An order type α is said to be weakly homogeneous (ℵ0 homogeneous) if for any x1 < x2 and y1 < y2 there exists an order preserving bijection f on α such that f(xi)= y i for i = 1, 2. The reverse of an order type a is denoted, as usual, by α*. We say that α is order invertible if α*≤α. J. Q. Longyear [5] has asked whether for a weakly homogeneous order type α such that no (non-trivial) interval of α is order invertible we may deduce that every interval of α contains a copy of ηω1 or (ηω1)*.

The twisted wreath product of two groups was first defined by B. H. Neumann ([1]) who used this construction to present a group-theoretic proof of a theorem due to Auslander and Lyndon. In this paper we present a complete characterization of the direct product decompositions of a restricted twisted wreath product of two groups A and B provided this product is not simply a semi-direct product of A by B.

Let Fn be the free group on {ai:i ∊ ℤ n} where the set of congruence classes mod n is used as an index set for the generators. The permutation (1, 2, 3, …, n) of ℤn induces an automorphism θ of Fn by permuting the subscripts of the generators. Suppose w is a word in Fn and let N(w) denote the normal closure of {wθi-1:l ≤i≤n}. Define the group Gn(w) by Gn(w)=Fn/N(w) and call wdi-1=l the relation (i) of Gn(w).

Throughout, topological spaces are assumed to be completely regular. C(X) (resp. C*(X)) will denote the ring of all (resp. all bounded) continuous real-valued functions. βX will denote the Stone-Cech compactification of X. In [2], Nel and Riorden defined C≠(X) to be the set of all f ∊ C(X) such that M(f) is real in the residue class ring C(X)/M for every maximal ideal M in C(X). C≠(X) is a subalgebra as well as a sublattice of C*(X).

The existence and uniqueness of strong global solutions of initial-boundary value problems for the quasilinear equation utt—∂σi(uxi)/∂xi—ΔNut= f is established for functions σi(ξ), i = 1, …, N, satisfying: σi,(ξ) ∊ C1(-∞, ∞), σi(0) = 0 and for some constant K0.

Let G be a finite group of automorphisms of the ring R, and let RG denote the ring of fixed points of G in R; that is, RG={x∊R|Xg = x,∀∊ G}. Let |G| denote the order of G. In this note, we prove the following:

Theorem.Assume that R has no nilpotent ideals and no |G|-torsion. Then if RG is semi-simple Artinian, R is semi-simple Artinian.

The following theorem is a well-known tool in the study of measurable functions:

Theorem. Let (M; μ) be a finite measure space and let (xn) be a sequence of functions in L1(M; μ) so that xn→0 in the norm of L1(M; μ). Then there is a sub-sequence so that xnk →0 pointwise almost everywhere on M.

It is well-known that odd dimensional spheres carry a normal contact structure [6], Blair, Ludden and Yano [2] have recently studied a more general structure whose non-trivial example is an even dimensional sphere. Recently, the present author introduced the notion of almost contingent structures [8] with a view to develop a unified theory of various existing structures on a differentiable manifold. It is the purpose of this paper to show that even as well as odd dimensional spheres carry an almost contingent structure. In the sequel, each manifold introduced is C∞, arcwise connected and satisfies the second axiom of countability.

The object of this paper is to consider the relationships between matrix rings and rings having classical quotient rings which are quasi-Frobenius X (QFX) rings. The main result of this paper is Theorem 12, which shows that if S is a ring with a QFX right classical quotient ring T, then T is isomorphic to a direct sum of a finite number of matrix rings over local rings Ui, while S is almost a direct sum of matrix rings over rings Ci, the Ui being right classical quotient rings of the Ci.

The aim of this paper is to generalize the main results of [1] to GLn(B) by means of proofs which are more conceptual and less computational. In addition, by means of the Stone space we will obtain results which are new even for the case n = 2. Finally we shall make some remarks of a categorical nature.

The author is especially interested in the subject because of the overlap here of many areas of mathematics. Concepts from topology, model theory, and category theory are all relevant.

The class of all combinatorial geometries of rank three shall coincide with the class of all pairs (V, S) such that V is a set and S is a collection of non-empty subsets of V such that each pair of distinct elements of V belong to exactly one member of S. (See [3].)

Consider a combinatorial geometry (V, S) of rank three.

This paper develops another canonical form for (0, 1)-matrices which may be used in the same spirit as the nearly decomposable matrix [5] or the k-nearly decomposable matrix [1], This form is intrinsic in each fully indecomposable matrix and does not require the replacement of any of its non-zero entries by 0's.

We are well-acquainted with the theorem about sequences which states that, the existence of

(1)

is sufficient to imply limk→∞ak = 0. Partially out of a growing interest in the theory of regularly varying sequences ([1]), and probably as an interesting problem, in and of itself, some mathematicians have tried to find conditions weaker than (1) that would guarantee limk→∞ak = 0.

Let R be an associative ring and denote by the injective hull of the right module RR. If can be endowed with a ring multiplication which extends the existing module multiplication, we say that is a ring and the statement that R is a ring will always mean in this sense.

It is known that is a regular ring (in the sense of von Neumann) if and only if the singular ideal of R is zero.

W-groups are treated in the more general context of near-ring modules. It is shown that they share properties of d.g. modules with respect to central series. In particular, results of Magomaev are obtained and used to show that the group-theoretic and module-theoretic central series coincide for W-groups.

Serfling (1968) has considered a central limit theorem in which assumptions are made concerning the expectation of variables conditioned on their distant predecessors. Dvoretsky (1972, theorem 5.3) has continued this investigation. Serfling showed that both martingales and φ-mixing sequences satisfied his conditions, and Dvoretsky extended this to Strong mixing sequences of random variables.

Let α be an open arc on the unit circle

and for z=reiθ, 0 ≤ r < 1, let

(1)

The function ω(z; α) is called the harmonic measure of the arc α with respect to the unit disc, (Nevanlinna 2); it is harmonic and bounded in the unit disc and possesses (Fatou) boundary values 1 and 0 at interior points of α and the complementary arc β respectively.

We introduce almost polar-dense lattices and prove that the generalized interval topology of an almost polar-dense, modular lattice is equivalent to its interval topology. Furthermore, for totally ordered sets, the converse holds: if the generalized interval topology is the interval topology, then the set is almost polar-dense.

The set of projections in the BAER*-semigroup of hemimorphisms on an orthomodular lattice L can be partially ordered such that the subset of closed projections becomes an orthocomplemented lattice isomorphic to the underlying lattice L. The set of closed projections is identical with the set of Sasaki-projections on L (Foulis [2]). Another interesting class of (in general nonclosed) projections, first investigated by Janowitz [4], are the symmetric closure operators. They map onto orthomodular sublattices where Sasaki-projections map onto segments of the lattice L.

We consider here the nonlinear differential equation

(1.1)

where x ∊l= [a, ∞). We will make the following assumptions

1. (A) f is continuous on [a, ∞) × Rn,

2. (B) solutions of initial value problems (I.V.P.'s) are unique and extend throughout [a, ∞).

3. (C)