The study of existence of a universal C*-completion of the *-algebra canonically associated to a Hecke pair was initiated by Hall, who proved that the Hecke algebra associated to (SL2($\mathbb{Q}$p), SL2($\mathbb{Z}$p)) does not admit a universal C*-completion. Kaliszewski, Landstad and Quigg studied the problem by placing it in the framework of Fell–Rieffel equivalence, and highlighted the role of other C*-completions. In the case of the pair (SLn($\mathbb{Q}$p), SLn($\mathbb{Z}$p)) for n ⩾ 3 we show, invoking property (T) of SLn($\mathbb{Q}$p), that the C*-completion of the L1-Banach algebra and the corner of C*(SLn($\mathbb{Q}$p)) determined by the subgroup are distinct. In fact, we prove a more general result valid for a simple algebraic group of rank at least 2 over a $\mathfrak{p}$-adic field with a good choice of a maximal compact open subgroup.

For a gas with a general equation of state the stream function for the flow past a flat plate is determined when the plate is held at an angle ½π–α to the stream. At infinity the Mach number M1 = 1 and free streamlines, on which M1 = 1, extend from the edges of the plate to infinity downstream. The method of solution requires the determination of the correct singularity corresponding to the free stream and the subsequent satisfying of the boundary conditions. The drag coefficient is determined in the form of a contour integral and the computational problems in evaluating the drag are discussed. The integral is evaluated for small values of α for a gas with an equation of state as given by Tomotika and Tamada(6).

The goodness of fit of hypothetical discriminant functions has been discussed by Bartlett(1), Kshirsagar(2, 3), Radcliffe(4) and Williams (5–8). The associated problem of obtaining confidence intervals for the coefficients of a single discriminant function or canonical variable has been dealt with by Bartlett (1).

The rates of condensation and evaporation of adsorbed particles when they interact with long-range forces are deduced from kinetic considerations, including also the case of adsorption of diatomic molecules with dissociation. The rate of evaporation is compared with experiment, and the variation of the dipole moment with θ required to make the theory agree with experiment is discussed.

Finally the formulae for the rates of condensation and evaporation are applied to the problem of the diffusion of gases through metals, and it is found that for the processes considered the effect of interaction does not alter the conclusion arrived at in an earlier paper concerning the diffusion equation at small p and at large p.

In this paper, it will be shown how it is possible, using Bernoulli polynomials of the first kind, to derive correction terms for any quadrature approximation in the form of a weighted sum over a finite interval. The terms derived will involve either the difference or the sum of derivatives taken at the ends of the range of integration, and they will be described as being of either Type A or Type B respectively. There will also be some discussion of the uses of such terms.

We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

Let E(s, Q) be the Epstein zeta function attached to a positive definite quadratic form of discriminant D < 0, such that h(D) ≥ 2, where h(D) is the class number of the imaginary quadratic field ${\mathbb{Q}} (\sqrt D)$. We denote by ${N_E}({\sigma _1},{\sigma _2},T)$ the number of zeros of $[E(s,Q)$ in the rectangle ${\sigma _1} < {\mathop{\rm Re}\nolimits} (s) \le {\sigma _2}$ and $T \le {\mathop{\rm Im}\nolimits} (s) \le 2T$, where $1/2 < {\sigma _1} < {\sigma _2} < 1$ are fixed real numbers. In this paper, we improve the asymptotic formula of Gonek and Lee for ${N_E}({\sigma _1},{\sigma _2},T)$, obtaining a saving of a power of log T in the error term.

In obtaining a solution of Laplace's equation in two dimensions by the method of conformal mapping, one first maps the points (x, y) of the Euclidean plane R2 into the algebra of complex numbers C by means of the real-linear function g: R2→C using the prescription g(x, y) = x + iy ≡ z. One then obtains solutions of Laplace's equation by allowing those mappings of C into itself that are expressed by analytic functions.

The inequality

for fεLp(− ∞, ∞)or Lp(0, ∞) (1≤p ≤ ∞), and its extension

for T an Hermitian or dissipative linear operator, in general unbounded, on a Banach space X, for xεX, have been considered by many authors. In particular, forms of inequality (1) have been given by Hadamard(7), Landau(15), and Hardy and Little-wood(8),(9). The second inequality has been discussed by Kallman and Rota(11), Bollobás (2) and Kato (12), and numerous further references may be found in the recent papers of Kwong and Zettl(i4) and Bollobás and Partington(3).

In this paper we obtain a cohomological splitting criterion for locally free sheaves on arithmetically Cohen–Macaulay surfaces with cyclic Picard group, which is similar to Horrocks' splitting criterion for locally free sheaves on projective spaces. We also recover a duality property which identifies a general K3 surface with a certain moduli space of stable sheaves on it, and obtain examples of stable, arithmetically Cohen–Macaulay, locally free sheaves of rank two on general surfaces of degree at least five in ${\mathbb P}^3$.

1. The elastic stability of a long flat plate when acted on by a shearing force along its edges has been discussed by Southwell and Skan*, and Leggett† has solved the shearing problem when the plate has a small constant curvature. The present paper deals with the elastic stability of a long corrugated plate which is acted on by a shearing force along its generators, the work being applicable to plates with any given number of bays forming the corrugations. It is supposed that the plate is thin and that the depth d of a bay is a small multiple of the semithickness h; little progress has been made without some such assumption as the second, which enables us to reduce the fundamental equations to a soluble form.

Let f be an entire transcendental function of finite order and Δ be a forward invariant bounded Siegel disk for f with rotation number in Herman's class $\mathcal{H}$. We show that if f has two singular values with bounded orbit, then the boundary of Δ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.

1. We write L for the class of integrable functions in (− ∞, ∞), V for the class of functions of bounded variation, and define A, A to be the classes of functions F(x) which may be expressed in the forms

respectively.

Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z and w; let $\mathcal{X}$ be the generic element of the family of surfaces in ℙ given by

\begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*}

$\mathcal{X}$

$\mathcal{X}$

$\mathcal{X}$

Recently, through a repeated application of one-dimensional finite integral transforms, Cinelli(1) gave a solution for the temperature distribution in a hollow circular cylinder of finite length. Since no new ideas or techniques are introduced, the extension claimed in (1) with regard to the finite Hankel transform technique employed in the transformation of the radial space variable in the hollow cylinder problem is trivial, in view of well-known works by Sneddon(2) and Tranter (3), to mention a few. The list of the finite Hankel transforms given in (1) for a variety of boundary conditions at r = a and r = b is the result of routine, algebraic manipulations well known from the general theory of eigenvalue problems specialized for the hollow cylinder. In this list a set of seemingly different series expansions is given for the inverse Hankel transform for each combination of boundary conditions at the two radial surfaces. In each case, the two expressions for inversion can readily be shown to be identical to each other when use is made of the frequency equation. One of the inversion forms is therefore unnecessary once the other is given. Furthermore, the general solution as given by equation (54) of Cinelli(1)does not satisfy his boundary conditions (27), (28), (29) and (30), unless these latter are homogeneous.

There is a close parallelism between the theories of convergence of directed nets and of filters, in which ‘subnet’ corresponds, in general, to ‘refinement’. With the standard definitions, however (1), pages 65 et seq., this correspondence is not exact, as there is no coarsest net converging to x0 of which all other nets with the same limit are subnets. (Suppose, for example, that a net X = {xj,: j ∈ J} in R1 has both the sequence-net S = {n−1; n = 1, 2,…} and the singleton-net {0} as subnets. Then (with an obvious notation), there exist

such that j0 ≥ jn for all n, while jn ≥ j0 for all n ≥ n0 say. But, given any j ∈ J, there exists n with jn ≥ j: it follows that jn ∈ j for all n ≥ n0 (independent of j); thus X cannot converge to 0. Even if nets with a last member are excluded, a similar result can be obtained by considering the net Y = {yθ; Θ an ordinal less than ω1}, where yθ = 0 for all Θ. If X has both Y and S as subnets we can show that (with a similar notation) there exists Θ0 such that Θ ≥ Θ0 implies jθ ≥ all jn, but also n0 such that n ≥ n0 implies ; the rest is as before.) Moreover, the theory of convergence classes, (l), pages 73 et seq., contains a condition (Kelley's condition (c)) whose analogue need not be separately stated for filters. These differences can be removed by adopting a wider definition of subnet, a course which does not seem unnatural, inasmuch as the standard definition is already wider than the ‘obvious’ one, and our proposed definition is equivalent to the standard one in the special case of sequences.

This paper derives some sufficient conditions for the global validity of projective relativity theory. The basic object of study is the infinitesimal bundle, which is a generalization of a principal circle bundle. A method ofconstructing infinitesimal bundles is developed. The main result is that ifspacetime is non-compact and simply connected, and has a finite homology basis, then projective relativity is globally valid.

It is trivial that a group all of whose elements except the identity have order two is Abelian; and F. Levi and B. L. van der Waerden(1) have shown that a group all of whose elements except the identity have order three has class less than or equal to three. On the other hand, R. Baer(3) has shown that if the fact that all the elements of a group have orders dividing n implies a limitation on the class of the group, then n is a prime. The object of the present note is to extend this result by showing that if M is a fixed integer there are at most a finite number of prime powers n other than primes, such that the fact that all the elements of a group have orders dividing n implies a limitation on the class of its Mth derived group.

1. Introduction. A considerable amount of attention has been paid to the problem of determining the conditions which decide whether a liquid heated from below is stable or unstable. The motion consequent upon the disturbance of an unstable ideal gas does not, however, seem to have been treated so far, and this problem forms the subject of the present paper. Heat conduction and viscosity are at first neglected, and we are therefore dealing with the small motions of a gas slightly disturbed from a position of equilibrium under the influence of gravity. The condition for the stability of such a gas is well known, namely, the temperature gradient must be less than the adiabatic gradient. Furthermore, it is known that there is a sharp distinction between slow large-scale (meteorological) and rapidly varying small-scale (acoustical) phenomena. The present paper confirms these points and derives the time scale of meteorological phenomena. Heat conduction and viscosity are then shown to set a lower limit to the dimensions of such disturbances, while the effect of the earth's rotation is shown to be negligible.