtain statistical properties of these partial minima using exact probabilistic methods and heuristic scaling techniques. The average number of partial minima, A, decays algebraically with the total number of points, A ~ N^{-(d-k)/k}, when 1<=k<d. Interestingly, there are k-1 distinct scaling laws characterizing the largest coordinates as the distribution P(y_j) of the jth largest coordinate, y_j, decays algebraically, P(y_j) ~ (y_j)^{-alpha_j-1}, with alpha_j=j(d-k)/(k-j) for 1<=j<=k-1. The average number of partial minima grows logarithmically, A ~ [1/(d-1)!](ln N)^{d-1}, when k=d. The full distribution of the number of minima is obtained in closed form in two-dimensions.Comment: 6 pages, 1 figur

Popularity-Driven Networking

We investigate the growth of connectivity in a network. In our model, starting with a set of disjoint nodes, links are added sequentially. Each link connects two nodes, and the connection rate governing this random process is proportional to the degrees of the two nodes. Interestingly, this network exhibits two abrupt transitions, both occurring at finite times. The first is a percolation transition in which a giant component, containing a finite fraction of all nodes, is born. The second is a condensation transition in which the entire system condenses into a single, fully connected, component. We derive the size distribution of connected components as well as the degree distribution, which is purely exponential throughout the evolution. Furthermore, we present a criterion for the emergence of sudden condensation for general homogeneous connection rates.Comment: 5 pages, 2 figure

A simple electrostatic model applicable to biomolecular recognition

An exact, analytic solution for a simple electrostatic model applicable to biomolecular recognition is presented. In the model, a layer of high dielectric constant material (representative of the solvent, water) whose thickness may vary separates two regions of low dielectric constant material (representative of proteins, DNA, RNA, or similar materials), in each of which is embedded a point charge. For identical charges, the presence of the screening layer always lowers the energy compared to the case of point charges in an infinite medium of low dielectric constant. Somewhat surprisingly, the presence of a sufficiently thick screening layer also lowers the energy compared to the case of point charges in an infinite medium of high dielectric constant. For charges of opposite sign, the screening layer always lowers the energy compared to the case of point charges in an infinite medium of either high or low dielectric constant. The behavior of the energy leads to a substantially increased repulsive force between charges of the same sign. The repulsive force between charges of opposite signs is weaker than in an infinite medium of low dielectric constant material but stronger than in an infinite medium of high dielectric constant material. The presence of this behavior, which we name asymmetric screening, in the simple system presented here confirms the generality of the behavior that was established in a more complicated system of an arbitrary number of charged dielectric spheres in an infinite solvent.Comment: 15 pages, 6 figure

Entropic Tightening of Vibrated Chains

We investigate experimentally the distribution of configurations of a ring with an elementary topological constraint, a ``figure-8'' twist. Using vibrated granular chains, which permit controlled preparation and direct observation of such a constraint, we show that configurations where one of the loops is tight and the second is large are strongly preferred. This agrees with recent predictions for equilibrium properties of topologically-constrained polymers. However, the dynamics of the tightening process weakly violate detailed balance, a signature of the nonequilibrium nature of this system.Comment: 4 pages, 4 figure

The K-Band Galaxy Luminosity Function

We measured the K-band luminosity function using a complete sample of 4192 morphologically-typed 2MASS galaxies with 7 < K < 11.25 mag spread over 2.12 str. Early-type (T -0.5) galaxies have similarly shaped luminosity functions, alpha_e=-0.92+/-0.10 and alpha_l=-0.87+/-0.09. The early-type galaxies are brighter, M_*e=-23.53+/-0.06 mag compared to M_*l=-22.98\pm0.06 mag, but less numerous, n_*e=(0.0045+/-0.0006)h^3/Mpc^3 compared to n_*l=(0.0101+/-0.0013)h^3/Mpc^3 for H_0=100h km/s Mpc, such that the late-type galaxies slightly dominate the K-band luminosity density, j_late/j_early=1.17+/-0.12. Our morphological classifications are internally consistent, consistent with previous classifications and lead to luminosity functions unaffected by the estimated uncertainties in the classifications. These luminosity functions accurately predict the K-band number counts and redshift distributions for K < 18 mag, beyond which the results depend on galaxy evolution and merger histories.Comment: submitted to ApJ, 25 pages, 6 figures, complete redshift survey. Table 1 included in sourc

Kinetics of Heterogeneous Single-Species Annihilation

We investigate the kinetics of diffusion-controlled heterogeneous single-species annihilation, where the diffusivity of each particle may be different. The concentration of the species with the smallest diffusion coefficient has the same time dependence as in homogeneous single-species annihilation, A+A-->0. However, the concentrations of more mobile species decay as power laws in time, but with non-universal exponents that depend on the ratios of the corresponding diffusivities to that of the least mobile species. We determine these exponents both in a mean-field approximation, which should be valid for spatial dimension d>2, and in a phenomenological Smoluchowski theory which is applicable in d<2. Our theoretical predictions compare well with both Monte Carlo simulations and with time series expansions.Comment: TeX, 18 page

Random Geometric Series

Integer sequences where each element is determined by a previous randomly chosen element are investigated analytically. In particular, the random geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments grow algebraically, n^beta(s) with beta(s)=2^s-1, while the typical behavior is x_n n^ln 2. The probability distribution is obtained explicitly in terms of the Stirling numbers of the first kind and it approaches a log-normal distribution asymptotically.Comment: 6 pages, 2 figure

Localisation Transition of A Dynamic Reaction Front

We study the reaction-diffusion process $A+B\to \emptyset$ with injection of each species at opposite boundaries of a one-dimensional lattice and bulk driving of each species in opposing directions with a hardcore interaction. The system shows the novel feature of phase transitions between localised and delocalised reaction zones as the injection rate or reaction rate is varied. An approximate analytical form for the phase diagram is derived by relating both the domain of reactants $A$ and the domain of reactants $B$ to asymmetric exclusion processes with open boundaries, a system for which the phase diagram is known exactly, giving rise to three phases. The reaction zone width $w$ is described by a finite size scaling form relating the early time growth, relaxation time and saturation width exponents. In each phase the exponents are distinct from the previously studied case where the reactants diffuse isotropically.Comment: 13 pages, latex, uses eps