We show that the asymptotic behaviour of the quantization error allows the definition of dimensions for probability distributions, the upper and the lower quantization dimension. These concepts fit into standard geometric measure theory, as the upper quantization dimension is always between the packing and the upper box-counting dimension, whereas the lower quantization dimension is between the Hausdorff and the lower box-counting dimension.

Geometric structure of classical field theory in Lagrangian formulation is investigated. Symmetry transformations with generators depending on higher-order derivatives are considered and the corresponding conservation laws are obtained.

If the components x1, x2,…, xk of a vector X have a non-singular multivariate normal distribution having a null vector of means and variance-covariance matrix Σ= σ2, the matrix R=[ρij] (where ρii = 1) is known in certain cases but σ2 is unknown. If s2 is an estimate of σ2 based on ƒ degrees of freedom and is distributed independently of X, the distribution of the vector t=x/s is known as the multivariate t-distribution. This distribution was first obtained by Dunnett and Sobel (6) and independently by Cornish (3). Dunnett, Sobel and Bechhofer(2) have discussed some practical applications of this distribution. Cornish (3) obtained this distribution while considering the pre-treatment to be given to certain types of replicated experiments. This distribution possesses some useful properties and makes it suitable as a basis for exact tests of significance in various problems, and Dunnett and Sobel (6), by providing tables of the probability integral, have taken the first step towards its use in practice. Cornish, in a later paper (4) considered the sampling distribution of statistics derived from the multivariate t-distribution and using this he obtained the well-known ((7), (8)) distribution of the sample regression coefficient of one variate with respect to another, when both have a bivariate normal distribution.

The object of this note is to give an example of an infinite locally finite p-group which has no proper characteristic subgroup except the unit group. (A group G is a locally finite p-group if every finite set of elements of G generates a subgroup of finite order equal to a power of the prime p.) It is known that an infinite locally finite p-group cannot be simple, for if it were it would satisfy the minimal condition for normal subgroups, and so have a non-trivial centre (see(1)). However our example shows that it can be characteristically-simple. Examples are known of locally finite p-groups with trivial centre ((2), (4)), and of locally finite p-groups coinciding with their commutator groups ((1), (5)). Since the centre and commutator subgroup of a group are characteristic subgroups our example will have both of these properties. We may remark that the direct product of a simple, or even of a characteristically-simple group with itself any number of times is also characteristically-simple, but by Corollary 2.1 our group cannot be so decomposed.

In [2] it was conjectured that the coloured Jones function of a framed knot K, or equivalently the Jones polynomials of all parallels of K, is sufficient to determine the Alexander polynomial of K. An explicit formula was proposed in terms of the power series expansion , where JK, k(h) is the SU(2)q quantum invariant of K when coloured by the irreducible module of dimension k, and q = eh is the quantum group parameter.

In this paper I show that the explicit formula does give the Alexander polynomial when K is any torus knot.

Multisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.

Dirac has suggested that the quantization of electric charge could be explained by the existence of magnetic monopoles. In view of this hypothesis, this paper investigates what theoretically would be the behaviour of such monopoles in a Wilson cloud chamber. The treatment, which for simplicity is basically classical, closely follows Bohr's work on the decrease of velocity and ionization properties of α- and β-particles, and expressions are derived for the rate of decrease of energy and the number of ion-pairs produced per centimetre by a monopole passing through a gas. These expressions are then discussed with particular reference to the case of heavy particles, and the main differences between them and the corresponding expressions for α-particles both as to range and ionization are indicated; these differences can be summarized by saying that monopoles have much shorter paths, but create many more ion-pairs per centimetre than α-particles. Also, the very sharp increase in the ionization at the end of the path of an electric particle is missing, the ionization for the monopole decreasing to a small amount near the end of the path.

1. Introduction. All spaces in this paper are Hausdorff. We recall that a space X is sequentially compact, if every countable subset of X contains a convergent sequence. Let us consider the three statements:

(1) Every compact space of cardinality ≤ ʗ contains a point of countable character.

(2) Every compact space of cardinality ≤ ʗ is sequentially compact.

(3) Every infinite compact space of cardinality ≤ ʗ contains a convergent sequence.

Nous ne considérerons, en général, que les sous-ensembles de l'intervalle [0, 1] et nous désignerons par et respectivement l'ensemble des nombres rationnels et l'ensemble des nombres irrationnels de cet intervalle.

Nous allons employer pour les fractions continues la notation

Étant donnée une suite de nombres naturels {zn}, nous dirons que le nombre correspond à la suite {zn} et vice versa, et nous écrironns z = v({zn}). Pareillement, nous dirons qu'un ensemble correspond à une famille Φ de suites de nombres naturels si l'on a E = v(Φ).

The ordinary theory of stability of plane parallel flows is considerably simplified by a result due to Squire (2) which says that if a velocity profile becomes unstable to a small three-dimensional disturbance at a given Reynolds number, then it will become unstable to a small two-dimensional disturbance at a lower Reynolds number. This result enables us to restrict investigation of the stability to the cases of two-dimensional disturbances.

It has been found by one of the authors (1) in collaboration with Dr H. Jones that a flow of heat in liquid He ii is accompanied by what seems to be a transfer of momentum. The effect can be seen when the channel through which the heat and liquid flow consists of a smooth-walled glass capillary, such as shown in Fig. 1a. Due to the high thermal conductivity of He ii, a considerable part of the heat put into the reservoir is carried down through the capillary to the bath. When a steady heat flow exists, a flow of liquid takes place in the opposite direction, and the level of the liquid in the reservoir is seen to be higher than that in the bath. Smooth capillaries, however, produce a rise in level of only 1 or 2 cm. at most, since the viscosity of the liquid is small and hydrostatic pressure pulls the accumulated liquid in the reservoir back through the capillary. When the heat flow is large, violent surging is observed in the reservoir, but there is no further rise in level.

Previous theories of the recombination of ions in gases are shown to be inapplicable to high pressures. An approximate quantitative theory which takes into account all the relevant phenomena is developed. The effect of both thermal agitation and mutual attraction in determining encounters of ions of opposite sign is considered, and the frequency of encounters which lead to recombination calculated. The criterion for an encounter to lead to recombination is that the drift of the ions towards each other due to their mutual attraction must be greater than their tendency to separate due to their Brownian movement. Excellent agreement is obtained with such experimental results as are available. Preferential recombination between an ejected electron and its parent ion is also discussed.

I am indebted to various members both of the Wills Physics Laboratory and of the Cavendish Laboratory for discussion of the problems dealt with in this paper.

During the course of recent work (6) on the zeros of the Bessel functions Jn(x) and Yn(x), it became evident that the theory of the asymptotic expansion of Bessel functions whose arguments and orders are of comparable magnitudes was incomplete. The existing expansions for large orders are those of Debye and Meissel, detailed derivations of both of which are given by Watson ((8), pp. 237–48).

The theoretical transition between basic properties of elastoplastic media at two levels of description is examined rigorously. At the micro-level the material response is heterogeneous, whereas at the macro-level it appears homogeneous. A broad class of constitutive relations is envisaged, and no restriction is placed on the magnitude of deformations and rotations at the micro-level. The investigation is concerned with quadratic differential forms that feature prominently in constitutive analyses, and is complementary to a previous study of bilinear differential forms. A principal objective is to access the transmissibility of measure-invariant inequalities from one level to the other.

As Armitage(1) has noted, this is a problem of some importance in dust particle counting where overlapping particles are usually counted as single particles owing to their small size and the difficulty of detecting overlapping. In his paper Armitage(1) obtained approximate formulae and also expansions for the expected number of clumps (as he called a self-overlapping group) (i) when the particles were all circular laminae (though of different sizes) and (ii) when the particles were similar rectangular laminae randomly oriented. His expansions were accurate to the first two terms, but he only gave approximate values for the third terms. However, by using a method similar to that of the author (5,6) in treating aggregates in random placings of points and to that of Hammersley (4) in treating one-dimensional problems analogous to those dealt with here, this paper gives exact formulae for any mixture of laminae of different shapes and sizes, the only restrictions being (i) that they are all convex (though they need not be symmetrical), (ii) that they are randomly oriented with no preferred direction, and (iii) that there are a large number of each kind. With these restrictions we shall show in addition that, surprisingly, the only relevant geometrical properties of the laminae are (i) their areas and (ii) their perimeter lengths, their shapes apart from this being immaterial.

We provide a new proof of the following results of H. Schubert: if $K$ is a satellite knot with companion $J$ and pattern $(\skew1\hat{V}, L)$ with index $k$, then the bridge numbers satisfy the following: $b(K) \geq k \cdot (b(J))$. In addition, if $K$ is a composite knot with summands $J$ and $L$, then $b(K) = b(J) + b(L) - 1.$

Suppose that events occur at random points on a line from t = −∞ to +∞, the probability of an event occurring between t and t + dt being λdt. If we select any interval of the line, say the interval [0, y], there will be a finite probability that it contains 0, 1, 2,…,r,…events; in fact, it is not difficult to show that these probabilities form a Poisson distribution, the probability that the interval contains r events being (see e.g. (1)). Consider the case when each event consists of an interval of length α (an event being characterized by its first point). What is the probability that the covered portion of the interval [0, y] lies between x and x + dx?

The behaviour of the conduction electrons in a vibrating metallic lattice is treated by the Born-Oppenheimer method. Non-adiabatic processes, due to certain terms usually neglected, are shown to correspond to scattering of electrons by the lattice vibrations, as in Bloch's theory of electrical resistivity, but the matrix elements are not the same as Bloch's except when energy is conserved in the process. This may have important consequences in Fröhlich's theory of superconductivity.

The spaces Lp(, φ) for 1 ≤ p ≤ ∞, where φ is a faithful semifinite normal trace on a von Neumann algebra , are defined in (10),(2),(14). The problem of determining the general form of an isometry of one such space into another has been studied in (i), (6), (9), (12), (5). Our main result, Theorem 2, is a characterization of such isometries for 1 ≤ p ≤ ∞, ≠ 2. The method of proof is based on that of (7), where isometries between Lp function spaces are characterized. The main step in the proof is Theorem 1, which gives the conditions under which equality holds in Clarkson's inequality.

Let Top be the category of compactly generated topological spaces and continuous maps. The category, Top, can be given the structure of a simplicially enriched category (or S-category, S being the category of simplicial sets). For A a small category, Vogt (in [22]) constructed a category, Coh (A, Top), of homotopy coherent A-indexed diagrams in Top and homotopy classes of homotopy coherent maps, and proved a theorem identifying this as being equivalent to Ho (TopA), the category obtained from the category of commutative A-indexed diagrams by localizing with respect to the level homotopy equivalences. Thus one of the important consequences of Vogt's result is that it provides concrete coherent models for the formal composites of maps and formal inverses of level homotopy equivalences which are the maps in Ho (TopA). The usefulness of such models and in general of Vogt's results is shown in the series of notes [14–17] by the second author in which those results are applied to give an obstruction theory applicable in prohomotopy theory.