Integrals, partitions, and cellular automata

We prove that

where is the decreasing function that satisfies , for . When is an integer and we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.

     

随机化交通灯的二维元胞自动机交通模型

元胞自动机交通模型以简单的规则反映交通系统中的多种因素，可以分析各种交通现象，且可在计算机上方便、高效地运作·Ｂｉｈａｍ－Ｍｉｄｄｌｅｔｏｎ－Ｌｅｖｉｎｅ模型（ＢＭＬ模型）实现了二维交通问题的元胞自动机模型的模拟研究·本文对ＢＭＬ模型作了改进，解除了该模型中关于交通灯同步变化的限制·在新模型中，每个路口的交通灯可以自由选定起始工作时间和变化节奏，于是可以更全面、准确地反映交通灯对交通系统性能的影响·本文还对新模型中出现的若干新效应作了解释·     

On connectivity,conductance and bootstrap percolation for a random k‐out,age‐biased graph

A uniform attachment graph (with parameter k), denoted G_n,k in the paper, is a random graph on the vertex set [n], where each vertex v makes k selections from [v ? 1] uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well‐studied random graphs: k‐out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of G_n,k to show that the conductance of G_n,k is of order . We also study the bootstrap percolation on G_n,k, where r infected neighbors infect a vertex, and show that if the probability of initial infection of a vertex is negligible compared to then with high probability (whp) the disease will not spread to the whole vertex set, and if this probability exceeds by a sub‐logarithmical factor then the disease whp will spread to the whole vertex set.     

Control parameters in Boolean networks and cellular automata revisited from a logical and a sociological point of view

The use of “control parameters” as applied to describe the dynamics of complex mathematical systems within models of real social systems is discussed. Whereas single control parameters cannot sufficiently characterize the dynamics of such systems it is suggested that domains of values of certain sets of parameters are appropriately denoting necessary conditions for highly disordered dynamics of social systems. Various of those control parameters permit a straightforward interpretation in terms of properties of social rules and structures. © 1999 John Wiley & Sons, Inc.     

Classifying cellular automata automatically: Finding gliders,filtering, and relating space‐time patterns,attractor basins,and the Z parameter

Cellular automata (CA) rules can be classified automatically for a spectrum of ordered, complex, and chaotic dynamics by a measure of the variance of input‐entropy over time. Rules that support interacting gliders and related complex dynamics can be identified, giving an unlimited source for further study. The distribution of rule classes in rule‐space can be shown. A byproduct of the method allows the automatic “filtering” of CA space‐time patterns to show up gliders and related emergent configurations more clearly. The classification seems to correspond to our subjective judgment of space‐time dynamics. There are also approximate correlations with global measures on convergence in attractor basins, characterized by the distribution of in‐degree sizes in their branching structure, and to the rule parameter, Z. Based on computer experiments using the software Discrete Dynamics Lab (DDLab), this article explains the methods and presents results for 1D CA. © 1999 John Wiley & Sons, Inc.     

基于元胞自动机的植物病虫害传播的计算机仿真

利用元胞自动机方法建立植物病虫害传播的数学模型。在此基础上,分别对两种不同病虫害来源的情况进行仿真。仿真结果表明,在参数给定的情况下,无论病虫来源于自身还是外界,植物病虫害的传播均在一定时间后达到稳定状态,不同状态元胞占有率相近;相同参数下,同病虫来源于自身相比,植物病虫从外界入侵时,植物被感染的变化率较低,病虫害传播路径较有规律,有利于病虫害源的确定和病虫害的治理。     

基于元胞自动机的供需网企业合作驱动建模与仿真

针对供需网企业合作驱动具有一定的复杂性这一现状,基于元胞自动机模型对三种状态的节点交互及供需网系统合作演变过程进行仿真,探讨了不同驱动作用下供需网企业合作行为的演化趋势。研究结果表明:供需网企业合作行为演化存在供需质交互频率阈值,当交互频率大于该阈值时,企业合作数量较高且会稳定在某一水平;提高企业的多功能合作执行力、外部力量等可以增加企业合作规模和数量,进而可以促进合作子系统的形成和合作平台的构建。     

Stress relaxation: Experiment,theory, and computer simulation

Experimental evidence on stress relaxation is analyzed first for a wide variety of classes of materials: metals and their alloys, synthetic and natural polymers, glasses and frozen non-polymeric organic liquids. Common features of curves (t) of relaxation of stress a as a function of time t are discussed, and the importance of the internal stress _i() noted. Theoretical approaches are then reviewed, with particular attention to the cooperative model and its modifications; that model corresponds well to the experimental results. Some simulation results obtained with the method of molecular dynamics are reported for ideal metal lattices, metal lattices with defects, and for polymeric systems. In agreement with both experiments and the cooperative theory, the simulated (log t) curves exhibit three regions: initial, nearly horizontal, starting at ₀; central, descending approximately linearly; and final, corresponding to _i. In agreement with the theory, the slope of the simulated central part is proportional to the initial effective stress ₀*= ₀– _i. The time range taken by the central part is strongly dependent on the defect concentration: the lower the defect concentration, the shorter the range. Imposition in the beginning of a high strain destroys largely the resistance of a material to deformation, resulting in low values of the internal stress _i. On the joint basis of experimental, theoretical, and numerical results, we explain the mechanism of stress relaxation in terms of deformations occuring in the immediate environment of the defects. Simulations show several common features in the behavior of metals and polymers. Apart from the defect concentration, the amount of free volumev _f is also important.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 591–606, September–Ocotober, 1995.     

Merging percolation on Zd and classical random graphs: Phase transition

We study a random graph model which is a superposition of bond percolation on Z^d with parameter p, and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so‐called “rank 1 case” of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters c ≥ 0 and 0 ≤ p < p_c, where p_c = p_c(d) is the critical probability for the bond percolation on Z^d. The phase transition is of second order as in the classical random graph. We find the scaled size of the largest connected component in the supercritical regime. We also provide a sharp upper bound for the largest connected component in the subcritical regime. The latter is a new result for inhomogeneous random graphs with unbounded kernels. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Density Delay and Organizational Survival: Computational Models and Empirical Comparisons

Research on the ecological dynamics oforganizational populations has demonstrated that competitiveconditions at the time of founding have enduring effects onorganizational survival. According to ecological theories,organizational life chances are systematically affected by density (the number of organizations in a population) at thetime of founding because the lower resource endowments thatcharacterize organizations appearing in periods of highpopulation density tend to become self-reinforcing, and—over time—amplify differences in mortality rates oforganizations founded under different conditions. However,credible arguments have been offered that could justify both positive and negative effects of the delayed effectsof population density on organizational mortality rates, andreceived empirical research in part reflects this ambiguity.To develop new insight into this issue and to explore theboundaries of received empirical results, in this study wepresent a computational model of organizational evolutionaccording to which the global dynamics of organizationalpopulations emerge from the iteration of simple rules oflocal interaction among individual organizations. We use the synthetic data produced by simulation to estimate eventhistory models of organizational mortality, and compare theparameter estimates with those reported in the most recentempirical studies of actual organizational populations. Theconclusions supported by the model qualify and extendreceived empirical results, and suggest that delayed effectsof density are highly sensitive the details of local structure of connections among members of organizationalpopulations.     

Multicolor particle systems with large threshold and range

In this paper we consider the Greenberg-Hastings and cyclic color models. These models exhibit (at least) three different types of behavior. Depending on the number of colors and the size of two parameters called the threshold and range, the Greenberg-Hastings model either dies out, or has equilibria that consist of debris or fire fronts. The phase diagram for the cyclic color models is more complicated. The main result of this paper, Theorem 1, proves that the debris phase exists for both systems.     

Epr-bohm experiment and Bell’s inequality: Quantum physics meets probability theory

Our main aim in this paper is to inform the physics community (and especially experts in quantum information) about investigations of the problem of the probabilistic compatibility of a family of random variables: the possibility of realizing such a family based on a single probability measure (of constructing a single Kolmogorov probability space). These investigations were started a hundred years ago by Boole. The complete solution of the problem was obtained by the Soviet mathematician Vorobiev in the 1960s. It turns out that probabilists and statisticians obtained inequalities for probabilities and correlations that include the famous Bell’s inequality and its generalizations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 99–115, October, 2008.     

Emulation and complementarity in one‐dimensional alternatives of the axelrod model with binary features

We investigate the one‐dimensional dynamics of alternatives of the Axelrod model (ξ_t) with k binary features and confidence parameter ε = 0, 1,…, k. Simultaneously, the simple Axelrod model is also critically examined. Specifically, for small and large ε, simulations suggest that the convergent model (ξ_t) is emulated by a corresponding attractive model (η_t) with the same parameters (conditional on bounded confidence). (η_t) is more mathematically tractable than (ξ_t), and the very definitions of the two qualitative behaviors of cyclic particle systems (fluctuation and fixation) are applicable in special cases. Moreover, we observe a complementarity: for not too small k and $\varepsilon \approx {k \over 2}$ , (η_t) fixates (each site has a final type independent of the possibly infinite size of the lattice), whereas (ξ_t) fluctuates (each site changes type at arbitrarily larger times t as the size of the lattice increases). © 2011 Wiley Periodicals, Inc. Complexity, 2011     

How to choose the simulation model for computer experiments: a local approach

In many scientific areas, non‐stochastic simulation models such as finite element simulations replace real experiments. A common approach is to fit a meta‐model, for example a Gaussian process model, a radial basis function interpolation, or a kernel interpolation, to computer experiments conducted with the simulation model. This article deals with situations where more than one simulation model is available for the same real experiment, with none being the best over all possible input combinations. From fitted models for a real experiment as well as for computer experiments using the different simulation models, a criterion is derived to identify the locally best one. Applying this criterion to a number of design points allows the design space to be split into areas where the individual simulation models are locally superior. An example from sheet metal forming is analyzed, where three different simulation models are available. In this application and many similar problems, the new approach provides valuable assistance with the choice of the simulation model to be used. Copyright © 2011 John Wiley & Sons, Ltd.     

New distribution formulas for classical gas, clusters, and phase transitions

We obtain a Bose-Einstein-type distribution for the classical vapor. We show that the analogue of the Bose condensate is the formation of clusters. We write a new PV diagram for interaction with the form of the Lennard-Jones potential using scattering theory and a strict constraint. We compare our results with experimental data. Dedicated to the memory of a dear friend, Tolya Karatsuba __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 250–272, November, 2008.     

Estimation and status prediction in a discrete mover‐stayer model with covariate effects on stayer's probability

A discrete‐time mover‐stayer (MS) model is an extension of a discrete‐time Markov chain, which assumes a simple form of population heterogeneity. The individuals in the population are either stayers, who never leave their initial states or movers who move according to a Markov chain. We, in turn, propose an extension of the MS model by specifying the stayer's probability as a logistic function of an individual's covariates. Such extension has been recently discussed for a continuous time MS but has not been considered before for a discrete time one. This extension allows for an in‐sample classification of subjects who never left their initial states into stayers or movers. The parameters of an extended MS model are estimated using the expectation‐maximization algorithm. A novel bootstrap procedure is proposed for out of sample validation of the in‐sample classification. The bootstrap procedure is also applied to validate the in‐sample classification with respect to a more general dichotomy than the MS one. The developed methods are illustrated with the data set on installment loans. But they can be applied more broadly in credit risk area, where prediction of creditworthiness of a loan borrower or lessee is of major interest.     

The Co‐evolution of cooperation and complexity in a multi‐player,local‐interaction prisoners' dilemma

Agents located on a 20 × 20 toroidal lattice play a Prisoners' Dilemma game with their Moore neighbors, adopting policies of cooperation and defection that depend only on their own action and the number of cooperators in the neighborhood in the last round of the game. These policies (“characters”) are encoded in 19‐bit strings, which are subjected to evolution according to a genetic algorithm, with selection based on the cumulative scores of the agents in the neighborhood over 10 rounds of the basic game. Simulations examine the evolution of the population of characters over 1000 generations. Even with selection disabled, the genetic algorithm organizes the population into a small number of surviving characters clustered in spatially homogeneous regions. Selection for fitness rapidly achieves uniform cooperation. The characters evolved cooperate on the initial play, continue to cooperate when five or more of their neighbors cooperate, tend to defect defensively when they have cooperated and most of their neighbors have defected, and switch back to cooperation when five or more neighbors cooperate. When selection operates at the level of the whole society, however, the diversity of the population rapidly collapses, a single character predominates, and the cooperativeness of the dominating character is a matter of chance, so that there is no systematic tendency to evolve cooperation. © 2001 John Wiley & Sons, Inc.     

Algorithms,genetics, and populations: The schemata theorem revisited

Models for the genetic evolution of natural populations have supplied the inspiration for the adaptive computer algorithms known as “genetic algorithms.” In its original form, a genetic algorithm simulates the evolution of a haploid population: Genetic variation is produced by mutation at a number of genetic loci in a chromosome, represented as a string of bits, and the variants in the various chromosomes are reshuffled by genetic recombination. In nature, the ability of a haploid individual to survive and reproduce is expressed as its fitness; in computer science the value of a string is measured by its ability to program a given task. The performance of genetic algorithms is evaluated in the “schemata theorem,” which we present and discuss in the context of the population genetics of multiple loci and propose a generalization of the theorem. © 1998 John Wiley & Sons, Inc.