Krickeberg (in (5) and (6)) showed that disintegration applied to invariant measures sometimes yields an integral representation which is useful in analysing the moment measures of point processes. His results, based on Bourbaki's disintegration theory, raised several questions. We refine the theory, using a more general disintegration theorem, and answer his questions by several examples. Finally we consider how far the enlarged theory is applicable in stochastic geometry.

The initial value problem for the equation

has been studied recently as a model for long waves in nonlinear dispersive systems. Benjamin, Bona and Mahony (2) introduced this equation as an alternative to the KdV equation of Korteweg-de Vries. Hence, it is referred to as the BBM equation. They studied solutions u(x, t) of the BBM equation for t ≥ 0 and x∈(− ∞, ∞), satisfying u(x, 0) = g(x). Bona and Bryant(1) carried through the study of the BBM equation for t ≥ 0 and x ∈ [0, ∞), satisfying u(x, 0) = g(x) and u(0, t) = h(t). The aim of this paper is to study the equation

where At and Bt are mappings defined on subsets of Banach spaces, especially when At is a second order elliptic operator and B is a differential operator of lower order, defined on an unbounded subset Ω of .

We prove that the following conditions are equivalent: X is an F space; every ideal of C(X) is flat; every submodule of a free C(X)-module is flat.

The famous theorem of Schnirelmann states that a constant c exists such that every integer greater than one may be expressed as the sum of at most c primes. Recently, Heilbronn, Landau and Scherk proved that this holds with c = 71. Probably the true value of c is 3. Another well-known theorem (Hilbert's solution of Waring's problem) is that every integer is the sum of a bounded number of positive kth powers. If we omit the restriction that all the kth powers are positive, the problem is referred to as the easier Waring problem and the proof of the result is then much simpler.

This is a brief study of the homology of cubical sets, with two main purposes.

First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes.

But cubical sets can also express topological facts missed by ordinary topology. This happens, for instance, in the study of group actions or foliations, where a topologically-trivial quotient (the orbit set or the set of leaves) can be enriched with a natural cubical structure whose directed cohomology agrees with Connes' analysis in noncommutative geometry. Thus, cubical sets can provide a sort of ‘noncommutative topology’, without the metric information of C*-algebras.

We show for sequences $\left(a_{n}\right)_{n \in \mathbb N}$ of distinct positive integers with maximal order of additive energy, that the sequence $\left(\left\{a_{n} \alpha\right\}\right)_{n \in \mathbb N}$ does not have Poissonian pair correlations for any α. This result essentially sharpens a result obtained by J. Bourgain on this topic.

Let V⊂S3 be a solid, knotted torus. Through the work of Birman and Menasco [2], the observation has been made that a satellite link L=C[midast ]P⊂V (with companion C, pattern P and essential torus T=∂V) falls into one of two broad categories: reverse string and non-reverse string. These categories are borne of the three embedding types identified in [2] and are distinguished by the existence of, or lack of, a meridional disc D⊂V with an orientation, whose point-intersections with the oriented satellite C[midast ]P are all similarly oriented. If no such D exists, then L is said to be a reverse string satellite.

If C[midast ]P is a non-reverse string satellite, it is known that the braid index b(C[midast ]P) is dependent only on b(C) and certain specified properties of the pattern and not on the choice f∈Z of framing parameter. In [2] it is conjectured that, if C[midast ]fP is a reverse string satellite (in which the framing parameter is f), the braid index b(C[midast ]fP) depends on the arc index α(C), properties of the pattern and also the framing. We study this dependence further via established upper and lower bounds for braid index. The upper bound comes from explicit closed braid diagrams of L, for which constructions are shown. The lower bound comes from the Homfly polynomial, via the Morton–Franks–Williams inequality [5, 8]. We will prove the following theorem.

(1) An apparatus has been devised in which the composition of the vapour phase passing over the catalyst can be varied at will, or two vapours can be used alternately as reactants.

(2) By using ethyl alcohol, and amyl alcohol from fusel oil alternately as reactants, the poisoning of a very active copper gauze catalyst has been followed and the temperature coefficient of the dehydrogenation of ethyl alcohol by the copper has been measured during the poisoning.

(3) The activity of the catalyst, poisoned by amyl alcohol, fell in accordance with the simple logarithmic law

(4) Confirmation was obtained that the rates of dehydrogenation of ethyl and amyl alcohols were equal.

(5) The temperature coefficient of dehydrogenation on the catalyst was unchanged throughout the poisoning, though the reaction velocity was reduced 30 times.

(6) The temperature coefficient was also shown to be unchanged during sintering, the reaction-velocity having been reduced 10 times.

(7) Equations are derived for the rate of poisoning of an exponential distribution of centres of activity when the poison attacks the more active centres selectively.

(8) The relation deduced

cannot usually be distinguished (except with special relations between the constants) from the simple logarithmic law of poisoning.

(9) Thus the exponential distribution normally poisons like a surface composed of homogeneous centres, the temperature coefficient being unchanged by the poisoning.

(10) The equation

describes the reaction-velocity as a function of time, and temperature, and gives adequate explanation of previous experimental results on the forms of the heating and cooling curves with “poisonous” alcohols.

(11) Frenkel has shown , where τ0 is the vibration frequency of the molecule perpendicular to the surface, and μ0 is the heat of desorption of one gram molecule. Thus if the heat of desorption is not small compared with the energy necessary to cause chemical disruption of the molecule, the previous equation becomes

the form of the equation being unaltered.

The differential equation of wave propagation is

where u is a variable describing the state of a medium, x, y, z are rectangular coordinates, and r is the time (the time scale is assumed to be such that waves are propagated with unit velocity).

Experience has shown that simple absorption measurements lead to quite trustworthy estimates of the maximum energies represented in the continuous spectra of many β ray bodies. For the interpretation of these experiments the range-energy relation for homogeneous particles is apparently all that is necessary. In the present paper a detailed analysis is given of the various factors which conspire to this useful result. Analytical simplifications are possible when the energy distribution in question extends beyond about 7 × 105 electron volts energy. Under this limitation it is shown that the empirical result follows from the known form of the absorption curve for homogeneous particles and the most general considerations regarding the characteristics of such measuring instruments as are regularly employed. From the analysis in one case, however, it appears that, despite numerical agreement between the results of different methods of investigation, caution is still necessary in any statement concerning the absolute nature of the high energy limits of continuous β ray spectra.

Recently, Conlon, Fox and the author gave a new proof of a relative Szemerédi theorem, which was the main novel ingredient in the proof of the celebrated Green–Tao theorem that the primes contain arbitrarily long arithmetic progressions. Roughly speaking, a relative Szemerédi theorem says that if S is a set of integers satisfying certain conditions, and A is a subset of S with positive relative density, then A contains long arithmetic progressions, and our recent results show that S only needs to satisfy a so-called linear forms condition.

This paper contains an alternative proof of the new relative Szemerédi theorem, where we directly transfer Szemerédi's theorem, instead of going through the hypergraph removal lemma. This approach provides a somewhat more direct route to establishing the result, and it gives better quantitative bounds.

The proof has three main ingredients: (1) a transference principle/dense model theorem of Green–Tao and Tao–Ziegler (with simplified proofs given later by Gowers, and independently, Reingold–Trevisan–Tulsiani–Vadhan) applied with a discrepancy/cut-type norm (instead of a Gowers uniformity norm as it was applied in earlier works); (2) a counting lemma established by Conlon, Fox and the author; and (3) Szemerédi's theorem as a black box.

In this paper G will denote a compact metric Abelian group having finite topological dimension, dim G. The discrete dual group of G will be denoted by Γ. The metric on G will be denoted by d(x, y), or |x-y| since translation-invariance is assumed.

We prove that if K/ℚ is a Galois extension of finite exponent and K(d) is the compositum of all extensions of K of degree at most d, then K(d) has the Bogomolov property and the maximal abelian subextension of K(d)/ℚ has the Northcott property.

Moreover, we prove that given any sequence of finite solvable groups {Gm}m there exists a sequence of Galois extensions {Km}m with Gal(Km/ℚ)=Gm such that the compositum of the fields Km has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in ℚ(d).

We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewicz.

Solutions to the Schrödinger, heat and stochastic Schrödinger equation with rather general potentials are represented, both in x- and p-representations, as integrals over the path space with respect to σ-finite measures. In the case of x-representation, the corresponding measure is concentrated on the Cameron–Martin Hilbert space of curves with L2-integrable derivatives. The case of the Schrödinger equation is treated by means of a regularization based on the introduction of either complex times or continuous non-demolition observations.

The fact that the prime ideal associated with a given irreducible algebraic variety has a finite basis is a pure existence theorem. Only in a few isolated particular cases has the base for the ideal been found, and there appears to be no general method for determining the base which can be carried out in practice. Hilbert, who initiated the theory, proved that the prime ideal defining the ordinary twisted cubic curve has a base consisting of three quadrics, and contributions to the ideal theory of algebraic varieties have been made by König, Lasker, Macaulay and, more recently, by Zariski. A good summary, from the viewpoint of a geometer, is given by Bertini [(1), Chapter XII]. However, the tendency has been towards the development of the pure theory. In the following paper we actually find the bases for the prime ideals associated with certain classes of algebraic varieties. The paper falls into two parts. In Part I there is proved a theorem (the Principal Theorem) of wide generality, and then examples are given of some classes of varieties satisfying the conditions of the theorem. In Part II we find the base for the prime ideals associated with Veronesean varieties and varieties of Segre. The latter are particularly interesting since they represent (1, 1), without exception, the points of a multiply-projective space.

The methods introduced into the theory of atomic spectra by Racah are applied to the calculation of electric quadrupole line strengths for transitions involving d and s electrons. The transitions considered are of the types dn−dn−1s, dn−1s−dn−1s and dn−1s−dn−2s2. Sum rules and rules for almost closed shells are obtained. Detailed numerical results are given for various transitions in and between the configurations d9 and d8s, and d7, d6s and d5s2.

1. I. N. Sneddon has recently considered(11) the two-dimensional problem of the stress distribution due to a force uniformly distributed along a strip in the interior of a semi-infinite elastic medium by integrating the distribution due to an isolated force. The method of solution for the isolated force problem differs from that originally used by Melan(9), but both writers employ real variables. A solution of this problem, using complex variables, has been given by Stevenson and a considerable saving of labour is revealed. A complex variable method of solution for problems of stress distributions in an elastic medium bounded by a plane face was, however, given earlier by Sen(10), although he only made applications to problems in which the force system was applied to the plane face. Sen's method is somewhat different from that used by Stevenson and gives results more directly for many problems, including the problem considered by Sneddon.

We prove the following two theorems:

(I) Let $n \geqslant 1$ be a (fixed) integer. Then we have for $\theta \in (0, \pi)$: \[ \sum\limits^n_{k=1}\frac{\cos (k\theta)}{k}\leqslant -\log\left(\sin\left(\frac{\theta}{2}\right)\right)+\frac{\pi-\theta}{2}+\sigma_n, \] with the best possible constant $\sigma_n = \sum\nolimits^n_{k=1}(-1)^k/k$.

(II) For even integers $n \geqslant 2$ and for $\theta \in (0, \pi)$ we have \[ \sum\limits^n_{k=1}\frac{\sin(k\theta)}{k}\leqslant\alpha(\pi-\theta), \] with the best possible constant $\alpha = 0.66 395\ldots$.

Our results refine inequalities due to C. Hyltén-Cavallius ‘11’ and P. Turán ‘23’, respectively.

Quasi-alternating links of determinant 1, 2, 3 and 5 were previously classified by Greene and Teragaito, who showed that the only such links are two-bridge. In this paper, we extend this result by showing that all quasi-alternating links of determinant at most 7 are connected sums of two-bridge links, which is optimal since there are quasi-alternating links not of this form for all larger determinants. We achieve this by studying their branched double covers and characterising distance-one surgeries between lens spaces of small order, leading to a classification of formal L-spaces with order at most 7.

In this paper we establish that a convex planar body C is centrally symmetric provided either one of the following conditions hold:

(1) Each line halving the circumference of the boundary γ of C is a diametral line.(A diametral line is a line intersecting C in a chord of maximal length in the family of parallel chords.)

(2) Each line halving the area of C is a diametral line.