Let $M$ be a real hypersurface in a complex space form $M_n(c)$, $c \not= 0$. In this paper, we prove that if the structure Jacobi operator $R_\xi$ is $\phi\nabla_{\xi}\xi$-parallel and $R_\xi$ commute with the Ricci tensor, then $M$ is a Hopf hypersurface provided that the mean curvature of $M$ is constant with respect to the structure vector field.

We express explicitly the Heckman-Opdam hypergeometric function for the root system of type A with a certain degenerate parameter in terms of the Lauricella hypergeometric function.

In this paper, we study topological properties of the subpower Higson coronas of proper metric spaces and show that the subpower Higson corona of the half open interval with the usual metric is an indecomposable continuum. Continuous surjections from Higson-type coronas onto a Higson-type compactifications of the half open interval are also constructed.

Methods for computing logarithmic vector fields along a semi-weighted homogeneous hypersurface with an isolated singularity are considered in the context of symbolic computation. The main idea of our approach is based on the concept of polar variety and of algebraic local cohomology. New algorithms are introduced for computing a set of generators of the modules of logarithmic vector fields. The keys of the resulting algorithms are a notion of parametric syzygy system and that of parametric local cohomology system.

We provide a sufficient condition for a proper metric space in order that its Higson compactification may be of Wallman type.

Johnson, Holroyd and Stahl [5] conjectured that the circular chromatic number of the Kneser graph is equal to the ordinary chromatic number. Chen completely confirmed the conjecture in [4]. Chen's alternative Kneser coloring theorem is a key lemma in his proof of Johnson-Holroyd-Stahl conjecture. Chen [4] and Chang, Liu and Zhu [3] proved the theorem by using Fan's lemma. In this paper, we prove Chen's alternative Kneser coloring theorem by using cohomology.

T. Ito defined an analog of the Arakawa-Kaneko zeta function to obtain relations among Mordell-Tornheim multiple zeta values. In this paper, we develop two things related to an analog of the Arakawa-Kaneko zeta function. One is to find an analog of the Arakawa-Kaneko zeta function of Miyagawa-type (defined by T. Miyagawa) and to obtain relations among Miyagawa-type multiple zeta values. The other is to find a class of zeta functions to which Ito's zeta functions of the case of general index are related.

The tail probability of the product of independent random variables is discussed. The class of random variables with regularly varying tails is known to be closed under multiplication and some explicit examples are given by Kifer-Varadhan [2]. We extend them.

There is a well-known formula for the Hurwitz zeta function which implies the functional equation for the Riemann zeta function. We give a new proof of that formula and recover certain integral representations for the Hurwitz and Riemann zeta functions.