Beardon [1] gave a procedure for constructing a polygon with prescribed angles. For each ordered set of angles Beardon's polygon is unique up to congruence. The polygon obtained this way has an inscribed circle. It is possible to obtain by means of these polygons a fundamental region for a non-Euclidean crystallographic (NEC) group with a given signature having equal angles in each of the cycles: see [5]. In [5] the minimal number of sides of a convex fundamental region of an NEC group is calculated, and regions are explicitly obtained achieving the bound.

In [6] and [9] the second author and Bill Richter showed that the natural ‘degree’ filtration on the homology of ΩSU(n) has a geometric realization, and that this filtration stably splits (as conjectured by M. Hopkins and M. Mahowald). The purpose of the present paper is to prove the real and quaternionic analogues of these results. To explain what this means, consider the following two ways of viewing the filtration and splitting for ΩSU(n). When n = ∞, ΩSU = BU. The filtration is BU(1)⊆BU(2)⊆… and the splitting BU≅ V1≤<∞is a theorem of Snaith[14]. The result for ΩSU(n) may then be viewed as a ‘restriction’ of the result for BU. On the other hand there is a well-known inclusion ℂPn−1. This extends to a map ΩΣℂPn−1→ΩSU(n), and the filtration (or splitting) may be viewed, at least algebraically, as a ‘quotient’ of the James filtration (or splitting) of ΩΣℂPn−1. It is now clear what is meant by the ‘real and quaternionic analogues’. In the quaternionic case, we replace BU by BSp=Ω(SU/SP), ΩSU(n) by Ω(SU(2n)/SP(n))and ℂPn−1 by ℍPn−1. The integral homology of Ω(SU(2n)/SP(n)) is the symmetric algebra on the homology of ℍPn−1, and may be filtered by the various symmetric powers. We show that this filtration can be realized geometrically, and that the spaces of the filtration are certain (singular) real algebraic varieties (exactly as in the complex case). The strata of the filtration are vector bundles over filtrations of Ω(SU(2n−2)/SP(n−1)), and the filtration stably splits. See Theorems (1·7) and (2·1) for the precise statement. In the real case we replace BU by Ω(SU/SO), Ω(SU(n)/SO(n)) and ℂPn−1 by ℝPn−1. Here integral homology must be replaced by mod 2 homology, and splitting is only obtained after localization at 2. (Snaith's splitting of BO in [14] can be refined [2, 8] so as to be exactly analogous to the splitting of BU:BO≅V1≤<∞MO(k).)

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.

In this paper we prove the following

Theorem. Each element of the group π3(S2) = ℤ can be rendered harmonic, i.e. admits a harmonic representative, provided that the domain S3is given a suitable riemannian metric.

The Jones polynomial VL(t) of a link L in S3 contains certain information on the homology of the 2-fold branched covering D(L) of S3 branched along L. The following formulae are proved by Jones[3] and Lickorish and Millett[6] respectively:

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

The series is said to be summable (R, k) to s, where k is a positive integer, if

converges for all values of h in some neighbourhood of the origin and if its sum tends to s as h → 0.

1. Introduction. Suppose that Ω is a region (i.e. a connected open set) in ࠶n, for some fixed n ≥ 1. We define (Γ, μ) to be a Fatou pair in Ω if

(a) Γ is a continuous family of boundary curves γw in Ω, one ending at each w ∈ ∂Ω,

(b) μ is a positive finite Borel measure on ∂Ω, and

(c) the conclusion of Fatou's theorem holds with respect to Γ and μ. Let us state (a) and (c) in more detail:

(a) The map (w, t) → γw(t) is continuous, from ∂Ω × [0, 1) into Ω, and

for every w in the boundary ∂Ω of Ω.

(c) For every f ∈ H∞(Ω) (the class of all bounded holomorphic functions in Ω), the limit

exists a.e. [μ].

A double sequence x = {xjk: j, k = 0, 1, …} of real numbers is called almost convergent to a limit s if

that is, the average value of {xjk} taken over any rectangle {(j, k): m ≤ j ≤ m + p − 1, n ≤ k ≤ n + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n. The notion of almost convergence for single sequences was introduced by Lorentz [1].

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.

When and ℬ are C*-algebras their algebraic tensor product ⊗ ℬ is a *-algebra in a natural way. Until recently, work on tensor products of C*-algebras has concentrated on norms α which make the completion ⊗α ℬ into a C*-algebra. The crucial role played by the Haagerup norm in the theory of operator spaces and completely bounded maps has produced some interest in more general norms (see [8; 12]). In this paper we investigate geometrical properties of algebra norms on ⊗ ℬ. By an ‘algebra norm’ we mean a norm which is sub-multiplicative: α(u.v) ≤ ≤ α(u).α(v).

Let B be the open unit ball of c0. We give a geometric characterization of the sequences {xn} ⊂ bB with the property that, given any bounded sequence {αn} ⊂ ℂ, there is a continuous function , analytic in B, such that f(xn) = αn for all n and such that supi∈B|f(x)| = supn∈N|αn|.

Dirichlet forms and their associated function spaces have been studied by a number of authors [4, 6, 7, 12, 15–18, 22, 25, 26]. Important motivation for the study has been the connection of Dirichlet forms with Markov processes [16–18, 25, 26]: for example, to every regular symmetric Dirichlet form, there is an associated Hunt process [13, 20]. This makes the theory of Dirichlet forms a convenient source of examples of Hunt processes. In the non-commutative setting, Markov fields have been studied by several authors [1–3, 14, 19, 24, 28]. It is therefore interesting to develop a non-commutative extension of the theory of Dirichlet forms and to study their connection with non-commutative Markov processes.

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.

We extend to a wider class of rearrangement invariant spaces a result of S. Guerre concerning the extraction of almost symmetric sequences from a given weakly null sequence in Lp.

The notion of statistical convergence was introduced by Fast[1] and has been investigated in a number of papers[2, 5, 6]. Recently, Fridy [2] has shown that k(xk–xk+l) = O(1) is a Tauberian condition for the statistical convergence of (xk). Existing work on statistical convergence appears to have been restricted to real or complex sequences, but in the present note we extend the idea to apply to sequences in any locally convex Hausdorif topological linear space. Also we obtain a representation of statistical convergence in terms of strong summability given by a modulus function, an idea recently introduced in Maddox [3, 4]. Moreover Fridy's Tauberian result is extended so as to apply to sequences of slow oscillation in a locally convex space, and we also examine the local convexity of w(f) spaces.

A celebrated theorem of Weierstrass states that any continuous real-valued function f defined on the closed interval [0, 1] ⊂ ℝ is the limit of a uniformly convergent sequence of polynomials. One of the most elegant and elementary proofs of this classic result is that which uses the Bernstein polynomials of f

one for each integer n ≥ 1. Bernstein's Theorem states that Bn(f) → f uniformly on [0, 1] and, since each Bn(f) is a polynomial of degree at most n, we have as a consequence Weierstrass' theorem. See for example Lorentz [9]. The operator Bn, defined on the space C([0, 1]; ℝ) with values in the vector subspace of all polynomials of degree at most n has the property that Bn(f) ≥ 0 whenever f ≥ 0. Thus Bernstein's Theorem also establishes the fact that each positive continuous real-valued function on [0, 1] is the limit of a uniformly convergent sequence of positive polynomials. This raises the following natural question: consider a compact Hausdorff space X and the convex cone C+(X):= {f ∈ C(X; ℝ); f ≥ 0}. Now the analogue of Bernstein's Theorem would be a theorem stating when a convex cone contained in C+(X) is dense in it. More generally, one raises the question of describing the closure of a convex cone contained in C(X; ℝ), and, in particular, the closure of A+:= {f ∈ A; f ≥ 0}, where A is a subalgebra of C(X; ℝ).

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.

A Riesz space tensor product of Archimedean Riesz spaces was introduced by D. H. Fremlin[2, 3]. His construction as well as a subsequent simplified version by H. H. Schaefer[10] depended on representation techniques and it is our aim to find a more direct way to prove the existence of the tensor product and to derive its properties. This tensor product proved to be extremely useful in the theory of positive operators on Banach lattices (see [3] and [10]) and should be considered as one of the basic constructions in the theory of Riesz spaces. It is therefore of interest to construct it in an intrinsic way. The problem to do this was already posed by Fremlin in [2]. In this paper we shall present two different approaches, the first of which is analogous to the formation of a free lattice generated by a given partially ordered set. (See [5], p. 41.) In the second one we first assume the Riesz spaces involved to have the principal projection property. In this case a simple method of construction by step-elements is available and the tensor product of arbitrary Archimedean Riesz spaces can then be obtained by embedding the spaces into their Dedekind completions. To complete the latter step we need results on the extension of Riesz bimorphisms which will be proved in §1. Both our approaches hinge on results about the tensor product of ordered vector spaces. It turns out that a unique tensor product for ordered vector spaces exists and is contained in the Riesz space tensor product. This is investigated in §2.