This note concerns a question of elementary calculus. Given a smooth composite function u = g o f [with values u(x) = g(f(x))], we write explicit formulae for its derivatives, of arbitrary order, in terms of derivatives of f and g. We consider (A) the general case,

in which E, F and G are Banach spaces, and U, V are open sets; (B) the finite-dimensional case E = ℝM and F = ℝN, where ℝM denotes real M-dimensional Euclidean space; and (C) the particular case of (B) (due to restricting g to part of an M-dimensional surface in ℝM + 1) in which N = M + 1 and u(x)= g(x, φ (x)), so that φ denotes a real-valued (scalar-valued) function of x = (x1, …, xM).

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.

If M is an arbitrary loopless matroid with ground set E(M) and rank function ρ, then let α(M) be the minimum size of a set of cocircuits of M, whose union is E(M), and let β(M) be the maximum size of a set of pairwise disjoint cocircuits of M. The following conjecture is based on Gallai's theorem that the vertex-stability and vertex-covering numbers of a graph G sum to the number of vertices of G (1).

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.

Let Σ denote the set of all series

of complex numbers. By a ‘summability method’, say A, we mean a function from some subset (the set of ‘A -summable series’) of Σ into the set of complex numbers. We will use the language generally associated with this definition, and will take for granted the case in which A is (C, 1). A summability method A will be called linear if, whenever a, b are A -summable, then so is λa + μb (where λ, μ are any complex constants) and if the. A -sums of a, b, λa + μb are then related in the natural way. We call A regular if, whenever a converges to σ, it is A -summable to σ. If A is a regular summability method, then any condition P on the series (1) will be called a Tauberian condition for A if any A -summable series which satisfies P is convergent.

Let = (Ai;i ∈ I) be an indexed family of sets. The family is said to contain the family = (Bj;j ∈ J) if there is a one-to-one map f from J into I such that Bj = Af(i) for all j in J. The indexed family = (Ai;i ∈ I) is said to be a Δ (λ)-family if |I| = λ and Ai ∩ Aj = Ak ∩ Ai for all pairs i, j and k, l from I. The family is said to be a (λ, k)-family if |I| = λ and |Ai| = k for all Ai.

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying

A subset, X, of R belongs to the family T(a) if |X| ≥ 2 and, for all x and y belonging to X, ax + (1 − a) y ∈ X, where a ∈ R.

I consider the problem of determining for which values of a > 1 all elements of T(a) are dense in R.

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 A = (aij) be a non-negative n × n matrix, that is, aij ≥ 0, i, j = 1, …, n; n > 1. We write A ≥ 0. Let r = r(A) be the spectral radius of A; assume r > 0 throughout to avoid trivial cases. Let be the mth derivative of r with respect to the element aij, all other elements of A being held constant.

If H is a Moufang loop, and x ∈ H, there are defined permutations of H, L(x):y ↦ xy and R(x): y ↦ yx. The group Gr (H), generated by these permutations for all choices of x, is called the multiplication group of H. It has a close connexion with the structure of H, as shown, for instance, in the papers of Albert(1). The purpose of this paper is to investigate the correspondence between groups and loops, so that group theoretic results may be applied to determine the structure of Moufang loops.

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.

In (1) it is remarked that theorem 5·10 is an analogue, for endomorphisms of a free group, of a result of A. Shenitzer ((3), theorem 2) about automorphisms. With this in mind, D. Solitar asked the author if there might also be an analogue for the corollary on page 276 of(3). There is such an analogue and, after a preliminary discussion of definitions and notation, we will describe it.

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

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.

A generalized m-manifold is a euclidean neighbourhood retract X that is a homol-ogy m-manifold, i.e. for which H*(X, X–x;Z) = H*(Rm, Rm–0;Z) for each point x of X. In a generalized m-manifold X, a pointy is called singular, or a singularity of X, if p has no neighbourhood homeomorphic to Rm. We shall denote the set of singular points of X by S(X).

In (1) we showed that a subgroup closed Fitting formation is a primitive saturated formation, and in (2) we showed that a subgroup closed and metanilpotent Fitting class is a formation. Whether or not a subgroup closed Fitting class is always a formation is a question that has plagued us ever since. The purpose of this paper is to prove

In (2), Erdös and Straus pose the following two conjectures.

If G is a finite group of order n and X is a subset of G consisting of pairwise non-commuting elements, then |X| ≤ n/2 + 1.

Let Y be a compact connected C∞ Riemannian manifold with negative sectional curvatures. Let G be a non-trivial subgroup of the fundamental group π1(Y). G is known to be cyclic if it is abelian (Preissmann (6)) or contains a subnormal abelian (hence cyclic) subgroup (Yau(9)). These results may be generalized as follows: Say that a group G is of type (α) if ∃a ∈ G, a ≠ e, such that for all b belonging to a set of generators for G we have ambn = bqap for some integers m, n, p, q with either m = p or n = q.