Abstract. Consider solving large sparse symmetric positive semidefinite linear systems. Since CG may not converge for inconsistent systems, we use MINRES. We will use right preconditioning for MINRES, since it is easier to preserve the equivalence of the problem, especially for the inconsistent case. We first introduce the algorithm for the right preconditioned MINRES. Then, we propose the right preconditioned MINRES using Eisenstat SSOR. Some numerical experiments indicate that the right preconditoned MINRES using Eisenstat SSOR is efficient and robust. Finally, we show that the residual norm can be further reduced by restarting the method.

Abstract. For the first time in the history of apportionment, the United States Supreme Court deliberated upon the constitutionality of the Huntington-Hill method, currently being used. Since around 1980, Balinski and Young have pointed out that Huntington-Hill's method favors small states while Webster's method has no bias in favor of small or large states. However the Court upheld the constitutionality of the Huntington-Hill method. Considering an assertion made by Ernst who was then an official at the U.S. Census Bureau and supported Huntington-Hill's method in the court, it is shown that his assertion partly lacks validity.

Abstract. This paper is concerned with a rounding error analysis of dot product by floating-point arithmetic. Floating-point arithmetic is fast performed on recent architectures. However, since the number of total bits in floating-point format is finite, rounding errors may occur in arithmetic operations. Recently, new analysis of rounding errors for dot product was developed. We review the results of the error analysis of dot product and revise them.

Abstract. In the present article we treat an environment-dependent model that describes immune responses against cancer cells. This model can explain theoretically that there exists a critical bound, the so-called saturation of immunity suggested in the tumourimmunology. We can also show that this model exhibits local extinction which just corresponds to the situation where cancer cells are expelled locally by immune actions of the effector group. Moreover, we discuss the survival and coexistence results in the standpoint of the theory of mathematical models.

Abstract. Texture image analysis plays a pivotal role in characterizing object properties. This technique numerically represents the spatial arrangement of the gray levels of digital images, and is used on problems of classifying image data sets. Second harmonic generation, an optical label-free technique based on nonlinear optical effects, is recognizing important for biomedical applications. Researches that integrate these two technologies are increasingly being used not only for advancing basic understanding of biological processes, but also for applications of clinical diagnosis. Here, we highlight recent advances in this combined research area, and describe how this approach is used in tackling medical problems.