Let f(z) be entire i.e. analytic in the finite whole plane Z. The order of f(z) is defined as

where A ray χ(θ) = {z = r·eiθ : 0 < r < + ∞} is called a Julia direction of f(z) if, in any open sector containing the ray, f(z) takes all values of Z, with at most one finite exceptional value, infinitely often.

The purpose of the present paper is to give a new proof to the triad excision theorem of Blakers and Massey [1], in case m≦2 and n≦2, by the aid of path spaces and in connection with a recent work of J. P. Serre [2].

Let X be a real prehomogeneous vector space under a reductive group G, such that X is an absolutely spherical G-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz–Bruhat functions on X against generalized matrix coefficients of admissible representations of $G(\mathbb {R})$ , twisted by complex powers of relative invariants. We establish the convergence of these integrals in some range, the meromorphic continuation, as well as a functional equation in terms of abstract $\gamma $ -factors. This subsumes the archimedean zeta integrals of Godement–Jacquet, those of Sato–Shintani (in the spherical case), and the previous works of Bopp–Rubenthaler. The proof of functional equations is based on Knop’s results on Capelli operators.

In this paper, we will prove that any $\mathbb{A}^{3}$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable $\mathbb{A}^{2}$-forms over a field $k$ extends to $\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing $\mathbb{Q}$.

Let S be a locally compact (not compact) Hausdorff space satisfying the second axiom of countability and let ℬ be the σ field of all Borel subsets of S and let A be the σ-field of all the subsets of S which, for each finite measure μ defined on (S, A), are in the completed σ field of ℬ relative to μ. We denote by C0 the Banach space of continuous functions vanishing at infinity with the uniform norm and Bk the space of bounded A-measurable functions with compact support in S.

Montel [10] proved in 1912 the following theorem: Let w = f(z) be an analytic function in the horizontal strip B : 0 < x < + ∞, 0 < y < 1 (z = x + iy) which is continuous on 0 < x < + ∞, 0 ≦ y < 1 and omits at least two values. If f (x) converges to a value w0 as x → + ∞, then f(z) converges to W0 as z tends to in 0 < x < + ∞, 0 ≦ y < 1 − ε for any ε such that 0 < ε < 1.

Cette mémoire sera consacrée à l’etude sur l’existence et la nonexistence des alg’bres de Dirichlet sur un ouvert de l’espace euclidien Rn à n(≧1) dimensions. La notion des algèbres de Dirichlet a été introduite par A. Beurling et J. Deny (cf. [1]).

Notations. The following notations will be used throughout this paper.

˛: the identity of a finite group.

Q: the rational number field.

P: an algebraic number field of finite degree, fixed as the ground field.

l: a prime number.

ζl: a primitive l-th root of unity.

It is well known that the quasi-exellent property (i.e. G-ring and J-II) ascends by finite type morphisms. On the other hand Greco showed in [2, Proposition 4.1 and Proposition 2.3] that G-property (i.e. formal fibers are geometrically regular) does not descend by finite type morphisms, although J-II property (i.e. any finite type algebra has regular locus open) does.

We let the notations be as in [3]. Then, in the category 6 of all bordered Riemann surfaces, the following inclusion diagram holds [3, Theorem 9]:

Let q a prime power and ${\mathbb F}_q$ the finite field of q elements. We study the analogues of Mahler’s and Koksma’s classifications of complex numbers for power series in ${\mathbb F}_q((T^{-1}))$ . Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.

In the present note, we prove that every simple CM-type is non-degenerate (i.e. the rank is maximal) if the dimension of corresponding abelian varieties is a prime. This follows directly from the argument of Tankeev [5], in which he has treated the 5-dimensional case.

Recently, S. G. Tankeev and K. A. Ribet have established similar results for more general types of abelian varieties (see [3], [4], [6]).

Let be a solvable group of order g. Let p be a prime and let g = pa g′ with (p,g′) = 1. In [4] we have tried to find sufficient conditions for to possess an irreducible character of p-defect 0, that is, a character whose degree is divisible by pa.

Let f(z) be a function analytic and bounded, |f(z)| < 1, in |z| < 1. Then, by Fatou’s theorem the radial limit f*(eiθ) = limr→1f(reiθ) exists almost everywhere on |z| = 1. Seidel [8, p. 208] and Calderón, González- Domínguez, and Zygmund [1] (see also [9, pp. 281-282]) proved the following: if f*(eiθ) is of modulus 1 almost everywhere on an arc a < θ < b of |z| = 1, then either f(z) is analytically continuable across this arc or the values f*(eiθ), a < d < b, cover the circumference |w| = 1 infinitely many times.

This paper is devoted to the problem of the realisation of homology classes of a PL-manifold by PL-submanifolds.

The present study is founded on the consideration of Thorn complexes M(PLk), M(SPLk) for PL-microbundles which is defined by R. Williamson [5]. We shall apply Thom’s method [4] to PL-manifolds.

The author is grateful to Professors R. Shizuma and K. Shiraiwa for their kind criticisms.

Let Ω be the projective limit space of a sequence of probability space Ωn which is a certain subset of (n − 1)-dimensional sphere with the usual uniform probability distribution on it. T. Hida [2], starting from a sequence of finite dimensional flows which are derived from some one-parameter subgroups of rotations of spheres, constructed a flow {Tt} on Ω as the limit of them. Observing his method, the concept of consistency of flows which approximate {Tt} seems to play an essential role in his work [2]. As will be made clear in the following sections, the concept of consistency is closely related to the projective limiting structure of our basic space Ω. The purpose of this paper is to determine all the flows on Ω which can be approximated in the sense of [2] by finite dimensional flows.

In this short note, we would like to point out that the following property (called ASSUMPTION REMOVABILITY in the present paper) is characteristic of the positive logics, the primitive logic LO, the positive predicate logics LP (intuitionistic) and LQ (classical):