In the first half of this paper we introduce a new method of examining the q-hook structure of a Young diagram, and use it to prove most of the standard results about q-cores and q-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q-weight and says how many of them are q-regular. In the case where q is prime, this tells us how many ordinary and q-modular irreducible representations of the symmetric group there are in a given q-block. None of the results of section 2 is original. In the next section we give a new definition, the p-power diagram, which is closely connected with the p-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be p-regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p. In this paper we derive combinatorial results involving the p-power diagram, and in a later article we investigate the relevant representation theory.

We study the question: Which ordinary irreducible representations of the symmetric group remain irreducible modulo a prime p?

Let Sλ be the Specht module corresponding to the partition λ of n. The definition of Sλ is ‘independent of the field we are working over’. When the field has characteristic zero, Sλ is irreducible, and gives the ordinary irreducible representation of corresponding to the partition λ. Thus we are interested in the problem of whether or not Sλ is irreducible over a field of characteristic p.

In (1) the authors introduced the idea of a syntopogenous preordered space (SPS), thereby generalizing simultaneously the concepts of ‘uniform preordered space’ and ‘proximity preordered space’. In the case of a classical topology the SPS condition was seen to be equivalent to the condition, ‘convex plus T1-preordered’. In the present paper we show that, in the general case, the SPS condition may be characterized as the sum of two generalizations of the logically independent conditions of convexity and T1-preorderedness.

Let Z2 denote the integer lattice in the plane, let A be a non-singular 2 x 2 matrix and let c ∈ R2. Then G = AZ2 + c is called a grid, and its determinant det G is defined to be det A. The grid G is called a subgrid if G ⊂ Z2.

The object of this paper is to prove that the following classes of graphs are edge-reconstructible: (i) 4-connected planar graphs with minimum valency 5; (ii) maximal planar graphs with minimum valency at least 4.

Let N take the values 1, 2, … A theorem of Dirichlet asserts that for any (x, y) in the Euclidean plane R2 the inequality

is soluble in integers q1, q2, p with 0 < max (|q1|, |q2|) ≤ N.

In his important paper (1), Lorentz defined the space f of almost convergent sequences, using the idea of Banach limits. If x ∈ l∞(R), the space of bounded real sequences, and

where the inf is taken over all sets n(1), n(2), …, n(r) of natural numbers, then a Banach limit L may be defined as a linear functional on l∞(R) which satisfies

Let α be a positive integer, and El, …, Eα Hadamard sets of positive integers. It is shown that E = E1 + … + Eα determines a Littlewood–Paley decomposition of Z.

Suppose that is a Hadamard set of positive integers such that nj+1/nj ≥ 2 for all j. Let α be a positive integer, and

We show that F(α) also determines a Littlewood-Paley decomposition of Z.

Of all sets lying within a given convex plane set S, which one gives the greatest ratio of area A to perimeter P? This was posed to one of us by E. Bombieri for the case when S is a square. He asserted that T. Bang used an easy upper bound on A/P in an elementary proof of the Prime Number Theorem. After solving this problem from scratch, we learned from M. Silver that Besicovitch (1) had solved a closely related problem, which we call Besicovitch's problem and which will be discussed shortly. Later, we found that the problem of maximizing A/P had occurred to Garvin (4, 5) and that he solved the problem when S is a triangle. Garvin inspired DeMar (2) to consider and solve Besicovitch's problem for the triangle and for polygons. DeMar found that Steiner (8, p. 166) had solved Besicovitch's problem for triangles, but Steiner was only interested in subsets which touched all sides of a polygon (8, p. 168). More recently, J. Wills has referred us to Herz and Kaapke (6) who use Besicovitch's results to solve our problem of maximizing A/P for polygons circumscribed about a circle.

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphism

in group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.

The purpose of this paper is to give a proof of the following splitting theorem in stable homotopy theory. We assume all spaces are localized at a fixed prime p. Let k be the symmetric group on {1, …, k}, Q(.) = lim ΩnΣn(.), and QkS0, k ∈ , denote the components of QS0.

In (2) Adams gave a splitting of complex K-theory with coefficients in the ring R(d) of rationals a/b such that b contains no prime p with p ≡ 1 (mod d). The splitting comes from a complete set of projection operators on K(X; R(d)). One of the operators is then used to obtain a stable, multiplicative idempotent ε on complex cobordism with coefficients in the same ring R(d) and hence a splitting of the representing spectrum MUR(d). However, the idempotent is initially defined over the rational numbers and work is needed to show that it actually gives an operation on MUR(d). Since Novikov (6) has shown that multiplicative cobordism operations are distinguished by their values on the generator ω ∈ MU2CP∞, it is natural to seek an explicit formula for ε(ω) which wi11 show that ε gives an operation over the subring R(d).

Let Yn ⇒ Y∞ be a sequence of random variables converging in distribution, or more generally a sequenceof random elements of a suitable metric space whose distributions are converging weakly. Let τn → ∞ be positive integer-valued random variables. If {τn} and {Yn} are independent, it is trivial that

In two recent papers (6, 10) a new formulation of the governing equations for finite plane-strain deformations of compressible isotropic elastic solids was presented. This has been used to obtaingeneral solutions to the field equations for a certain class of strain–energy functions, and then closed-form solutions to a large class of boundary-value problems.

This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.

Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and ⅙π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, and

where g is the acceleration due to gravity. Krasovskii's set of solutions is included in the set that we obtain.