This paper is devoted to studies on interacting particle systems (IPSs) based on correlation inequalities, for examples, Harris-FKG inequality, submodularity, and BFKL inequality. In particular, BFKL inequality is a new type of correlation inequality and a refinement of Harris-FKG inequality and submodularity. As a typical example of IPSs, here we focus on one-dimensional contact process. By using above correlation inequalities, we can easily and systematically obtain lower bonds on critical value and upper bounds on survival probability of contact process, compared with other methods, i. e., edge process technique and Harris lemma method. So we call this method correlation inequality method. However there are some limitations on the method in the present stage. The contact process is attractive and two-state process. Therefore we are studying whether or not Harris-FKG inequality and/or BFKL inequality hold even in non-attractive and/or multi-sate IPSs. Concerning non-attractive Domany-Kinzel model, Monte-Carlo simulations suggest that Harris-FKG inequality is correct. As for multi-state IPSs, for example, 3-state cyclic particle system and successional model in two dimensions, both Harris-FKG type inequality and BFKL type one would be correct according to results on Monte-Carlo simulations.

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The paper of this article is to describe a proximal point algorithm (PPA) for solving the nonlinear complementarity problem (NCP). PPA was originally proposed by Martinet and further developed by Rockafellar for monotone operator problems. It is known to have nice convergence properties under appropriate criteria for approximate solutions of subproblems. However, it is not always easy to solve subproblems or to check those criteria. Moreover, PPA has been applied mainly to monotone problems. In this article, we present a PPA for the NCP involving a Po function and establish its global convergence under appropriate conditions by using the results derived from the reformulation approach for NCP. Moreover, without assuming the local uniqueness of the solution, we show that the rate of convergence is superlinear in a genuine sense.

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Recent explosion of genomic data has provided us a new field of mathematics in Genome Informatics. Genomic sequence analysis requires flexible and feasible symbolic processing of sequence matching, structure annotation, and function prediction. Once we have got structure or function of genes, we have the opportunity of applying it to new sequences or new genes using sequence similarity. However, since most sequences daily determined will be new, we sometimes can not predict their function only by sequence matching. Thus we must abstract and give bias to some parts in sequences. Here we illustrate how multiple sequence alignment, homology search, extraction of characteristic sequences, gene structure prediction, and gene function prediction provide clues to bring us insight into medical science and biology. We also show how gene expression in each tissue is measured by experiments, e. g. Microarray, what kind of mathematics is required for prediction of gene-network and gene-expression. Finally, we draw what is the promising topics of the new era of Human Genome Project, e. g. SNPs, for both mathematics and medical science.

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[in Japanese]

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