Harmonic functions with certain boundedness properties on a given open Riemann surface R attain their maxima and minima on the harmonic boundary ΔB of R. The significance of such maximum principles lies in the fact that the classification theory of Riemann surfaces related to harmonic functions reduces to a study of topological properties of Δ(cf. [11; 8; 3; 12].

For the corresponding problem in higher dimensions we shall first show that the complement of ΔR with respect to the Royden boundary ΓR of a Riemannian N-space R is harmonically negligible: given any non-empty compact subset E of ΓR – ΔR there exists an Evans superharmonic function v, i.e., a positive continuous function on R* = R ∪ ΓR, superharmonic on R, with v = 0 on ΔR, v ≡ ∞ on E, and with a finite Dirichlet integral over R.

Suppose that

(1.1)

and define the bounded self-adjoint operators H and J on the Hilbert space L2(0, 1) by

(1.2)

the integral being a Cauchy principal value

It is seen that

(1.3)

or, equivalently,

(1.4)

Since (Cƒ, ƒ) = π–1|(ƒ, ϕ)|2 ≧ 0, A is semi-normal. (For a discussion of such operators, see [4].)

Let A denote the Banach algebra of functions analytic in the open unit disc D and continuous in . If f and its first m derivatives belong to A, then the boundary function f(eiθ) belongs to Cm(∂D). The space Am of all such functions is a Banach algebra with the topology induced by Cm(∂D). If all the derivatives of/ belong to A, then the boundary function belongs to C∞(∂D), and the space A∞ all such functions is a topological algebra with the topology induced by C∞(∂D). In this paper we determine the structure of the closed ideals of A∞ (Theorem 5.3).

Beurling and Rudin (see e.g. [7, pp. 82-89; 10]) have characterized the closed ideals of A, and their solution suggests a possible structure for the closed ideals of A∞.

It has been pointed out repeatedly in the literature that the methods of integral geometry (a mathematical theory founded by Wilhelm Blaschke and considerably extended by several mathematicians) provide highly suitable means for the solution of problems concerning “geometrical probabilities“ [2; 6; 12; 15]. The possibilities for the application of these integral geometric results to the evaluation of probabilities, satisfying certain conditions of invariance with respect to a group of transformations which acts on the probability space, are obviously not yet exhausted. In this article, such applications are presented. First, some concepts and notation are introduced (§1). In the next section we derive some integral geometric relations (§ 2). These results are generalizations of known systems of formulae and they are valid in the k-dimensional Euclidean space. In § 3, we determine mean-value formulae for the fundamental characteristics of point-sets, generated by randomly placed convex bodies.

Representations of harmonic functions by means of integrals taken over the harmonic boundary ΔR of a Riemann surface R enable one to study the classification theory of Riemann surfaces in terms of topological properties of ΔR (cf. [6; 4; 1; 7]). In deducing such integral representations, essential use is made of the fact that the functions in question attain their maxima and minima on ΔR.

The corresponding maximum principle in higher dimensions was discussed for bounded harmonic functions in [3]. In the present paper we consider Dirichlet-finite harmonic functions. We shall show that every such function on a subregion G of a Riemannian N-space R attains its maximum and minimum on the set , where ∂G is the relative boundary of G in R and the closures are taken in Royden's compactification R*. As an application we obtain the harmonic decomposition theorem relative to a compact subset K of R* with a smooth ∂(K ∩ R).

In this paper we settle (with a counterexample) the question raised by Clifford and Preston in [2, p. 275], concerning maximal group homomorphic images of semigroups. We also consider the question in a more general context and characterize all such examples. The notation and definitions follow [1; 2].

By a type of semigroups we mean a class of semigroups, closed under isomorphisms and containing the one-element semigroup. If S is any semigroup and is a type, then a semigroup S* is defined, in [1, p. 18], to be a maximal homomorphic image of S having type if

(i) ,

(ii) S* is a homomorphic image of S, and

(iii) whenever and T is a homomorphic image of S, then there exists a homomorphism from S* onto T.

Let {zk},0 < |zk| < 1, be a given sequence of points in the open unit disc D = {z: |z| < 1} and let E be its set of limit points on the unit circle T. In this note we consider the problem of finding conditions on the sequence {zk} which will ensure the existence of a function f analytic in D satisfying

(A)

and whose derivative f′ belongs to the Hardy class H1 or, alternatively, whose derivatives of all orders are bounded in D. We shall prove the following two theorems.

THEOREM 1. If

(1)

(2)

and

(3)

then there is a function f analytic in D which satisfies (A) and its derivative f′ belongs to H1.

Let F be the free group on n generators x1, …, Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x1, x2, …, xn) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω1, ω2, …, ωm ∈ F with m < n, and K1, …, Km are conjugacy classes of a finite group G, then the number of x ∈ Gn with ωi(x) ∈ Ki for each i, is divisible by |G|. Solomon proved this by constructing a suitable equivalence relation on Gn.

Another recent application of an unusual equivalence relation in group theory is in Brauer's paper [1], where he gives an elementary proof of the Frobenius theorem on solutions of xk = 1 in a group.

Our aim in this paper, generally stated, is to formulate an abstract concept of the notion of the sum of a numerical series. More particularly, it is a study of the type of sequence space called “sum space”. The idea of sum space arose in connection with two distinct problems.

1.1 The Köthe-Toeplitz dual of a sequence space T consists of all sequences t such that st ∈ l1 (absolutely summable sequences) for each s∈T. It is known that if cs or bs is used in place of l1, an analogous theory of duality for sequence spaces can be developed (cf. [2]). What other spaces of sequences can play a rôle analogous to l1? This problem is treated in [6].

1.2. Let {xn, fn} be a complete biorthogonal sequence in (X, X*), where X is a locally convex linear topological space and X* is its topological dual space.

In [3] Schafer has defined generalized standard rings as rings satisfying the identities

(1)

(2)

(3)

and observed that these identities imply (y,y, (x, z)) = 0 and if the characteristic is not three, (x, y, x2) = 0. Schafer determined the structure of simple, finite-dimensional generalized standard algebras of characteristic not two or three by showing that they must be either commutative, Jordan, or alternative.

Previously one of us [2] had studied accessible rings, which are defined by the identities (x,y,z) + (z,x,y) – (x,z,y) = 0 and ((w,x), y,z) = 0.

Let V be an irreducible algebraic variety of dimension > 1 defined over a field k in an affine n-space over k, and let H be the generic hyperplane defined by u0 + u1X1 + … + unXn = 0, where u0, u1, …, un are indeterminates over k. It is well known that:

(1) if V is normal over k, then V ∩ H is normal over k(u0, …, un) (see [6]), and

(2) if P is in the intersection V ∩ H, then P is absolutely simple on V ∩ H over k(u0, …, un) if and only if P is absolutely simple on V over k (see [2; 5]).

In this paper we prove:

(1′) if V is factorial over k, then V ∩ H is also factorial over k(u0, …, un) (Theorem 3), and

(2′) if P is in V ∩ H, then P is normal on V ∩ H over k(u0, …, un) if and only if P is normal on V over k (Theorem 2).

Let S be a finite set and let S1, S2, …, St be subsets of S, not necessarily distinct. Does there exist a set of distinct representatives (SDR) for S1, S2, …, St? That is, does there exist a subset {a1, a2, …, at} of S such that ai ∊ Si, 1 ≦ i ≦ t, and ai ≠ aj if i ≠ j? The following theorem of Hall [2; 6, p. 48] gives the answer.

THEOREM. The subsets S1, S2, …, St have an SDR if and only if for each s, 1 ≦ s ≦ t, |Si1 ∪ Si1 ∪ … ∪ Sis| ≧ s for each s-comhination {i1, i2, …, is} of the integers 1, 2, …, t.

(Here and below, |A| denotes the number of elements in A.)

Let G be a locally compact abelian group with Bohr compactification Ga. Then [3, Theorem 1.2] any subset F of G compact in Ga is necessarily compact in G; alternatively, any closed non-compact subset F of G has its closure F– in Ga ≠ F; hence F –\F ≠ ø. One of our aims in the present note is to give a result (Corollary 6) which asserts that F –\F has no points which are Gδs, so that F–\F is a perfect set. Another aim is to give an extension of a cited result of [3] in which commutativity and local compactness are essentially irrelevant, and to unify the proofs.

The Hewitt realcompactification vX of a completely regular Hausdorff space X has been widely investigated since its introduction by Hewitt [17]. An important open question in the theory concerns when the equality v(X × Y) = vX × vY is valid. Glicksberg [10] settled the analogous question in the parallel theory of Stone-Čech compactifications: for infinite spaces X and Y, β(X × Y) = βX × β Y if and only if the product X × Y is pseudocompact. Work of others, notably Comfort [3; 4] and Hager [13], makes it seem likely that Glicksberg's theorem has no equally specific analogue for v(X × Y) = vX × vY. In the absence of such a general result, particular instances may tend to be attacked by ad hoc techniques resulting in much duplication of effort.

In this paper, we shall construct a vector space, called the (G, σ) space, which generalizes the tensor space, the Grassman space, and the symmetric space. Then we shall determine a necessary and sufficient condition that the (G, σ) product of the vectors x1, x2, …, xn is zero.

1. Let G be a permutation group on I = {1, 2, …, n} and F, an arbitrary field. Let σ be a linear character of G, i.e., σ is a homomorphism of G into the multiplicative group F* of F.

For each i ∈ I, let Vi be a finite-dimensional vector space over F. Consider the Cartesian product W = V1 × V2 × … × Vn.

1.1. Definition. W is called a G-set if and only if Vi = Vg(i) for all i ∊ I, and for all g ∊ G.

In this paper we explore the extent to which embedding and isomorphism questions about a Noether lattice ℒ can be reduced to questions about simpler structures associated with ℒ.

In § 1, we use a variation of Dilworth's congruence approach [2] to associate a collection of semi-local Noether lattices with a given Noether lattice ℒ. We show that these semi-localizations determine ℒ to within isomorphism (Corollary 1.5); thus embedding and isomorphism questions about ℒ are largely reduced to the semi-local case.

In § 2, we consider the influence on a semi-local Noether lattice ℒ of the substructure ∂ ℒ consisting of all elements, all of whose associated primes are maximal. Here we find that if ∂ ℒ can be embedded in a semi-local Noether lattice ℒ*, then ℒ can be embedded in an extension of ℒ*.

Let Pn(α,β) be the Jacobi polynomial of degree n, order (α,β), α,β > – 1, defined by

[9, p. 67], and let Rn(α,β)(x) = Pn(α,β)(x)/Pn(αβ)(1). Then for n ≧ m,

where

Since Rn(α, β)(l) = 1, it follows that

(1)

It is known that if (the ultraspherical case) or if α = β + 1, then α = β + 1, then g(k, n, m) ≧ 0.

Let denote the variety of all groups which are extensions of a nilpotent-of-class-c group by a nilpotent-of-class-d group, and let denote the variety of all metabelian groups. The main result of this paper is the following theorem.

THEOREM. Let be a subvariety of which does not contain . Then every -group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent. In particular, a finitely generated torsion-free -group is a nilpotent-by-finite group.

This generalizes the main theorem of Ŝmel′kin [4], where the same result is proved for subvarieties of , where is the variety of abelian groups. See also Lewin and Lewin [2] for a related discussion.

The theory of gentle perturbations was introduced by Friedrichs [3] as a tool to study the perturbation theory of the absolutely continuous spectrum of a self-adjoint operator H0 and developed in an abstract form by Rejto [7; 8]. Two examples of gentle structures are well knowTn. In the first of these, the gentle operators have Hölder continuous complex or operator-valued kernels, and in the second, the kernels are Fourier transforms of L1 functions [4].

The gentle structure has traditionally been verified in the case when H0 is in its spectral representation, that is, when H0 is the simple differentiation operator. This is not the natural setting for the second example mentioned above where one should consider the simple differentiation operator in a suitable L2-space and perturbations with L1 kernels.

Dans ce travail, R désignera toujours un anneau commuta tif ayant un élément unité 1. Les R-algèbres considérées seront supposées associatives. Si A est une R-algèbre, nous supposerons toujours 1 · a = a quel que soit a ∊ A. Si B ⊂ A, nous désignerons par Ann(B) l'ensemble des r ∊ R tels que rB = {0}.

Soit A une R-algèbre commutative (avec ou sans élément unité). Nous désignerons par EA l'ensemble des éléments idempotents de A. Si l'on définit pour e, f ∊ EA,

alors EA devient un treillis distributif relativement complémenté dont 0 est le plus petit élément [2].

Nous nous intéresserons aux R-algèbre commutatives engendrées par leurs éléments idempotents.