After a review of the work of Lawvere and Tierney, it is shown that every topos may be exactly embedded in a product of topoi each with 1 as a generator, and near-exactly embedded in a power of the category of sets. Several metatheorems are then derived. Natural numbers objects are shown to be characterized by exactness properties, which yield the fact that some topoi can not be exactly embedded in powers of the category of sets, indeed that the “arithmetic” arising from a topos dominates the exactness theory. Finally, several, necessarily non-elementary, conditions are shown to imply exact embedding in powers of the category of sets.

Let G1 and G2 be two locally compact abelian groups and let 1 ≤ p ∞. We prove that G1 and G2 are isomorphic as topological groups provid∈d there exists a bipositive or isometric algebra isomorphism of M(Ap (G1)) onto M(Ap (G2)). As a consequence of this, we prove that G1 and G2 are isomorphic as topological groups provided there exists a bipositive or isometric algebra isomorphism of Ap (G1) onto Ap (G2). Similar results about the algebras L1 ∩ Lp and L1 ∩ C0 are also established.

The result of Magill concerning differentiable maps on the reals; namely, that all automorphisms of this semigroup are inner; is shown to be untrue in the case of analytic maps on the complexes. Also a characterization of the inner automorphisms of the analytic maps is given.

A sharply doubly transitive group which acts on a set of at least two elements is isomorphic to the group of affine transformations on a system S. This statement is true if S is replaced by either strong pseudo-field or pseudo-field. The additive system of a strong pseudo-field is a loop while the additive system of a pseudo-field need not be a loop. We show that any pseudo-field is either a strong pseudo-field or can be obtained from a strong pseudo-field in a nice way. Every near-field is a strong pseudo-field. The converse is an open question.

A method is present for simplifying the calculation of steady Stokes flows. It is applicable for domains which are essentially half spaces and utilizes a special vector stream potential to decouple the boundary conditions. The results are applied to estimate the effect of no-shear zones on a shear flow.

The object of this paper is to investigate under very general conditions the existence and mean-square stability of a random solution of a class of stochastic integral equations in the form

for t ≥ 0, where a random solution is a second order stochastic process {x(t; w) t ≥ 0} which satisfies the equation almost certainly. A random solution x(t; w) is defined to be stable in mean-square if E[|x(t; w)|2] ≤ p for all t ≥ 0 and some p > 0 or exponentially stable in mean-square if E[|x(t; w)|2] ≤ pe-at, t ≥ 0, for some constants ρ > 0 and α > 0.

Suppose G is a finite solvable p′-group admitting the elementary abelian p–group A as an operator group. if n = max{nilpotent length of CG(X)| X ∈ A#} and |A| ≥ pn+2, then the nilpotent length of G is n.

Let κ and λ be cardinal numbers. Take any family A = {Aν; ν ∈ N} where each Aν is a product Aν = Bν x Cν with |Bν = |Cν| = Nα, such that if B x CAμ x Aν (for μ ≠ ν) then |B|, |c| < λ. We investigate under what conditions on α, κ, λ and |N| there will be a set T with 1 ≤ |T∩Aν| < κ for each ν.

It is shown that a finite group G is a normal subgroup of the group of units of the group ring of G over the ring of integers modulo n if and only if G is abelian or n = 2 and G is isomorphic to the symmetric group on 3 letters.

A locally convex topological vector (LCTV) space E is said to have property V (Dieudonné property) if for every complete separated LCTV space F, every unconditionally converging (weakly completely continuous) operator T: E → F is wsakly compact. First, an investigation of the permanence of property V is given. The permanence of the Dieudonné is analogous. Relationships between property V and the Dieudonné property are then given.

Operators are defined that yield basic Galois closure operations for almost every category. These give rise to a new and more general approach for characterization of epireflective subcategories, and construction of epireflective hulls. As a by-product, satisfactory characterizations of classes of perfect morphisms and ω-extendable epimorphisms are obtained. Detailed proofs and examples will appear elsewhere.

Let G = H0 > H1 > … > Hr = 1 and G = K0 > K1 > … > Kr =1 be two chief series of the finite soluble group G. Suppose Mi complements Hi/Hi+1. Then Mi also complements precisely one factor Kj/Kj+1, of the second series, and this Kj/Kj+1 is G-isomorphic to Hi/Hi+1. It is shown that complements Mi can be chosen for the complemented factors Hi/Hi+1 of the first series in such a way that distinct Mi complement distinct factors of the second series, thus establishing a one-to-one correspondence between the complemented factors of the two series. It is also shown that there is a one-to-one correspondence between the factors of the two series (but not in general constructible in the above manner), such that corresponding factors are G-isomorphic and have the same number of complements.

Certain topological spaces X may bear various uniform structures compatible with the topology of X; to each uniform structure there corresponds a completion of X, that is, a complete space Z containing X as a dense subspace. For compact completions, there has been extensive study of the relationship between X and the possible remainders Z\X. This paper begins a study of the more general, and apparently easier, problem of the relationship between X and its not necessarily compact remainders. We find that for spaces X admitting a complete metric, every space Y which satisfies certain conditions obviously necessary for Y to be the remainder of a completion of X in fact occurs as such a completion.

A projective plane has characteristic three if in every ternary ring coordinatizing it all elements ≠ 0 of the additive loop have order 3. We show that if a finite plane of characteristic three is coordinatized by a near-field, then the plane is desarguesian.

The notion of quadratic form over a field of characteristic two is extended to an arbitrary division ring of characteristic two with an involution of the first kind. The resulting isometry groups are shown to have a simple normal subgroup and the structure of the factor group is calculated. It is indicated how one may define and analyse all the classical groups in a unified manner by means of quadratic forms.

This paper gives simple proofs of two theorems of Steck concerning the distribution of sample distribution functions.

In this note we examine the relationship of a distributive lattice to its lattice of ideals. Our main result is that a distributive lattice and its lattice of ideals share exactly the same collection of finite sublattices. In addition we give a related result characterizing those finite distributive lattices L which can be embedded in a lattice L′ whenever they can be embedded in its lattice of ideals T(L′).

In this paper it is proved that non-abelian free groups are residually (x, y | xm = 1, yn = 1, xk = yh} if and only if min{(m, k), (n, h)} is greater than 1, and not both of (m, k) and (n, h) are 2 (where 0 is taken as greater than any natural number). The proof makes use of a result, possibly of independent interest, concerning the existence of certain automorphisms of the free group of rank two. A useful criterion which enables one to prove that non-abelian free groups are residually G for a large number of groups G is also given.

The asymptotic expansion of an integral of the type , is derived in terms of the large parameter t. Functions Φ(k) and ψ(k) are assumed analytic, and ψ(k) may have zeros at a stationary phase point. The usual one dimensional stationary phase and Airy integral terms are found as special cases of a more general result. The result is used to find the leading term of the asymptotic expansion of the double integral. A particular two dimensional Φ(k) relevant to surface water wave problems is considered in detail, and the order of magnitude of the integral is shown to depend on the nature of ψ(k) at the stationary phase point.

The concepts of circulant and backcirculant matrices are generalized to obtain incidence matrices of subsets of finite additive abelian groups. These results are then used to show the existence of skew-Hadamard matrices of order 8(4f+1) when f is odd and 8f + 1 is a prime power. This shows the existence of skew-Hadamard matrices of orders 296, 592, 1184, 1640, 2280, 2368 which were previously unknown.

A construction is given for regular symmetric Hadamard matrices with constant diagonal of order 4(2m + 1)2 when a symmetric conference matrix of order 4m + 2 exists and there are Szekeres difference sets, X and Y, of size m satisfying x є X ⇒ −xє X, y є Y ⇒ −y єY.