The Brauer–Manin obstruction is a concept which has been very effective in finding counter-examples to the Hasse principle, that is, sets of polynomial equations which have solutions in every completion of the rational numbers but have no rational solutions. The standard way of calculating the Brauer–Manin obstruction involves listing all the p-adic solutions to some accuracy, at finitely many primes p; this is a process which may be time-consuming. The result described in this paper shows that, at some primes, we do not need to list all p-adic solutions, but only those lying over a closed subset; and, at other primes, we need only to list solutions modulo p.

In this paper we present results which relate the linearization of a G-equivariant vector field at an equilibrium to the linearization of the vector field obtained by projection on the orbit space of the group action. This allows us to propose a notion of hyperbolicity for an equilibrium of the projected vector field which is compatible with the orbit space structure. It provides an alternative to the notion of normal hyperbolicity for relative equilibria (flow invariant group orbits) developed by Field [9, 10].

It has long been realised that there are difficulties in understanding the continuous β-ray spectra emitted by certain radioactive bodies, if the conservation of energy, as applied to nuclear processes, is to be retained. In the case of radium E, for example, the energy of the disintegration β particles varies from quite low values to about a million volts, the mean being nearly 400,000 volts.

Let d(n) denote the number of positive divisors of n, and let f(x) be a polynomial in x with integer coefficients, irreducible over ℤ. Erdös[3] showed that there exist constants λ1, λ2 (depending on f) such that

Problems dealing with the generation of internal waves at the surface separating two fluids involves the consideration of different types of singularities in one of the two fluids. In this paper the velocity potentials describing line and point multipoles are obtained for the case when each fluid is of finite constant depth, neglecting effects of surface tension at the surface of separation.

1. Consider the constant-coefficient fifth-order differential equation:

It is known from the general theory that the trivial solution of (1·1) is unstable if, and only if, the associated (auxiliary) equation:

has at least one root with a positive real part. The existence of such a root naturally depends on (though not always all of) the coefficients a1, a2,…, a5. For example, if

it is clear from a consideration of the fact that the sum of the roots of (1·2) equals ( – a1) that at least one root of (1·2) has a positive real part for arbitrary values of a2,…, a5. A similar consideration, combined with the fact that the product of the roots of (1·2) equals ( – a5) will show that at least one root of (1·2) has a positive real part if

for arbitrary a2, a3 and a4. The condition a1 = 0 here in (1·4) is however superfluous when

for then X(0) = a5 < 0 and X(R) > 0 if R > 0 is sufficiently large thus showing that there is a positive real root of (1·2) subject to (1·5) and for arbitrary a1, a2, a3 and a4.

Let

be regular in U = {|z| < 1}. Suppose that the values of f(z) all lie in a domain D in the w-plane. If certain geometrical restrictions are made on D we can deduce growth conditions on the maximum modulus

the means

and the coefficients αn. Good bounds for M(r, f) have been obtained under various conditions on D.

In (1), J. F. Adams showed that for p odd, the Adams E2-term for a sphere, , is zero for s ≤ t ≤ (2p − 1)s − 1, while is the one-dimensional vector space generated by where corresponds to the Bockstein.

Consideration is given to the flow of a micropolar liquid down an inclined plane. The steady state is analysed and Yih's technique is employed in an investigation of the stability of this flow with respect to long waves. Detailed calculations are given for thin films and it is shown that the micropolar properties of the liquid play an important role in the stability criterion.

An apparatus for the production of beams of fast positive ions is described. Measurements of the cross-section for capture of an electron by a positive ion from a neutral gas atom are given on the case of He+ ions in helium and protons in helium and hydrogen. These are compared with the theory given by Massey and Smith, and as far as possible with previous experiments.

Recent papers by J. R. Ellis and the author contain formulae for the fields of moving dipoles which are expressed in rather different forms in the two papers, and it appears that the author's results are more general than Ellis's. Here, the results are re-derived from a consideration of the Hertzian six-vector potential for a distribution of moving dipoles, and a comparison is made with the previous forms. It is found that Ellis's results are essentially as general as the author's. The expressions for the potentials and the field tensors are treated as weak functions in Temple's sense, and very few restrictions have to be imposed on the dipole moments and velocities, which may be expressed either as functions of coordinate time or as functions of proper-time.

We use certain derivations of a polynomial ring, that do not leave any proper non-zero ideal invariant, to construct simple non-holonomic modules over the nth Weyl algebra. This approach extends to the rings of differential operators of other smooth affine varieties, like smooth quadric surfaces.

An ordered topological space is a set X endowed with a topology τ and a partial order ≤. We shall denote such a space by (X, τ), it being understood that (unless otherwise stated) the symbol ≤ is used to denote the partial order on X. An account of some of the properties of these spaces can be found in (5), (6) and (7).

It is well known (1) that for an unsaturated hydrocarbon of the type CnHn+2 (where n is the number of carbon atoms in the linear chain), the theory of molecular orbitals applied to calculate the binding energy of the mobile π-electrons leads to the secular determinant, of continuant type,

where β is the resonance integral between adjacent carbon centres and x stands for the quantity generally written E0 − E, E being the energy of a mobile electron, which we seek to calculate. The equation for the allowed energies E is, of course, simply Pn = 0.

A brief account is given of the fundamental properties of a new generalization ((1), (2)) of Einstein's gravitational theory. The field equations are then solved exactly for the case of a static spherically symmetric gravitational and electric field due to a charged particle at rest at the origin of the space-time coordinates. This solution provides information about the gravitational field produced by the electric energy surrounding a charged particle and yields the Coulomb potential field. The solution satisfies the required boundary conditions at infinity, and it reduces to the Schwarzschild solution of general relativity when the charge is zero.

In this paper, we study the structure of certain infinite-dimensional alternative algebras admitting an inner product and an involution. These algebras, called alter-native H*-algebras in the sequel, are the alternative analogues of the H*-algebras of Ambrose [1].

The exchange forces between two heavy particles (neutrons, protons) have been treated on the basis of the meson theory by Yukawa, Sakata and Taketani (5), and by Fröhilch, Heitler and Kemmer (2), assuming the existence of positive and negative mesons only. The calculations have been extended by Kemmer (3) for a meson field which includes in addition neutral particles (neutretto's). The first order approximation of the exchange potential, i.e. the potential due to the emission and subsequent absorption of one meson only, gives qualitatively the correct interaction between the unclear particles.

Classifying (unordered) sets by the elementary (first order) properties of their automorphism groups was undertaken in (7), (9) and (11). For example, if Ω is a set whose automorphism group, S(Ω), satisfies

then Ω has cardinality at most ℵ0 and conversely (see (7)). We are interested in classifying homogeneous totally ordered sets (homogeneous chains, for short) by the elementary properties of their automorphism groups. (Note that we use ‘homogeneous’ here to mean that the automorphism group is transitive.) This study was begun in (4) and (5). For any set Ω, S(Ω) is primitive (i.e. has no congruences). However, the automorphism group of a homogeneous chain need not be o-primitive (i.e. it may have convex congruences). Fortunately, ‘o-primitive’ is a property that can be captured by a first order sentence for automorphisms of homogeneous chains. Hence our general problem falls naturally into two parts. The first is to classify (first order) the homogeneous chains whose automorphism groups are o-primitive; the second is to determine how the o-primitive components are related for arbitrary homogeneous chains whose automorphism groups are elementarily equivalent.

Following the techniques of ordinary homotopy theory, a theoretical treatment of proper homotopy theory, including the known proper homotopy groups, is provided within Baues's theory of cofibration categories.

Let Sw+2(Γ0(N)) be the vector space of cusp forms of weight w + 2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the Eichler–Shimura–Manin isomorphism theorem to Γ0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2(Γ0(2)). As an application of main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of Sw+2(Γ0(2)).