Many of the properties of analytic functions can be proved in a purely topological manner, so that such properties are then valid for the larger class of functions which are topologically equivalent to analytic functions. The importance of such functions has been recognized fairly recently, particularly in the theory of partial differential equations, where certain solutions have been shown to possess the topological properties of analytic functions, i.e., interiority and continuity, but not necessarily the analytic properties of complex differentiability and integrability.

In this paper we treat the problem of determining the rate of growth of entire functions which are solutions of first order algebraic differential equations whose coefficients are arbitrary entire functions (i.e. equations of the form Ω(z, y, dy/dz) = 0, where Ω(z, y, dy/dz) = is a polynomial in y and dy/dz, whose coefficients fkJ(z) are entire functions).

Let S(n) be the matrix of a positive definite quadratic form and

Define

1.

Here the sum is over unimodular matrices which lie in a complete set of representatives for the equivalence relation with P unimodular and having block form

We describe all centrioles in irreducible simply connected pointed symmetric spaces of compact type in terms of the root system of the ambient space, and we study some geometric properties of centrioles.

Let $A$ be a truncated polynomial ring over a complete discrete valuation ring ${\mathcal{O}}$, and we consider the additive category consisting of $A$-lattices $M$ with the property that $M\otimes {\mathcal{K}}$ is projective as an $A\otimes {\mathcal{K}}$-module, where ${\mathcal{K}}$ is the fraction field of ${\mathcal{O}}$. Then, we may define the stable Auslander–Reiten quiver of the category. We determine the shape of the components of the stable Auslander–Reiten quiver that contain Heller lattices.

If D ⊂ ℂn is a pseudoconvex domain and X ⊂ D a closed analytic subset, the famous theorem B of Cartan-Serre asserts, that the restriction operator r : (D) → (X) mapping each function F to its restriction F|X is surjective. A very important question of modern complex analysis is to ask what happens to this result if certain growth conditions for the holomorphic functions on D and on X are added.

In his lecture notes ([1, pp. 33-35], [2, pp. 145-152]), M. Eichler reduced ‘quadratic’ Hilbert modular forms of dimension —k {k is a positive integer) to holomorphic automorphic forms of dimension — 2k for the reproduced groups of indefinite ternary quadratic forms, by means of so-called Eichler maps.

In [1] L. Fuchs and I. Halperin have proved that a regular ring R is isomorphic to a two-sided ideal of a regular ring with identity. ([1] Theorem 1). Their methed is to imbed the regular ring R in the ring of all pairs (a, p) with a ∊ R and p from a suitable commutative regular ring S with identity such that R is an algebra over S. Thus S may be seen as the ring of R — R endomorphisms of the additive group of R. The following question is naturally raised: Is it true that the ring of all R-R endomorphisms of a rugular ring is a commutative regular ring? The main purpose of this paper is to answer this question affirmatively. (Theorem 1). After established this theorem we can follow the method in [1] to solve the problem in the title.

In [1] S. Kobayashi showed that the connected components of the set of zeros of a Killing vector field on a Riemannian manifold (Mn,g) are totally geodesic submanifolds of (Mn,g) of even codimension including the case of isolated singular points. The purpose of this short note is to give a simple proof of the corresponding result for conformal vector fields on compact Riemannian manifolds. In particular we prove the following

In this paper it is shown that there is a unimodal Levy process with oscillating mode. After the author first constructed an example of such a self-decomposable process, Sato pointed out that it belongs to the class of semi-stable processes with β < 0. We prove that all non-symmetric semi-stable self-decomposable processes with β < 0 have oscillating modes.

In this paper, the commutators of singular integrals with rough kernels are considered. By the method of block decomposition for kernel function and Fourier transform estimates, some new results about the Lp(ℝn) boundedness for these commutators are obtained.

For prime l we study the structure of the 2-part of the ideal class group Cl of ℚ(ζl). We prove that Cl ⊗ ℤ2) is a cyclic Galois module for all l < 10000 with one exception and compute the explicit structure in several cases.

In this paper, we improve a result of the third paper of this series, that is we show

THEOREM. Let K be a nilpotent extension of the rational number field Q with Galois group Γ, and G a Γ- stable finite subgroup of GLn(0K). Then G is of A-type.

Probability measure valued diffusion processes have been discussed by many authors, in connection with population genetics. Most papers studying probability measure valued diffusions are mainly concerned with the ones describing single locus models. In this paper, we will discuss a measure valued diffusion describing an n locus model. Random sampling, mutation and gene conversion, a kind of interaction between loci, which was introduced and investigated by T. Ohta in [5], [6], will be taken into consideration.

Tahara [1] has concretely determined the conjugate classes of finite subgroups of GL(3, Z). The group W5 in his list of groups of order 24 is in fact of order 12 and consists of the same matrices as the group W6 in his list of groups order 12. Hence, there are only 10 conjugate classes of subgroups of order 24 in GL(3, Z) and the total number of conjugate classes of finite subgroups is reduced to 73.

In [I-S2], we gave an explicit form of zeta functions associated to the space of symmetric matrices. In this paper, the case of L-functions is treated. In the case of definite symmetric matrices, we show the ratinality of special values of these L-functions.

Our object in this paper is to prove the following result.

THEOREM. Let G be a finite group satisfying the following conditions:

• (*) G has subgroups L1, L2 such that L1 ≃ SL(2, q1), L2 ≃ SL(2, q2), [L1 L2] = 1, , where j is an involution, and |C(j): L1L2| = 2.

• (**) where n2 = 1,

In this paper we deal with the problem (C) in § 4 of [4]. Let Ik be the Shimura mapping in [4] of Sk(4N, χ) into k-1(N′ χ2) (see p. 458). The problem (C) can be stated as follows: I3(f) is a cusp form if and only if ‹f, h› = 0 for all h ∈ U, where U is the vector space spanned by every theta series of S3(4N, χ) associated with some Dirichlet character.

Let k be either a local or a global field, and K be a finite Galois extension of k with g = Gal (K/k). Let L be a Galois extension of K which is also Galois over k. Such an extension is called central if Gal(L/iT) lies inside the centre of Gal(L/K). Clearly L is abelian over K. Next set L* = L∩K · kab where kab is the maximal abelian extension of k in its algebraic closure. This is the genus field of L over K/k.

We define weight changing operators for automorphic forms on Grassmannians, that is, on orthogonal groups, and investigate their basic properties. We then evaluate their action on theta kernels, and prove that theta lifts of modular forms, in which the theta kernel involves polynomials of a special type, have some interesting differential properties.