An integral domain D is said to be an essential domain if D is an intersection of valuation rings that are localizations of D. D is called a v-multiplication ring if the finite divisorial ideals of D form a group. Griffin has shown [2, pp. 717-718] that every v-multiplication ring is an essential domain, and that an essential domain having a defining family of valuation rings {Vα} which is of finite character (i.e., every nonzero element of D is a non-unit in at most finitely many Vα) is necessarily a v-multiplication ring. It is noted in [4, p. 860] that any localization of a v-multiplication ring is again a v-multiplication ring. In this vein, Joe Mott has asked whether a localization of an essential domain must again be an essential domain. An example of an essential domain that is not a v-multiplication ring is given in [4], however it can be seen for this example that each localization is again an essential domain [6].

If G is a polycyclic group and k an absolute field then every irreducible kG-module is finite dimensional [10], while if k is nonabsolute every irreducible module is finite dimensional if and only if G is abelian-by-finite [3]. However something more can be said about the infinite dimensional irreducible modules. For example P. Hall showed that if G is a finitely generated nilpotent group and V an irreducible kG-module, then the image of kZ in EndkGV is algebraic over k [3]. Here Z = Z(G) denotes the centre of G. It follows that the restriction Vz of V to Z is generated by finite dimensional kZ-modules. In this paper we prove a generalization of this result to polycyclic group algebras.

We introduce some terminology.

Throughout this paper “space” means “T1 topological space.“

1. The concept of an ortho-base was introduced by W. F. Lindgren and P. J. Nyikos.

Definition 1. A base of a space X is called an ortho-base provided that for each subcollection either is open or is a local base of a point x ∊ X [17].

Ortho-bases are related to interior-preserving collections which have been known for some time.

Definition 2. A collection of open sets of a space X is called interior-preserving provided that the intersection of any subcollection is open. A space X is called orthocompact provided that each open cover has an open interior-preserving refinement.

It was proved in [17], in particular, that each space with an ortho-base is orthocompact, and each orthocompact developable space (which is the same as a non-archimedean quasi-metrizable developable space [4]) has an ortho-base.

Let and denote infinite dimensional Hilbert spaces and let denote the space of all bounded linear operators from to . For A in and B in , let τAB denote the operator on defined by τAB(X) = AX – XB. The purpose of this note is to characterize the semi-Fredholm domain of τAB (Corollary 3.16). Section 3 also contains formulas for ind(τAB – λ). These results depend in part on a decomposition theorem for Hilbert space operators corresponding to certain “singular points” of the semi-Fredholm domain (Theorem 2.2). Section 4 contains a particularly simple formula for ind(τAB – λ) (in terms of spectral and algebraic invariants of A and B) for the case when τAB – λ is Fredholm (Theorem 4.2). This result is used to prove that (τBA) = –ind(τAB) (Corollary 4.3). We also prove that when A and B are bi-quasi-triangular, then the semi-Fredholm domain of τAB contains no points corresponding to nonzero indices.

The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [1, 2, 9, 10, 13-16]. Standard defining relation [7, 18] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function:

with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials Rn(α,β)(x), normalized to unity at x = 1, are more convenient to use:

(1.1)

Roughly speaking, the linearization problem consists of finding the coefficients g(k, m, n; α,β) in the expansion

(1.2)

A chaque c.w. complexe X, 1-connexe, de type fini, est associé une algèbre différentielle graduée commutative (ΛZ, d) qui est un modèle dans le sens de Sullivan [10] et représente donc le type d'homotopie rationnelle de X.

ΛZ est munie d'une seconde graduation Z = ⊕q≧0Zq. La différentielle d se décompose alors sous la forme et d0(Zp) ⊂ (Λ2Z)p–1 et di(Zp) ⊂ Zp–i, i ≧ 1. En particulier d0 est purement quadratique tandis que di est du premier degré, i ≧ 1.

L'algèbre ΛZ et la partie quadratique d0 ne dépendent que des nombres de Betti de X. Les différents types d'homotopie rationnelle de nombres de Betti donnés sont donc paramétrés par les déformations di. La partie d1 exprime la structure multiplicative de l'algèbre de cohomologie rationnelle de X, les parties d2, d3, d4. … les structures algébriques plus complexes du type d'homotopie rationnelle.

Let o be a quasi semilocal semihereditary ring, i.e., o is a commutative ring with 1 which has finitely many maximal ideals {Ai|i ∊ I} and the localization oAi at any maximal ideal Ai is a valuation ring. We assume 2 is a unit in o. Furthermore * denotes an involution on o with the property that there exists a unit θ in o such that θ* = –θ. V is an n-ary free module over o with f : V × V → o a λ-Hermitian form. Thus λ is a fixed element of o with λλ* = 1 and f is a sesquilinear form satisfying f(x, y)* = λf(y, x) for all x, y in V. Assume the form is nonsingular; that is, the mapping M → Hom (M, A) given by x → f( , x) is an isomorphism. In this paper we shall write f(x, y) = xy for x, y in V.

A lattice L is finitely presented (or presentable) if and only if it can be described with finitely many generators and finitely many relations. Equivalently, L is the lattice freely generated by a finite partial lattice A, in notation, L = F(A). (For more detail, see Section 1.5 of [6].)

It is an old “conjecture” of lattice theory that in a finitely presented (or presentable) lattice the elements behave “freely” once we get far enough from the generators.

In this paper we prove a structure theorem that could be said to verify this conjecture.

THEOREM 1. Let L be a finitely presentable lattice. Then there exists a congruence relation θ such that L/θ is finite and each congruence class is embeddable in a free lattice.

1. Introduction. The reduction of a matrix to its Jordan normal form is an unstable operation in that both the normal form itself and the reducing mapping depend discontinuously on the elements of the original matrix. For example, the matrix trivially reduces to itself in Jordan form, but there are arbitrarily small perturbations of this matrix that reduce to the form which is certainly not a small perturbation of the original matrix, and moreover the reducing mapping is not a small perturbation of the identity. In [1], Arnol'd derives the simplest possible normal forms to which families of matrices may be linearly reduced in a ‘stable’ manner. In this paper, we consider a ‘topological' version of the problem, using the classification of matrices up to topological conjugacy given in [8] and the classification of linear dynamical systems up to orbital equivalence given in [9].

Although the theory of complex Banach algebras is by now classical, the first systematic exposition of the theory of real Banach algebras was given by Ingelstam [5] as late as 1965. More recently, further attention to real Banach algebras was paid in 1970 [1], where, among other things, the (real) standard algebras on finite open Klein surfaces were introduced. Generalizing these considerations, real uniform algebras were studied in [7] and [6].

In the present paper, an attempt is made to develop the theory of real function algebras (see Section 1 for the definition) along the lines of the complex function algebras. Although the real function algebras are not structurally different from the real uniform algebras introduced in [7], they are easier to deal with since their elements are actually (complex-valued) functions.

If A is a group acting on a set X and x ∈ X, we denote the stabilizer of x in A by CA(x) and let Γ(x) be the set of elements of X fixed by CA(x). We shall say the action of A is partitive if the distinct subsets Γ(x), x ∈ X, partition X. A special example of this phenomenon is the case of a semiregular action (when CA (x) = 1 for every x ∈ X so the induced partition is a trivial one). Our concern here is with the case that A is a group of automorphisms of a finite group G and X = G#, the set of non-identity elements of G. We shall prove that if A is nilpotent, then except in a very restricted situation, partitivity implies semiregularity.

R. Matsuda has shown that a group ring is a Krull domain if and only if the coefficient ring is a Krull domain and the group is a torsion-free abelian group satisfying the ascending chain condition (ace) on cyclic subgroups [6]. D. F. Anderson has used this to obtain a partial determination of when a semigroup ring is a Krull domain, and under certain circumstances to describe the divisor class group of such a ring ([1], [2]). Using some of Anderson's techniques, but taking a different approach, we arrive at a complete answer of a different nature to these questions. We call a semigroup satisfying the major new conditions arising a Krull semigroup, and define its divisor class group.

In particular, every abelian group is the divisor class group of such a ring, and it follows that every abelian group is the divisor class group of a quasi-local ring, which seems to be a new result.

In the first part of the paper, given a continuous map f from a Hausdorff topological space X onto a Hausdorff topological space Y, we consider the reciprocal map f* from Y into the collection of closed subsets of X, which maps y ∈ Y to . is endowed with the pseudotopological structure of convergence of closed sets. We will use the filter description of this convergence, as defined by Choquet and Gähler [2], [5], which is equivalent to the “topological convergence” of sets as introduced by Frolík and Mrówka [4], [10]. These notions in fact generalize the convergence of sequences of sets defined by Hausdorff [6]. We show that the continuity of f* is equivalent to the openness of f. On f*(Y), the set of fibers of f, we consider the pseudotopological structure induced by the closed convergence on .

It is a well known and useful fact [4] that every (partially) ordered set P has a linear extension L (that is, a totally ordered set (chain) on the same underlying set as P and satisfying a ≦ b in L whenever a ≦ b in P). It is just as well known that an ordered set P can be embedded in an ordered set P′ which, in turn, has a complete linear extension L′ (that is, a linear extension in which every subset has both a supremum and an infimum); just take L′ to be the “completion by cuts” of L. However, an arbitrary ordered set P need not, itself, have a complete linear extension (for example, if P is the chain of integers or, for that matter, if P is any noncomplete chain). It is natural to ask which ordered sets have a complete linear extension?

Two aspects, and their connections, of the problem of enumerating solutions to certain systems of congruences are explored in this paper. Although slightly more general cases are mentioned, the basic object of study is a system of diagonal equations

where d1, d2, …, dt are positive integers and the coefficient matrix [aij] has entries from Fp = GF(p), and for which solutions x = (x1, x2 …, xt) ∈ Fpt are sought. Speaking loosely, such a system usually has approximately pt–n solutions in the sense that the difference between pt–n and the correct value becomes small in comparison with pt–n as p becomes large. A parameter is introduced which measures the extent to which the matrix [aij] is non-singular over Fp. The effect of this parameter on the size of the error term (the difference between the number of solutions to (1) and pt–n) is the first aspect of the enumeration problem to be treated.

Let Δ denote the unit disk in the complex plane C. The space BMO has been extensively studied by many authors (see [3] for a good exposition of this topic). Recently, the subspace BMOA (Δ) has become a topic of interest. An analytic function f, in the Hardy class H2(A), belongs to BMOA (Δ) if

(1)

where

It is known (see [3, p. 96]) that (1) is equivalent to

(2)

In 1889, A. A. Markov proved the following inequality:

INEQUALITY 1. (Markov [4]). If pn is any algebraic polynomial of degree at most n then

where ‖ ‖A denotes the supremum norm on A.

In 1912, S. N. Bernstein established

INEQUALITY 2. (Bernstein [2]). If pn is any algebraic polynomial of degree at most n then

for x ∈ (a, b).

In this paper we extend these inequalities to sets of the form [a, b] ∪ [c, d]. Let Πn denote the set of algebraic polynomials with real coefficients of degree at most n.

THEOREM 1. Let a < b ≦ c < d and let pn ∈ Πn. Then

for x ∈ (a, b).

Lorsque est un composé de deux corps quadratiques réels, nous pouvons calculer explicitement un système fondamental d'unités de L4 (cf. [3], [2] et [5]), en considérant les unités fondamentales η2, δ2 et ∈2 des trois sous-corps quadratiques L2, F2 et K2 de L4.

Let G be an amenable locally compact group. We will use the terminology of [3] and denote by VN(G) the Von Neumann algebra of the regular representation and by A(G) its predual, which is the algebra of the coefficients of the regular representation. The Von Neumann algebra VN(G) is, in a natural fashion, a module with respect to A(G) [3].

The algebra of bounded linear operators on VN(G), which commute with the action of A(G), has been studied in [6] and in [1]. If UCB(Ĝ) is the space of the elements of VN(G) of the form vT, for some v in A(G) and some T in VN(G) (see for instance [4]), in [6] and in [1] it is proved that, for any amenable locally compact group there exists an isometric bijection between and UCB(Ĝ)*.

In [8] Jha raised the following problem.

(*) Let Γ be a spread whose components are subspaces of V2n(GF(q)). Suppose G ≦ Aut Γ leaves a set of q + 1 components invariant while acting transitively on Γ\Δ.

Find the possibilities for Γ or, more generally, the possibilities for (G, Γ, n, q).

Many special cases of (*) have been settled. For instance, Cohen et al [1] have shown that if G fixes two non-zero points of V, that do not both lie in the same component of Γ, then Γ is the spread associated with either a Hall plane or the Lorimer-Rahilly plane of order 16 (LR-16) [14], [18].

Another such result is given in [8]; there it is shown that if q is a prime number and G is a one-dimensional projective unimodular group then Γ is the spread associated with one of the following translation planes:

• (1) the Desarguesian planes of order 4, 8, or 9;

• (2) the nearfield plane of order 9;

• (3) LR-16;

• (4) the translation plane JW-16, obtained by transposing the slope maps of LR-16 [19].