We investigate the Green function, the Poisson kernel and the Martin kernel of circular cones in the symmetric stable case. We derive their sharp estimates. We also investigate properties of the characteristic exponent of these estimates. We prove that this exponent is a continuous function of the aperture of the cone.

Our aim in this paper is to deal with maximal functions for Lebesgue spaces with variable exponent approaching 1.

It is shown that the Lack-of-Fit test can be considered as the likelihood ratio test on the mean structure for some linear model. The asymptotic expansion of the null distribution of the test statistic is derived up to order $n^-1$ under nonnormality. A certain robustness against nonnormality is also investigated.

In this paper, we consider an analytic kind of structure on the ideal boundary of a Riemann surface, which is finer than the topological one, and show that the set of the natural equivalence classes of mutually quasiconformally related such structures admits a complex Banach manifold structure.

Chillingworth found an algorithm for determining whether a given element of the fundamental group of a surface contains simple closed curves. We extend the theory to ‘open’ curves on a punctured surface.

In this paper we consider the Cauchy problem for the parabolic system arising in biology. By the method of the analytic semigroup developed in Osaki and Yagi, Yagi we show existence, uniqueness and non-negativity of global solutions.

In this paper we investigate the ring $A_r(R)$ of arithmetical functions in r variables over an integral domain R with respect to the unitary convolution. We study a class of norms, and a class of derivations on $A_r(R)$. We also show that the resulting metric structure is complete.

A smooth closed manifold is said to be an almost sphere if it admits a Morse function with exactly two critical points. In this paper, we characterize those smooth closed manifolds which admit Morse functions such that each regular fiber is a finite disjoint union of almost spheres. We will see that such manifolds coincide with those which admit Morse functions with at most three critical values. As an application, we give a new proof of the characterization theorem of those closed manifolds which admit special generic maps into the plane. We also discuss homotopy and diffeomorphism invariants of manifolds related to the minimum number of critical values of Morse functions; in particular, the Lusternik-Schnirelmann category and spherical cone length. Those closed orientable 3-manifolds which admit Morse functions with regular fibers consisting of spheres and tori are also studied.