Rigid-plastic finite element method has been widely utilized not only for research and development in metal forming but also for process operation design in practice. In this situation, two items are noted to consider an adaptive rigid-plastic finite element methodology to accurate, reliable metal forming simulation. The first item is a nuisance of lack of knowledge about the mixed finite elements with consideration of incompressibility condition. In addition to discussion on the element pairs satisfying the LBB condition and the constraint relation as an engineering patch test, three dimensional composite element is used for practical resizing forging to evaluate the validity and effectiveness of the present rigid-plastic analysis. The second is a hot discussion whether the Lagrangian approach is still valid to actual metal forming analysis or the Eulerian approach must be selected as a new frontier. After making some comments on the trade-off between two approaches, three methods are introduced in propelling the Eulerian approach. Among them, the Arbirary Lagragian Eulerian(ALE) method is employed to demonstarte how to deal with the tool-work interaction in practical forward extrusion process.

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The connectivity augmentation problem asks to add to a given graph the smallest number of new edges so that the edge- (or vertex-) connectivity of the graph increases up to a specified value k. The problem is first studied by K. P. Eswaran and R. E. Tarjan in 1976, and both type of connectivity augmentation problems for k=2 are shown to be polynomially solvable. Afterwards, in 1987 T. Watanabe and A. Nakamura proved that the problem of making a given graph k-edge-connected by adding the smallest number of edges can be solved in O(k^2(kn+m)n^4) time for general k, where n and m are the number of vertices and edges in the input graph, respectively. Recently, a significantly faster O(nm log n+n^2 log^2 n) time algorithm for solving this problem is proposed by H. Nagamochi and T. Ibaraki by applying L. Lovasz's edge-splitting theorem. This note first reviews this alogorithm and then shows how to modify the algorithm to solve the edgeconnectivity augmentation problem for a graph with real-weighted edges.

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We have investigated how a simplified cognitive system perceives a simplified world which consists of a set of n dimensional binary patterns. We can find a clear mathematical structure in the underlying cognitive mechanism. In this paper, we consider two different types of cognitive judgments of patterns, i. e., goodness and similarity judgments of patterns, from the viewpoint of the Transformational Structure Theory. The theory can explain how cognitive judgments of patterns are performed and can predict ordinal relations between rated goodness and between rated similarities, respectively. The prediction has been supported by many experimental results. Moreover, the Transformational Structure Theory has the following distinctive characteristics: it can be formulated in a dual way for the two qualitatively different types of cognitive judgments of patterns, and it can give a multiple and cognitive definition of "pattern". We expect that mathematical method is applicable and will produce interesting results in the study of fundamental human cognitive phenomena.

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