Let 0 < λ1 < λ2 < … be an increasing sequence of positive reals tending to infinity, and let ν1, ν2 be two (finite) signed measures on ℝ+ such that

for all k. It was originally proved by Müntz (12) that, if

then ν1 = ν2, and, conversely, if this condition fails, then there exist distinct ν1 and ν2 satisfying (1).

In chapter 7 of (2) conditions were given for a ring to be embeddable in a skew field; in particular, it was shown that any semifir has a universal field of fractions, over which all full matrices can be inverted. This was generalized in two different directions, by Bergman (in a letter to one of the authors in 1971) and by Dicks and Sontag(7). Dicks and Sontag characterized those rings having a field of fractions in which all full matrices are inverted; they showed that this is equivalent to Sylvester's law of nullity, and further showed that this forces the ring to have weak global dimension not exceeding 2 and all finitely generated projective modules to be free. Bergman on the other hand investigated weakly semihereditary rings having a rank function on projective modules which takes values in the natural numbers. He showed that there was a homomorphism from any such ring to a field of fractions in which every full map between finitely generated projective modules is inverted. Weakly semihereditary rings with a rank function to the natural numbers are the analogue of semifirs and so it is natural to look for a characterization of rings with a rank function on projective modules such that all full maps between projective modules become invertible in a suitable field of fractions. We shall find that, as before, this is the case if and only if Sylvester's law of nullity holds with respect to the rank function, for maps between projective modules. Further, the ring must have weak global dimension at most two. This is the content of Sections 2 and 3.

The object of the present paper is to discuss the Fourier expansion of the Riesz potential. For this purpose a new definition of the electromagnetic potentials, depending upon an arbitrary parameter α is given. It is shown that this definition is a generalization of the Wentzel potentials in the α-plane, whereas that given by Fremberg (3) is a generalization of the Maxwell potentials. The analysis is applied to the problem of eliminating, in a straightforward way, the longitudinal part of the potential describing the electromagnetic field. The problem of the quantization of the field, based on its Fourier expansion, will be considered in another paper. The recent work of Tomonaga, Schwinger and Dyson, and the regularization process of Pauli has lifted the theory of quantum electrodynamics to a much higher level of rigour and fruitful applicability. All the same, a further study of Riesz potential seems to us of some interest in this field.

This paper is concerned with the unsteady hydromagnetic boundary-layer flow past a semi-infinite flat plate when the oncoming free stream is perturbed by an arbitrary function of time and the applied magnetic field is parallel to the plate far away from it. Following Lighthill, the two-dimensional boundary-layer equations are separated into those representing steady and unsteady parts of the flow and they have been solved in sequence. For the unsteady part of the motion two types of solutions are obtained, one for large times and the other for small times. Also the quasi-steady solution is obtained in terms of the steady solution. The skin friction and the tangential magnetic field at the plate are calculated.

L. M. Milne-Thomson has recently described a method of solving linear finite difference equations by using processes some-what analogous to those employed by Heaviside.

The spinor representations of the rotation group in a six-dimensional space with indefinite metric are shown to be four-component spinors, which become the usual Dirac spinors when the formalism is restricted to a four-dimensional subspace. Eriksson's work on the five-dimensional Lorentz group is found to result from a restriction of the six-dimensional treatment to a five-dimensional subspace, and the algebraic significance of Eriksson's work is thereby clarified.

This paper contains the derivation of a metric for flat space-time associated with the supertranslations of the Bondi-Metzner group, and a description of the geometry of the null hypersurfaces on which the metric is based.

The hydrogen discharge has for some time been established as a convenient source of continuous spectrum in the ultra-violet. The continuum extends from nearly 4000 Å. to the limit of a quartz spectrograph and provides a very good background for absorption measurements. A number of tubes have been described for producing this spectrum, but the following advantages are claimed for the form here described.

A recent result of Leung (Proceedings of the American Mathematical Society, 2015) states that the Banach algebra ℬ(X) of bounded, linear operators on the Banach space X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓ1 contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓp and X = (⊕n∈$\mathbb{N}$ ℓ1n)ℓp whenever p ∈ (1, ∞).

One of the topics covered by most textbooks on partial differential equations is the classification problem for second-order equations in the plane. In a typical treatment, it is shown that under some smoothness conditions on the coefficients, hyperbolic and parabolic equations can be reduced locally to normal form (uniformized) by an elementary procedure. For elliptic equations, the procedure fails unless an extraneous hypothesis (analyticity of the coefficients) is introduced. It is then pointed out that a different and much deeper method (essentially the general uniformization theorem) is effective for the elliptic case and even yields a global result. It is striking, but not surprising in view of recent developments on generalized solutions, that the alternate (global) procedure for elliptic equations requires much less smoothness of the coefficients than is needed for a sensible local result in the other cases.

We show that a positive braid knot has maximal topological 4-genus exactly if it has maximal signature invariant. As an application, we determine all positive braid knots with maximal topological 4-genus and compute the topological 4-genus for all positive braid knots with up to 12 crossings.

By defining new Bryant-type vector fields for foliations on a Riemannian manifold we find necessary and sufficient conditions that a foliation produces $p$-harmonic morphisms. Two applications are given. First, we characterize one-parameter conformal actions using $p$-harmonic morphisms. Then we classify $p$-harmonic morphisms on a constant curvature space with one-dimensional fibres by studying bi-minimal distributions. We also give a description of all conformal foliations which have minimal fibres on a Riemannian manifold in terms of $p$-harmonic morphisms.

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense that

Set qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)

where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence of

to zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

This note exhibits an infinite independent set of laws for monoids (viz. (xpyp)2 = (ypxp)2, p prime) and a variety of groups defined by one law (which is x2y2 = y2x2) such that the smallest containing variety of monoids is not finitely definable.

A theory of quadratic forms is developed over R(t), the field of rational functions with real coefficients, instead of over an algebraic number field: some results are analogous to the classical ones, but many, in particular the non-finiteness of the class number, whose more detailed treatment will be the subject of a later paper, are not analogous.

This paper considers the problem of expressing the magnetic scalar potential associated with the steady flow of an electric current around a thin circular wire, in terms of ‘local’ toroidal coordinates. This potential is known to be a multi-valued function of position and so cannot be expressed directly in terms of fundamental (i.e. single-valued) solutions of Laplace's equation. It is shown that the potential can however be expressed quite simply, as an infinite series of multi-valued toroidal harmonics and that this series is rapidly convergent in the neighbourhood of the current-ring.

The first purpose of our paper is to show how Hooley’s celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy–Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation.

We give new, short proofs of the presentations for the partition monoid and its singular ideal originally given in the author's 2011 papers in Journal of Algebra and International Journal of Algebra and Computation.

A result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

A large class of initial-boundary problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit representations are presented for the generalised Dirichlet to Neumann maps. Namely, the determination of the unknown boundary values when an essential set of initial and boundary data is given.