Evolutionary games in a generalized Moran process with arbitrary selection strength and mutation

By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2×2 games in a well-mixed population with a finite size is investigated. In the model, the individuals' payoff accumulating from games is mapped into fitness using an exponent function. Both selection strength β and mutation rate ε are considered. The process is an ergodic birth-death process. Based on the limit distribution of the process, we give the analysis results for which strategy will be favoured when ε is small enough. The results depend on not only the payoff matrix of the game, but also on the population size. Especially, we prove that natural selection favours the strategy which is risk-dominant when the population size is large enough. For arbitrary β and ε values, the 'Hawk--Dove' game and the 'Coordinate' game are used to illustrate our model. We give the evolutionary stable strategy (ESS) of the games and compare the results with those of the replicator dynamics in the infinite population. The results are determined by simulation experiments.     

A stochastic derivation of the Sivashinsky equation for the self-turbulent motion of a free particle

Within the framework of the Kershaw approach and of a hypothesis on spatial stochasticity, the relativistic equations of Lehr and Park, Guerra and Ruggiero, and Vigier for stochastic Nelson mechanics are obtained. In our model there is another set of equations of the hydrodynamical type for the drift velocityv _i(x _j,t) and stochastic velocityu _i(x _j,t) of a particle. Taking into account quadratic terms in l, the universal length, we obtain from these equations the Sivashinsky equations forv _i(x _j,t) in the caseu _i →0. In the limit l →0, these equations acquire the Newtonian form.     

An analytic approach to the measurement of nestedness in bipartite networks

We present an index that measures the nestedness pattern of bipartite networks, a problem that arises in theoretical ecology. Our measure is derived using the sum of distances of the occupied elements in the incidence matrix of the network. This index quantifies directly the deviation of a given matrix from the nested pattern. In the simplest case the distance of the matrix element a_i,j is d_i,j=i+j, the Manhattan distance. A generic distance is obtained as d_i,j=(i^χ+j^χ)^1/χ. The nestedness index is defined by ν=1−τ, where τ is the “temperature” of the matrix. We construct the temperature index using two benchmarks: the distance of the complete nested matrix that corresponds to zero temperature and the distance of the average random matrix where the temperature is defined as one. We discuss an important feature of the problem: matrix occupancy ρ. We address this question using a metric index χ that adjusts for matrix occupancy.     

Integrals involving Euler-Bernoulli flexure and Saint-Venant torsion eigenfunctions

Tables of the integral ∝₀^LX_i(x)θ_j(x) dx where X_i(x) and θ_j(x) are Euler-Bernoulli and Saint-Venant eigenfunctions respectively are presented for 1?i, j?5 for beams with combinations of clamped, pinned and free ends. These integrals arise in application of the Rayleigh-Ritz and Ritz-Galerkin methods to free vibration and dynamic stability problems involving coupled torsion and bending.     

Observation of the vortex-glass transition in YBa2Cu3O7 by phase-resolved current-voltage measurements

Abstract. We report on investigations of the current-dependent ac impedance Z at 17 Hz of a YBa₂Cu₃O₇ thinfilm microbridge in magnetic fields from 1 to 8 T. A phasesensitive lock-in technique was applied and supplemented by a pulsed method for high current densities. Above the vortex-glass transition temperature T _g a constant Z(j) is observed with low current densities j, and a power-law behavior at high currents. Below T _g, the curvature of the Re Z vs. j curve is negative in a log-log plot. The Re Z(j) isotherms can be collapsed onto two universal scaling functions according to the predictions of the vortex-glass model. Close to and below T _g, a phase lag between current and voltage appears, which saturates to finite values at low j and to zero at high j. The phase lag within the constant-Z(j) region remains below the critical value, predicted theoretically by Dorsey, if T > T _g and approaches 90° at T < T _g indicating the freezing of the vortices into the dissipation-free vortex-glass state.     

Promotion of cooperation induced by nonuniform payoff allocation in spatial public goods game

A nonuniform payoff allocation mechanism is proposed for spatial public goods games where individuals are nodes on a scale-free network. Each individual is assigned a weight k_i ^α, where k_i is the degree of individual i and α is an adjustable parameter that controls the degree of diversity in individuals’ profits. During the evolution progress, the allocation of payoff on individual i is assumed to be proportional to its weight. Individuals synchronously update their strategies according to the stochastic rule with a fixed noise level. It is found that there exists an optimal value of α which yields the highest level of cooperation. Other pertinent quantities, including the payoff and the probability of finding a node playing as cooperator versus the degree, are also investigated computationally and analytically. Our results suggest that a suitable degree of diversity among individuals can promote the emergence of cooperation.     

The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation KM + MK = r s^T we introduce a scalar function S^{(i, j)} = s^T K^j (I + M)^?1Kⁱr which is defined as same as in discrete case. S^{(i, j)} satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on r and s, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of S^{(i, j)} defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property S^{(i, j)} = S^(j,i). The solution M provides t function by t = ∣I + M∣. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.     

A phenomenological theory of possible sequences of ferrotoroidal phase transitions in boracites

A phenomenological theory of the sequence of two second-order phase transitions with close temperatures is considered; such transitions occur in the Ni-Br boracite. The thermodynamic potential is written as a function of polarization P _i, magnetization M _i, and toroidal moment T _i vectors and fields E _i and H _i; T _i is treated as an order parameter. It is assumed that only one coefficient of T _i ² passes through zero as T decreases. The possibility of a sequence of two proper ferrotoroidal phase transitions along the T ₁ and T ₂ components is demonstrated. Spontaneous T _i, P _i, and M _i vector values and equations for susceptibility tensors (dielectric χ _ij = dP _i/dE _j, magnetic k _ij = dM _i/dH _j, and magnetoelectric α_ij = dP _i/dH _j = dM _j/dE _i) were obtained for three phases. Some of these values have well-defined anomalies in the vicinity of transitions. All possible sequences of ferrotoroidal phase transitions in boracites are considered. Depending on two potential coefficient values, these sequences may consist of one, two, or three such transitions.     

Note on conservation laws based upon traceless energy-momentum tensors in flat space-time

In this note we prove the following theorem. If in a flat space-time with metric g_ij(x) treferred to general coordinates xⁱ a vector ξⁱ(x) satisfies (Tⁱ_jξ^j)_;i=0 (semicolon denotes covariant differentiation) for all energy-momentum tensors of the set $\text{{}\text{T}{}^{\mathrm{ij}}\text{T}{}^{\mathrm{i}}{}_{\mathrm{j;i}}\text{=0;g}{}_{\mathrm{ij}}\text{T}{}^{\mathrm{ij}}\text{=0}$; T^ij = T^ji; T^iju_iu_j > 0 (where u_i is a time-like vector)}, then the vector ξⁱ defines a conformal motion. This theorem, which may be considered as a converse (in flat space-time) to a well-known result of Trautman, is a generalization of a result obtained by J. T. ?opuszański and J. Szczucka-Soko?owska [Reports on Mathematical Physics 11 (1977), 153] in which they assumed the vector ξⁱ was a polynomial in Minkowski coordinates.     

Passenger's fluctuation and chaos on ferryboats

We study the fluctuation of shipping passengers on a few ferryboats which shuttle between an origin and a destination repeatedly. We present the dynamical model for the ferryboats. The model is described in terms of nonlinear maps defined from the vectors T_i(n) and W_i(n), i=1, 2, …, N for N ferryboats where T_i(n) is the arrival time of ferryboat i at the origin on trip n and W_i(n) the number of passengers waiting at the origin on trip n. We clarify the fluctuations of shipping passengers and tour time for the ferry schedule. It is found that the dynamical transitions among the regular, periodic, and chaotic motions occur by varying the ferry's capacity F_max, headway T_min, and loading parameter ΓΠ. Even if the second ferryboat follows the leader (first ferryboat) keeping the constant headway, the passengers shipping on the second fluctuate highly when the parameters take specific values.     

Spouge's Conjecture on Complete and Instantaneous Gelation

We investigate the stochastic counterpart of the Smoluchowski coagulation equation, namely the Marcus–Lushnikov coagulation model. It is believed that for a broad class of kernels, all particles are swept into one huge cluster in an arbitrarily small time, which is known as a complete and instantaneous gelation phenomenon. Indeed, Spouge (also Domilovskii et al. for a special case) conjectured that K(i, j)=(ij) , >1, are such kernels. In this paper, we extend the above conjecture and prove rigorously that if there is a function (i, j), increasing in both i and j such that _j=1 1/(j(i, j))< for all i, and K(i, j)ij(i, j) for all i, j, then complete and instantaneous gelation occurs. Evidently, this implies that any kernels K(i, j)ij(log(i+1)log(j+1)) , >1, exhibit complete instantaneous gelation. Also, we conjuncture the existence of a critical (or metastable) sol state: if lim _i+j ij/K(i, j)=0 and _{i, j=1} 1/K(i, j)=, then gelation time T _g satisfies 0<T _g<. Moreover, the gelation is complete after T _g.     

Degree Complexity of Birational Maps Related to Matrix Inversion

For a q × q matrix x = (x _{i, j} ) we let ${J(x)=(x_{i,j}^{-1})}For a q × q matrix x = (x _{i, j} ) we let J(x)=(x_i,j^-1){J(x)=(x_{i,j}^{-1})} be the Hadamard inverse, which takes the reciprocal of the elements of x. We let I(x)=(x_i,j)^-1{I(x)=(x_{i,j})^{-1}} denote the matrix inverse, and we define K=I°J{K=I\circ J} to be the birational map obtained from the composition of these two involutions. We consider the iterates Kⁿ=K°?°K{K^n=K\circ\cdots\circ K} and determine the degree complexity of K, which is the exponential rate of degree growth d(K)=lim_n?￥( deg(Kⁿ) )^1/n{\delta(K)=\lim_{n\to\infty}\left( deg(K^n) \right)^{1/n}} of the degrees of the iterates. Earlier studies of this map were restricted to cyclic matrices, in which case K may be represented by a simpler map. Here we show that for general matrices the value of δ(K) is equal to the value conjectured by Anglès d’Auriac, Maillard and Viallet.     

Cooperation and charity in spatial public goods game under different strategy update rules

Human cooperation can be influenced by other human behaviors and recent years have witnessed the flourishing of studying the coevolution of cooperation and punishment, yet the common behavior of charity is seldom considered in game-theoretical models. In this article, we investigate the coevolution of altruistic cooperation and egalitarian charity in spatial public goods game, by considering charity as the behavior of reducing inter-individual payoff differences. Our model is that, in each generation of the evolution, individuals play games first and accumulate payoff benefits, and then each egalitarian makes a charity donation by payoff transfer in its neighborhood. To study the individual-level evolutionary dynamics, we adopt different strategy update rules and investigate their effects on charity and cooperation. These rules can be classified into two global rules: random selection rule in which individuals randomly update strategies, and threshold selection rule where only those with payoffs below a threshold update strategies. Simulation results show that random selection enhances the cooperation level, while threshold selection lowers the threshold of the multiplication factor to maintain cooperation. When charity is considered, it is incapable in promoting cooperation under random selection, whereas it promotes cooperation under threshold selection. Interestingly, the evolution of charity strongly depends on the dispersion of payoff acquisitions of the population, which agrees with previous results. Our work may shed light on understanding human egalitarianism.     

Harness Processes and Non-Homogeneous Crystals

We consider the Harmonic crystal, a measure on with Hamiltonian H(x)=∑ _i,j J _i,j (x(i)−x(j))²+h∑ _i (x(i)−d(i))², where x, d are configurations, x(i), d(i)∈ℝ, i,j∈ℤ ^d . The configuration d is given and considered as observations. The ‘couplings’ J _i,j are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite temperature and to find the unique ground state configuration m corresponding to the Hamiltonian.     

On-shell T-matrices in multiple scattering

The transition operator T for the scattering of a particle from N potentials V_j(x) can be expanded into a series featuring the transition operators t_j associated with the individual potentials. For V_j(x) both absolutely and square integrable in x, we show, using an analytic continuation argument, that if T is on-shell, i.e. in 〈k|T(k₀²±i0)|k′〉, |k| = |k′| = k₀, then each t_j is also on-shell.     

The residual interaction of bound nucleons-two-nucleon matrix elements deduced from transfer experiments

Matrix elements for the effective two-nucleon interaction have been deduced from the population of multiplets near closed shells as observed in direct transfer reactions. In the evaluation, the limited purity of such multiples was taken into consideration, typically by weighting the observed fractions of the two-nucleon configurations by their spectroscopic strenghts and by using the resulting energy centroids. In a few cases, off-diagonal matrix elements are available from empirical wave funcitons. The systematic errors for particle-particle matrix elements extracted directly and those obtained from Pandya transformations were found to go in opposite directions. In some cases, this feautre of the empirical mehtod could be used to suggest upper and lower “bounds” for the extracted matrix elements. Diagonal matrix elements for the empirical residual interaction show a number of features suggestive of an underlying simplicity in the interaction of bound nucleons. Within experimental uncertainties (of about 10% for T=0 matrix elements) the monopole parts of the matrix elements are fit well with a simple A?0.75 dependence, and the data available to date do not reveal any significant monopole dependence on the quantum numbers of the interacting nucleons. The usefulness of scaling is suggested. Generally, diagonal matrix elements E_J(j₁, j₂) normalized by the extracted A-dependent monopole strength agree within expected experimental uncertainties whether derived from particle-particle or particle-hole multiples and whether extracted from the beginning or the end of a major shell. For values J≠0, the diagonal E_J(j²) matrix elements seem to follow two universal functions which depend on the semi-classical coupling angles θ₁₂, but are otherwise independent on j. For j₁≠j₂ several “typical” functions ?(θ₁₂) can be constructed which fit subsets of the data and differ in a predictable way. The general features of the bound-nucleon interaction appear consistent with those of theoretical matrix elements based on a number of short-range model forces or on calculations using the G matrix approach to deal with realistic free nucleon forces. For the latter, the available theoretical numbers for j₁=j₂ agree well with the T=1 set, but they differ quantitatively from the observed matrix elements for T=0, sometimes by many (experimental) standard deviations.     

Head-on collision of dust-ion-acoustic solitons in electron-dust-ion quantum plasmas

In this paper, we study the head-on collision between two dust-ion-acoustic (DIA) solitons in quantum electron-dust-ion plasma. Using the extended Poincaré–Lighthill–Kuo (PLK) method, we obtain the Korteweg–de Vries (KdV) equations, the phase shifts and the trajectories after the head-on collision of the two DIA solitons. We investigate the effect of quantum diffraction parameters for electrons and ions (H _e, H _i), the Fermi temperature ratio (σ) and the dust charged number density (n _d0) on the phase shifts. Different values of μ?=?z _d0(n _d0/n _i0) and μ _d?=?z _d0(m _i/m _d) are taken to discuss the effects on phase shifts, where z _d0 denotes the dust charge number, n _j0 represents the equilibrium number density and m _j is the mass of the jth species (j?=?e, i, d for electrons, ions and dust particles, respectively). It is observed that the phase shifts are significantly affected by the plasma parameters.     

A sequence of two ferrotoroidal phase transitions in nickel-bromine boracite Ni3B7O13Br

The phenomenological theory of a sequence of two second-order phase transitions in Ni-Br boracite is presented. Two different components of the toroidal moment vector T _i are the order parameters of these transitions. Expressions are derived for the temperature dependences of the spontaneous values of T _i , polarization P _i , and magnetization M _i and the dielectric χ_ij=dP _i /dE _j , magnetic k _ij =dM _i /H _j , and magnetoelectric α _ij=dP _i /dH _j =dM _j /dE _i susceptibilities. Some of these susceptibilities display sharp temperature peaks in the vicinity of phase transitions.     

Measurement of the coupling strength distribution in ferromagnetic (F)/antiferromagnetic (AF) exchange bias systems

We propose a method for determination of the distribution function P(j) of the coupling energy density j in polycrystalline textured ferromagnetic (F)/antiferromagnetic (AF) film systems. P(j) governs the entire film coupling J and the exchange bias field H_e and was not measurable until now. The method is verified by torquemetry in a high magnetic field and by reversing its rotation sense. The transition to a new magnetic steady state after rotation reversal is analyzed within a Stoner–Wohlfarth model including thermal relaxation. This transition is completed earlier for strongly coupled grains than for grains with smaller j, which is reflected in the torque curves. We determined P(j) for a sputtered NiFe(16 nm)/IrMn(0.8 nm) film at T=50 K in the hysteretic range of coupling energies and found that P strongly decreases for increasing j.     

An Inhomogeneous Contact Process Model for Speciation

We propose the following model on $\mathbb{Z}^{+}$ for speciation and extinction. A species at site i gives birth to a new species at site j at rate λp(i,j) where i and j are nearest neighbors. A death at site i occurs at rate δ _i . We show that the existence of a phase transition in λ depends critically on the value of the limit of $\frac{p(n,n+1)}{\delta_{n}}$ .