Automorphism towers were first considered in (19) with the case of finite groups. Recently polycyclic and extremal groups were considered in (7) and (12) respectively. The purpose of this paper is to consider automorphism towers of groups in a class of almost soluble groups which contains all polycyclic and all extremal groups.

Let e be an integer greater than 1 and let pbe a prime such that p ≡ 1 (mod e). Criteria for 2 to be a residue of degree e modulo p have been obtained in various forms for e = 2, 3, 4 and 5. Thus, Euler and Lagrange proved that 2 is a quadratic residue mod p if and only if p ≡ ± 1 (mod 8). Gauss showed that 2 is a cubic residue mod p if and only if p is representable as p = l2 + 27m2 with integers l and m, and that 2 is a quartic residue mod p if and only if p is representable as p = 12 + 64m2. Lehmer (5) and Alderson (1) have found similar but more complicated conditions for 2 to be a quintic residue mod p. There are analogous results about 3. For example, it follows from quadratic reciprocity that 3 is a quadratic residue mod p if and only if p ≡ ± 1 (mod 12), and Jacobi(4) showed that 3 is a cubic residue mod p if and only if 4p is representable as l2 + 35m2.

The polynomial invariants of a certain classical linear group of order 336 arise naturally in studying error-correcting codes over GF(7). An incomplete description of these invariants was given by Maschke in 1893. With the aid of the Poincaré series for this group, found by Edge in 1947, we complete Maschke's work by giving a unique representation for the invariants in terms of 12 basic invariants. A conjecture is made concerning the relationship between the Poincaré series and the degrees of the basic invariants for any linear group. A partial answer to this conjecture, due to E. C. Dade, is given.

In a Noetherian commutative ring with identity, every ideal can be expressed (not necessarily uniquely) as a finite intersection of primary ideals (called a primary decomposition). This note is concerned with powers of ideals generated by subsets of an R-sequence in a local ring R (i.e. a Noetherian commutative ring R with identity possessing a unique maximal ideal m) and with a decomposition of such ideals.

The purpose of this note is to construct a non-compact 3-manifold M with the following properties:

(i) M is orientable and irreducible

(ii) ∂M is connected and compact and its inclusion in M is a homotopy equivalence

(iii) M is not homeomorphic to ∂M × R+.

(Throughout this paper, all manifolds will be PL and all embeddings will be PL and locally flat.)

This note describes a new, and possibly more informative, proof of the fact that any closed orientable 3-manifold is a branched cover over the 3-sphere.

A family ℳ of closed circular discs in the plane is called a saturated family or a saturated system of circles if (i) the infimum r of the radii of the discs in ℳ is positive and (ii) every closed disc of radius r in the plane intersects at least one disc in ℳ. For a saturated family ℳ, we denote by S the point-set union of the interiors of the members of ℳ and by S(l) the part of S inside the circular disc of radius l centred at the origin. We define the lower density ρℳ of the saturated family ℳ as

where V(S(l)) denotes the Lebesgue measure of the set S(l).

Let ℛ be a von Neumann algebra, with predual ℛ*, acting on a Hilbert space ℋ; G a locally compact group with left Haar measure m, and α a representation of G on aut (ℛ), the group of all *-automorphisms of ℛ, i.e. α is a group homomorphism from G to aut (ℛ). We shall show that if ℋ is separable, then very weak measurability assumptions on the representation α produce strong continuity properties. This will be used to obtain results on the extension of representations from a C*-algebra to its weak closure, giving a much simpler proof of a result of Aarnes ((1), theorem 8, p. 31), and on continuity of tensor products of representations. The main result was suggested by the analogous theory concerning unitary representations of locally compact groups, and its proof employs ideas frequently used in that context. (See, for example, (5), theorem 22.20 (b), p. 347.)

Let F be the finite field of q = pn elements and let F0 be its prime subfield; thus, card F0 = p. For polynomials f ∈ F[x] and non-principal additive characters η of F A. Weil (1) proved the estimate

The object of this paper is to complete the half proved theorem 2 of (1).

Let be a primitive fifth root of unity. Any element of Z[ζ] is a polynomial f(ζ) in ζ of degree ≤ 3 since 1 + ζ + ζ2 + ζ3 + ζ4 = 0. The units of Z[ζ] are ± ζi(1 + ζ)i or better still , with 0 ≤ i ≤ 4, j ε Z, where is the fundamental unit of the maximal real subfield of Q(ζ).

In many mechanical and other problems the following equation

is reached, where Jν(α) and Yν(α) are Bessel functions of the first and second kind of any real order ν and β is a positive parameter.

For example, equation (1) is reached in the case of determining the critical load Pcr, for a simply supported strut with a variable inertia moment by a power law, where the power m is any real number.

In this paper the authors consider a problem, frequently encountered in diffraction theory, of expanding a given function f(r) in terms of a set of orthogonal functions which is not complete. The expansion in question can be written as

where r < a and

It is proved that the elliptic ball criterion is a necessary condition on the matrix M(t) for the linear feedback control equation f(D)x + BM(t)g(D)x = 0 to have a special kind of quadratic Lyapunov function. That it is also a sufficient condition has already been proved elsewhere. These two facts lead to a comparison theorem which enables the existence of a quadratic Lyapunov function for one equation to be deduced from that for another equation.

The main purpose of the note is to compare necessary and sufficient conditions for weak ergodicity of finite inhomogeneous Markov chains given by Doeblin (3) and Hajnal (4), the former paper being little known; and more generally to expand on the nature and consequences of Doeblin's approach as compared to Hajnal's in some detail. A consequence is some insight into the relation between various ‘coefficients of ergodicity’.

Let S be n dimensional Euclidean space and let T be a division of S into cells. Assume that each cell must be either white or black at any time t. At time 0 the cell at the origin, α0, is black and all other cells are white. Let G be some stochastic growth process which tends to change white cells with black neighbours into black cells. Let C(t) be the black shape at time t. For a family, F, of such growth processes we prove the following theorem.

Lagrangian coordinates are used in conjunction with the self-similarity hypothesis in order to examine the problem of the vertical entry at constant speed of a wedge and also of a cone into an incompressible fluid initially at rest. Certain known properties of the free surface are recovered in a very direct and simple manner and new exact results concerning the inclination of the free surface and the angle of contact with the rigid surface are obtained.

It is shown how the solution for the velocity potential may be determined when the effect of surface tension is included in the linearized theory of surface waves over a sloping beach. In particular, two independent standing wave solutions are found, both of which have finite amplitude at the shoreline. The results agree with those of previous writers when the surface tension force tends to zero.