Groups associated to a knot in S3 which classify the knot have been constructed by Conway and Gordon [4], Simon [12] and Whitten[16]; see also [1] and [19] for certain special classes of knots. In this note we show that for a knot K in S3 of the groups π1(S3/K)/〈malb〉 classify the knot; here m resp. l denote a meridian resp. longitude of the knot and 〈〉 denotes normal closure. This is based on Thurston's orbifold geometrization theorem [13–15] and the following result (see also [8, 19]):

Theorem 1. Let O1 and O2 be good closed irreducible 3-orbifolds which possess a decomposition into geometric pieces (along Euclidean 2-suborbifolds, see [2, 15]) and have infinite (orbifold-) fundamental group. Then O1 and O2 are diffeomorphic (as orbifolds) if and only if π1 O1 and π1 02 are isomorphic.

Suppose that K is a number field and n = [K: ℚ]. Write S∞(K) for the set of archimedean places of K, i.e. the set of all embeddings σ: K → ℂ. Suppose that is a linear form in t variables x1, …, xt, where the aj ∈ K. This gives rise to a norm form N(x), where

All groups G considered in this paper are finite and all representations of G are defined over the complex numbers. The reader unfamiliar with projective representations is referred to [4] for basic definitions and elementary results.

We show how to partition certain classes of finite projective planes into equicardinal arcs. Several partitions of this kind are to be found in the recent literature and they have aroused a certain amount of interest on two counts, as we shall shortly see.

An automorphism σ(G) of a graph G is a permutation of the set of its vertices which preserves adjacency. Under the operation of composition the automorphisms of G form a group Aut(G). The graph G is called asymmetric if Aut(G) is trivial, and symmetric otherwise.

Throughout this paper A denotes a Noetherian local ring with maximal ideal m and M denotes a finitely generated A-module. Moreover stands for the ith local cohomology functor with respect to m (cf. [10]). We refer to [15] for unexplained terminolog.

Let f1, …, fs: ℝn+1→ℝ be homogeneous polynomials, and let

The Euler characteristic χ(X) is an important topological and geometric invariant of X.

The Brown–Grossman proper homotopy groups at an end of a locally compact Hausdorff space, X, were introduced by E. M. Brown in 1974. In this note we define and study ‘global’ versions of these groups and compare these global groups with the groups ‘at infinity’. We also obtain several interlocking exact sequences relating these groups with the Hurewicz and Steenrod homotopy groups of X.

An earlier recognition theorem of one of the authors is extended to the bounded case, and conditions yielding concordance of triangulations are established.

The theory of generalized symmetric spaces was begun by P. J. Graham and A. J. Ledger in 1967 in their paper [1]. Other authors who contributed to this topic were, e.g. F. Brickel, A. Deicke, A. S. Fedenko, A. Gray, V. G. Kac, M. Obata, B. Pettit, K. Sekigawa and J. Wolf; more recently also J. A. Jiménez, A. R. Razawi, C. Sánchez, M. Sekizawa, M. Toomanian, G. Tsagas and L. Vanhecke. A systematic exposition is given in the book by O. Kowalski[3, 4]. The author classifies all generalized symmetric Riemannian spaces in dimension n ≤ 5, using a systematic method.

In this paper, we prove that if X is a strictly convex Lk-UR space, then X is an L-kR space. However, the converse need not be true. Also, for each k ≥ 2, there exists a Lk-UR space which is URED but is not L-(k – 1)R.

Let X, Y be Banach spaces and G a closed subspace of Y. Let L(X, Y) be the space of bounded linear operators from X into Y. In this paper we investigate when L(X, G) is proximal in L(X, Y). Further, we discuss the related problem of proximinality of L∞(T, G) in L∞(T, Y). We improve results obtained by Light and Cheney in this direction.

The object of this paper is to make Lp estimates for the n-linear form

defined on n-tuples of functions (ø1, …, øn) on the sphere Sn−1 inℝn.

Let D be the unit disc in the complex plane ℂ with centre 0 and let ∂D be its boundary. By Möb (D) we denote the group of all Möbius transformations which leave D invariant. A Fuchsian group G acting on D is a discrete subgroup of Möb (D). The limit set of G is in ∂D. We decompose ∂D into the following three disjoint sets:

We determine the rate of growth of the expected number of real zeros of a class of random polynomials whose coefficients form a stationary uniformly mixing sequence.

The purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem (CLT). As is well known, the latter states that for IID random variables Y1, Y2, … on a probability space (Ω, , P) with we have weak convergence of the distributions of to the standard normal distribution on ℝ. We recall that weak convergence of finite measures μn on a metric space S to a finite measure μ on S is defined to mean that

for all bounded, continuous real functions on S. Equivalently, one may require the validity of (1·1) only for bounded, uniformly continuous real functions, or even for all bounded measurable real functions which are μ-a.e. continuous.

We describe the spectrum of a self-adjoint pseudo-differential operator on L2 (– ∞, ∞). We show that the essential spectrum coincides with the interval ([1, ∞) and give a lower bound for the lowest eigenvalue in (– ∞, 1). A sufficient condition for the existence of an eigenvalue in (– ∞, 1) is also given.

The quantal quartic oscillator, characterized by two real and two complex conjugate transition points (simple zeros), is studied by means of the phase-integral method developed by Fröman and Fröman, and various quantization conditions are obtained. The main results, obtained in §3·3 and §4, are summarized below.

A correction to the generalized Bohr–Sommerfeld quantization condition, due to the complex conjugate transition points, is obtained from estimates of the. F-matrix for a path passing above the complex conjugate transition points. This quantization condition is closely related to the quantization condition for a double-oscillator. The correction is shown to be rigorously valid when the distance between the real transition points is large compared to the distance between the complex conjugate ones, which may lie at an unspecified distance from each other.

At sufficiently large distances from the cluster of four transition points the solutions of the Schrödinger equation can be characterized by the Stokes constants between anti-Stokes lines emerging from the cluster in six directions towards infinity. There are initially six non-trivial such Stokes constants in the problem under consideration, but, from the results in previous papers by N. Fröman and the present authors, these Stokes constants are linked by a number of algebraic relations, so that one or two Stokes constants suffice to describe the solutions far away from the cluster of transition points. The quantization condition is expressed in terms of these Stokes constants. Approximate values of the relevant Stokes constants are given in the three limiting cases when all four transition points coalesce (and the first-order phase-integral approximation is used), when the distance between the real transition points is sufficiently large (extreme double-oscillator), and in the Bohr-Sommerfeld limit.