Let G bea finite group, let C1, …, Cn be the cyclic subgroups of G and let 1i be the identity linear character of Ci (1 ≤ i ≤ n). It is a well-known result of Artin ((l), 39·1) that a character x of a linear representation of G over the rational numbers may be written

where a1, …, an are integers. In this note, we establish an analogue of this result for characters of projective representations over the rational numbers.

1. The integral equation

where x, f (x), and λ are real and α positive, may be regarded as a differential equation of order α. Suppose for example that α is a positive integer p, that f (x) tends to 0, when x → ∞, with sufficient rapidity, and that

Then, if we integrate repeatedly by parts, and write z for fp (x), (1·1) becomes

The only solutions are finite combinations of exponentials.

This paper develops the metric theory of simultaneous inhomogeneous Diophantine approximation on a planar curve with respect to multiple approximating functions. Our results naturally generalize the homogeneous Lebesgue measure and Hausdorff dimension results for the sets of simultaneously well-approximable points on planar curves, established in Badziahin and Levesley (Glasg. Math. J., 49(2):367–375, 2007), Beresnevich et al. (Ann. of Math. (2), 166(2):367–426, 2007), Beresnevich and Velani (Math. Ann., 337(4):769–796, 2007) and Vaughan and Velani (Invent. Math., 166(1):103–124, 2006).

In this paper upper bounds are established for the principal solution of the differential equation

and its derivative, for unrestricted values of the complex variable t and the complex parameter μ. The results may have little interest in their own right, but they are of great value in developing the asymptotic theory of linear second-order differential equations in a domain containing two turning points. Equation (1·1) is the simplest example of a differential equation of this type.

By a covering presentation of a 3-manifold, we mean a labelled link (i.e., a link with a monodromy representation), which presents the 3-manifold as the simple 4-fold covering space of the 3-sphere branched along the link with the given monodromy. It is known that two labelled links present a homeomorphic 3-manifold if and only if they are related by a finite sequence of some local moves. This paper presents a method for constructing topological invariants of 3-manifolds based on their covering presentations. The proof of the topological invariance is shown by verifying the invariance under the local moves. As an example of such invariants, we present the Dijkgraaf–Witten invariant of 3-manifolds.

In 1968, Amemiya and Kōmura ((1), p. 275) gave an example of a separable and incomplete Montel space. Their example depended upon constructing a subspace of ω, the countable product of real lines, which had several properties and in particular had codimension in ω. There is an error on page 276, line 5 of the proof of the Hilfssatz concerning this construction. The constructed subspace has codimension one. A more delicate construction is needed here; in fact, an example given by Webb ((7), p. 360) is sufficient to make this correction.

Nicholson (3) stated that the Legendre functions of the second kind, Qn(x), are orthogonal between − 1 and 1. This is incorrect in general, and one purpose of this paper is to give the correct value of

when m ╪ n. Nicholson's value of the integral when m = n, namely,

is correct.

1. In this paper I discuss the expression of m + 1 general forms F, F′, …, F(m), each of order n and homogeneous in r + 1 variables z0, z1, …, zl, …, zr, as the sums each of the nth powers of (the same) h + H linear forms in these variables. I take h > 0 of these linear forms to be undetermined, namely the forms

whose coefficients are undetermined; and I take the remaining H ≥ 0 linear forms to be assigned, namely the forms

whose coefficients are assigned.

The solution for the flow of an incompressible fluid past an infinitely long wedge with a finite sloping edge (a finite wedge) is generalized by the hodograph method. In the flow thus obtained the axis of symmetry and a sloping edge of the wedge are again part of one streamline. It becomes possible to describe the flow of an ideal gas past a finite wedge if the hypothesis is made that the first singularity on this streamline, along the sloping edge, corresponds to the shoulder of the wedge. For a given wedge, with gradually increasing velocity at infinity upstream, the singularity appears at first at subsonic velocity. Beyond a certain critical velocity at infinity the singularity is always associated with the speed of sound. The hypothesis thus implies that put forward by Maccoll(9) and supported by Busemann(l). A qualitative examination shows that the solution reproduces experimentally known features of the flow of compressible fluid past a finite wedge.

Let Aut G and Inn G denote the group of all automorphisms of the group G and the subgroup of all inner automorphisms of G, respectively. A group G is said to be complete if it has trivial centre and Aut G = Inn G. Examples of such groups abound and they have been the object of study for many years. Following Heineken (8) we call a group G semicomplete if Aut G = Inn G.

Duskin [1, 2] showed that gerbes over a Grothendieck topos E can be interpreted by certain internal groupoids of E which he called bouquets. He proved that for any bouquet B of E the fibred category F ors(B) of B-torsors is a gerbe over E, and conversely that any gerbe G gives rise to a bouquet B satisfying G ≅ F ors(B). We want to give here a somewhat different approach to these results, based on a construction of gerbes given in [3].

For each integer n ≥ 0, we give a distinct closed model category structure to the categories of spaces and of simplicial sets. Recall that a non-empty map is said to be a weak equivalence if it induces isomorphisms on the homotopy groups for any choice of base point. Putting the condition on dimensions ≥ n, we have the notion of a weak n-equivalence which is at the base of the nth closed model category structure given here.

We give an elementary construction of the compact real form of the Lie algebra 2. This construction exhibits the group 23·L3(2) as a group of automorphisms. We also show that there is a unique 14-dimensional real Lie algebra invariant under the action of this group.

When a stream is flowing along a wall against an adverse pressure gradient, the skin friction at the wall decreases to zero, at which point or at least very near it the flow seems to leave the surface altogether. Between the main stream and the wall there may be a slow back-flow. Upstream of separation the influence of the wall on the flow is satisfactorily explained by the boundary-layer theory, which among other things presupposes that this influence is confined to a narrow region near the wall. Such a theory is clearly inadequate downstream, however, because this layer rapidly increases in width and, further, the main stream is seriously affected, and consideration of the Navier-Stokes equations would seem to be necessary before a full theoretical explanation of separation could be given. Apart from a note by Dean (2) on the relations between the pressure and velocity at separation, there has, however, been no relevant attack on these equations.

Let θ be an algebraic number of degree n, with conjugates θ(1), …, θ(n), where θ(1), …, θ(r) are real and θ(r+j), θ(r+s+j) are complex conjugates for j = 1, …, s. [Here r ≥ 0, s ≥ 0, r + 2s = n.] Let ω1, …, ωn be a basis for the integers of k(θ), and set

In [1] Chan and Williams considered a one-dimensional diffusion of the form

where F is a strictly increasing continuous function with F(0) = 0 and ε is a decreasing deterministic function such that ε(0) is finite and ε(t) ↓ 0 as t↑ ∞, and gave necessary and sufficient conditions for Yt →0 a.s. as t→∞.

1. In a paper read to this Society Mr F. P. White has derived a remarkable chain of theorems in [n] which, for n = 2, becomes Clifford's chain. White's method is so elegant that, as he has restricted himself to curves going through two given points, it may be permissible to shew that his line of argument applies equally to curves going through three or more given points, thus producing other chains of theorems analogous to Clifford's. For this purpose it is necessary to consider the successive projections, not of a simplex, but of a figure of n, n − 1, n − 2,…vertices in [n].

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 < r1 < r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequality

has a solution in integers x1, x2, …, xr.

The heat of adsorption of particles which occupy two sites on a surface is examined by determining the statistical equilibrium. In particular, equations which are appropriate to immobile films are obtained. The heats of adsorption calculated from these equations are compared with the experimental values obtained for the heat of adsorption of hydrogen on a tungsten surface, and with the values which were obtained using an empirical method.

Let Sj: ℝd → ℝd for j = 1, . . ., N be contracting similarities. Also, let (p1,. . ., pN, p) be a probability vector and let ν be a probability measure on ℝd with compact support. We show that there exists a unique probability measure μ such thatThe measure μ is called an in-homogenous self-similar measure. In this paper we study the asymptotic behaviour of the Fourier transforms of in-homogenous self-similar measures. Finally, we present a number of applications of our results. In particular, non-linear self-similar measures introduced and investigated by Glickenstein and Strichartz are special cases of in-homogenous self-similar measures, and as an application of our main results we obtain simple proofs of generalizations of Glickenstein and Strichartz's results on the asymptotic behaviour of the Fourier transforms of non-linear self-similar measures.