A tree is a connected graph that has no cycles. If x is any endnode of a tree, then the limb determined by x is the unique path that joins x with the nearest node other than x that does not have degree two in the tree; let l(x) denote the length of this path. (For definitions and results not given here see [2] or [3].) Different endnodes determine different limbs with one exception; when the tree is a path then both endnodes determine the same limb, namely, the tree itself. Our object here is to investigate the distribution of the length of limbs of trees Tn chosen at random from the set of nn-2 trees with n labelled nodes; in particular, it will follow from our results that the length of the longest limb in most trees Tn is approximately log n when n is large.

This paper uses systems of image sources to construct suitable generalised Green's functions for considering small amplitude short surface waves due to an oscillating immersed sphere. A sphere of radius a is half-immersed in a fluid under gravity and is making vertical oscillations of small constant amplitude and period 2π/σ about this position. It is required to find the fluid motion, and in particular the virtual mass and wavemaking coefficients. For sufficiently small amplitudes the motion depends non-trivially on only a single dimensionless parameter

where g is the gravitational acceleration. This was shown by Ursell in an unpublished U. S. Navy report in which the methods of an earlier paper (Ursell, 1949) were adapted from cylindrical to spherical symmetry and resolved the conflict between Havelock (1955) and Barakat (1962) in favour of the former. Ursell showed that the virtual mass coefficient is

i.e. infinitely increasing initially so that Barakat's results must be incorrect near N = 0. However, although existence is proved for all N, the same difficulty arises as with the heaving cylinder, namely computation is only practical for values of N up to about 1. In the cylindrical case, Ursell (1953) developed a method for finding the asymptotic solution for large N and here it will be adapted, surprisingly perhaps, to deal with the spherical case. Ursell (1954) then published a formal solution which gave the same virtual mass as the rigorous treatment and if this formal method is applied to the heaving sphere, the virtual mass coefficient obtained is

Every n-dimensional manifold admits an embedding in R2n by the result of H. Whitney [11]. Lie groups are parallelizable and so by the theorem of M. W. Hirsch [5] there is an immersion of any Lie group in codimension one. However no general theorem is known which asserts that a parallelizable manifold embeds in Euclidean space of dimension less than 2n. Here we give a method for constructing smooth embeddings of compact Lie groups in Euclidean space. The construction is a fairly direct one using the geometry of the Lie group, and works very well in some cases. It does not give reasonable results for the group Spin (n) except for low values of n. We also give a method for constructing some embeddings of Spin (n), this uses the embedding of SO(n) that was constructed by the general method and an embedding theorem of A. Haefliger [3]. Although this is a very ad hoc method, it has some interest as it seems to be the first application of Haefliger's theorem which gives embedding results appreciably below twice the dimension of the manifold. The motivation for this work was to throw some light on the problem of the existence of low codimensional embeddings of parallelizable manifolds.

In a recent paper [1] I made the following

Definition. The function f: ℤ+ ∪ {0} → ℤ is a pseudo-polynomial if

for all integers n ≥ 0, k ≥ 1.

Let x1, …, xn be linear forms in u1, …, un with real coefficients, and (for simplicity) determinant 1. Given a form (that is, a homogeneous polynomial) F(x1, …, xn), we can ask the following question: do there exist, for arbitrary real α1, …, αn, integers ul, …, un such that

where C is a suitable number independent of α1, …, αn and of the particular linear forms x1 …, xn? In two well-known cases this is true: namely when

and when

where 1 ≤ r ≤ n − 1.

If L is a set of disjoint closed line segments in En, let E(L) denote the end set of L, i.e. the set of end points of members of L.

In [1, 2] V. L. Klee and M. Martin proved the following lemma:

IfLis a disjoint set of closed line segments in E2 such that E(L) is compact, then E(L) has zero 2-measure.

This note examines one particular feature of the motion produced by an infinitely long insulating circular cylinder moving parallel to its axis in a conducting fluid permeated by a uniform transverse magnetic field. The particular aspect examined is that of determining, for large values of the Hartmann number, the flow in the neighbourhood of those points on the cylinder where the applied field is tangential.

The problem of determining the fields in these regions is very similar to that of calculating the acoustic potential in the neighbourhood of glancing incidence for a plane wave normally incident on a sound soft circular cylinder. An elegant method of treating this latter problem has been given by Jones [1] and the purpose of this note is to indicate the way in which Jones' method has to be modified to treat the magnetohydrodynamic problem. The basis of the approach is to determine a solution which satisfies the boundary conditions in the tangential regions and this can be achieved by conventional integral representation methods.

If K is a set in n-dimensional Euclidean space En, n ≥ 2, with a non-empty interior, then a point p of the interior of K is called a pseudo centre of K provided each two-dimensional flat through p intersects K in a section centrally symmetric about some point, not necessarily coinciding with p. A pseudo centre p of K is called a false centre if K is not centrally symmetric about p. Rogers [5] showed that a convex body (compact convex set with interior points) with a pseudo centre necessarily has a true centre of symmetry. But, as each interior point of an ellipsoid is a pseudo centre, the true centre need not necessarily coincide with the pseudo centre. Rogers conjectured that, for n ≥ 3, a convex body K with a false centre is necessarily an ellipsoid. In this paper we prove this conjecture.

This note is concerned with those linear viscoelastic materials which are known to be compatible with thermodynamics in a sense to be denned below. It is proved that equality of its instantaneous and equilibrium elastic moduli is a necessary and sufficient condition for such a material to be elastic.

2-Benzylidene-2,3-dihydrophenalen-i-one with ethyl acetoacetate and sodium ethoxide yields four products depending on the concentration of the alkali. One of these, ethyl 7a,8,9,10-tetrahydro-10-oxo-8- phenyl-7H-benz[de]-anthracene-9-carboxylate has been converted into 8-phenyl-7H-benz[de]anthracene and its reduction with sodium borohydride or hydrogen in the presence of platinum oxide or Raney nickel presents points of interest.

Formulae for the calculation of the stress intensity factor at the tip of a Griffith crack and for the normal component of the surface displacement are derived for a stressfree crack in an elastic solid in which there is an asymmetrical distribution of body forces. Particular distributions of point forces are considered in detail.

The effect of the presence of singularities on the method of stationary phase for a certain integral of two variables is examined. The singularities lie on two straight lines whose positions depend upon parameters. For certain values of the parameters the straight lines can pass through the point of stationary phase.

The asymptotic development, which is uniform with respect to the parameters, is determined. It is found that it can be expressed in terms of a function which, in certain circumstances, reduces to a Fresnel integral. The main properties of this function are derived in an appendix.

Explicit formulae are given for the dominant asymptotic terms in the cases when the point of stationary phase is approached by (i) neither line, (ii) only one of the lines, (iii) both lines.

On treatment with t-amyl-alcoholic sodium t-amyloxide, sulphones of the type TsCHAr.C6H4.NPhCH2R lose the elements of toluenesulphinic acid giving the Schiff base ArCH2.C6H4.N = CHR, which can be hydrolysed to ArCH2.C6H4.NH2.

We consider, for a given ordered set E with minimum element O, the semigroup Q of O-preserving isotone mappings on E and examine necessary and sufficient conditions under which an element fε Q is such that the left [resp. right] annihilator of f in Q is a principal left [resp. right] ideal of Q generated by a particular type of idempotent. The results obtained lead us to introduce the concept of a Baer assembly which we use to extend to the case of a semilattice the Baer semigroup co-ordinatization theory of lattices. We also derive a co-ordinatization of particular types of semilattice.

A theory of the random walk with “persistence” of movement of a point in a three-dimensional cubic lattice is presented from which explicit expressions for the moments of the distribution function for the displacements of an ensemble of points after N steps for any arbitrary initial average velocity are derived. The results are applied to the problem of small angle multiple scattering of particles on their passage through a material medium, and formulae for the mean square of the lateral displacements are obtained which, in first approximation, have the form of the expressions, generally used for evaluating the experimental results but, in higher approximation, indicate a deviation from this relationship for greater thickness of matter.

Another approach to the same problem of multiple scattering is further presented which is based on Kramer's stochastic differential equation for the distribution function for the position and velocities of an ensemble of particles in phase space. By this method formulae for the mean square of the scatter angles, the lateral displacements and the correlation products between these are derived. The first of these expressions shows again characteristic deviations from the usual ones for greater thickness of matter, the second coincides essentially with the expression obtained from the random walk theory.

Models which treat molecules as graphs provide the first approximation to practically all chemical and physical theories of polymers. This paper, worded so as to aim at intelligibility by both chemists and mathematicians, deals with statistical effects of correlations on distributions of molecules in their graph-like states. The correlations are formally equivalent to fertility correlations in family trees, and the theory covers the range of correlation to the point where a man's fertility expectations depend on how many brothers he has. Chemically, this describes the ‘second shell substitution effect’ which includes certain steric hindrance situations; mathematically, the problem is solved by restoring the Markovian nature of a conventional Galton-Watson process. Long-range correlations are an inevitable and troublesome aspect of the structure of gels and this work represents a step towards improving models for real gels.

p-Toluenesulphinic acid, with formaldehyde or an aromatic aldehyde, attack the nucleus of a secondary aromatic amine, giving a 4-alkylamino-benzyl (or -benzhydryl) p-tolyl sulphone.

§ 1. Let L2(0, ∞) denote the Hilbert space of Lebesgue measurable, integrable-square functions on the half-line [0, ∞).

Integral operators of the form

acting on the space L2 (0, ∞) occur in the theory of ordinary differential equations; see for example the book by E. C. Titchmarsh [4; § 2.6]. It is important to establish when operators of this kind are bounded; see the book by A. E. Taylor [3; § 4.1 and §§4.11, 4.12 and § 4.13].