Let T be an automorphism on a probability space (Ω, B, P). Given a finite partition α of Ω, into disjoint measurable sets A1, A2, · · ·, An, its entropy is

(D.1)

Let

(1) p(x,D)u = f

be a system of partial difierential equations. We shall say that p (x,D) is hypoelliptic if the distribution solution u is in C∞ wherever f∈C∞ (cf. section 2, Definition 5.)

Here we shall give a sufficient condition for the hypoellipticity of overdetermined systems with variable coefficients.

Suppose that Ω is a simply connected domain in the w-plane, w = u + iv, and that w∞ is an accessible boundary point of Ω located at w = ∞.

In this article we shall prove the uniqueness and existense of a weak solution for the Cauchy problem of linear parabolic equations with discontinuous and unbounded coefficients

(1.1)

Let S be a simply connected domain in the w + u + iv plane and let ∂S denote its boundary which we assume passes through w= ∞. Suppose that the segment L= {u ≧ u0; v = 0} of the real axis lies in S and that w∞ is the point of ∂ S accessible along L. Let z = z(w) = x(w) + iy(w) map S in a (1 — 1) conformal way onto ∑ = {z = x + iy: — ∞ < x < + ∞ } so that . The inverse map is w = w(z) = u(z) + iv(z). S is said to possess a finite angular derivative at w∞ if z(w) — w approaches a finite limit (called the angular derivative) as w→w∞ in certain substrips of S.

In this paper, we investigate several classes of groups, among which the most general is defined as follows:

Definition 1.1. A finite group G is a SR-group if it contains a subgroup P1 of order 3 satisfying:

• (a) A/S3-subgroup P2 of NGP1 is elementary of order 9;

• (b) NG(P2)/P2 acts semi-regularly by conjugation on the conjugates of P1 contained in P2.

Let G be a group and let k be a field. A K-representation ρ of G is a homomorphism of G into the group of non-singular linear transformations of some finite-dimensional vector space V over k. Let K be the field of fractions of the symmetric algebra S(V) of V, then G acts naturally on K as k-automorphisms. There is a natural inclusion map V→K, so we view V as a k-subvector space of K. Let v1, v2, · · ·, vn be a basis for V, then K is generated by v1, v2, · · ·, vn over k as a field and these are algebraically independent over k, that is, K is a rational field over k with the transcendence degree n. All elements of K fixed by G form a subfield of K. We denote this subfield by KG.

Although there are several constructions of meromorphic functions with prescribed asymptotic sets [e.g., 5,6], it is usually difficult to determine or prescribe the nature of the asymptotic paths used in these constructions. On the other hand, there are several other constructions of meromorphic functions with prescribed asymptotic paths [e.g., 1, 10, 12], but the extent of the asymptotic values for these functions cannot always be restricted to the values approached along the given paths. Gross [3] has accomplished both results by prescribing paths for every value in the extended complex plane.

1. Introduction. Let G be a rank 3 permutation group of even order on a finite set X, |X| = n, and let Δ and Γ be the two nontrivial orbits of G in X×X under componentwise action. As pointed out by Sims [6], results in [2] can be interpreted as implying that the graph = (X, Δ) is a strongly regular graph, the graph theoretical interpretation of the parameters k, l, λ and μ of [2] being as follows: k is the degree of , λ is the number of triangles containing a given edge, and μ is the number of paths of length 2 joining a given vertex P to each of the l vertices ≠ P which are not adjacent to P. The group G acts as an automorphism group on and on its complement = (X,Γ).

Recently, R. Rentschler [4] proved that if k is a field of characteristic 0, any action of the additive group Ga on the affine plane is equivalent to an action of the form t(x, y) = (x, y + tf(x), where t∈Ga(k), (x, y)∈k2 and f∈k[X].

Let (ω: Z2r→Cr be an isomorphism of the free additive group of rank 2r into the complex vector n-space such that the quotient group T = Crω/(Z2r) is compact, i.e., Tω is a complex torus.

D’après D. Courrège [1], l’opérateur de convolution associé en valeur principale à un noyau de Caldéron-Zygmund homogène de classe C0,μ(∑n) applique continûment dans Cp,λ(Rn) si 0 < λ < μ ≤ 1, et l’opérateur de convolution associé à un noyau homogène de degré m—n et de classe Cm,μ(∑n) applique continûment dans Cp+m,λ(Rn) si 0 < λ < μ ≤ 1.

Soit X un groupe abélien localement compact, et on désigne par dx sa mesure de Haar. On notera respectivement CK et MK les ensembles des fonctions finies, continues dans X, à support compact, et des fonctions mesurables, bornées dans X, à valeurs réelles et à support compact. (resp. ) est le sous-ensemble des fonctions non-négatives de CK (resp. MK).

Let f(z) be a k-valued algebroid function in |z| <. ∞ and

(1) F(z, f) ≡ A0(z)fk + A1(z)fk-1 + · · · Ak(z) = 0

be its defining equation such that the coefficients Ai(z) (i = 0, 1, · · ·, k) are entire functions without any common zero and the left hand side is irreducible.

The notions of tame and wild ramification lead us to make the following definition.

Definition. The quotient field extension of an extension of discrete rank one valuation rings is said to be fiercely ramified if the residue class field extension has a nontrivial inseparable part.

We should like to study three dimensional algebraic tori in the same way as Voskresenskiĭ does in [14] and [15]. To do so, it is necessary to determine all finite subgroups of GL(3,Z) up to conjugacy.