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## Asymptotic Behavior (asymptotic + behavior)
## Selected Abstracts## Two Competing Models of How People Learn in Games ECONOMETRICA, Issue 6 2002Ed HopkinsReinforcement learning and stochastic fictitious play are apparent rivals as models of human learning. They embody quite different assumptions about the processing of information and optimization. This paper compares their properties and finds that they are far more similar than were thought. In particular, the expected motion of stochastic fictitious play and reinforcement learning with experimentation can both be written as a perturbed form of the evolutionary replicator dynamics. Therefore they will in many cases have the same asymptotic behavior. In particular, local stability of mixed equilibria under stochastic fictitious play implies local stability under perturbed reinforcement learning. The main identifiable difference between the two models is speed: stochastic fictitious play gives rise to faster learning. [source] ## Use of fluorinated maleimide and telechelic bismaleimide for original hydrophobic and oleophobic polymerized networks JOURNAL OF POLYMER SCIENCE (IN TWO SECTIONS), Issue 10 2008Aurélien SoulesAbstract The syntheses of original fluorinated maleimide and telechelic bismaleimide bearing C6F13 and C6F12 groups, respectively, and their use as reactive additives in photopolymerizable formulations of telechelic poly(propylene glycol) bismaleimide (PPGBMI) are presented. Fluorinated maleimide was synthetized in five steps in 63% overall yield from C6F13C2H4I precursor, whereas the fluorinated bismaleimide was prepared in six steps in 14% overall yield from IC6F12I. These latter led to fluorinated azido and diazido intermediates that were reduced into the fluorinated amine and diamines in two steps. The condensation of amine and diamine onto maleic anhydride offered an amic acid and a diamic acid, which were subsequently cyclized into fluorinated maleimide and bismaleimide. Formulations of telechelic PPGBMI containing a low concentration of these fluorinated maleimide and bismaleimide were UV cured and the surface properties of the resulting films were investigated. A deep modification of the surface properties was noted when the monomaleimide was used. In all the cases, a selective enrichment of the fluorinated monomer at the film surface was observed. The dependence of the surface properties on the fluorinated maleimide and bismaleimide concentrations were also studied, and showed an asymptotic behavior of the contact angle with only 1.5 wt % of fluorinated maleimide additive, whatever the conditions. This monomaleimide led to better hydrophobic and oleophobic properties of the resulting material than that containing the telechelic one. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 3214,3228, 2008 [source] ## A mathematical model of immune competition related to cancer dynamics MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2010Ilaria BrazzoliAbstract This paper deals with the qualitative analysis of a model describing the competition among cell populations, each of them expressing a peculiar cooperating and organizing behavior. The mathematical framework in which the model has been developed is the kinetic theory for active particles. The main result of this paper is concerned with the analysis of the asymptotic behavior of the solutions. We prove that, if we are in the case when the only equilibrium solution if the trivial one, the system evolves in such a way that the immune system, after being activated, goes back toward a physiological situation while the tumor cells evolve as a sort of progressing travelling waves characterizing a typical equilibrium/latent situation. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## On the shape of the fringe of various types of random trees MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2009Michael DrmotaAbstract We analyze a fringe tree parameter w in a variety of settings, utilizing a variety of methods from the analysis of algorithms and data structures. Given a tree t and one of its leaves a, the w(t,,a) parameter denotes the number of internal nodes in the subtree rooted at a's father. The closely related w,(t,,a) parameter denotes the number of leaves, excluding a, in the subtree rooted at a's father. We define the cumulative w parameter as W(t) = ,aw(t,,a), i.e. as the sum of w(t,,a) over all leaves a of t. The w parameter not only plays an important rôle in the analysis of the Lempel,Ziv '77 data compression algorithm, but it is captivating from a combinatorial viewpoint too. In this report, we determine the asymptotic behavior of the w and W parameters on a variety of types of trees. In particular, we analyze simply generated trees, recursive trees, binary search trees, digital search trees, tries and Patricia tries. The final section of this report briefly summarizes and improves the previously known results about the w, parameter's behavior on tries and suffix trees, originally published in one author's thesis (see Analysis of the multiplicity matching parameter in suffix trees. Ph.D. Thesis, Purdue University, West Lafayette, IN, U.S.A., May 2005; Discrete Math. Theoret. Comput. Sci. 2005; AD:307,322; IEEE Trans. Inform. Theory 2007; 53:1799,1813). This survey of new results about the w parameter is very instructive since a variety of different combinatorial methods are used in tandem to carry out the analysis. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Positivity and time behavior of a linear reaction,diffusion system, non-local in space and time MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2008Andrii KhrabustovskyiAbstract We consider a general linear reaction,diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle and positivity of the solution and investigate its asymptotic behavior. Moreover, we give an explicit expression of the limit of the solution for large times. In order to obtain these results, we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution to a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction,diffusion system. Using this and the facts that the diffusion equation on manifolds satisfies the maximum principle and its solution converges to a easily calculated constant, we can obtain analogous properties for the original system. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Asymptotic analysis of solutions of a radial Schrödinger equation with oscillating potential MATHEMATISCHE NACHRICHTEN, Issue 15 2006Sigrun BodineAbstract We are interested in the asymptotic behavior of solutions of a Schrödinger-type equation with oscillating potential which was studied by A. Its. Here we use a different technique, based on Levinson's Fundamental Lemma, to analyze the asymptotic behavior, and our approach leads to a complete asymptotic representation of the solutions. We also discuss formal simplifications for differential equations with what might be called "regular/irregular singular points with periodic coefficients". (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Elliptic and parabolic problems in unbounded domains MATHEMATISCHE NACHRICHTEN, Issue 1 2004Patrick GuidottiAbstract We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi-explicit representation formulas via operator-valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Optimal empty vehicle redistribution for hub-and-spoke transportation systems NAVAL RESEARCH LOGISTICS: AN INTERNATIONAL JOURNAL, Issue 2 2008Dong-Ping SongAbstract This article considers the empty vehicle redistribution problem in a hub-and-spoke transportation system, with random demands and stochastic transportation times. An event-driven model is formulated, which yields the implicit optimal control policy. Based on the analytical results for two-depot systems, a dynamic decomposition procedure is presented which produces a near-optimal policy with linear computational complexity in terms of the number of spokes. The resulting policy has the same asymptotic behavior as that of the optimal policy. It is found that the threshold-type control policy is not usually optimal in such systems. The results are illustrated through small-scale numerical examples. Through simulation the robustness of the dynamic decomposition policy is tested using a variety of scenarios: more spokes, more vehicles, different combinations of distribution types for the empty vehicle travel times and loaded vehicle arrivals. This shows that the dynamic decomposition policy is significantly better than a heuristics policy in all scenarios and appears to be robust to the assumptions of the distribution types. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008 [source] ## On a class of multiple failure mode systems NAVAL RESEARCH LOGISTICS: AN INTERNATIONAL JOURNAL, Issue 2 2002Michael V. BoutsikasAbstract The primary objective of this work is to introduce and perform a detailed study of a class of multistate reliability structures in which no ordering in the levels of components' performances is necessary. In particular, the present paper develops the basic theory (exact reliability formulae, reliability bounds, asymptotic results) that will make it feasible to investigate systems whose components are allowed to experience m , 2 kinds of failure (failure modes), and their breakdown is described by different families of cut sets in each mode. For illustration purposes, two classical (binary) systems are extended to analogous multiple failure mode structures, and their reliability performance (bounds and asymptotic behavior) is investigated by numerical experimentation. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 167,185, 2002; DOI 10.1002/nav.10007 [source] ## Energy properties preserving schemes for Burgers' equation, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2008R. AnguelovAbstract The Burgers' equation, a simplification of the Navier,Stokes equations, is one of the fundamental model equations in gas dynamics, hydrodynamics, and acoustics that illustrates the coupling between convection/advection and diffusion. The kinetic energy enjoys boundedness and monotone decreasing properties that are useful in the study of the asymptotic behavior of the solution. We construct a family of non-standard finite difference schemes, which replicate the energy equality and the properties of the kinetic energy. Our approach is based on Mickens' rule [Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.] of nonlocal approximation of nonlinear terms. More precisely, we propose a systematic nonlocal way of generating approximations that ensure that the trilinear form is identically zero for repeated arguments. We provide numerical experiments that support the theory and demonstrate the power of the non-standard schemes over the classical ones. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] ## The game chromatic number of random graphs RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2008Tom BohmanGiven a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number ,g(G) is the minimum k for which the first player has a winning strategy. In this study, we analyze the asymptotic behavior of this parameter for a random graph Gn,p. We show that with high probability, the game chromatic number of Gn,p is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008 [source] ## Nodes of large degree in random trees and forests RANDOM STRUCTURES AND ALGORITHMS, Issue 3 2006Bernhard GittenbergerWe study the asymptotic behavior of the number Nk,n of nodes of given degree k in unlabeled random trees, when the tree size n and the node degree k both tend to infinity. It is shown that Nk,n is asymptotically normal if and asymptotically Poisson distributed if . If , then the distribution degenerates. The same holds for rooted, unlabeled trees and forests. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006 [source] ## Coin flipping from a cosmic source: On error correction of truly random bits RANDOM STRUCTURES AND ALGORITHMS, Issue 4 2005Elchanan MosselWe study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x , {0, 1}n, and k parties, who have noisy version of the source bits yi , {0, 1}n, when for all i and j, it holds that P[y = xj] = 1 , ,, independently for all i and j. That is, each party sees each bit correctly with probability 1 , ,, and incorrectly (flipped) with probability ,, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on balanced functions fi: {0, 1}n , {0, 1} such that P[f1(y1) = , = fk(yk)] is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The function fi may be thought of as an error correcting procedure for the source x. When k = 2,3, no error correction is possible, as the optimal protocol is given by fi(yi) = y. On the other hand, for large values of k, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to k, n, and ,. Our analysis uses tools from probability, discrete Fourier analysis, convexity, and discrete symmetrization. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005 [source] ## Organic glasses: cluster structure of the random energy landscape ANNALEN DER PHYSIK, Issue 12 2009S.V. NovikovAbstract An appropriate model for the random energy landscape in organic glasses is a spatially correlated Gaussian field. We calculated the distribution of the average value of a Gaussian random field in a finite domain. The results of the calculation demonstrate a strong dependence of the width of the distribution on the spatial correlations of the field. Comparison with the simulation results for the distribution of the size of the cluster indicates that the distribution of an average field could serve as a useful tool for the estimation of the asymptotic behavior of the distribution of the size of the clusters for "deep" clusters where value of the field on each site is much greater than the rms disorder. We also demonstrate significant modification of the properties of energetic disorder in organic glasses at the vicinity of the electrode. [source] ## Diffusion-Influenced Reversible Trapping Problem in the Presence,of,an,External Field CHEMISTRY - AN ASIAN JOURNAL, Issue 1-2 2006Soohyung ParkAbstract We investigate the field effect on the diffusion-influenced reversible trapping problem in one dimension. The exact Green function for a particle undergoing diffusive motion between two static reversible traps with a constant external field is obtained. From the Green function, we derive the various survival probabilities. Two types of trap distribution for the many-body problem are considered, the periodic and random distributions. The mean survival probability is obtained for the crossing-forbidden case for the two types of trap distribution. For the periodic distribution it decays exponentially. For the random trap distribution, similar to the irreversible case, there exists a critical field strength at which the long time asymptotic behavior undergoes a kinetic transition from the power law to exponential behaviors. The difference between equilibrium concentrations for the two types of trap distribution due to the fluctuation effect of trap concentration vanishes as the field strength increases. [source] ## Dyson's nonintersecting Brownian motions with a few outliers COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 3 2009Mark AdlerConsider n nonintersecting Brownian particles on , (Dyson Brownian motions), all starting from the origin at time t = 0 and forced to return to x = 0 at time t = 1. For large n, the average mean density of particles has its support, for each 0 < t < 1, on the interval ±,2nt(1 , t). The Airy process ,,(,) is defined as the motion of these nonintersecting Brownian motions for large n but viewed from the curve ,, : y = ,2nt(1 , t) with an appropriate space-time rescaling. Assume now a finite number r of these particles are forced to a different target point, say a = ,0,n/2 > 0. Does it affect the Brownian fluctuations along the curve ,, for large n? In this paper, we show that no new process appears as long as one considers points (y, t) , ,, such that 0 < t < (1 + ,),1, which is the t -coordinate of the point of tangency of the tangent to the curve passing through (,0,n/2, 1). At this point the fluctuations obey a new statistics, which we call the Airy process with r outliers ,,(r)(,) (in short, r-Airy process). The log of the probability that at time , the cloud does not exceed x is given by the Fredholm determinant of a new kernel (extending the Airy kernel), and it satisfies a nonlinear PDE in x and ,, from which the asymptotic behavior of the process can be deduced for , , ,,. This kernel is closely related to one found by Baik, Ben Arous, and Péché in the context of multivariate statistics. © 2008 Wiley Periodicals, Inc. [source] ## Relativistic diffusions and Schwarzschild geometry COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2007Jacques FranchiThe purpose of this article is to introduce and study a relativistic motion whose acceleration, in proper time, is given by a white noise. We deal with general relativity and consider more closely the problem of the asymptotic behavior of paths in the Schwarzschild geometry example. © 2006 Wiley Periodicals, Inc. [source] ## Existence of an asymptotic velocity and implications for the asymptotic behavior in the direction of the singularity in T3 -Gowdy COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2006Hans RingströmThis is the first of two papers that together prove strong cosmic censorship in T3 -Gowdy space-times. In the end, we prove that there is a set of initial data, open with respect to the C2 × C1 topology and dense with respect to the C, topology, such that the corresponding space-times have the following properties: Given an inextendible causal geodesic, one direction is complete and the other is incomplete; the Kretschmann scalar, i.e., the Riemann tensor contracted with itself, blows up in the incomplete direction. In fact, it is possible to give a very detailed description of the asymptotic behavior in the direction of the singularity for the generic solutions. In this paper, we shall, however, focus on the concept of asymptotic velocity. Under the symmetry assumptions made here, Einstein's equations reduce to a wave map equation with a constraint. The target of the wave map is the hyperbolic plane. There is a natural concept of kinetic and potential energy density; perhaps the most important result of this paper is that the limit of the potential energy as one lets time tend to the singularity for a fixed spatial point is 0 and that the limit exists for the kinetic energy. We define the asymptotic velocity v, to be the nonnegative square root of the limit of the kinetic energy density. The asymptotic velocity has some very important properties. In particular, curvature blowup and the existence of smooth expansions of the solutions close to the singularity can be characterized by the behavior of v,. It also has properties such that if 0 < v,(,0) < 1, then v, is smooth in a neighborhood of ,0. Furthermore, if v,(,0) > 1 and v, is continuous at ,0, then v, is smooth in a neighborhood of ,0. Finally, we show that the map from initial data to the asymptotic velocity is continuous under certain circumstances and that what will in the end constitute the generic set of solutions is an open set with respect to C2 × C1 topology on initial data. © 2005 Wiley Periodicals, Inc. [source] ## On a wave map equation arising in general relativity COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2004Hans RingströmWe consider a class of space-times for which the essential part of Einstein's equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1 + 1 nonlinear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations as t , ,. For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to 0 as t,1/2 as t , ,. In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behavior need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the nonhomogeneous case, one gets the following asymptotic behavior in the upper half-plane (after applying an isometry of hyperbolic space if necessary): 1The solution converges to a point. 2The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary). 3The solution goes to infinity along a curve y = const. 4The solution oscillates around a circle inside the upper half-plane. Thus, even though the solutions become spatially homogeneous in the sense that the spatial variations die out, the asymptotic behavior may be radically different from anything observed for spatially homogeneous solutions of the equations. This analysis can then be applied to draw conclusions concerning the associated class of space-times. For instance, one obtains the leading-order behavior of the functions appearing in the metric, and one can conclude future causal geodesic completeness. © 2004 Wiley Periodicals, Inc. [source] |