1. In this paper we find explicitly the base for the prime ideal associated with any irreducible Vd−1 on a Segre variety, or a Veronesean variety, Vd. This work extends that of an earlier paper (1) in which the base was found when the Vd−1 is a complete intersection of Vd with one primal. As particular examples we can write down the base for any irreducible curve on a quadric surface, a Veronese surface or a Del Pezzo surface. We also show how the base for any prime ideal changes under a Veronesean transformation.

The present author recently published what he believed to be a proof and extension of Campbell's theorem on fluctuations. More recently a paper has been published containing a criticism of this proof. It therefore seems desirable that some parts of the proof be amplified. Further explanation of the foundations of the theory seems required, in order that it shall not appear that the theory ignores the randomness of the events, and a more detailed account of the derivation of the mean value of functions of pairs of events (i.e. sums of the type ) is necessary.

General methods of successive approximations for problems in the theory of large elastic deformations have been proposed by Signorini (9), Misicu (5), Rivlin (6), Green and Spratt (4) and Rivlin and Topakoglu (7). Rivlin has applied his method to the problem of torsion of cylinders as far as terms of the second order. A solution of the problem of second order effects in the torsion of incompressible cylinders of arbitrary cross-section has been given by Green and Shield (3) in terms of complex potential functions, and the complete details of the stress distribution can be obtained once certain integral equations are solved. Blackburn and Green (l) have considered the second order effects in the deformation of compressible cylinders by forces on the ends. For the problems of torsion and bending by couples they obtained details of the displacements and stress distributions in terms of two complex potential functions which satisfy a certain boundary condition. The second order torsion problem has also been discussed by Sheng(8).

This paper is concerned with the exponential sums

(formula here)

and with Heilbronn's type exponential sums

(formula here)

where (and below) p is prime, m, n, a and b are integers with m > n > 0 and p [nmid ] ab, and er(x) = exp(2πix/r) as usual.

Let p (x) be an essentially positive function defined in the interval 0 ≤ x ≤ π. We consider the non-linear partial differential equation

for the boundary conditions u (x, t) = 0,

for x = 0 and x = π,

An account is given here of the measurements of the mass absorption coefficients of iron, nickel, copper and aluminium over a wave-length range 0·705 Å.U. to 1.932 Å.U.

An investigation of the random or stochastic integral equations of the form

and

is presented, where ω ∈ Ω, the supporting set of the probability measure space (Ω, A, P). The existence and uniqueness of a random solution, a second-order stochastic process, of the equations is considered. Several theorems utilizing fixed point theorems and successive stochastic approximations give sufficient conditions for the existence of a random solution.

A recent result of the first two authors has bridged the gap between the Timantype pointwise estimate and the norm estimate for polynomial approximation in C[– 1, 1]. Here we show that, when considering the second order modulus of smoothness, the aforementioned result holds with polynomials that preserve monotonicity, convexity and the values of the function at the end points of the interval [–1, 1]. This yields a generalization of results by Telyakovskii, Gopengauz, DeVore, Yu and the third author. Moreover, the construction here does not depend on the construction for the non-constrained approximation by polynomials.

It was proved by Davenport (2) that a cubic form in n variables with rational coefficients always represents zero if n > 32; and this condition was later (3) relaxed to n ≥ 29. The object of the present paper is to establish conditions under which a cubic form will represent every rational number other than zero, for rational values of the variables. The main result is as follows.

It is a problem of considerable interest in the theory of surfaces to determine the irregular non-singular surface of minimum order, not referable to a scroll; in previous investigations the author has discussed the regularity or referability of surfaces in higher space, reaching the conclusion that all non-singular surfaces of order n ≤ 10 in S4 are regular or referable, with the possible exception of the surface of order n = 10 and sectional genus π = 6, which may be elliptic (pg = 0, pa = −1) or hyperelliptic (pg = 1, pa = −1). In their memoir on hyperelliptic surfaces, Enriques and Severi have obtained for the irregular hyperelliptic surface of general moduli a model 6F10 of minimum order, situated in S4, with the characters n = 10, π = 6. Using transcendental methods, Comessatti has constructed a class of irregular hyperelliptic surfaces the properties of which he has examined in detail; this class includes a member 6Π10 which is a special case of 6F10; and since Comessatti has shown that Π10 is without singularities, so also is F10, whence it follows that F10 is a solution of the proposed problem.

We recall from (3) that a group G is (centrally) eremitic if there exists a positive integer e such that, whenever an element of G has some power in a centralizer, it has its eth power. The eccentricity of an eremitic group G is the least such positive integer e.

In ((4), Theorem A) we proved that if A is a torsion free Abelian normal subgroup of a finitely generated group G with G/A nilpotent, then G has a subgroup of finite index with eccentricity 1. In this note we use a much simpler method to prove a stronger result.

Hobson has given a proof of this theorem in its fullest generality. The present note gives an alternative for part of Hobson's argument. The theorem may be stated in two forms. If f(x) is a function of x, monotone when a ≤ x ≤ b, and φ(x) is integrable over the same range, then

where a ≤ X ≤ b,

(ii) the same holds with a < X < b except in some trivial cases where f(x) is constant in the open interval a < x < b. The form (ii) is not mentioned by Hobson.

We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are given by explicit pairs of Eschenburg spaces. To deduce the second statement from the first, we extend our earlier work on the stable converse soul question and show that it has a positive answer for a class of spaces that includes all Eschenburg spaces.

We denote by M(G) the space of bounded regular borel measures on a non-discrete locally compact Abelian topological group G. Under the convolution product and norm of total mass M(G) becomes a complex commutative banach algebra. We denote by Φ the space of m.l.f.s (multiplicative linear functionals) on M(G) and by σ the subset of Φ consisting of functionals symmetric in the sense that

where * denotes the usual involution on M(G). These matters are discussed more fully in Williamson (1).

In this paper we consider a model of a population in which the individuals are diploid and are of two sexes. The individuals have genotypes aa, Aa and AA and we suppose that mutation is occurring in both directions. By supposing that each step in the process is the result of a single death followed by a birth instead of supposing (as in Wright's models (3)) that all the individuals die together at the same instant, to be replaced by a new generation, the mathematical details are much simplified. In this way it is possible to deal exactly with more complicated situations than that of the population of haploid individuals used by Wright in his study of the effects of mutation in a finite population. Moreover, the present model not only involves zygotes and two sexes but is more realistic for those natural populations in which the generations are overlapping.

The formulae considered in this paper deal with the multiple tangent lines and planes to surfaces in three dimensions; the functional method employed is that of a previous paper, in which a more restricted discussion was given for surfaces in higher space.

This paper is an excerpt from a Rayleigh essay submitted at the University of Cambridge in January 1970. We reproduce it now as it gives a general construction of a family of Moufang loops to which all bar one of the finite subloops of the Cayley algebra belong. These subloops were classified up to isomorphism in the original essay, but are classified up to equivalence under the action of the group of symmetries of in Boddington and Rumynin [1]. Explicitly, given a group G with an element a such that a2=1 in its centre, we construct a Moufang group in which G has index 2. will be non-associative unless G is abelian.

Let X be a CW-complex with finite skeletons and with H*(X, Z) free of p-torsion, where p is some odd prime. We shall prove two theorems on the action of the Steenrod powers

This paper is an application of O. G. Sutton's theory of eddy diffusion to a case for which it was not originally developed; the theory is used to predict the dispersion of a cloud of falling droplets released from a point source at great height above the surface of the earth. Some experimental and theoretical values are given for the widths of ground area covered by liquid drops released from sources situated at heights between 1000 and 5000 ft., and agreement between theory and experiment is seen to be quite reasonable. This paper therefore provides further support for the use of O. G. Sutton's theory in problems of this type. The theory has practical applications, e.g. in the spraying of crops from aeroplanes.

The absolute measurement of fast neutron flux presents several difficult problems. Few methods have yet been described in the literature, although the experimental techniques developed by several authors for the detection of fast neutrons (Baldinger, Huber and Staub(7), Barshall and Kanner(9), Amaldi, Bocciarelli, Ferretti and Trabacchi (3), Gray (19), Barshall and Battat(8)) may easily be adapted to this type of measurement. It is, however, most important to have available methods of measuring fast neutron flux to permit the determination of cross-sections for nuclear processes induced by fast neutrons, and several such methods have been developed in the Cavendish Laboratory in recent years. They are the subjects of separate papers (Bretscher and French (13), Kinsey, Cohen and Dainty (21), Allen (l), Allen and Wilkinson (2)). The main purpose of the present paper is to describe the results of experiments carried out to compare these methods in order to test the validity of the assumptions implicit in the individual methods.