Let (A, G, α, τ) be a twisted covariant system such that G/Nτ is abelian, and let Nτ ⊆ M ⊆ H be closed subgroups of G. We show that inducing covariant representations from A ⋊α, τM to A ⋊α, τH is in a certain sense dual to restricting representations from (A⋊α, τG)⋊άM┴ to (A⋊α, τG)⋊άH┴, and that, similarly, restricting representations from A⋊α, τH to A⋊α, τM is dual to inducing representations from (A⋊τG)⋊άH┴ to (A ⋊α, τG)⋊άM┴.

Conditions are given on the real coefficients p, q and r and the weight w, for the fourth order formally symmetric differential expression

to have the properties of being strong limit-2 and Dirichlet at ∞, when considered in the weighted Hilbert space, . These extend existing results due to both W. N. Everitt and V. Krishna Kumar and cover an expression which is important in the study of certain orthogonal polynomials.

The following theorem was proved by Milloux:

Theorem A. Suppose that f (z) is regular and that | f(z) | < 1 in the unit circle. Suppose also that the set of points in the circle | z | ≤ r′ < 1 at which

cannot be enclosed in circles, the sum of whose radii is equal to 2eh−1. Then

where K is a numerical constant.

The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.

In this paper we show that the Castelnuovo–Mumford regularity of the associated graded module with respect to an m-primary ideal I is effectively bounded by the degree of nilpotency of I. From this it follows that there are only a finite number of Hilbert-Samuel functions for ideals with fixed degree of nilpotency.

Let 〈a1, . . ., am ∣ wn〉 be a presentation of a group G, where n ≥ 2. We define a system of codimension-1 subspaces in the universal cover, and invoke Sageev's construction to produce an action of G on a CAT(0) cube complex. We show that the action is proper and cocompact when n ≥ 4. A fundamental tool is a geometric generalization of Pride's C(2n) small-cancellation result. We prove similar results for staggered groups with torsion.

Theorem. if a ≥ 2, b ≥ 1, and a + b ≥ ℵ0, then every collection of more than ab sets each of cardinal not exceeding b contains a subcollection of more than ab sets every two of which have the same intersection.

The non-linear equations governing the convective flow of heat, beyond marginal stability, in a fluid sphere containing a uniform distribution of heat sources, and cooled at its surface, are expressed in the Boussinesq approximation as a perturbation of the steady-state conduction solution. A steady-state axisymmetric poloidal flow is assumed, and the equations for an approximate solution, involving the first spherical harmonic only, are obtained using a special case of the general evolution criterion of Glansdorff and Prigogine. The solution is found numerically using an iterative procedure, the results showing good agreement with those obtained by Stuart's ‘shape assumption’ for values of the Rayleigh number just above the critical value for the onset of marginal stability.

We study quadratic forms defined on an adelic vector space over an algebraic extension K of the rationals. Under the sole condition that a Siegel lemma holds over K, we provide height bounds for several objects naturally associated to the quadratic form, such as an isotropic subspace, a basis of isotropic vectors (when it exists) or an orthogonal basis. Our bounds involve the heights of the form and of the ambient space. In several cases, we show that the exponents of these heights are best possible. The results improve and extend previously known statements for number fields and the field of algebraic numbers.

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

To investigate invariants of links derived from their diagrams, the recent new polynomial invariants of links play important roles. Murasugi6, 7, Kauffman3 and Thistlethwaite 9 independently showed that the number of crossings in a proper connected alternating diagram of a link is the minimal-crossing number of the link and that the writhe of the diagram is invariant. Murasugi 8 also determined the minimal-crossing number of torus links. In 5, Lickorish and Thistlethwaite introduced the concept of an adequate link diagram and showed that the number of crossings in an adequate diagram of a semi-alternating link is the minimal-crossing number of the link. They also determined the minimal-crossing number of almost all Montesinos links. In this paper we show that for some links represented by plats and braids which are not adequate, the numbers of crossings in the diagrams are the minimal-crossing numbers of the links.

This note is devoted to the construction of a 〈Q, EΦ〉-closed Fischer class which is not a formation. It provides a negative answer to the Questions 3–7 raised by Cossey in (1). All groups considered are finite and soluble.

We prove that the coefficients of the mock theta functions

\begin{eqnarray*} f(q) = \sum_{n=1}^{\infty} \frac{ q^{n^2}}{(1+q)^2 (1+q^2)^2 \cdots (1+q^n)^2 } \end{eqnarray*}

\begin{eqnarray*} \omega(q)=1+\sum_{n=1}^\infty \frac{q^{2n^2+2n}}{(1+q)^2(1+q^3)^2\cdots (1+q^{2n+1})^2} \end{eqnarray*}

Some time ago Besicovitch(l) gave an example of a linear Gδ-set E of Besicovitch dimension 1, with the property that E – A also has dimension 1, for each Fσ.-set A contained in E. This shows that an Fσ-set contained in E cannot be a very close approximation to E, but it leaves plenty of scope for Fσ-sets contained in E to be reasonably good approximations to E.

A collineation in a space of paths is defined as a point transformation which carries paths into paths. Such transformations were first studied by L. P. Eisenhart and M. S. Knebelman. Subsequently J. Douglas introduced the geometry of K-spreads and E. T. Davies has shown that the results for an affine space of paths can be extended to these more general spaces and written in a very elegant form by the aid of Lie derivation.

Let F = (F1, …, Fn-1): (ℝn, 0)→(ℝn-1, 0) and G:(ℝn, 0)→(ℝ, 0) be germs of analytic mappings, and let X = F-1(0). Assume that 0 ∈ ℝn is an isolated singular point in X, i.e. 0 ∈ ℝn is isolated in {x ∈ X|rank[DF(x)] < n-1}. Hence a germ of X/{0} at the origin is either void or a finite disjoint union of analytic curves. Let b denote the number of branches, i.e. connected components, of X/{0} and let b+ (resp. b-, b0) denote the number of branches of X/{0} on which G is positive (resp. G is negative, G vanishes). The problem is to calculate the numbers b, b+, b-, b0 in terms of F and G.

The differential equation

defining the function of the amplitude sL is integrated by an iterative procedure. An expression for the phase

is obtained in terms of integrals of ωL and its derivatives. The method is applied to the numerical calculation of Coulomb functions and the accuracy of the results is discussed. Recurrence relations are used to obtain øL, L ≽ 1.

Some years ago M. Jackson(5) obtained the sum of a particular 3H3 series which generalized the theorems of Whipple and Watson on sums of a 3F2 series, and later she(6) deduced the sum of a particular well-poised 6H6( – 1) series. In this note sums of certain particular bilateral hypergeometric series of the same type are given. They are believed to be new. In the sequel, the sum of a particular 8Ψ8 series has been obtained. The existence of a particular case of this sum was pointed out earlier by Slater and Lakin (11).

In this paper, we introduce two new invariants that are closely related to Milnor's curvature-torsion invariant. The first, a particularly natural invariant called the spiral index of a knot, captures the number of local maxima in a knot projection that is free of inflection points. This invariant is sandwiched between the bridge and braid index of a knot, and captures more subtle properties. The second invariant, the projective superbridge index, provides a method of counting the greatest number of local maxima that occur in a given projection. In addition to investigating the relationships among these invariants, we use them to classify all those knots for which Milnor's curvature-torsion invariant is 6π.

The object of the present note is to evaluate some integrals involving Meijer's G-function, in which the argument of the G-function contains a factor where m and n are positive integers and t is the variable of integration. Two different forms of the general result have been obtained, one for m > n and the other for m < n. The value of the corresponding integral when m = n is also obtained. For the definition, properties and the behaviour of the G-function, see (2), §§ 5·3, 5·31 and (5), § 18.