We consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is qn. We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1, 2, …, b, where b = O(n/log n), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a Poisson-Dirichiet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.

The distribution of the total (or ‘second’) curvature of a stationary random Gaussian surface is derived on the assumption that the squares of the surface slopes are negligible. The distribution is found to depend on only two parameters, derivable from the fourth moments of the energy spectrum of the surface. Each distribution function satisfies a linear differential equation of the third order, and the distribution is asymmetrical with positive skewness, in general. A special case of zero skewness occurs when the surface consists of two intersecting systems of long-crested waves.

We prove an existence theorem for a steady planar flow of an ideal fluid past an obstacle, containing a bounded symmetric vortex pair and approaching a uniform flow at infinity. The vorticity is a rearrangement of a prescribed function.

The stream function ψ for the flow satisfies the equation

−Δψ=ϕ∘ψ

in a region bounded by the line of symmetry, where ϕ is an increasing function that is unknown a priori.

The result is obtained from a variational principle in which a functional related to the kinetic energy is maximised over all flows whose vorticity fields are rearrangements of a prescribed function.

We define equivariant Chern–Schwartz–MacPherson classes of a possibly singular algebraic $G$-variety over the base field $\mathbb{C}$, or more generally over a field of characteristic 0. In fact, we construct a natural transformation $C^G_*$ from the $G$-equivariant constructible function functor $\cal{F}^G$ to the $G$-equivariant homology functor $H^G_*$ or $A^G_*$ (in the sense of Totaro–Edidin–Graham). This $C^G_*$ may be regarded as MacPherson's transformation for (certain) quotient stacks. The Verdier–Riemann–Roch formula takes a key role throughout.

An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).

A considerable volume of knowledge is now available on random fluctuations (noise) as regards the behaviour in amplitude. Familiar names in this field are those of Uhlenbeck and Ornstein(7), Fürth(1) and Rice (4), although very many others have made valuable contributions. A particular class of problem, of considerable practical importance, exists when the frequency spectrum is limited to a relatively narrow range. The resulting noise has then the character of a more or less regular oscillation modulated randomly in amplitude and phase. In this case, if we write the fluctuation in the form

(where R(t) and θ(t) are variables changing slowly in comparison with sinω0t), it is clear that the magnitude of the envelope R(t) and the phase θ(t) are now the significant quantities. Rice (4), among others, has made a study of the statistical properties of R, deriving in particular the correlation function R(t) R(t + τ) in terms of the characteristics of the (power) spectrum ω(f). Fürth and the writer (2) have extended this work and carried out a collateral experimental investigation.

A pair of identical circular discs, held at equal and opposite potentials, forms a condenser whose capacity C depends on the ratio ε of separation against diameter. The determination of an asymptotic expansion for C when ε is small poses an axisymmetric boundary-value problem for harmonic functions that has engaged the attention of numerous investigators over a long span of time. It is a simple matter to construct a Fredholm integral equation of the first kind for the charge density ± σ on the discs, in terms of which the potential field and the capacity are implicitly determined, but the equation is unsuitable if ε ≪ 1. Integral equations of the second kind and of the dual variety have also been proposed as a means of securing a more manageable formulation of the boundary-value problem. An elementary approximation follows from the hypothesis that the charge density is almost the same as though the discs were of infinite extent, except for a region close to the edges, and leads to the result C ∼ l/8ε as ε → 0. Kirchhoff considerably improved on this crude estimate by suggesting a plausible edge correction which yields two further terms for C, of orders log ∈ and a constant, respectively, and his results have been rigorously established by the more refined analysis of Hutson. In the present work an integral equation of the first kind for the distribution of potential off the discs is derived and utilized to obtain an approximation for C when ε is small, reproducing the result of Kirchhoff and Hutson. Furthermore, an estimate of the error provides explicit details regarding the next term in the asymptotic expansion of C, which is of the order ε(log ε)2.

An exact solution is given for the slow viscous flow caused by the translation of two equal spheres in contact with equal velocities perpendicular to their line of centres. An asymptotic theory is presented for solving the problem when two equal spheres of radius a almost in contact rotate with equal and opposite angular velocities. The problem when the spheres touch is shown not to be well posed; the forces acting on the spheres are shown to be O(1) and the couples of the form α log∈ + β where α and β are independent of the minimum clearance 2∈α between the spheres and have been determined explicitly. The relevance of the results to the free settling of two spheres in a viscous fluid under the influence of gravity is discussed.

My aim in this paper is to give an abstract characterization of the C∞ spaces described in (6) or (9), and to develop some of the remarkable special properties of these spaces. Although the subject is in some ways highly specialized, inextensible and sequentially inextensible spaces seem common enough (they include all spaces of the forms Rx and L0) to be worth studying, and I have already employed them in the proof of more general results (1).

In the first section I set out those properties that can be described in simple Riesz space terms; much of this work has already been published in slightly different forms. In the second part I go on to questions that arise when we impose a topology on an inextensible Riesz space. Finally, in the third section, I discuss some problems, arising from the work before, which are related to the famous measurable cardinal problem.

A high-speed recording counter circuit is described which has proved very reliable. The basic circuit element employs two triodes in a symmetrical circuit which has two stable conditions. The circuit is triggered alternately from one stable state to the other by the applied impulses acting through a circuit comprising two “Westector” cuprous oxide rectifiers and an inductance.

It is easily possible to make a counter operate accurately at a speed of 20,000 random impulses a minute, provided that the associated recording meter operates in less than 1/100 sec.

A “watch-dog” thyratron recording circuit is also described. This is useful where the slow speed of the mechanical recording meter would introduce loss.

We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the domain 0 < x < L, 0 < t < T, with L and T positive finite constants. We present a general method for identifying well-posed problems, as well as for constructing an explicit representation of the solution of such problems. This representation has explicit x and t dependence, and it consists of an integral in the k-complex plane and of a discrete sum. As illustrative examples we solve some two-point boundary value problems for the equations iqt + qxx = 0 and qt + qxxx = 0.

1. During the past decade a variety of integral transformations have been extended to various classes of distributions. However, one important integral transformation, which has defied such generalization, is the Kontorovich–Lebedev transformation. The objective of this note is to alter that situation.

In this paper we obtain a representation of elastic wave fields (i.e. solutions of the equations of motion of classical elastokinetics) in terms of three functions satisfying scalar wave equations. Although the form of the representation implies no restriction upon the choice of coordinate system or upon the shape of the elastic body, it is found that the result can only be applied advantageously to initial-boundary-value problems having spherical polar coordinates as a natural frame of reference. The representation is shown to be complete in the sense that every (sufficiently smooth) elastic wave field in a homogeneous, isotropic body bounded by two concentric spheres can be expressed in the given form.

Under conditions of axial symmetry the representation generates wave fields which can be decomposed into poloidal and toroidal constituents, the former arising from two scalar wave functions and comprising both dilatational and rotational waves, and the latter being associated with a single scalar wave function and a state of pure shear of the elastic solid. Finally, the representation is used to obtain a formal solution describing the elastic pulse generated in an infinite body by the application of time-dependent tractions to the surface of a spherical cavity.

The nth symmetric product of a topological space, X, is defined to be the quotient of the Cartesian product Xn by the action of the symmetric group which permutes the factors. Even if X is a manifold, this product is, in general, not a manifold. The purpose of this note is to determine these products when X is the circle, S1, and to show that they are manifolds with boundary.

The Penrose transform is an integral geometric method of interpreting elements of various analytic cohomology groups on open subsets of complex projective 3-space as solutions of linear differential equations on the Grassmannian of 2-planes in 4-space. The motivation for such a transform comes from interpreting this Grassmannian as the complexification of the conformal compactification of Minkowski space, the differential equations being the massless field equations of various helicities. This transform is a cornerstone of twistor theory [22, 24, 30], but the methods generalize considerably as will be explained in this article. Closely related is the Ward correspondence[28], a non-linear version of a special case of the Penrose transform. It also admits a rather more general treatment. The object of this article is to explain the general case and the natural connection between the Penrose and Ward approaches.

It is a well-known fact that the non-conservation of energy during β decay may be explained by assuming the existence of light neutral particles, neutrinos, which are ejected from the nucleus at every act of β emission and which enable all the conservation laws to be fulfilled. A study of the energy distribution of recoil atoms during β decay might furnish some data for testing this hypothesis. However, in order to determine the energy distribution of recoil atoms consider-able experimental difficulties have to be overcome. The energy of recoil atoms of ordinary radioactive substances is of the order of magnitude of 1 volt, i.e. of the same order of magnitude as the adsorption energy of atoms on a surface. Thus, on escaping from the'surface on which they are adsorbed, they will lose an amount of energy of the same order of magnitude as that which they possess, and their velocity distribution will therefore be completely distorted. This difficulty may be met by employing an artificial, light radioactive substance, whose recoil atoms may have energies some ten times greater than 1 volt. For this reason active carbon C, obtained by bombarding boron with deuterons according to the reaction 10B+2H→11C+n, was employed as a radioactive substance. Moreover, the fact that the emission of positrons from 11C is not accompanied by the emission of γ-rays has the advantage of simplifying the interpretation of the results.

An inverse AI for an arbitrary matrix A was first given by Moore (4). Since the application to solution of linear equations only depended on the property A AI A = A, Bjerhammar (2) used this equation to define the set of generalized inverses AI. If A is regular, then only the regular inverse A−1 satisfies this definition. If A is a generalized inverse of AI, so that AI = AI AAI, then AI is a reciprocal inverse.

It is generally agreed that instants are mathematical constructions, not physical entities. If, therefore, there are instants, they must be classes of events having certain properties. For reasons explained in Our Knowledge of the External World, pp. 116–20, an instant is most naturally defined as a group of events having the following two properties:

(1) Any two members of the group overlap in time, i.e. neither is wholly before the other.

The object of this paper is to investigate the influence of the initial values on the behaviour of the one-dimensional unsteady isentropic motion of a perfect gas, and in particular the occurrence of singularities. For this purpose it is expedient to develop an unfolding procedure in the theory of partial differential equations of hyperbolic type. This not only resolves a difficulty in the classical treatment of the initial value problem associated with this type of equation, but also simplifies the presentation.

The geometrical properties of the singularities (branch and limit lines, edges of regression, etc.) occurring in two-dimensional steady flow have been well discussed already, and their counterparts in one-dimensional unsteady motion have been indicated by Stocker and Meyer(5). However, here a method for finding the explicit analytic connexion between the occurrence of such singularities and the prescribed initial conditions is developed.

Assuming quite general initial conditions, two examples are considered. In the first, a conjecture of Riemann is verified; in the second a special type of periodic solution is discussed, and it is shown that this solution cannot be continued to all values of t > 0. Finally, this latter result is shown to be true for arbitrary periodic initial conditions.