If a locally compact group G acts as a measure preserving transformation group on a Lebesgue space X, then there is a naturally induced unitary representation of G on L2(X), and one can study the action on X by means of this representation. The situation in which the representation has discrete spectrum (i.e., is the direct sum of finite dimensional representations) and the action is ergodic was examined by von Neumann and Halmos when G is the integers or the real line [7], and by Mackey for general non-abelian G [10].

The following is our main result.

(A) THEOREM. Let (M, g) be a closed connected Einstein space, n = dim M ≧ 2 (with constant scalar curvature R). Let K0 be the lower bound of the sectional curvature. Then either (M, g) is isometrically diffeomorphic to a sphere and the first nonzero eigenvalue ƛ1of the Laplacian fulfils

Let p be a prime. The algebraic K-theory of Z/p2 is unknown. However it is easy to show that Ki(Z/p2) is finite if i > 0 and that it differs only in its K-torsion from Ki(Z/p) which was computed in [2]. To proceed further one surely needs the mod p (co-)homology of GLZ/p2. There is an exact sequence

Let D ⊂⊂ Cn be a bounded domain with smooth boundary ∂D, and let F be a bounded holomorphic function on D. A generalization of the classical theorem of Fatou says that the set E of points on ∂D at which F fails to have nontangential limits satisfies the condition σ (E) = 0, where a denotes surface area measure. We show in the present paper that this result remains true when σ is replaced by 1-dimensional Lebesgue measure on certain smooth curves γ in ∂D. The condition that γ must satisfy is that its tangents avoid certain directions.

For partially ordered sets X and F let Yx denote the set of all order-preserving maps of X to Y partially ordered by f ≦ g if and only if f(x)≦ g (x) for each x ∈ X [1; 4; 6]. If X is unordered then Yx is the usual direct product of partially ordered sets, while if both X and Y are finite unordered sets then Yx is the commonplace exponent of cardinal numbers. This generalized exponentiation has an important vindication especially for those partially ordered sets that are lattices.

Let K be a field complete for a proper valuation (absolute value) v. It is classic that a finite-dimensional K-vector space E admits a unique Hausdorff topology making it a topological K-vector space, and that that topology is the “cartesian product topology” in the sense that for any basis c1 …, cn of E, is a topological isomorphism from Kn to E [1, Chap. I, § 2, no. 3; 2, Chap. VI, § 5, no. 2]. It follows readily that any multilinear mapping from Em to a Hausdorff topological K-vector space is continuous. In particular, any multiplication on E making it a K-algebra is continuous in both variables. If for some such multiplication E is a field extension of K, then by valuation theory the unique Hausdorff topology of E is given by a valuation (absolute value) extending v.

Let π: X’ —” X be a double cover of 2n-dimensional Poincaré duality (PD) spaces. The double cover is a fibering so it is classified by a map f: X → RP1+1(l ≫ n). If the homotopy class of f contains a representative which is Poincaré transverse [5] to RPl ⊂ RPl+1, we say that w is Poincarésplit-table.

Blocking sets in projective spaces have been of interest for quite some time, having applications to game theory (see [6; 7]) as well as finite nets and partial spreads (see [5]). In [4] Bruen showed that if B is a blocking set in a projective plane of order n, then

The classical Pontrjagin-van Kampen structure theorem states that any locally compact abelian (LCA) group G can be written as the direct product of a vector group Rm (where R denotes the additive group of real numbers with the usual topology, and m is a non-negative integer) and an LCA group H which contains a compact open subgroup. This important theorem, which van Kampen deduced from the work of Pontrjagin, was first stated and proved in [5, p. 461].

Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F-representation of G of smallest possible degree often cannot be lifted to an ordinary representation of G, though it can of course be lifted to an ordinary representation of some central extension of G. It is a natural question to ask whether by considering non-central extensions, it is possible in some cases to decrease the smallest degree of a faithful projective representation.

The genesis of this paper is the following well known result in field theory: Let R denote a field of characteristic p ≠ 0, and let denote a subfield of R such that for some e sufficiently large. Then R is isomorphic to the tensor product (over ) of primitive extensions of if and only if there exists a finite set Γ of -higher derivations on R such that is the field of constants of Γ. A proof of this theorem can be found in [6].

Suppose throughout that c is a fixed positive integer, that

and that

Let Ω C Cn be a bounded domain; let H∞ (Ω) be the uniform algebra of bounded analytic functions on 12; and let ∑ (Ω) be the maximal ideal space of H∞ (Ω). In the weak-* topology of (H∞ (Ω))*, ∑ (Ω) is a compact Hausdorf space in which Ω is embedded in a natural fashion, so that to every g ∈ H∞ (Ω) there corresponds the Gelfand transform ĝ ∈ C(∑ (Ω)); ĝ|Ω = g.

Definition. [1] A base B for the topology of a space S is centered provided every perfectly decreasing filterbase F in B is regular and either F is free or F converges. A centered base which contains no free perfectly decreasing filterbase is said to be complete.

Let E be an n-dimensional real affine space, V its vector space of translations and A(E) the affine group of E. Suppose that (. , .) is a nondegenerate symmetric bilinear form on F of signature (n — 1, 1), O(V) its orthogonal group and S(V) its group of similarities.

Let I be an interval, and let for 1 ≦ j ≦ I < ∞ be abutted subintervals such that . Let τj be a linear differential expression defined on Ij. In this paper we study densely defined operators associated with

(0.1)

In the recent paper [8] Phillips has shown that for each prime p there are 2ℵ0 non-isomorphic 2-generatecl p-groups. This same result was obtained independently by S. Jeanes and J. S, Wilson (unpublished) who show that the groups constructed in [1] have 2ℵ0 non-isomorphic images. The groups in both of these proofs all have infinite exponent. In this paper we show that, for large enough primes p, there are 2ℵ0 non-isomorphic 2-generated groups of exponent p2.

This paper originated in an attempt to carry over the results of [3] from the case of a field of characteristic different from two to that of semilocal rings. To carry this out, we reverse the point of view of [3] and do assume a full knowledge of the theory of Witt rings of classes of nondegenerate symmetric bilinear forms over semilocal rings as given, for example, in [10; 11]. It turns out that the rings WT of [3] are just the residue class rings of W(C), the Witt ring of a semilocal ring C, modulo certain intersections of prime ideals.

Volume decreasing properties of harmonic mappings of space forms were investigated by S. S. Chern and S. I. Goldberg [3] and the author. In a previous paper [6], a step toward generalization of the results was made proving the following theorem:

Theorem. Let ƒ: M —> N be a harmonic mapping of n-dimensional Riemannian manifolds, with C ≦ 0. Suppose the scalar curvature of M is not less than — S, and the Ricci curvature of N is not greater than —S/n, where S ≧ 0 and S > 0 are constants. Then, if u has a maximum on M,

i.e. ƒ is volume decreasing up to a constant.

Let Mn(C) be the vector space of all w-square complex matrices. Denote by (• , •) the standard inner product in the space Cn of complex n-tuples. For a matrix A ∈ Mn(C) and an n-tuple c = (c1,… , cn) ∈ Cn, define the c-numerical range of A to be the set

(1)