To a certain Volevič system of singular partial differential equations, called a Fuchsian system, all the solutions of the homogeneous equation in a complex domain are constructed and parametrized in a good way, without any assumption on the characteristic exponents.

Fundamental calculations on singular invariant hyperfunctions on the $n \times n$ square matrix space and on the $2n \times 2n$ alternating matrix space are considered in this paper.By expanding the complex powers of the determinant function or the Pfaffian function into the Laurent series with respect to the complex parameter, we can construct singular invariant hyperfunctionsas their Laurent expansion coefficients. The author presents here the exact orders of the poles of the complex powers and determines the exact supports of the Laurent expansion coefficients. By applying these results, we prove that every quasi-relatively invariant hyperfunction can be expressed as a linear combination of the Laurent expansion coefficients of the complex powers and that every singular quasi-relatively invariant hyperfunction is in fact relatively invariant on the generic points of its support. In the last section, we give the formula of the Fourier transforms of singular invariant tempered distributions.

Modifications of balayage spaces are studied which, in probabilistic terms, correspond to killing and transitions (creation of mass combined with jumps). This is achieved by a modification of harmonic kernels for sufficiently small open sets. Applications to coupling of elliptic and parabolic partial differential equations of second order are discussed.

We give an explicit determinant formula for a class of rational solutions of a $q$-analogue of the Painlevé V equation. The entries of the determinant are given by the continuous $q$-Laguerre polynomials.

In this paper we first prove some criteria for the propagation of algebraic dependence of dominant meromorphic mappings from an analytic finite covering space $X$ over the complex $m$-space into a projective algebraic manifold. We study this problem under a condition on the existence of meromorphic mappings separating the generic fibers of $X$. We next give applications of these criteria to the uniqueness problem of meromorphic mappings. We deduce unicity theorems for meromorphic mappings and also give some other applications. In particular, we study holomorphic mappings into a smooth elliptic curve $E$ and give conditions under which two holomorphic mappings from $X$ into $E$ are algebraically related.

Let $p$ be an odd prime number. By using Iwasawa theory, we shall construct cyclotomic fields whose maximal real subfields have class group with arbitrarily large $p$-rank and conductor with only four prime factors.

For rank-one symmetric spaces $M$ of the compact type all Kähler structures $F^\lambda$, defined on their punctured tangent bundles $T^0M$ and invariant with respect to the normalized geodesic flow on $T^0M$, are constructed. It is shown that this class $\{F^\lambda \}$ of Kähler structures is stable under the reduction procedure.