In this paper we are concerned with questions about the knottedness of a closed curve of given length embedded in Z3. What is the probability that such a randomly chosen embedding is knotted? What is the probability that the embedding contains a particular knot? What is the expected complexity of the knot? To what extent can these questions also be answered for a graph of a given homeomorphism type?

We use a pattern theorem due to Kesten 12 to prove that almost all embeddings in Z3 of a sufficiently long closed curve contain any given knot. We introduce the idea of a good measure of knot complexity. This is a function F which maps the set of equivalence classes of embeddings into 0, ). The F measure of the unknot is zero, and, generally speaking, the more complex the prime knot decomposition of a given knot type, the greater its F measure. We prove that the average value of F diverges to infinity as the length (n) of the embedding goes to infinity, at least linearly in n. One example of a good measure of knot complexity is crossing number.

Finally we consider similar questions for embeddings of graphs. We show that for a fixed homeomorphism type, as the number of edges n goes to infinity, almost all embeddings are knotted if the homeomorphism type does not contain a cut edge. We prove a weaker result in the case that the homeomorphism type contains at least one cut edge and at least one cycle.

Let R be a commutative ring. We say that R satisfies the ascending chain condition for principal ideals, or that R has property (M), if each ascending sequence (a1) ⊆ (a2) ⊆ … of principal ideals of R terminates. Property (M) is equivalent to the maximum condition on principal ideals; that is, under the partial order of set containment, each collection of principal ideals of R has a maximum element. Noetherian rings, of course, have property (M), but the converse is not true; for if R has property (M) and if {Xλ} is a set of indeterminates over R, then the polynomial ring R[{Xλ}] has property (M). Krull domains, and hence unique factorization domains, have property (M).

The wave functions and energies of both the normal and surface electronic states of a finite linear chain are determined in terms of the overlap integrals by the approximation of tight binding, it being assumed that the interaction of atomic s-states with p-states can be neglected. It is shown that the existence of surface states depends on the ratio of two overlap integrals being greater than unity, and reasons are given for expecting this to be so.

It is well known that Hubert's function of a homogeneous ideal in the ring of polynomials K[x0, …, xm], where K is a field and x0, …, xm are independent indeterminates over K, is, for large values of r, a polynomial in r of degree equal to the projective dimension of (1). Samuel (4) and Northcott (2) have both shown that if the field K is replaced by an Artin ring A, is still a polynomial in r for large values of r. Applying this generalization Samuel (4) has shown that in a local ring Q the length of an ideal qρ, where q is a primary ideal belonging to the maximal ideal m of Q, is, for sufficiently large values of ρ, a polynomial in ρ whose degree is equal to the dimension of Q.

Consideration is given to the steady flow of a micropolar liquid near a rigid boundary. It is shown that while micro-inertia itself is not important, nevertheless, some properties of the microstructure do play a vital rôle in determining the structure of the boundary layer. The Kármán—Polhausen method is used to provide an estimate of the shear stress at the boundary, and detailed calculations are given for flow near stagnation. The important significance of the standard length of the micropolar liquid is demonstrated for this flow, as for other flows in this medium.

Thomas's differential equation for the average field inside a heavy atom is analogous to Emden's equation for the polytropic equilibrium of a star. Emden's result that the total gravitational potential energy of a star is calculable once the differential equation has been solved is adapted to give the total electrostatic energy, and hence the total energy of binding, of an atom built on Thomas's model. This should be equal to the sum of the successive ionisation potentials. The total energy is found to be proportional to N, where N is the atomic number. The values found agree with Hartree's calculations of the successive ionisation potentials of certain atoms.

Let G be a simple adjoint algebraic group over an algebraically closed field K. We are concerned to describe the conjugacy classes of unipotent elements of G. G operates on its Lie algebra g by means of the adjoint action and we may consider classes of nilpotent elements of g under this action. It has been shown by Springer (11) that there is a bijection between the unipotent elements of G and the nilpotent elements of g which preserves the G-action, provided that the characteristic of K is either 0 or a ‘good prime’ for G. Thus we may concentrate on the problem of classifying the nilpotent elements of g under the adjoint G-action.

In this paper, I shall establish the sufficiency of certain conditions on an ideal A of a local ring Q, and on a set {g1 …,gk} of elements of Q generating a proper ideal G, for the ideals A and G to be analytically disjoint. Hence I shall establish an upper bound for the analytic spread of A.

The maximal ideal of Q will be denoted throughout by M, and it will be assumed that the field Q/M is infinite.

This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. This theory generalizes and unifies the classical Hamiltonian formalism of particle mechanics as well as the many pre-symplectic 2-forms used by Bridges. In this theory, solutions of a partial differential equation are sections of a fibre bundle Y over a base manifold X of dimension n+1, typically taken to be spacetime. Given a connection on Y, a covariant Hamiltonian density [Hscr ] is then intrinsically defined on the primary constraint manifold P[Lscr ], the image of the multisymplectic version of the Legendre transformation. One views P[Lscr ] as a subbundle of J1(Y)*, the affine dual of J1(Y)*, the first jet bundle of Y. A canonical multisymplectic (n+2)-form Ω[Hscr ] is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original partial differential equation as well as the Euler–Lagrange equations of the corresponding Lagrangian. Furthermore, we show that the n+1 2-forms ω(μ) defined by Bridges are a particular coordinate representation for a single multisymplectic (n+2)-form and, in the presence of symmetries, can be assembled into Ω[Hscr ]. A generalized Hamiltonian Noether theory is then constructed which relates the action of the symmetry groups lifted to P[Lscr ] with the conservation laws of the system. These conservation laws are defined by our generalized Noether's theorem which recovers the vanishing of the divergence of the vector of n+1 distinct momentum mappings defined by Bridges and, when applied to water waves, recovers Whitham's conservation of wave action. In our view, the multisymplectic structure provides the natural setting for studying dispersive wave propagation problems, particularly the instability of water waves, as discovered by Bridges. After developing the theory, we show its utility in the study of periodic pattern formation and wave instability.

This note is concerned with infinite sequences whose terms are chosen from a finite set of symbols. A segment of such a sequence is a set of one or more consecutive terms, and a repetition is a pair of finite segments that are adjacent and identical. A non-repetitive sequence is one that contains no repetitions.

The onset of turbulence in many systems appears disorganized and unpredictable. However, the present detailed study of a well-known model reveals a series of well-defined transitions from steady motion to highly non-periodic behaviour. The bifurcation structure of this simple model, which describes both a reversing disc dynamo and Benard convection in thin fluid loops, is characteristic of a class of systems with subcritical instabilities. It is found that the bifurcation curve for this instability does not have a stable branch and that non-periodic and linearly stable steady solutions coexist. The boundary between transient and non-periodic behaviour is marked by a non-uniformity in the number of oscillations between reversals. This non-uniformity, not previously observed, is a striking corroboration of the model's relationship to the geodynamo. Other features of reversal in the disc dynamo such as the presence of two time scales in the oscillations are also exhibited by geomagnetic fields. In addition to a geometric description of the transitions, a one-dimensional mapping first constructed by Lorenz for the system beyond marginal stability, is extended to the subcritical regime. This type of mapping, which can be interpreted in terms of invariant surfaces of the system, may be of value in dissecting more general systems with subcritical Hopf bifurcations.

Many stochastic problems arise in physics where we have to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E, and this distribution varies with another parameter t (which may be continuous or discrete; if t represents time or thickness it is of course continuous). This variation occurs because of transitions characteristic of the stochastic process under consideration. If the E-space were discrete and the states represented by E1, E2, …, then it would be possible to define a function

representing the probability that there are ν1 particles in E1, ν2 particles in E2, …, at t. The variation of π with t is governed by the transitions defined for the process; ν1, ν2, … are thus stochastic variables, and it is possible to study the moments or the distribution function of the sum of such stochastic variables

with the help of the π function which yields also the correlation between the stochastic variables νi.

By electrical analysis of the output from a hot-wire anemometer, it is possible to measure rapidly and accurately all the quantities appearing in the theoretical equations for the decay of isotropic turbulence. The technique for these measurements is described, and possible extensions and limitations of the method are discussed. Measurements have been made of the second and fourth derivatives of the longitudinal double-velocity correlation at the origin, the third derivative of the longitudinal triple-velocity correlation at the origin, the statistical distribution in time of the velocity fluctuations, and the integral scale of turbulence.

Assuming that the solute and solvent molecules in a solution can be regarded as occupying sites on a regular lattice and that the potential energy arises from interactions between molecules which occupy closest neighbour sites, the vapour-pressure equations have been determined for solutions in which each solute molecule contains three groups (or submolecules) and occupies three lattice sites in such a way that successive submolecules occupy closest neighbour sites on the lattice while each solvent molecule occupies only one site. The vapour-pressure equations are compared with those which have already been obtained by Fowler and Guggenheim for the case in which each solute molecule contains two submolecules in order to determine the effect on the form of the vapour-pressure equations of the number of submolecules in each solute molecule. This enables the determination of the vapour-pressure equations when each solute molecule contains n submolecules and occupies n lattice sites in such a way that successive submolecules occupy closest neighbour sites on the lattice. In this latter case the vapour-pressure equations are

These equations are used to determine the osmotic pressure of solutions of long-chain polymers, and it is found that in the region of osmotic interest, the osmotic pressure is given by an equation of the form

where c g. of solute per 100 c.c. of solution is the concentration. It is shown that this equation can be written approximately

which is the quadratic relation which has usually been fitted to osmotic measurements. To this approximation π/c plotted as a function of c gives a straight line of which the intercept on the π/c axis determines the molecular weight of the polymer molecule and the gradient determines the number of submolecules in each polymer molecule.

Bernstein's theorem states that the only complete minimal graphs in R3 are the hyperplanes. We shall produce evidence in favour of some conjectural generalizations of this theorem for the cases of spacelike hypersurfaces of constant mean curvature in Minkowski space and in de Sitter space. The results suggest that the class of asymptotically simple space-times admitting a complete spacelike hypersurface of constant mean curvature may well be considerably smaller than the general class of asymptotically simple space-times.

We study processes in which units (particles) associate into clusters, which are then also capable of dissociation. Such processes are discussed generally in section 2, where a stochastic kinetic equation (10) is proposed which bridges the gap between the conventional kinetic equations (8) and the statistical equilibrium concept of the Gibbs distribution.

In section 4 we consider the equilibrium behaviour of a process for which the association rate of two units which are already bound to j and k other units respectively has the form (21). This is very much more general than the equi-reactive bond model usually discussed. The principal results are given in Theorem 1; from a single pair of equations (28) and (29) based on the Hj of (21) one can determine critical points, expected number of bonds, the distribution and moments of cluster size, and most other quantities of interest. This is without reference to any other consideration, such as kinetic or stoichiometric relations.

Some particular cases are worked through in section 5. The classic Flory-Stockmayer results for units with f equi-reactive sites are obtained systematically and economically, with all parameters in terms of physically given quantities. Another type of example seems to indicate the existence of a second critical point.

Corresponding results for the case of several types of unit are stated and illustrated in section 6.

In what follows we shall derive some properties of the gravitational field of an isolated, axially symmetric, uniformly rotating mass of perfect fluid in a steady state, according to the general theory of relativity. Several exact models describing rotating fluids are known in Newtonian mechanics, the Maclaurin and Jacobi ellipsoids ((6)) being perhaps the most interesting. In general relativity, no such exact solution is known in its entirety, although Kerr ((4)) has exhibited a certain vacuum solution possessing features that one might expect of a space-time exterior to some rotating body. Throughout this paper we shall have Kerr's solution in mind. The question that we shall keep before us is whether a perfect fluid interior can be matched to any given exterior field. Our main results exhibit the class of all possible fluid boundaries, given the exterior field, and some relations between the pressure, density, 4-velocity, and interior metric tensor.

In this paper I shall demonstrate certain algebraic properties of ideals of finite homological dimension in local rings.

In the first section, I show that no non-zero ideal of finite homological dimension in a local ring can be of zero grade (this is stated by Auslander and Buchsbaum in the appendix to (l), but I cannot find a proof in the literature). Using this result, together with the complex defined by a matrix, which is described by Eagon and Northcott in (2), I prove that an ideal of homological dimension one in a local ring Q may always, for some integer n, be described as the set of determinants of matrices obtained by adjoining to a certain (n–l)× n matrix with elements in the maximal ideal of Q another row with elements arbitrarily chosen in Q. (This result was established in (3), under the additional condition that Q should be a domain.)

Penrose (4) gave a necessary and sufficient condition for the consistency of the linear matrix equation AXB = C and also its complete class of solutions. A necessary and sufficient condition for the equations AX = C, XB = D to have a common solution was given by Cecioni (3) and an expression for the general common solution by Rao and Mitra ((6), p. 25). In the present paper, we obtain a necessary and sufficient condition for the equations A1XB1 = C1 and A2XB2 = C2 to have a common solution and also an expression for the general common solution. This result isuseful in computing a constrained inverse of a matrix, a concept originallyintroduced by Bott and Duffin(2) and recently extended by Rao and Mitra(7) who consider more general constraints with the object of bringing together the various generalized inverses and pseudoinverses under a common classification scheme.

In fundamental papers Bernstein (3) and Loève(8) have proved central limit theorems for wide classes of dependent variables. Their theorems are stated in terms of conditional distributions. In the case of dn-dependent variables (see § 3) they assume the existence, as the ‘conditioning’ variates vary, of finite upper bounds for certain conditional absolute moments higher than the second. More recently, Hoeffding and Robbins (7) have proved central limit theorems for m-dependent variables with finite third absolute moments, and Moran(10) has given a direct generalization of the Lindeberg-Lévy theorem for stationary discrete linear scalar processes.