The geometrical configuration consisting of five straight lines tangent to the stream line (ψ = const.), isobar (p = const.), isocline (θ = const.), isovel (q = const.) and isopycnic (ρ = const.) at a point P of the physical plane in the plane flow of a fluid has a number of interesting properties. The study of this geometric configuration for a wide class of ideal fluids (which includes polytropic gases and incompressible fluids), in particular its properties at sonic points, is the purpose of this paper.

The classical theory of electromagnetism with both electric and magnetic charges is discussed. Following Gamblin(l) we proceed from an action principle and investigate the resulting particle equations. A consistent theory in which electrically and magnetically charged particles interact amongst themselves and with each other, without any constraints, is shown to be possible, in contrast to the conclusions of Gamblin. The introduction of constraints is considered.

I. Lattice points in the domain | x |α + | y |α ≤ 1

Theorem 4. Let G be the star domain

where α > 0. Then, when α tends to zero,

Proof. The linear substitution

changes G into the similar domain

and so

.

This paper continues the study initiated in [9] of the Markov branching process (MBP) with a density-independent catastrophe component. The population size process (Xt) was investigated in the subcritical and critical cases in [11], and the present paper is concerned with a parallel investigation of the supercritical case. We refer the reader to [11] for definitions and notation, and we will not repeat them here.

Considerable advances have been made in recent years in our knowledge of the γ-ray emission of radioactive bodies. It has been established quite definitely that series of definite frequencies are emitted, forming a characteristic line spectrum of the nucleus, and methods of measuring the frequencies have been devised so that at present the γ-ray spectra of most of the radioactive bodies are well known.

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

The canonical form of the Riemann tensor is found for certain non-flat empty space-times. Some invariant physical properties of these space-times are investigated. The status of Mach's principle in general relativity theory is briefly discussed in the light of these examples.

When the differential analyser (1) is being used it is often necessary to supply to the machine information in the form of a functional relation between variables occurring in the equation. This is usually done by keeping a pointer on a graph on an input table. The x-coordinate lead screw of the input table is driven by the machine, and the y-coordinate lead screw is rotated by hand so as to keep the pointer on the given curve. The disadvantages of this method, namely the necessity for the continuous attention of an operator and the possibility of personal errors, would be obviated by the use of an automatic follower instead of a hand control.

In this paper we consider the probability distribution function of a Gibbs measure supported on a self-conformal set given by an iterated function system (devil's staircase) applied to a compact subset of ${\mathbb R}$. We use thermodynamic multifractal formalism to calculate the Hausdorff dimension of the sets Sα0, Sα∞ and Sα, the set of points at which this function has, respectively, Hölder derivative 0, ∞ or no derivative in the general sense. This extends recent work by Darst, Dekking, Falconer, Kesseböhmer and Stratmann, and Yao, Zhang and Li by considering arbitrary such Gibbs measures given by a potential function independent of the geometric potential.

An ordinary differential equation for Hermite-Bell polynomials is derived. Bell's error concerning orthogonality of his polynomials is corrected.

Suppose that $P(D)$ is a linear differential operator with complex coefficients and $\{f_k\}_{k \in \mathbb{Z}}$ is a two-sided sequence of complex-valued functions on $\mathbb{R}$ such that $f_{k+1} \,{=}\, P(D)f_k$ with the following growth condition: there exist $a\,{\in}\,[0,1)$ and $N\,{\geq}\,0$ such that $|f_k(x)|\,{\leq}\,M_{|k|}\exp (N|x|^a)$, where the sequence $\{M_k\}_{k=0}^{\infty}$ satisfies that for any $\varepsilon\,{>}\,0$, the sequence $\{{M_k}/{(1+\varepsilon)^{k}}\}_{k=0}^{\infty}$ has a bounded subsequence. Then $f_0$ is an entire function of an exponential growth. This result is a generalization of those of Roe, Burkill and Howard.

In this paper we shall consider contractible 2-dimensional cell complexes K with one 0-cell v, one l-cell a, and one 2-cell D, where the boundary of D is attached to a by a word in a and a−1. In order for such a complex K to be contractible, the attaching word in a and a−1 must have total degree ± 1. The simplest example occurs when the attaching word is a; in this case K is a 2-cell, and is of no interest since K is collapsible. If any other word of degree ± 1 is used, the resulting K is contractible but not collapsible.

Let f(t) be defined and measurable for t > 0, and suppose that it has non-vanishing moment constants

where the integrals are Cauchy-Lebesgue integrals, or possibly if f(t) is null for t > t0 but not for t < t0 (a convention to which we adhere throughout).

The differential equation

defining the function of the amplitude sL is integrated by an iterative procedure. An expression for the phase

is obtained in terms of integrals of ωL and its derivatives. The method is applied to the numerical calculation of Coulomb functions and the accuracy of the results is discussed. Recurrence relations are used to obtain øL, L ≽ 1.

We compute the asymptotic behaviour of the number of partitions of large vectors (n1, n2) of ℤ+2 in the critical regime n1 ≍ √n2 and in the subcritical regime n1 = o(√n2). This work completes the results established in the fifties by Auluck, Nanda and Wright.

In the construction of the seismological tables it has been found that the times of several pulses at a given distance show a well-marked maximum frequency for each, about which the bulk of the observations are spread with a standard deviation of 1·5−3 sec. For others there was only a vague concentration, spread over about half a minute (Jeffreys and Bullen, 1936–40). The theoretical times of the latter type could all be calculated from those of the former by using the principle of stationary time for small variations of the path, and the observed times might be either shorter or longer than the theoretical ones. There was a complete association between the type of distribution of the residuals and the nature of the path. Whenever the time as calculated was a minimum for all small variations of the path, the distribution of residuals was of the former type; when it was stationary without being either a true minimum or a maximum it was of the latter.

A Haken sphere $S$ in a 3-manifold $M$ with a Heegaard splitting $V\cup_F W$ is a separating 2-sphere in $M$ which intersects the Heegaard surface $F$ only in an essential circle in $F$. We describe how any two Haken spheres in the genus 2 Heegaard splitting of connected sum of two lens spaces are related, which extends Scharlemann–Thompson's corresponding result on Haken spheres in genus 2 Heegaard splitting for $S^3$.

The Atiyah–Singer equivariant signature formula implies that the products of isometrically inequivalent classical spherical space forms with the circle are not homeomorphic, and in fact the same conclusion holds if the circle is replaced by a torus of arbitrary dimension. These results are important in the study of group actions on manifolds. Algebraic $K$-theory yields standard classes of counterexamples for topological and smooth analogs of spherical spaceforms. The results of this paper characterize pairs of nonhomeomorphic topological spherical space forms whose products with a given torus of arbitrary dimension are homeomorphic, and the main result is that the known counterexamples are the only ones that exist. In particular, this and basic results in lower algebraic $K$-theory show that if such products are homeomorphic, then the products are already homeomorphic if one uses a 3-dimensional torus. Sharper results are established for important special cases such as fake lens spaces. The methods are basically surgery-theoretic with some input from homotopy theory. One consequence is the existence of new infinite families of manifolds in all dimensions greater than three such that the squares of the manifolds are homeomorphic although the manifolds themselves are not. Analogous results are obtained in the smooth category.

In (general) elastico-viscous liquids the response to stress at any instant will depend on previous rheological history, the equations of state needed to describe the rheological properties of a typical material element at any instant t being expressible in the form of a (properly invariant†) set of integro-differential equations relating its deformation, stress and temperature histories (as defined by a metric tensor (of a convected frame of reference), a stress tensor and the temperature measured in the element as functions of previous time t'( < t)) together with the time lag (t – t') and physical constant tensors associated with the element (1). Thus in any type of oscillatory motion a rheological history will necessarily be a function of the frequency of the forcing agent, the rheological history of a number of different types of elastico-viscous liquids in some simple shearing oscillatory flows being a rather simple oscillatory history (see, for example, (2–4)). It is, therefore, to be expected that a liquid with elastic properties will behave somewhat differently from any inelastic viscous liquid when subjected to any kind of oscillatory motion, and it is for this reason that oscillatory motions have been used extensively to detect and measure the elastic properties of liquids (see, for example, (2–5)).