The Banach space of bounded continuous real or complexvalued functions on a topological space X is denoted C(X). An averaging operator for an onto continuous function ϕ : X → Y is a bounded linear projection of C(X) onto the subspace ﹛ƒ ∈ C(X) : f is constant on each set ϕ -1(y) for y ∈ Y﹜. The projection constant p(ϕ) for an onto continuous map ϕ is the lower bound for the norms of all averaging operators for ϕ ﹛p(ϕ) = ∞ if there is no averaging operator for ϕ).

For certain classes of singular symmetric differential operators L of order 2n, this paper considers the problem of determining sufficient conditions for L to be of limit point type or of limit circle type. The operator discussed here is defined by

In the last few years many results have appeared which deal with questions of how various algebraic properties of the symmetric elements of a ring with involution, or the subring they generate, affect the structure of the whole ring. If the ring has an identity, similar questions may be posed by making assumptions about the symmetric units or subgroup they generate. Little seems to be known about the special units which exist in rings with involution, although several questions of importance have existed for some time. For example, given a simple ring with appropriate additional assumptions, is the unitary group essentially simple? Also, what can be said about the structure of subspaces invariant under conjugation by all unitary or symmetric units (see [7])?

We assume familiarity with the notation and contents of Conway [2] and Edge [3]. That the Mathieu group is a subgroup of the simple group PSU (6, 22) appears to have been first recognized by Conway and is consequent upon his identification of ·222 with PSU (6, 22). Although we know of no proof of this identification in the literature, several proofs exist in the folklore of the subject: for example, N. Patterson showed one of the authors a proof that depends on ·222 being a Fischer group, hence on consideration of order, isomorphic to PSU (6, 22). There is another proof which relies on McLaughlin's work on rank three groups.

In this paper we consider the problem of extending the Weierstrass division theorem to quasianalytic local rings of germs of functions of k real variables which properly contain the local ring of germs of analytic functions. After some background material (§ 2) and some technical preliminaries (§ 3), we show by examples that the so-called generic division theorem fails in such rings if k ≧ 1 and that the Weierstrass division theorem fails in such rings if k ≧ 2 (§ 4).

Let R be a ring with unit element and G a finite group. We denote by RG the group ring of the group G over R and by U(RG) the group of units of this group ring.

The study of the nilpotency of U(RG) has been the subject of several papers.

Soit Q(x) une fonction positive paire continûment différentiable pour — ∞ < x < ∞ telle que xQ’ (x) soit croissante pour x > 0 et que Q‘(x) → ∞ lorsque x → co. Désignons par Qn l'unique racine positive de l'équation xQ‘(x) = n. L'approximation polynomiale des fonctions continues relativement au poids e-Q(x) a été étudiée par de nombreux mathématiciens. Le résultat suivant est une conséquence d'un théorème de l'un des auteurs (voir Freud [1]): si F(x) est une fonction réelle continue pour —∞ < x < ∞ qui satisfait une condition de Lipschitz d'ordre a où 0 < a < 1, il existe une suite de polynômes ﹛Pn(x)﹜ où deg Pn ≦ n telle que

In 1897 E. Borel proved a general theorem which implied as a special case the following result equivalent to his celebrated generalization of Picard's theorem [2]: If f1 … ,fm are entire functions such that for each, C then the functions exp f1, … , exp f/m are linearly independent over C. In 1929 R. Nevanlinna [6] extended Borel's theorem to consider arbitrary C-linearly independent meromorphic functions < ϕi, … , < ϕm satisfying < ϕ1 + … + ϕm = 1.

It has been known for some time but does not seem to be anywhere in the literature that the variety of all ortholattices is generated by its finite members (see (4.2) of this paper). This is well known to imply that the word problem for free ortholattices is solvable. On the other hand, it is also known that the solution obtained this way is of no practical use. The main purpose of this paper is to present a workable solution.

The object of this paper is to prove the following theorem, a special case of which was previously explored in [1].

THEOREM. Let R be any associative ring with the property that

(†) for each x,y ∊ R, there exist integers m,n ≧ I for which xy = ymxn.

Our method using CH is a blend of two earlier constructions (Hajnal-Juhász [2] and Ostaszewski [4]) of hereditarily separable (HS), regular, non-Lindelöf, first countable spaces. [4] produces a much better space than ours in § 1 ; it has all of our properties except that it is not realcompact (which is probably more interesting), and it is countably compact as well; however, the construction works only under ◇, which implies the continuum hypothesis (CH) but is not equivalent to it.

A partially ordered set P has the fixed point property if every order-preserving map f : P → P has a fixed point, i.e. there exists x ∊ P such that f(x) = x. A. Tarski's classical result (see [4]), that every complete lattice has the fixed point property, is based on the following two properties of a complete lattice P:

• (A) For every order-preserving map f : P → P there exists x ∊ P such that x ≦ f(x).

• (B) Suprema of subsets of P exist; in particular, the supremum of the set {x|x ≦ f(x)} ⊂ P exists.

Definition. An orthogonal design in order n and of type (u1, … , us) on the commuting variables x1, . . . , xs is an n X n matrix, X, with entries from the set ﹛0, ±x1, … , ±xs﹜ such that

Suppose (P, △) is an undirected graph without loops or multiple edges. We will denote by △ (x) the vertices adjacent to x and . Let (G, P) be a transitive permutation representation of a group G in a, set P, and Δ be a non-trivial self-paired (i.e. symmetric) orbit for the action of G on P X P. We identify △ with the set of all two subsets ﹛x, y﹜ with (x, y) in △. Then we have a graph (P, Δ) with G ≦ Aut (P, △), transitive on both P and △.

G. T. Whyburn, in 1926, began the development of cyclic element theory for Peano continua. This theory proved fruitful in the study of Peano spaces and a comprehensive development of the theory for metric spaces was presented in [6]. An excellent history of the theory is to be found in [4]. In [7] and [5] the generalization of cyclic element theory to more general spaces was begun. However, in each of these papers only basic definitions were set forth and fundamental results obtained. In this paper, we concern ourselves primarily with connected and locally connected Hausdorff spaces, developing the cyclic element theory initiated in [7] and demonstrating that the theory has many of the applications to connected and locally connected Hausdorff spaces that the classical theory has to Peano spaces.

Let S be a subset of some linear topological space. The set S is said to be m-convex, m ≧ 2, if and only if for every m-member subset of S, at least one of the line segments determined by these points lies in S. A point x in S is said to be a point of local convexity of S if and only if there is some neighborhood N of x such that if y, z Є N ⌒ S, then [y, z] ⊆ S. If S fails to be locally convex at some point a in S, then q is called a point of local nonconvexity (lnc point) of S.

Suppose that f(x) is integrable L in every finite interval [0, X], and that δ > 0. Define

If A is a (linear) transformation acting on a (finitedimensional, non-zero, complex) Hilbert space H the family of (linear) subspaces of H which are invariant under A is denoted by Lat A. The family of subspaces of H which are invariant under every transformation commuting with A is denoted by Hyperlat A. Since A commutes with itself we have Hyperlat A ⊆ Lat A. Set-theoretic inclusion is an obvious partial order on both these families of subspaces. With this partial order each is a complete lattice; joins being (linear) spans and meets being set-theoretic intersections. Also, each has H as greatest element and the zero subspace (0) as least element. With this lattice structure being understood, Lat A (respectively Hyperlat A) is called the lattice of invariant (respectively, hyper invariant) subspaces of A.

A remarkable theorem of Herstein [1, Theorem 2] of which we have made several uses states: If R is a semiprime ring of characteristic different from 2 and if U is both a Lie ideal and a subring of R then either U ⊂ Z (the centre of R) or U contains a nonzero ideal of R. In a recent paper [3] Herstein extends the above mentioned result significantly and has proved that if R is a semiprime ring of characteristic different from 2 and V is an additive subgroup of R such that [V, U] ⊂ V, where U is a Lie ideal of R, then either [V, U] = 0 or V ⊃ [M, R] ≠ 0 where M is an ideal of R. In this paper our object is to prove the following.

Let X be a σ-finite measure space and let Tk, k any integer, be a group of positive linear transformations in Lp(X) such that

with C independent of / and k. From now on / will be a positive function in Lp(X) and we will use the following notation: