Three formalisms have been considered for the ‘extraordinary’ field introduced by Podolsky in his generalized electrodynamics.

The c-representation (tildons) for negative energy particles (§ 3), leads to an unstable vacuum in electron hole theory and must be discarded (§ 7).

The d-representation suggested by Podolsky is a hole theory for bosons. This leads at once to inconsistencies and it is impossible to introduce a Fock representation (§ 6).

The least unsatisfactory is the b-representation (tilde photons) used by PKMG. Tilde photons can be created at high energies but decay immediately by pair creation. The introduction of tilde photons does not interfere with any agreement already established between experiment and ordinary quantum electrodynamics. Evidence might be obtained for the value of the constant a from consideration of the 2S level shift of the hydrogen atom and from the theory of showers where the deviation from the usual theory is greatest (§ 7).

However, the formalism involves the introduction of an indefinite metric for the tilde photon field and consequently ‘negative probabilities’ for states in which an odd number of tilde photons are present. Approximate results can be obtained by assuming that ‘negative probabilities’ in the theory are positive probabilities in the real world. This leads to a small but fundamental error and it does not seem possible that a theory in this form can have any deep seated validity (§ 5).

The same analysis applies to the extension of the Podolsky theory to the Proca field by Green (5). In order that the energy should be positive definite it is necessary to introduce an indefinite metric for one of the two superposed meson fields. Fundamentally the difficulties of interpretation are the same but in practice they will be more obvious because the coupling constants are greater.

Let S be a smooth cubic surface over a finite field $\mathbb{F}$q. It is known that #S($\mathbb{F}$q) = 1 + aq + q2 for some a ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton–Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton–Dyer's tables on cubic surfaces over finite fields.

In a previous paper, under the same title, I considered the problem of how far apart two consecutive primes can be. The present paper is concerned with the opposite question. How near together can large primes lie? The published literature on this subject is scanty and, though interesting, is mainly negative in character. It appears to be very difficult to give any answer that is not trivial, or that is at all illuminating.

A method is described for measuring thermal conductivities of metals at liquid helium temperatures. It is shown that the method of determining the temperature gradient by directly measuring the temperature at two points along a rod is probably free from systematic error, but that the end of a heated rod in direct contact with a helium bath is not at the temperature of the surface of the bath, even in the He ii region. A description is also given of a technique for using mercury as a low-temperature solder, tight to He ii. The results for copper are in reasonable agreement with Makinson's theory of metallic conduction. Tentative extrapolation to very low temperatures in the presence of fairly large magnetic fields shows that pure copper is a reasonably good material for thermal transfer in demagnetization experiments. The results for german silver seem to indicate that only the lattice conductivity is important since it follows a T3 law, and that scattering takes place with a mean free path of the order of 10−3 cm.

We give presentations of the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In particular, we show the volume conjecture for these knots, which states that the leading terms of the expansions present the hyperbolic volume and the Chern--Simons invariant of the complements of the knots. As higher coefficients of the expansions, we obtain a new series of invariants of these knots.

A non-trivial part of the proof is to apply the saddle point method to calculate the asymptotic expansion of an integral which presents the Kashaev invariant. A key step of this part is to give a concrete homotopy of the (real 3-dimensional) domain of the integral in ℂ3 in such a way that the boundary of the domain always stays in a certain domain in ℂ3 given by the potential function of the hyperbolic structure.

In an earlier paper [2], I gave explicit rules for calculating the Brauer group of a Del Pezzo surface of degree 3 or 4. Recent developments have made it desirable to extend these to the construction of the corresponding Azumaya algebras, so as to be able to calculate the associated Brauer–Manin obstruction to weak approximation or even to the Hasse principle. That is the main purpose of this note, which needs to be read in conjunction with [2]. The special case of diagonal cubic surfaces has already been treated in [1], though the process described there is not an algorithm in the strict sense; however, in numerical cases it is probably more efficient than the algorithm described in this note would be.

In [1] P. Cohn suggested the construction of a localization of a ring with respect to a class of square matrices. Let us briefly recall the definitions.

The inequality is

which is established by an elementary argument and is shown to lead directly to the evaluation of the integral

and to the expression of sin x as an infinite product.

Let L, M be Lie algebras over the same field F and let ℒ(L), ℒ(M) denote their lattices of subalgebras. By an L-isomorphism (lattice isomorphism) of L onto M we mean an isomorphism θ: ℒ(L) → ℒ(M); that is, θ a bijective mapping from ℒ(L) onto ℒ(M) such that

and

for all subalgebras A, B of L (where A ∪ B denotes the subalgebra of L generated by A and B). We shall write A* for θ(A), the image of A ∈ ℒ(L) under an L-isomorphism from L onto M = L*. Then we may ask how closely related are L and L*. In particular, are such important concepts as semisimplicity, solvability, nilpotency and ideals preserved by L-isomorphisms?

A classical fact is that Seifert manifolds with non-empty boundary are covered by surface bundles over the circle S1 and closed Seifert manifolds may or may not be covered by surface bundles over S1. Some closed graph manifolds are not covered by surface bundles over S1 ([LW] and [N]). Thurston asked if complete hyperbolic 3-manifolds of finite volume are covered by surface bundles [T]. J. Luecke and Y. Wu asked if graph manifolds with non-empty boundary are covered by surface bundles over S1 ([LW]). In this paper we prove:

THEOREM 0·1. Each graph manifold with non-empty boundary is finitely covered by a surface bundle over the circle S1.

Let G be an Abelian group. A binary relation ≥ denned in G is called an order of G if for each x, y, z ε G,

(i) x ≥ y or y ≥ x (and hence x ≥ x);

(ii) x ≥ y and y ≥ x ⇒ x = y,

(if x ≥ y and x ≠ y, we write x > y);

(iii) x ≥ y and y ≥ z ⇒ x = z;

(iv) z ≥ y ⇒ x + z ≥ y + z.

We will here prove the following special case of an (unpublished) conjecture for a polynomial of degree ndue to L. Ilieff:

Theorem. If all zeros of the cubic polynomial p3(z) lie in |z| ≤ 1, then at least one zero of p3′(z) lies in or on the boundary of a circle of radius unity around each zero of p′3(z).

We modify arguments used in the study of Universal Taylor Series and extend them from simply connected domains to general domains in $\bb {C}$ to give a new simple and natural proof of the fact that all holomorphic functions are generically non extendable.

In this paper we shall show how a large number of higher-order cohomology operations (see (4)) may be calculated in suitable spaces by means of Chern characters: the method is an extension of that used by Adams in (1) to relate ‘mod p reductions’ of Chern characters and certain primary operations.

Continued fractions were generalised to more than one dimension by Jacobi and others: later Perron gave an account of the existing state of the subject with a detailed discussion of periodic fractions. Quite recently the subject has been attacked afresh by Mr. Maunsell

Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S2)n. We show that if Xn is embedded antipodally in (S2)n then the natural Sn-action on (S2)n induces an Sn-representation on the image of H∗(Xn). This representation is the Springer representation. Our construction admits an elementary (and geometrically natural) combinatorial description, which we use to prove that the Springer representation on H∗(Xn) is irreducible in each degree. We explicitly identify the Kazhdan-Lusztig basis for the irreducible representation of Sn corresponding to the partition (n/2, n/2).

A modified kernel is devised which avoids the irregular values which plague the natural integral equations for the exterior problems of harmonic elastic waves in a homogeneous isotropic medium.

The actual distribution of a homogeneous group of electrons by semicircular magnetic focusing is calculated for different types of source.

The dependence of maximum intensity on the radius of curvature of the path of electrons is investigated and the approximate power laws assigned to different cases. The results can be used in the measurement of the intensity of β-ray line spectra.

Tables of numerical values of effective solid angle subtended by the photographic plate at the source are given for some special cases.

The results of the above calculations have been verified experimentally.

The basic state considered in this paper is a parallel flow of a jet-like character with the centre of the jet being at or near a free surface which is horizontal. Stationary surface gravity waves may exist on such a flow, and a number of examples are looked at for small amplitude waves. Explicit solutions are given for ‘top-hat’ profile jets and for two-dimensional flows. Asymptotic solutions are developed for stationary waves of large wave-number.

Given a totally finite measure space (S, S, μ) and two μ-integrable, non-negative functions f(x) and φ(x) defined in S, such that when

then

we define correlated sampling as the technique of estimating

by sampling an estimator function

where ξ is uniformly distributed in S with respect to μ (i.e. for any T ∈ S, p(T) = μ(T)/μ(S) is the probability that ξ lies in T): and importance sampling as estimating L by sampling the estimator function

where η is distributed in S with probability density φ(x)/Φ

Then, clearly,

It follows that υ(ξ) and ν(η) are both unbiased estimators of L, and that their variances can both be made to approach zero arbitrarily closely by making φ(x) a sufficiently close approximation to f(x).