The two-dimensional, irrotational, linear theory used in the investigation of the propagation of monochromatic surface gravity waves in a region of varying depth is considered. Uniqueness of the velocity potential is established for bottom profiles satisfying certain convexity conditions. These include the majority of profiles of physical interest.

Necessary and sufficient conditions for sequence-to-sequence or sequence-to-function summability method to include (R, λ, α), when 1 < α ≤ 2, are given. Also, for suitably restricted sequences λ, necessary and sufficient conditions for a series-to-sequence or series-to-function summability method to include (R, λ, α) for 1 < α ≤ 2 are given. These results are obtained by showing that a certain sequence {δj} (j ≥ 0) is a Schauder-basis in Rλα(N) for each α, 1 < α ≤ 2.

Much attention has been focused on the excitation of atomic spectra by electrons whose velocities do not greatly exceed the critical value, and it is now possible to form a rough idea of the behaviour of the “excitation function,” or probability that an electron of given velocity shall excite on impact, when the energy of the electron is only slightly greater than that required for excitation. But the form of this excitation function for much higher velocities is still in considerable doubt.

The properties p-convexity and q-concavity are fundamental in the study of Banach sequence spaces (see [L-TzII]), and in recent years have been shown to be of great significance in the theory of the corresponding Schatten ideals ([G-TJ], [LP-P] and many other papers). In particular, the notions 2-convex and 2-concave are meaningful in Schatten ideals. It seems to have been noted only recently [LP-P] that a Schatten ideal has either of these properties if the underlying sequence space has. One way of establishing this is to use the fact that if (E, ‖ ‖E) is 2-convex, then there is another Banach sequence space (F, ‖ ‖F) such that ‖x;‖ = ‖x2‖F for all x ε E. The 2-concave case can then be deduced using duality, though this raises some difficulties, for example when E is inseparable.

We give a complete description of the additive Dyer-Lashof operations on the Hopf ring with coefficients in . We re-prove a result of Ravenel and Wilson [9] giving the operations on the element and give formulae for the operations on the other generators.

Introduction. The object of this note is to give a proof of the following theorem on the solution of an important functional equation.

Let f(t) be a function periodic in 2π, and absolutely integrable in the interval (0, 2π). We write

The notion of the order of convergence of iterative processes may well have been known to Gauss or even Newton, but to the author's knowledge it was first considered in detail in 1870 by Schröder(5). The subject was later investigated independently by Hartree(4) and Bodewig(1). Hartree alone mentions, briefly, the extension from one to several variables, but gives no detailed analysis. Domb and Fisher(2) consider a particular type of iterative process in several variables, which they call ‘degenerate’, in which all the variables tend to the same value. We shall follow essentially Hartree's treatment, in considering the solution of r simultaneous equations in r variables and applications to a generalization of the Newton-Raphson process.

A basic problem of the theory of Markov processes is to find a semigroup S of operators on the bounded Borel measurable functions on ℝd, such that S is given by a transition function, and

for all f belonging to the domain of a given second-order elliptic differential operator D. A process is constructed from the transition function associated with S, and then one hopes to realize the process as one with continuous paths [2].

1. A paper with the same title was published with Professor H. F. Baker in 1920; that paper was severely compact and would seem to deserve elucidation, more detail, and some revision. We use a symbolism which, while leaving the proofs geometrical, readily converts itself into ordinary analysis, as we shall show in § 12.

In these Proceedings, Vol. xx (1920), pp. 198–204, the writer ventured (l.c. p. 203) on some remarks as to the reason why Sir G. Darwin and Mr Jeans had obtained discordant results for the stability of the pear-shaped figure of equilibrium, suggesting that this was due to a different mode of expansion of the functions involved. Sir G. Darwin used an infinite series of Lamé functions; Mr Jeans' method was equivalent to using the early parts from an expansion, of which every term, when expressed as an integral series of Lamé functions, would be an infinite series. At that time there was difficulty in obtaining Liapounoff's papers; since then, by the kindness of M. Belopolsky, of Pulkova, the whole of the four parts of Liapounoff's publication “Sur les figures d'équilibre peu différentes des ellipsoïdes d'une masse liquide homogène douèe d'un mouvement de rotation,” in all over 750 large folio pages, have become available; and these are now in the market. It is particularly interesting to see that the fourth part (1914) is devoted precisely to that change in the method of development which would arise in passing from Sir G. Darwin's expansion to the other expansion referred to above. And it is only by this change that Liapounoff is able to give the general proof of a form of expression of his results—in terms of polynomials and not infinite series—upon which his theorem of instability is made to depend. The careful consideration of the convergence of his expansions, which adds so greatly to the length of Liapounoff's papers, supplies materials for the proof that the expansion used in Mr Jeans' paper can be placed on a sure foundation, while Sir G. Darwin's expansion requires an estimation of the remainder.

In various communications published in the Journal of the Chemical Society the author has discussed the effect of different nuclei on the absorption spectra of various compounds. The problem has been further investigated with a number of substances derived from aniline, o-and m-cresols, and α-and β-naphthylamines, and this communication is to give an account of the results. A condensed Cd-spark was used as the source of radiant energy, and alcoholic solutions of the substances were examined in the usual way.

The problem of the double integral in the calculus of variations, when expressed in the parametric notation, was first fully discussed by Kobb. In this paper Kobb finds conditions for the minimising of the double integral

taken over the surface

Numerical integration of the partial differential equations shows that rigid boundaries promote the development of a shock from a finite-amplitude disturbance. A simple step-by-step method agrees with one that also utilizes the characteristics.

The theorem given by Professor Mordell in the preceding paper is capable of a slight generalisation. I first prove that the series

where α, β, γ, … are positive, converges only when κ > Σ (a′/α), the summation referring to those exponents a′ of a, b, c, … for which a ≥ 0; or, if there are no such exponents, the series converges only when κ > a/α, k > b/β, …

Let p be a prime and let f(x) be a quartic polynomial with integral coefficients. I consider the problem of estimating the least non-negative non-residue k of f(x) (modp) (I omit the mod p hereafter), for large primes p, so f(x) ≡ r has a solution for

but not for r = k. The same problem for cubics has been considered by Mordell ((1)), who showed that

as p → ∞, where the constant implied in the O-symbol is independent of the coefficients of the cubic. In fact a more detailed examination of Mordell's proof gives the better estimate

It is the purpose of this paper to show that this same estimate also holds for quartic polynomials.

Let S(Rn) be the space of Schwartz class functions. The dual space of S′(Rn), S(Rn), is called the temperate distributions. In this article, we call them distributions. For 1 ≤ p ≤ ∞, let FLp(Rn) = {f:∈ Lp(Rn)}, then we know that FLp(Rn) ⊂ S′(Rn), for 1 ≤ p ≤ ∞. Let U be open and bounded in Rn−1 and let M = {(x, ψ(x));x ∈ U} be a smooth hypersurface of Rn with non-zero Gaussian curvature. It is easy to see that any bounded measure σ on Rn−1 supported in U yields a distribution T in Rn, supported in M, given by the formula

In this note we prove the following result due to Milnor ((5), Theorem 4·1).

Let G be a locally compact group and L1(G) be the group algebra of G. We show that G is abelian or compact if every continuous automorphism of L1(G)** maps L1(G) onto L1(G) This characterizes all groups with this property and answers a question raised by F. Ghahramani and A. T. Lau in [7]. We also show that if G is a compact group and θ is any (algebra) isomorphism from L1(G)** onto L1(H)**, then H is compact and θ maps L1(G) onto L1(H).