Abstract. The original Parrondo game, denoted as AB3, contains two independent games: A and B. The winning or losing of A and B game is defined by the change of one unit of capital. Game A is a losing game if played continuously, with winning probability p = 0.5 − ǫ, where ǫ = 0.003. Game B is also losing and it has two coins: a good coin with winning probability p g = 0.75 − ǫ is used if the player's capital is not divisible by 3, otherwise a bad coin with winning probability p b = 0.1 − ǫ is used. Parrondo paradox refers to the situation that the mixture of A and B game in a sequence leads to winning in the long run. The paradox can be resolved using Markov chain analysis. We extend this setting of Parrondo game to involve players with one-step memory. The player can win by switching his choice of A or B game in a Parrondo game sequence. If the player knows the identity of the game he plays and the state of his capital, then the player can win maximally. On the other hand, if the player does not know the nature of the game, then he is playing a (C,D) game, where either (C=A, D=B), or (C=B,D=A). For player with one-step memory playing the AB3 game, he can achieve the highest expected gain with switching probability equal to 3/4 in the (C,D) game sequence. This result has been found first numerically and then proven analytically. Generalization to AB mod(M ) Parrondo game for other integer M has been made for the general domain of parameters p b < p = 0.5 = p A < p g . We find that for odd M , Parrondo effect does exist. However, for even M , there is no Parrondo effect for two cases: initial game is A and initial capital is even, or initial game is B and initial capital is odd. There is still a possibility of Parrondo effect for the other two cases when M is even: initial game is A and initial capital is odd, or initial game is B and initial capital is even. These observations from numerical experiments can be understood as the factorization of the Markov chains into two distinct cycles. Discussion of these effects on games is also made in the context of feedback control of the Brownian ratchet.
We construct a method of time warping in quasiperiodic time series analysis using genetic algorithm in order to extract the instantaneous phase difference between a template signal and a testing signal. Contrast to previous studies, which involves correlation estimations to determine the shape similarity of two signals taken from the quasiperiodic time series, time warping perform the comparison of the two signals by first constructing a discrete set of M points formed from uniformly sampled values of the template signal f(t). The discrete set of sample values of the testing signal, g(t), which contains N points, will be interpolated to form a continuous function so that the difference between the template signal at those M points and the corresponding testing signals are minimize to best preserve the mapping of the two signals. The result of this optimization procedure produces a phase shift function that relates the time t in the testing signal to the time t in the template signal. Due to the numerous choices in the partitioning of the time domain of the two signals, genetic algorithm is found to be effective in extracting this phase shift function. We apply this theoretical tool of time warping using genetic algorithm to analyze the electrocardiographic (ECG) signals, with the aim to investigate if central apneic and obstructive apneic episodes can be differentiated from non-apneic episodes. Detailed statistical analysis of the phase shift from real ECG data of sleep apnea patient indicates that the difference of both magnitude and phase of the signals can be used to differentiate apneic events from nonapneic events.
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