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.

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.

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 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.

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.

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.

A graph Γ is called symmetric if its automorphism group is transitive on its vertices and edges. Let p be an odd prime, Z(p) the field of integers modulo p, and Z*(p) = (a ∈ Z(p) | a ≠ 0}, the multiplicative subgroup of Z(p). This paper gives a simple proof of the equivalence of two statements:

(1) Γ is a symmetric graph with p vertices, each having degree n ≥ 1;

(2) the integer n is an even divisor of p − 1 and Γ is isomorphic to the graph whose vertices are the elements of Z(p) and whose edges are the pairs {a, a+h} where a ∈ Z(p) and h ∈ H, the unique subgroup of Z*(p) of order n.

In addition, the automorphism group of Γ is determined.

The dynamic response of a prestressed incompressible Mooney material is studied by investigating plane wave propagation and the response of the material to impulsive lines of force. The choice of an initial deformation which is axially symmetric gives a particularly simple form for the secular equation for the plane wavefront velocities. The speeds of propagation and the amplitudes of the two permissible transverse waves are found and necessary and sufficient conditions for there to exist two real wave speeds in all directions are established. The simple form of the secular equation enables the response of the material to concentrated disturbances to be readily solved using Fourier transforms. The motions caused by a line of impulsive forces is examined in some detail.