Topological properties of typical human organs, such as the circulatory system, the liver tissue and the lung airway, are explained. Next, a computer simulation of constructing a lung airway is introduced, which has been made by the group of this author. An algorithm of this simulation is based on the function of the lung and some anatomical data. The former is to distribute air equally to all parts of the lung with minimum energy consumption. Nine rules are assumed as an algorithm to generate successive branching within a given outer boundary from a given initial condition (main trunk, trachea). The resulting airway system is quite similar to the real one. As an application of this result, a new method of diagnosis of lung cancer and inflammation is proposed. As another application of the present method airway systems within a sphere and a cube are constructed. A hope for future development of a new field "computational anatomy" is expressed.

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Preconditioning is often used to improve the convergence of the residual norm of iterative methods for large sparse linear systems of equations. ILU decomposition is one of the most popular preconditioner. Though it's not easy to implement the ILU preconditioner on the MIMD parallel machines, the block divided method is able to parallelize it partly by pipelining process. Recently, new preconditioners, quite different from ILU decomposition, are proposed. These preconditioners are assembled by the eigenvectors gathered up Arnordi process of GMRES(m) method. In this paper, we apply the parallel ILU decomposition and the deflation technique to GMRES(m) method, and analize the performance of each algorithm. We also give the brief introduction of the approximate inverse preconditioner. At last, we show the numerical experiments of the boundary value problems of elliptic partial differential equations.

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A brief survey on the spectral geometry of a finite or infinite graph is given. After the adjacency matrix, discrete Laplacian and discrete Green's formula are introduced, the spectral geometry of finite graphs, particularly, estimation of the first positive eigenvalue in terms of the Cheeger constant, examples of isospectral or cospectral graphs and the Faber=Krahn type inequality are discussed. For infinite graphs, spectrum of the discrete Laplacian, the heat kernel and Green kernel are estimated. Finally, a relation between the finite element method for the Dirichlet boundary eigenvalue problem and the eigenvalue problem of the adjacency matrix for a graph is given.

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