Let K be a knot in the 3-sphere S3; let m be a meridian and 1 a longitude of K. In [1] we gave a detailed study of the ‘π-orbifold group’ O(K): π1(S3\K)/〈m2〉 of the knot, where 〈m2〉 is the subgroup normally generated by the square of the meridian. In particular, we found out to what extent the π-orbifold group classifies knots (and links) and determines the symmetry group of a knot. In the present paper, we consider the groups Gn(K): = π1(S3\K)/〈ln〉. Our main results are the following.

Let Γ be a Kleinian group of finite covolume and denote by Γ(2) the subgroup generated by {γ2:γ ∈ Γ}. In [9] the trace field of Γ(2) was shown to be an invariant of the commensurability class of Γ. In [8] this field was termed the invariant trace field of Γ and further properties of this field were studied. Following the notation of [8] we denote the invariant trace field of Γ by k(Γ).

Results of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

Based on ideas from semigroup theory, Fountain and Gould [2, 3, 4] introduced a notion of order in a ring which need not have an identity. In some important cases of rings with identity, e.g. if the larger ring is a semisimple artinian ring, this notion coincides with the classical one. The most important result of Fountain and Gould (see [4]) is a Goldie-like characterization of two-sided orders in a regular ring with minimum condition on principal one-sided ideals. In addition, for the same class of rings, a generalization of the Faith–Utumi theorem has been proved by Gould and Petrich[7]. The methods of these papers seem not to work for one-sided orders.

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

Let R be a Cohen-Macaulay local ring of dimension d, multiplicity e and embedding dimension v. Abhyankar [1] showed that v − d + 1 ≤ e. When equality holds, R is said to have minimal multiplicity. The purpose of this paper is to study the preservation of this property under the formation of Rees algebras of several ideals in a 2-dimensional Cohen-Macaulay (CM for short) local ring. Our main tool is the theory of joint reductions and mixed multiplicities developed by Rees [9] and Teissier[12].

In this paper we investigate the minimal number of charts (spaces homeomorphic to open subsets of ℝ3) needed to cover a closed 3-manifold M3. We prove that this number is two if the Bockstein βω1(M3)∈H2(M3;ℤ) of the first Stiefel-Whitney class of M3 is zero and three if it is not. We state several properties equivalent to βω1(M3) = 0, for example the condition of being covered by two orientable subspaces, and the condition that all torsion elements of H1(M3;ℤ) be represented by orientation-preserving loops. We also show that the non-orientable 3-manifolds which can be covered with two charts are precisely those manifolds which can be described as sewn-up r-link exteriors. These are manifolds obtained by removing from S3 the interiors of two disjoint handlebodies of genus r and then identifying the two boundary components by a homeomorphism.

For any regular covering p:M→M of 3-dimensional manifolds M, M with M a homology 3-sphere we construct a regular covering p′: M′ → M′ of 3-manifolds M′, M′ with the same group of covering transformations and a degree 1 map f:M → M′ so that M′ is a homology 3-sphere, M′ (and hence M′) is irreducible and does not contain incompressible tori, and the regular covering p:M→M is induced from the regular covering p′: M′ → M′ by the map f. Assuming Thurston's geometrization conjecture it follows that M′ (and hence M′) is either hyperbolic or Seifert fibred.

In the present paper we suggest as a cobordism invariant of an immersed or embedded submanifold in Euclidean space the singularity set of its projection to a hyperplane. A similar approach has been employed by Banchoff[1] and Koschorke[6]; see also [15]. We consider the range of dimensions n ≤ 3k where n is the dimension and k is the codimension. We prove that in this range (1) our singularity invariant is complete modulo 2-torsion, and (2) modulo-torsion, it can take any value from Thorm's oriented cobordism group of corresponding dimension for k even, while for k odd this invariant is always trivial.

Let G denote a discrete group and let us say that G is C*-simple if the reduced group C*-algebra associated with G is simple. We notice immediately that there is no interest in considering here the full group C*-algebra associated with G, because it is simple if and only if C is trivial. Since Powers in 1975 [26] proved that all nonabelian free groups are C*-simple, the class of C*-simp1e groups has been considerably enlarged (see [1, 2, 6, 7, 12, 13, 14, 16, 24] as a sample!), and two important subclasses are so-called weak Powers groups ([6, 13]; see Section 4 for definition and examples) and the groups of Akemann-Lee type [1, 2], which are groups possessing a normal non-abelian free subgroup with trivial centralizer.

One of the most celebrated theorems of mathematical analysis is the intermediate value theorem of Bolzano which, in a simple case, states that a real-valued continuous map f of a closed interval [a, b], such that f(a) and f(b) have different signs, has a zero in (a, b). Recently, Shih in [5] observed that without loss of generality we may suppose that a 7 < 0 < b and f(a) < 0 < f(b) and, consequently, the condition f(a).f(b) < 0 becomes x.f(x) > 0 for x∈∂Ω where ∂Ω denotes the boundary of the interval Ω = (a, b); then the conclusion is that f has at least one zero in ω. It is a remarkable fact that Shih extends this form of Boizano's theorem to analytic maps in ℂ [5] and, subsequently, in ℂn [6]. He proved that if Ω is a bounded domain in ℂn containing the origin, is continuous in and analytic in Ω and Re for z∈∂Ω, then f has exactly one zero in Ω. In this paper we extend Shih's result to Banach spaces.

Given a convex symmetric body C ⊂ ℝn we put a(C) = |C| sup |P|−1 where the supremum extends over all parallelepipeds containing C and |A| denotes the volume of a set A ⊂ ℝn. Let an = inf {a(C): C ⊂ ℝn}. We show that

which slightly improves the estimate due to Dvoretzky and Rogers [6]. In every dimension n we construct a convex symmetric polytope Wn such that the unit Euclidean ball is the ellipsoid of maximal volume inscribed into Wn and the volume of every parallelepiped containing Wn is greater than

for large n which shows ‘the limit’ to the Dvoretzky Rogers method for bounding an below. We present an alternative proof of the result of I. K. Babenko [l] that . We show that , and that a local minimum of the function C→a(C) for C ⊂ ℝn is attained only at an equiframed convex body (that is, a body such that every point of its boundary belongs to a parallelepiped of minimal volume containing the body).

In this paper, we are concerned with establishing a characterization of any compact, convex set K in a normed space A in an arbitrary topos with natural number object. The characterization is geometric, not in the sense of categorical logic, but in the intuitive one, of describing any compact, convex set K in terms of simpler sets in the normed space A. It is a characterization of the compact, convex set in the sense that it provides a necessary and sufficient condition for an element of the normed space to lie within it. Having said this, we should immediately qualify our statement by stressing that this is the intuitive content of what is proved; the formal statement of the characterization is required to be in terms appropriate to the constructive context of the techniques used.

The maximal proper prime filters together with the ultrafilters of zero sets of any metrizable compact topological space are shown to have a compact Hausdorff topology in which the ultrafilters form a discrete, dense subspace. This gives a general theory of compactifications of discrete versions of compact metrizable topological spaces and some of the already known constructions of compact right topological semigroups are special cases of the general theory. In this way, simpler and more elegant proofs for these constructions are obtained.

In [8], Pym constructed compactifications for discrete semigroups which can be densely embedded in a compact group. His techniques made extensive use of function algebras. In [4] Helmer and Isik obtained the same compactifications by using the existence of Stone ech compactifications. The aim of this paper is to present a general theory of compactifications of semitopological semigroups so that Helmer and Isik's results in [4] are a simple consequence. Our proofs are different and are based on filters which provide a natural way of getting compactifications. Moreover we present new insights by emphasizing maximal proper primes which are not ultrafilters.

We start by defining filters of zero sets (called z-filters) on a given topological space X, and their convergence. In the case of compact metrizable topological spaces, we establish the connections between proper maximal prime z-filters on X and zultrafilters in β(X\{x})\(X\{x}) where β(X\{x}) is the Stone-ech compactification of X\{x}. We then define a topology on the set of all prime z-filters on X such that the subspace of all proper maximal primes is compact Hausdorff. We denote by the set of all proper maximal prime z-filters on X together with the z-ultrafilters and show that when X is a compact metrizable cancellative semitopological semigroup, is a compact right topological semigroup with dense topological centre. Also, when is considered for a compact Hausdorff metrizable group, the semigroup obtained is exactly the same (algebraically and topologically) as the semigroup obtained in [4]. Hence the result in [4] is just a consequence of the general theory presented in this paper.

We discuss the possible cofinalities of ⊆ *-chains in ℕ which are ⊆ *-unbounded in their topological closures.

1. The problem. Throughout this paper, the power set ℕ of the set ℕ of natural numbers will be given its usual compact metrizable topology corresponding to the product topology of {0,1}N. We say that a ⊆ * b if a/b is finite.

Let 0 < p,pi ≤ ∞, 0 < q,qi < ∞ (i = 1, 2) such that

Let E be a quasi-Banach lattice which fails to contain c0 and whose α-convexity constant is equal to 1 for some 0 < α < ∞. Then for every f∈H(E(q)) there exist g∈Hp, 0(E(q0)), h∈Hp1(E(q1)) such that

Consequently, E is q-concave for some finite q if and only if E is uniformly H1-convexifiable in the sense of [24]. Analogous results are also obtained for symmetric spaces of measurable operators. Another result proved in the paper says that if E is a symmetric quasi-Banach function space on (0, ∞) having the analytic Radon–Nikodym property then LE(M, τ) also possesses this property for any semifinite von Neumann algebra (M, τ).

Some general observations about stability of periodic solutions of Hamiltonian systems are presented as well as stability results for the periodic solutions that exist near a collision of pure imaginary eigenvalues. Let I = ∮ p dq be the action functional for a periodic orbit. The stability theory is based on the surprising result that changes in stability are associated with changes in the sign of dI / dw, where w is the frequency of the periodic orbit. A stability index based on dI / dw is defined and rigorously justified using Floquet theory and complete results for the stability (and instability) of periodic solutions near a collision of pure imaginary eigenvalues of opposite signature (the 1: – 1 resonance) are obtained.

We prove the following result: if a Banach space E does not contain l1 and F has the (RDPP), then E ⊗nF has the same property, provided that L(E, F*) = K(E, F*). Hence we prove that if E ⊗n F has the (RDPP) then at least one of the spaces E and F must not contain l1. Some corollaries are then presented as well as results concerning the necessity of the hypothesis L(E, F*) = K(E, F*).

Valiron[3] showed that if f(z) is an entire function for which

where A(r) = log(sup{|f(z)|:|z| = r}), then there is a function x(r) = rρ(r) satisfying both

A characterization of the topological spaces that possess a bicomplete fine quasi-uniformity is obtained. In particular we show that the fine quasi-uniformity of each sober space, of each first-countable T1-space and of each quasi-pseudo-metrizable space is bicomplete. Moreover we give examples of T1-spaces that do not admit a bicomplete quasi-uniformity.

We obtain several conditions under which the semi-continuous quasi-uniformity of a topological space is bicomplete and observe that the well-monotone covering quasiuniformity of a topological space is bicomplete if and only if the space is quasi-sober.