In a recent note (1) I sharpened existing theorems, which state, roughly speaking, that unless a symmetric convex body K has an excessively small homogeneous minimum, it must have a reasonably small inhomogeneous minimum; in other words, if the translates of K centred at points of the integer lattice form an efficient packing, then by expanding them we may derive an efficient covering. In this note I will prove the converse, that a bad packing leads to a bad covering; the result is a much less useful one, as the worst case occurs when we foolishly try to pack rather spiky bodies point to point. A result of this kind has previously been proved by Mahler (4), but his result is less precise from our present point of view; a full account is given by Cassels(3).

Let [Sfr ]n and [Afr ]n denote the symmetric and alternating groups of degree n∈ℕ respectively. Let p be a prime number and let F be an arbitrary field of characteristic p. We say that a partition of n is p-regular if no p (non-zero) parts of it are equal; otherwise we call it p-singular. Let SλF denote the Specht module corresponding to λ. For λ a p-regular partition of n let DλF denote the unique irreducible top factor of SλF. Denote by ΔλF=DλF↓[Afr ]n its restriction to [Afr ]n. Recall also that, over F, the ordinary quiver of the modular group algebra FG is a finite directed graph defined as follows: the vertices are labelled by the set of all simple FG-modules, L1, ..., Lr, and the number of arrows from Li to Lj equals dimFExtFG(Li, Lj). The quiver gives important information about the block structure of G.

A version of Corollary 2 for Hilbert space was orally communicated to the author by C. Pearcy. His proof employed facts about the numerical range, inner products, and orthogonality, and thus made use of structure peculiar to Hilbert space. A Banach algebra with identity is a more comfortable setting for these results, and there it is much clearer just what is important in each proof.

The Ising model of a two-dimensional ferromagnetic crystal containing defects is considered. It is shown that the combinatorial method is particularly appropriate to the investigation of the two-dimensional defect crystal and how it may be readily modified to calculate the partition function for such a crystal. It is found that the cooperative behaviour of the infinite perfect crystal, as indicated by the occurrence of a logarithmic singularity in the specific heat as a function of temperature, is not destroyed in the defect crystal; the singularity persists, though at a progressively lower temperature as the density of defects is increased, until a high density of defects is reached. The value of this limiting density of defects, above which the cooperative behaviour is destroyed, is calculated.

Throughout this note, A will denote a (commutative, Noetherian) local ring (with identity) having maximal ideal m and dimension d. Let x1, …, xd be a system of parameters (s.o.p.) for A. A (not necessarily finitely generated) A-module M is said to be a big Cohen–Macaulay A-module with respect to x1, …, xd, if x1, …, xd is an M-sequence. In the last ten or fifteen years there has been substantial interest in such modules, initially stemming from M. Hochster's discoveries that, if A contains a field as a subring, and x1, …,xd is any s.o.p. for A, then there exists a big Cohen-Macaulay A-module with respect to x1, …,xd, and that the existence of such modules has important consequences for the local homological conjectures in commutative algebra: see [6].

In a recent paper, Fenn (3) has established the following theorem: if ø is a piecewise linear involution on Sn without fixed points, if f: Sn → Sn and g: are continuous and the degree of f is odd, then there are points x and y on Sn such that f(x) = ø(f(y)) and g(x) = g(y). The Borsuk–Ulam theorem is the special case of this in which f is the identity and ø corresponds to reflexion in the origin. Since an infinite-dimensional version of the Borsuk–Ulam theorem is known, involving compact maps (see, for example, page 72 of (2)), it is natural to ask whether Fenn's result can also be extended to general Banach spaces, and in this paper we give such an extension when ø is reflexion in the origin. More precisely, we prove that if B is the closed unit ball in a Banach space X and f, g: B → X are compact, with deg (I − f, B, 0) odd (this is the Leray–Schauder degree and I is the identity map) and (I − g) (B) contained in a proper, closed subspace of X, then there exist x, y on the boundary ∂B of B and apositive numberα such that α(I − f) (x) = − (I − f) (y) and (I − g) (x) = (I – g) (y).

We construct examples of closed non-Haken hyperbolic 3-manifolds with a Heegaard splitting of arbitrarily large distance.

The theory of reductive monoids has been well developed by Renner and the author, cf. [8, 13]. This paper concerns the finite reductive monoids MF where M is a reductive monoid and F[ratio ]M→M is a Frobenius map. We show that the Deligne–Lusztig characters RT, θ of finite reductive groups GF have natural extensions to MF. We accomplish this by first showing that the action of a finite monoid (by partial transformations) on an algebraic variety gives rise to a virtual character of the monoid via étale cohomology. We then find the correct analogue of the algebraic set L−1(U) (where L is the Lang map) for M. The action of MF on this algebraic variety gives rise to virtual characters RT¯, θ where T is an F-stable maximal torus of G, T¯ the Zariski closure of T in M and θ an irreducible character of the finite commutative monoid T¯F. We go on to show that any irreducible character of MF is a component of some RT¯, θ.

Let G be a simply primitive permutation group on a set Ω of order p2, where p is a prime (necessarily odd). In theorem 27·2 of (9), Wielandt states without proof:

Theorem A. (i) ¦G¦ is not divisible by p3;

(ii) if G has a pair of Sylow p-subgroups with nontrivial intersection, then G has an imprimitive subgroup of index 2 which is the direct product of two intransitive groups.

The propagation of electric currents through long cables has often been treated theoretically and the solutions, first obtained by Kelvin and by Heaviside, have moulded the development of transcontinental telegraphy and telephony. These solutions presume that the source of electrical disturbance is situated at one point in the cable and this is the condition appropriate to wired communication, but in the more recent process of wireless communication we have an analogous problem which requires different terminal conditions.

Mendes Lopes and Pardini showed that minimal general type surfaces of Albanese dimension one have slopes K2/χ dense in the interval [2,8]. This result was completed to cover the admissible interval [2,9] by Roulleau and Urzua, who proved that surfaces with fundamental group equal to that of any curve of genus g ≥ 1 (in particular, having Albanese dimension one) give a set of slopes dense in [6,9]. In this note we provide a second construction that complements that of Mendes Lopes–Pardini, to recast a dense set of slopes in [8,9] for surfaces of Albanese dimension one. These surfaces arise as ramified double coverings of cyclic covers of the Cartwright–Steger surface.

Consider an ideal A of a commutative ring R; denote by Ai the integral closure of A in R.

1. Introduction. It is well known (see (5) and (6)) that if T is a 2-group containing an involution t such that |CT(t)| = 4, then T is dihedral, semidihedral or cyclic of order 4. Fomin(1) has studied 2-groups T containing an involution t such that |CT(t)| ≤ 8. He showed in particular that such a group has sectional 2-rank at most 4. In this paper we will prove the following:

Theorem A. Let T be a 2-group containing an involution t such that |CT(t)| ≤ 16. Then T has sectional 2-rank at most 6.

Let X, Y be complex manifolds with smooth (C∞) boundaries ∂X, ∂Y. We give conditions which ensure that a smooth map Φ: ∂X → ∂Y has an extension to a holomorphic map X → Y. Let J denote the complex structure on X. We say that Φ satisfies the ‘tangential Cauchy-Riemann equation ∂¯bΦ = 0’if the differential dΦ restricted to the complex subspace Tp(∂X) ∩ JTp(∂X) of the tangent space Tp(∂X) is complex linear at all points p ∈ ∂X. Clearly this is a necessary condition for the existence of a holomorphic extension. A further necessary condition is that there exists no topological obstruction to extension, hence we assume that a smooth extension φ: X → Y is given and we shall look for a holomorphic map f: X → Y with the same boundary values.

In this paper the condition for the existence of a secondary flow in a straight non-circular pipe is determined according to the modified vorticity transfer theory, with Goldstein's assumed form for the tensor It is shown that a secondary motion arises if the mixture length is not constant on the curves along which | grad u | is constant, u being the velocity parallel to the pipe axis.

In problems of turbulent flow treated by means of the modified vorticity transfer theory, the quantity where is the mean value of the square of the velocity fluctuation and p the mean pressure, plays a part analogous to the pressure in laminar flow. In two-dimensional flow through a channel the theory shows the existence of a gradient of across the channel from the central plane to each wall. Qualitative arguments such as are used to explain the existence of a secondary flow for laminar motion in a curved pipe are applied here to show that a secondary flow is to be expected near the short sides in the turbulent flow through a straight rectangular pipe of large length/breadth ratio.

The equations to determine the secondary flow through an almost circular elliptic pipe are discussed, the mixture length being assumed constant on ellipses similar to and concentric with the pipe section. For a first approximation the problem is reduced to the numerical solution of three simultaneous ordinary linear differential equations.

D. R. Hughes stated the following conjecture: If π is a finite projective plane satisfying the condition: (C)π contains a collineation group δ inducing a doubly transitive permutation group δ* on the points of a line g, fixed under δ, then the corresponding affine plane πg is a translation plane.

Let f[ratio ]X→[Copf]ℙp be a finite complex analytic map into complex projective space, with dimX<p. We obtain a result on the equivalence of low homotopy groups between the image of f and [Copf]ℙp, the level of comparison is a function of p, the maximal number of preimages of f and how bad the singularities of X are. This global result is deduced from a generalisation of a theorem of H. Hamm on the local structure of singularities, see [7].

Let A be an A*-algebra and let denote its auxiliary norm closure. The multiplier algebras of dual A*-algebras of the first kind have been studed by Tomiuk [12], [13] and Wong[15]. In this paper we study the double multiplier algebra of A*-algebras of the first kind. In particular, we prove that, if A is an A*-algebra of the first kind, then the double multiplier algebra M(A) of A is *-isomorphic and (auxiliary norm) isometric to a subalgebra of M(), extending in the process some results established by Tomiuk[12]. We also consider the embedding of the double multiplier algebra of A in **, when the latter is regarded as an algebra with the Arens product, and, in particular, we show that, for an annihilator A*-algebra, M(A) is *-isomorphic and (auxiliary norm) isometric to **.

In this paper we are investigating boundedness and certain stability properties of differential systems in spaces, utilizing the generalization of Bellman's Lemma which was formulated by one of the authors (8).

If light of a frequency which corresponds to an energy greater than the ionisation potential falls on an atom, an electron may be ejected and energy absorbed. To calculate the absorption coefficient, or the rate of absorption of energy per unit intensity of incident radiation for a given frequency, one must first choose a model for the atom. If we confine ourselves to the inner K electrons there will be two electrons in this shell for the heavier atoms, and a fairly good model of the atom is obtained by considering each electron to be moving independently in a central field of force due to the charged nucleus: i.e. we neglect electronic interaction and assume that the wave functions for the system are hydrogenic. Some writers make a partial correction for this neglect of interaction by modifying the central charge through the introduction of a screening factor which is so chosen that the minimum calculated energy required to remove one of the K electrons will agree with the experimental value provided by the K absorption edge. In general, however, the approximation is fairly good, and this is particularly so in the interior of a star where the atoms are highly ionised. It is not so good when the atom is bound as in a metal, and, of course, most of the laboratory work has been carried out on atoms in this bound state.