Interdisciplinary Information Sciences
Online ISSN : 1347-6157
Print ISSN : 1340-9050
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Volume 17 , Issue 3
Showing 1-4 articles out of 4 articles from the selected issue
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GSIS SELECTED LECTURES: Exploring Collaborative Mathematics
  • Kanako SUZUKI
    Volume 17 (2011) Issue 3 Pages 131-153
    Released: November 30, 2011
    JOURNALS FREE ACCESS
    In 1952, A. M. Turing proposed the notion of “diffusion-driven instability” in his attempt of modeling biological pattern formation. Following his ingenious idea, many reaction-diffusion systems have been proposed later on. On the other hand, Turing patterns can be explained by some cellular automata. Cellular automata are theoretical models which consist of a regular grid of cells, and they exhibit the complex behavior from quite simple rules. In this paper, we describe the mathematical properties of reaction-diffusion systems modeling pattern formation, in particular, Turing patterns. Moreover, we explain ideas which connect differential equations with cellular automata.
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  • Keiji MIURA
    Volume 17 (2011) Issue 3 Pages 155-174
    Released: November 30, 2011
    JOURNALS FREE ACCESS
    In this paper, we review the maximum likelihood method for estimating the statistical parameters which specify a probabilistic model and show that it generally gives an optimal estimator with minimum mean square error asymptotically. Thus, for most applications in information sciences, the maximum likelihood estimation suffices. Fisher information matrix, which defines the orthogonality between parameters in a probabilistic model, naturally arises from the maximum likelihood estimation. As the inverse of the Fisher information matrix gives the covariance matrix for the estimation errors of the parameters, the orthogonalization of the parameters guarantees that the estimates of the parameters distribute independently from each other. The theory of information geometry provides procedures to diagonalize parameters globally or at all parameter values at least for the exponential and mixture families of distributions. The global orthogonalization gives a simplified and better view for statistical inference and, for example, makes it possible to perform a statistical test for each unknown parameter separately. Therefore, for practical applications, a good start is to examine if the probabilistic model under study belongs to these families.
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  • Takehisa HASEGAWA
    Volume 17 (2011) Issue 3 Pages 175-195
    Released: November 30, 2011
    JOURNALS FREE ACCESS
    We review some studies on complex networks. The complex networks often have structural characteristics: the power-law degree distribution; the logarithmic size dependence of the average path length; and the high clustering coefficient. Several network models with such properties are introduced. We also consider the percolation models on the scale-free networks to show how the critical probability and critical behavior change with network topology.
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  • Yota OTACHI
    Volume 17 (2011) Issue 3 Pages 197-216
    Released: November 30, 2011
    JOURNALS FREE ACCESS
    In these lecture notes, we study problems of designing low-congestion subnetworks of given networks. We apply some newest results in Graph Minor Theory to the problem, and obtain efficient (even best possible) algorithms.
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