Showing papers on "Polynomial chaos published in 1996"
TL;DR: In this paper, the authors define a generating function called the Wiener polynomial, whose derivative is a q-analog of the original Wiener index, and compute it for some common graphs.
Abstract: rn The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some common graphs. We then find a formula for the Wiener polynomial of a dendrimer, a certain highly regular tree of interest to chemists, and show that it is unimodal. Finally, we point out a connection with the Poincar6 polynomial of a finite Coxeter group. 0 1996 John Wiley & Sons, Inc.
150 citations
TL;DR: In this paper, the problem of a medium with two layers separated by an interface randomly fluctuating in space was formulated using the second-moments characteristics of the interface spatial fluctuations, and the Karhunen-Loeve and polynomial chaos expansions were used to transform the problem into a computationally tractable form.
Abstract: This paper addresses the problem of a medium with two layers separated by an interface randomly fluctuating in space. The medium is subjected to an in-plane strain field simulating the effect of a surface foundation. The second-moments characteristics of the interface spatial fluctuations are used to formulate the problem. The Karhunen-Loeve and the polynomial chaos expansions are utilized to transform the problem into a computationally tractable form, thus resulting in a system of linear algebraic equations to solve. The difficulty in this problem stems from the geometric nature of the randomness, resulting in a stiffness matrix that is nonlinear in the randomness. This leads to a nonlinear stochastic problem, the solution of which is accomplished by relying on the polynomial chaos representation of stochastic processes.
72 citations
Journal Article•
12 citations
TL;DR: In this paper, the authors investigated the appearance of order in chaos with Gaussian white noise based on the Poincare return map constructed from the three-variables ODE of the Belousov-Zhabotinsky reaction.
Abstract: The mechanism and condition of appearance of order in chaos with Gaussian white noise is investigated based on the Poincare return map constructed from the three-variables ODE of the Belousov-Zhabotinsky reaction. When the noise is added to the chaos having the bifurcation parameter near period-three oscillation, it occasionally happens that topological entropy is constant but the Kolmogorov entropy decreases. Analysis on three-times iterated map reveals that this phenomenon, named “noise-induced convergence”, is caused by an increase of the length of laminar phase and the subsequent change of the invariant density. The same phenomenon is observed in m-times iterated map of the chaos of the logistic map. We consider that “noise-induced convergence” is characteristic of intermittent chaos.
7 citations
20 May 1996
TL;DR: According to this research result, the appearance of 1/f noise at the cause of chaos condition before and to randomness process from order generated with chaos growth is recognized.
Abstract: A novel phenomenon of 1/f/sup /spl alpha// noise related with chaos-chaos transition caused by non-linear process has been investigated in the simulation of simplest equation. The symmetry-broken problem appears as chaos-chaos transition caused by cross-talk coupling between two oscillatory chaotic state. In order to elucidate the fluctuation properties of generative complexity generated with chaos growth, we are proposed an appearance of 1/f noise at chaotic process. In according to this research result, we are recognized the appearance of 1/f noise at the cause of chaos condition before, and to randomness process from order generated with chaos growth.
7 citations
11 Dec 1996
TL;DR: A recursive in time Wiener chaos representation of the optimal nonlinear filter is derived for a continuous time diffusion model with uncorrelated noises.
Abstract: For pt. I see SIAM Journal on Control and Optimization (1997). A recursive in time Wiener chaos representation of the optimal nonlinear filter is derived for a continuous time diffusion model with uncorrelated noises. The existing representations are either not recursive or require a prior computation of the unnormalized filtering density, which is time consuming. An algorithm is developed for computing a recursive approximation of the filter, and the error of the approximation is obtained. When the parameters of the model are known in advance, the online speed of the algorithm can be increased by performing part of the computations off line.
4 citations
TL;DR: In this proposed method, the time history of a discrete dynamical system containing the return map, which generates the intermittency chaos, is simulated and during convergence (nonperiodically) to the local optimal solution, the global optimal solution is searched without trapping into theLocal optimal solution.
Abstract: The nonlinear global optimization methods developed so far can be divided into deterministic methods and stochastic methods. Various approaches proposed recently in which chaos, which is a probabilistic phenomenon and is generated by a deterministic dynamical system, is used to solve nonlinear global optimization problems. Chaos is used in these methods because fluctuations away from the local solution and global optimal solution can be searched without trapping into the local solution. the method of unconstrained global optimization proposed here uses intermittency chaos. In this proposed method, the time history of a discrete dynamical system containing the return map, which generates the intermittency chaos, is simulated and during convergence (nonperiodically) to the local optimal solution, the global optimal solution is searched without trapping into the local optimal solution. the validity of the proposed method is confirmed by applying it to a standard test problem.
1 citations
TL;DR: For a continuous map f from a real compact interval I into itself, the authors showed that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold.
Abstract: For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x, y) ∈ I2 for which lim infn→∞ |f(x)− f(y)| = 0 and lim supn→∞ |f(x) − f(y)| > 0. We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ2 denotes the Lebesgue measure on the square and Ch(f) is the set of points (x, y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ2(C(f)) > 0 need not imply λ2(Ch(f)) > 0. We use these results to propose some plausible definitions of “complete” and “observable” chaos.