As is well known, there is an algorithm for deciding if a system of linear equations with coefficients from the set Z of integers has a solution in integers. The purpose of this paper is to answer the following question: does this remain true if ‘linear’ is replaced by ‘quadratic’?

For numbers of increasing real functions f(x) with new integral inequalities. They generalize classical results. The proofs are short and simple being based on sequences.

In this paper, a new family of factors of (2, 3, 7) is obtained which contains groups found in three other papers. It is shown that for all n, there exists an m such that there are at least n isomorphism types of Hurwitz groups of order m. Finally, presentations for all groups considered are obtained.

This note approaches the immersion/embedding problem for real projective n-space Pn in Rn+k by viewing the Smale invariant of the induced immersion Sn → Pn → Rn + k as an obstruction to extending to an immersion of Pn + 1 in Rn + k. First steps in this direction were taken in (8), by studying the elementary problem of extending axial maps.

Let G bea finite group, let C1, …, Cn be the cyclic subgroups of G and let 1i be the identity linear character of Ci (1 ≤ i ≤ n). It is a well-known result of Artin ((l), 39·1) that a character x of a linear representation of G over the rational numbers may be written

where a1, …, an are integers. In this note, we establish an analogue of this result for characters of projective representations over the rational numbers.

An element B ∈ Bn, the braid group on n strings, which can be written as

in terms of the standard generators of Bn is called a split braid. It is easy to see that the resulting closed braid is the connected sum of the closed braids Ĉ and on k + 1 and n − k strings respectively (figure 1).

Compact right topological groups appear naturally in topological dynamics. Some continuity properties of the one arising as an enveloping semigroup from the distal function are considered here (and, by way of comparison, the enveloping semigroups arising from two almost automorphic functions are discussed). The continuity properties are established either explicitly or by citing a theorem which is proved here and gives some characterizations of almost periodic functions. One characterization is proved using the result (essentially due to W. A. Veech) that a distal, almost automorphic function is almost periodic. A proof of this last result is also given.

This paper proves that any knot is concordant to a prime knot; it thus solves Problem 13 of (3). In doing so it makes an exploration of a fairly general method of proving that a knot is a prime. Throughout, the word ‘knot’ means a knot of S1 in S3 (orientations being here irrelevant); occasionally reference will be made to the idea of a knotted arc spanning a 3-ball.

The purpose of this paper is to construct an operad ℋ∞ with the good properties of both the little convex bodies partial operad and the little cubes operad used in May's theory of E∞ ring spaces or multiplicative infinite loop spaces ((6), chapter VII). In (6) ℋ can then be used instead of and , and the theory becomes much simpler; in particular all partial operads can be replaced by genuine ones. The method used here is a modification of that which May suggests on (6), page 170, but cannot carry out.

In an earlier paper I considered the problem of determining those subgroups of SL (2, ℤ) consisting of all matrices

satisfying a linear congruence

Let A be a subring of the complex numbers containing 1, and Γ a subgroup of the modular group of finite index. We say that a modular form on Γ is A-integral if the coefficients of its Fourier expansion at infinity lie in A. We denote by Mk(Γ,A) the A-module of holomorphic A-integral modular forms of weight k, and by M(Γ, A) the graded algebra of A-integral modular forms on Γ.

Let X1, X2, … be independent and identically distributed (i.i.d.) random variables and let Xn, r denote the rth largest of X1, X2, …, Xn (so that Xn, r is the (n − r + l)th-order statistic of X1, X2, …, Xn). It is well known that if Xn, l/cn→1 in probability or with probability 1, for some sequence of constants cn, then Xn, r/cn→1 for each r ≥ 1. Therefore if r(n) → ∞ sufficiently slowly, Xn, r(n)/cn→1 for the same sequence of constants cn. In this paper we study behaviour of this type in considerable detail. We find necessary and sufficient conditions on the rate of increase of r(n), n ≤ 1, for the limit theorem Xn, r(n)/cn→1 to hold, and we investigate the rate of convergence in terms of a central limit theorem and a law of the iterated logarithm (LIL). The LIL takes a particularly interesting form, and there are five distinctly different modes of behaviour.

The method of antithetic variates introduced by Hammersley and Morton (2) is one of the most widely used Monte Carlo techniques for estimating an unknown parameter θ. The basis for this method was established by Hammersley and Mauldon(l).in the case of unbiased estimators with the form

where each of the variates ξj is required to have a uniform marginal distribution over the unit interval [0,1]. By assuming that n = 2 and that the gj are bounded Borel functions, Hammersley and Mauldon showed that the greatest lower bound of var (t) over all admissible joint distributions for the variates ξj can be approached simply by arranging an appropriate strict functional dependence between the ξj. Handscomb(3) extended this result to the case of n > 2 bounded antithetic variates gj(ξj). In many experiments involving distribution sampling or the simulation of some stochastic process over time, the response functions gry are unbounded. This paper further extends the antithetic-variates theorem to include the case of n ≥ 2 unbounded antithetic variates gj(ξj) each with finite variance.

A number of formulae of an integral geometric character are presented for the most general planar, homogeneous line-segment process, one in which a given segment length and orientation may depend upon (a) the point process of segment midpoints and (b) the lengths and orientations of other segments. The sense in which these formulae have a probabilistic/statistical interpretation is made precise. For the general process, two interpretations are given; one requires the theory of Palm distributions whilst the other depends upon ergodic results. When additional structure for the process is assumed, the integral geometric formulae lead to interesting, non-intuitive sampling formulae.

In our previous consideration in (1) of the constant-coefficient fifth-order differential equation:

an attempt was made to identify (though not exhaustively) different sufficient conditions on a1,…,a5 for the instability of the trivial solution x = 0 of (1·1). It was our expectation that the conditions so identified could be generalized in some form or other to equations (1·1) in which a1,…,a5 were not necessarily constants, thereby giving rise to instability theorems for some non-linear fifth-order differential equations; and this turned out in fact to be so except only for the case:

with R0 = R0(a1, a2, a3, a4) > 0 sufficiently large, about which we were unable at the time to derive any worthwhile generalization to any equation (1·1) in which a1, …,a5 are not all constants.

Atkinson and Peletier (2,3) have considered similarity solutions of the differential equation

where the function k(s) is defined, real and continuous for s ≥ 0 and k(s) > 0 if s > 0 (in (2) k(0) = 0 is also assumed). In particular they look for similarity solutions of the form u(x, t) = f(η) where η = x(t+l)−½ with boundary conditions f(0) = A and . They show that if k(s) satisfies the condition

then for any A > 0 there is a unique similarity solution which is non-negative and has compact support in [0, ∞). They also show in (2) that

is a necessary condition for the solution to have compact support. In (3) they prove existence of similarity solutions when

and show that in this case the similarity solution has the property that f(η) > 0 for all η > 0.

The two-dimensional, irrotational, linear theory used in the investigation of the propagation of monochromatic surface gravity waves in a region of varying depth is considered. Uniqueness of the velocity potential is established for bottom profiles satisfying certain convexity conditions. These include the majority of profiles of physical interest.

We obtain two approximate representations of a one-particle Gibbs state, both of which become asymptotically exact in trace norm as m → ∞. The second representation is an integral of pure coherent states over phase space, and can therefore be regarded as a classical approximation to the Gibbs state. We also obtain a version of the second representation applicable to the microcanonical ensemble.

The convexity of the internal energy, regarded as a function of any tensor strain measure, determines the stability of a material in a particular loading environment. Such convexity criteria reflect properties of the material alone and therefore have the alternative interpretation of constitutive inequalities. An assessment of the relative strengths of these criteria is therefore of interest because it provides an ordering of the constitutive inequalities. Existing comparison theorems, allowing such an assessment, suppose that work-conjugate measures of stress and strain are coaxial. The main content of this paper is a theorem which makes no such supposition and so enables a comparison of criteria of convexity, and an ordering of constitutive inequalities, for arbitrarily anisotropic materials.