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We study a discrete-time multi-type Wright-Fisher population process. The mean-field dynamics of the stochastic process is induced by a general replicator difference equation. We prove several results regarding the asymptotic behavior of the model, focusing on the impact of the mean-field dynamics on it. One of the results is a limit theorem that describes sufficient conditions for an almost certain path to extinction, first eliminating the type which is the least fit at the mean-field equilibrium. The effect is explained by the metastability of the stochastic system, which under the conditions of the theorem spends almost all time before the extinction event in a neighborhood of the equilibrium. In addition to the limit theorems, we propose a maximization principle for a general deterministic replicator dynamics and study its implications for the stochastic model.

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In this paper, we consider extensions of Spivey's Bell number formula wherein the argument of the polynomial factor is translated by an arbitrary amount. This idea is applied more generally to the r-Whitney numbers of the second kind, denoted by W(n, k), where some new identities are found by means of algebraic and combinatorial arguments. The former makes use of infinite series manipulations and Dobinski-like formulas satisfied by W(n, k), whereas the latter considers distributions of certain statistics on the underlying enumerated class of set partitions. Furthermore, these two approaches provide new ways in which to deduce the Spivey formula for W(n, k). Finally, we establish an analogous result involving the r-Lah numbers wherein the order matters in which the elements are written within the blocks of the aforementioned set partitions.

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We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced in [24], and is mathematically equivalent to an endemic variation of the Reed-Frost epidemic model introduced in [28]. Two types of heuristic approximation are frequently used for models of this type in applications, a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. In this paper we prove several results concerning the exact relation between the avalanche model and these limits, including rates of convergence and rigorous bounds for common characteristics of the model.

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We study the distributional properties of horizontal visibility graphs associated with random restrictive growth sequences and random set partitions of size n. Our main results are formulas expressing the expected degree of graph nodes in terms of simple explicit functions of a finite collection of Stirling and Bernoulli numbers.

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The main theme of this paper is the enumeration of the order-isomorphic occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence f r v ( k , n ) , the number of n-array k-ary words that contain a given pattern v exactly r times. In addition, we study the asymptotic behavior of the random variable Xn , the number of pattern occurrences in a random n-array word. The two topics are closely related through the identity P ( X n = r ) = 1 k n f r v ( k , n ) . In particular, we show that for any r ≥ 0, the Stanley-Wilf sequence ( f r v ( k , n ) ) 1 ∕ n converges to a limit independent of r, and determine the value of the limit. We then obtain several limit theorems for the distribution of Xn , including a central limit theorem, large deviation estimates, and the exact growth rate of the entropy of Xn . Furthermore, we introduce a concept of weak avoidance and link it to a certain family of non-product measures on words that penalize pattern occurrences but do not forbid them entirely. We analyze this family of probability measures in a small parameter regime, where the distributions can be understood as a perturbation of a uniform measure. Finally, we extend some of our results for words, including the one regarding the equivalence of the limits of the Stanley-Wilf sequences, to pattern occurrences in permutations.

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We revisit staircases for words and prove several exact as well as asymptotic results for longest left-most staircase subsequences and subwords and staircase separation number. The latter is defined as the number of consecutive maximal staircase subwords packed in a word. We study asymptotic properties of the sequence hr,k (n), the number of n-array words with r separations over alphabet [k] and show that for any r ≥ 0, the growth sequence (hr,k ,(n))1/n converges to a characterized limit, independent of r. In addition, we study the asymptotic behavior of the random variable S k ( n ) , the number of staircase separations in a random word in [k] n and obtain several limit theorems for the distribution of S k ( n ) , including a law of large numbers, a central limit theorem, and the exact growth rate of the entropy of S k ( n ) . Finally, we obtain similar results, including growth limits, for longest L-staircase subwords and subsequences.

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We revisit the model of the ballistic deposition studied in [5] and prove several combinatorial properties of the random tree structure formed by the underlying stochastic process. Our results include limit theorems for the number of roots and the empirical average of the distance between two successive roots of the underlying tree-like structure as well as certain intricate moments calculations.

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We consider a discrete-time random motion, Markov chain on the Poincaré disk. In the basic variant of the model a particle moves along certain circular arcs within the disk, its location is determined by a composition of random Möbius transformations. We exploit an isomorphism between the underlying group of Möbius transformations and ℝ to study the random motion through its relation to a one-dimensional random walk. More specifically, we show that key geometric characteristics of the random motion, such as Busemann functions and bipolar coordinates evaluated at its location, and hyperbolic distance from the origin, can be either explicitly computed or approximated in terms of the random walk. We also consider a variant of the model where the motion is not confined to a single arc, but rather the particle switches between arcs of a parabolic pencil of circles at random times.

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In this paper we obtain bounds on the probability of convergence to the optimal solution for the compact genetic algorithm (cGA) and the population based incremental learning (PBIL). Moreover, we give a sufficient condition for convergence of these algorithms to the optimal solution and compute a range of possible values for algorithm parameters at which there is convergence to the optimal solution with a predefined confidence level.

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The compact Genetic Algorithm (cGA) is an Estimation of Distribution Algorithm that generates offspring population according to the estimated probabilistic model of the parent population instead of using traditional recombination and mutation operators. The cGA only needs a small amount of memory; therefore, it may be quite useful in memory-constrained applications. This paper introduces a theoretical framework for studying the cGA from the convergence point of view in which, we model the cGA by a Markov process and approximate its behavior using an Ordinary Differential Equation (ODE). Then, we prove that the corresponding ODE converges to local optima and stays there. Consequently, we conclude that the cGA will converge to the local optima of the function to be optimized.

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