In this paper, we will prove that if A is a C*-algebra with an effective coaction ε by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-algebras.

The unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this paper we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equals its unknotting number, if and only if the given knot appears in an unknotting sequence of a torus knot.

Let A1 denote the algebra generated by one pair of canonical operators. It is shown that the realizations of sl(2) in A1 divide into two non-empty classes determined by the spectrum of the semisimple element of sl(2) in A1.

We first give a brief introduction to Hall–Littlewood functions; we follow closely the notation used in Macdonald [3].

Let λ = (λ1,…,λm) be a be a partition of n; that is λ1 + … + λm = n, λ1 ≥ λ2 ≥ … ≥ λm >0. We shall sometimes write l(λ) for m and refer to l(λ) as the length of λ and we shall write |λ| for ∑λi. Let x1, x2, … be an infinite set of indeterminates and t an indeterminate independent of the xi (i = 1,2, …). Let Pλ(x;t) = Pλ(x1, x2, …t) and Qλ(x,t) = Qλ(xl, x2, …t) be the Hall–Littlewood P- and Q-functions defined as in Macdonald.

The isothermal flow of gas past a gravitating point-sink has been studied for four values of the undisturbed gas velocity, all of which are supersonic. From these studies, estimates have been made of the rates of accretion of an isothermal gas by a star moving with a supersonic velocity. These estimates are of the same order as those obtained by earlier workers who used simpler mathematical models.

1. The object of this paper is to discuss conditions of validity of the Parseval formulae for Fourier integrals:

where the transforms are defined by ordinary convergence; we shall not be concerned with the more elegant theory in which they are given by convergence in mean.

We investigate the steady rotation of an insulating body of revolution in an unbounded electrically conducting fluid permeated by a uniform axial applied magnetic field. The assumptions of a small magnetic Reynolds number (Rm ≪ 1, i.e. the weakly conducting situation) and negligible inertia forces compared with the magnetic forces (R/M2 ≪ 1) permit us to suppress the inflow at the poles and outflow at the equator, which normally occurs for a non-conducting viscous fluid ((12), pp. 436–439). Thus in the case of the sphere, we find an exact solution of the reduced equations in terms of an infinite series of Legendre polynomials of order 1 with coefficients which are the ratios of modified spherical Bessel functions. This is the canonical problem by which results for arbitrary bodies of revolution are obtained.

R. A. Millikan and C. F. Eyring have shown that the auto-electronic emission is independent of the temperature of the cathode up to 1000° K. At higher temperatures a thermionic emission takes place and this masks the auto-emission. W. S. Pforte and the author have recently shown that the thermionic emission is increased by the field in accordance with a relation derived by Schottky, so that when both thermionic and auto-electronic emissions are present it is only necessary to subtract the thermionic current, as given by the Schottky relation, from the total measured current to obtain the value of the auto-emission at the temperature considered. In this way the thermionic and auto-emissions can be separated.

An exact solution in finite terms is derived within the limitations of the classical thin-plate theory, for the problem of a thin circular plate acted upon normally by hydrostatic pressure distributed over the area of a concentric ellipse, and subject to boundary conditions covering the usual rigidly clamped and simply supported boundaries.

In his attempt to give the irreducible characters of the Weyl groups of type A, B and D, Mayer [3] found a unified theory using their common structure as reflection groups. In this paper, we give the characters of the complex imprimitive reflection groups. We are able to give an algorithm which allows us to calculate the irreducible components of the principal character of a ‘Weyl’ subgroup induced up to the whole group. This is a generalization of the algorithm given by Mayer.

By means of purely qualitative arguments which do not depend on any particular model, the general scheme of stable nuclei and the isotopic breadth of nuclei with odd charge number are explained.

The breadth of the isobaric region can be obtained if the numerical values of certain energies are known. Though these can be estimated only very roughly, the values for the breadth of the isobaric region obtained in this way are in good agreement with the experimental values. The increase in the breadth of the isobaric region from light nuclei to heavier nuclei can be explained, but no plausible explanation has been found for the fact that the breadth decreases again for the heaviest nuclei.

A preliminary investigation is described in which a type of gold leaf source suitable for use with an expansion chamber is developed. Such a source is necessary for the detailed examination of β particles of energy less than about 50,000 electron volts by the Wilson method. Some results obtained with sources of radium D and thorium B + C of this type are presented in outline.

The writer would like here to acknowledge the help and encouragement he derived from many discussions with Professor Sir Ernest Rutherford and with Dr J. Chadwick during the course of the investigation, and further to express his thanks to Mr G. R. Crowe for assistance in the preparation of some of the sources.

1. There has been considerable debate on the conditions under which a given function f(r) can be expanded as a series of the orthogonal functions where υ1, υ2, … are the zeros of the Hankel function regarded as a function of its order. Here k, a are positive constants and 0 < a ≤ r < ∞. The formal expansion can be written as

where

A very great deal has been published upon the subject of the electrical counter and its mode of action. Most explanations are incomplete in that they neglect some of the factors involved. The question of the self-restoring property of the system has also received attention, but this property has not been properly related to the general problem of intermittency in discharge tubes.

The local change in an oriented link diagram which replaces by k positive half-twists is called a tk move. For k even, the local change replacing by is called a tk move. For an unoriented diagram define a k-move, replacing by for any k. The following conjecture was stated in [14] and [10].

1. The results. Let l be an odd prime, r ≥ 1, and let

be the quaternion group of order 4lr, as given by generators and relations. Throughout N is a tamely ramified normal number field with Galois group Gal (N/Q) = H (a ‘quaternion field’), and its ring of integers. We are interested in the structure of as a module over the integral group ring ZH. Deriving, first, certain classgroup invariants for locally free ZH-modules, we shall then determine those for the module in terms of the arithmetic invariants of N/Q. When 1 ≡ – 1 (mod 4), this yields again a Galois module interpretation of Artin root numbers quite analogous to that in (2). On the other hand for l ≡ 1 (mode 4), we shall get a weak ‘normal integral basis theorem’. The original impetus for this work came from computations of J. Queyrut, who – in different language – obtained these results in the case l = 3, r = 1 (cf. (7)). The tools, we are using, come from the general theory developed in recent years with such concrete applications in mind, and it is perhaps of interest to see how the various ‘strands’, on root numbers (cf. (3), (4)), on locally free modules (cf. (5)), and on Galois module structure (cf. (6)) are here pulled together. For technical reasons, we shall impose on N the slight further restriction, that l be non-ramified, although our results would remain true without this. Both the statements and the proofs of the theorem depend on ideas contained in (5) and (6). The reader who is prepared to take them for granted should, however, be able to read the present paper independently of those papers.

A multiset consists of elements, but the notion of a multiset is distinguished from that of a set by carrying information of how many times each element occurs in a given multiset. In this work we will investigate the notion of iterative multisets, where multisets are iteratively built up from other multisets, in the context Martin–Löf Type Theory, in the presence of Voevodsky’s Univalence Axiom.

In his 1978 paper, “the type theoretic interpretation of constructive set theory” Aczel introduced a model of constructive set theory in type theory, using a W-type quantifying over a universe, and an inductively defined equivalence relation on it. Our investigation takes this W-type and instead considers the identity type on it, which can be computed from the univalence axiom. Our thesis is that this gives a model of multisets. In order to demonstrate this, we adapt axioms of constructive set theory to multisets, and show that they hold for our model.

This paper addresses the question of how well the exterior of a knot determines the knot. It is well known that for a knotted sphere pair (Sn+2, Sn), n ≥ 2, a given exterior corresponds to at most two distinct knots (1), (4), (10); examples of distinct knots with the same exterior are given in (2), (6). For a knotted ball pair (Bn+2, Bn), n ≥ 2, the situation may be more complicated. In fact, given an arbitrary positive integer N, Hitt-Sumners (7), (8) constructed N distinct ball pairs with the same exterior, n ≥ 4.

Throughout this paper we deal only with complex and semi-simple algebras. Let B be such an algebra. We denote the socle of B as SB. B is a modular annihilator algebra if B/SB is a radical algebra, i.e. if every element of B is quasi-regular modulo the socle of B; see (1) or (12). Now assume that B is a modular annihilator algebra and a Banach algebra. Then any semi-simple closed subalgebra of B is a modular annihilator algebra by ((4), Cor. to Theorem 4·2,). It is not true, however, that any semi-simple subalgebra A of B is a modular annihilator algebra, even when A is a Banach algebra in some norm. We give a simple example to illustrate this. Let A be the algebra of all complex functions f, continuous on the closed unit disk D in the complex plane, analytic in the interior of D, and such that f(0) = 0. A is a Banach algebra in the usual sup norm over D. Now consider the norm on A defined by

Let B be the completion of A in this norm. A has an involution * defined by and also ‖ff*‖ = ‖f‖2 for all f ∈ A. Therefore B is a B*-algebra. It is not difficult to verify that the only non-zero multiplicative linear functionals on A which are continuous with respect to the norm ‖·‖, are the point evaluations at 1/n, n = 1, 2 …. It follows that every non-zero multiplicative linear functional on B is an extension of one of these point evaluations to B. Thus B can be identified with the algebra of all complex sequences which converge to zero.

1. Statement of problem and the method of Barnes and Swinnnerton-Dyer. 1·1. Let ξ1, ξ2, … ξn De n real, homogeneous linear forms in the real variables x1, x2,…, xn and of determinant Δ ǂ 0. Let y1y2, …, yn be any real numbers. We identify the n-tuplet (x1x2, …, xn) with the point P, coordinates (x1, x2, …, xn), of n-dimensional space.We write P≡P0 (mod 1) if x1≡y1, x2≡y2, …, xn≡yn (mod 1), and P0 is the point (y1, …, yn).