We have obtained the best constants of discrete Sobolev inequalities on the truncated regular M-hedron for M = 4, 6, 8. Giving appropriate indices on vertices of polyhedra and introducing discrete Laplacians A, we have obtained Green matrices G(a) = (A + aI)^<-1> (a > 0) and pseudo Green matrices G_*. (Pseudo) Green matrices are reproducing kernels by setting appropriate vector spaces and inner products. By applying Schwarz inequality to the reproducing relations, two types of discrete Sobolev inequalities are obtained. Diagonal values of (pseudo) Green matrices, which are identical, are best constants of these inequalities.

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This paper discusses the measure of biases of apportionment methods. Generally biases mean that large states are more favored than small states or the opposite. However it is not easy to define the large states or small states rigorously. In this paper a new bias measure is developed which does not require the rigorous definition of large or small states. We show by computer simulation that the new bias measure can measure bias in apportionment methods as correctly as the existing measures.

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It is difficult to detect chaoticity of limit sets in systems with zero- or small positive Lyapunov exponents, called weakly chaotic systems. New indicators detecting weak chaoticity are proposed by use of the second derivative and the logarithmic function. It is shown that these new indicators can detect the weak chaos in such as the Froeschle map in which the Arnold diffusion and the Chirikov diffusion occurs according to perturbation parameter, the Boole transformation and the S-unimordal function in which sub-exponential behavior and a zero Lyapunov exponent appear.

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In the presented paper, we discuss 0/1 occurrence rate per bit of outputs by the logistic map over integers. In order to obtain pseudorandom bits, it is necessary to extract bits from outputs by the map. However, bit positions of extractions from outputs may influence randomness in a pseudorandom number generator with the map. After we show a trend of occurrence rates per bit of outputs on the map, we give a theoretical guideline about bit positions of extractions for the pseudorandom number generator. Last, we confirm that the given guidelines can give outputs with randomness.

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We survey theories and techniques of the implementation of several modular forms and related topics. In particular, we focus on a recent progress of the implementation of Siegel modular forms that is an important generalization of elliptic modular forms.

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The set of rational points on an Abelian variety defined over a number field forms a finitely generated Abelian group and is called the Mordell-Weil group. For elliptic curves, various algorithms on the computation of the Mordell-Weil groups are known. Recently, these algorithms were generalized to the Jacobian varieties of hyperelliptic curves. In this article, we give a survey on the computation of the Mordell-Weil groups of the Jacobian varieties of hyperelliptic curves.

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