A set A ⊂ ωω is called compactly if, for every compact K ⊂ ωω, A ∩ K is . Consider the proposition that every compactly set is . (AD implies that it is true, ZFC + CH implies that it is false.) We are concerned here with whether this is consistent with ZFC, particularly when n = 1. In the case of sets (that is, analytic sets), this consistency question is due to Fremlin (see [7], page 483, problem 18). Kunen and Miller [3] have proved the following two theorems.

Formal propositional logic describing the laws of constructive (intuitionistic) reasoning was first proposed in 1930 by Heyting. It is obtained from classical pro-positional calculus by deleting the Law of Excluded Middle, and it is usually referred to as Heyting's (intuitionistic) propositional calculus ([9], §§23, 19) (we write HPP in short). Formal logic involving predicates and quantifiers based on HPP is called Heyting's (intuitionistic) predicate calculus ([9], §§31, 19) (we write HPR in short).

Let Γ be a subgroup of PSL2 (ℤ) of finite index. In this note we are concerned with the arithmetic nature of the Fourier coefficients of holomorphic Eisenstein series on Γ.

Bollobás and Erdös[1] have posed the problem:

If a is irrational, show that for 1 ≤ i < j ≤ p the number of integers t with 1 ≤ t ≤ p such that {(t–i)2a} < d and {(t–j)2a} < d, 0 < d < 1, is d2p + o(p) uniformly in i, j.

In [6] the first author introduced some combinatorial concepts involving Young diagrams corresponding to partitions with distinct parts and applied them to the projective representations of the symmetric group Sn. A conjecture concerning the p-block structure of the projective representations of Sn was formulated in terms of these concepts which corresponds to the well-known, but long proved, Nakayama ‘conjecture’ for the p-block structure of the linear representations of Sn. This conjecture has recently been proved by Humphreys [1].

Diagrams have been used in group theory by numerous authors, and have led to significant results (see [4] and the references cited there). The idea of applying diagrams to semigroups seems to be more recent [3, 7, 8]. In the present paper we discuss semi group diagrams and use them to obtain results concerning the word problem for one-relator semigroups. The word problem for one-relator groups has been solved by Magnus [6], but the analogous question for semigroups remains open. We are not able to solve the problem in full generality, but have obtained some partial results.

In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebra K[S] of an arbitrary semigroup S over a field K in the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties of S or, more precisely, by some groups derived from S. For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).

In this paper, we use the Shintani decomposition, known to number theorists, to describe an effective method of finding a resolution of the cusps of a Hilbert modular variety, in any dimension.

Let p denote a fixed prime and let Pn(pr) denote the cofibre of the degree pr map on Sn–1. We consider π*P3(pr) and show that the map q in the cofibre sequence

induces a split epimorphism on the p-primary component of π2pS3 if p > 2. That analogous maps q: Pn(pr) →Sn, n ≥ 4, induce split epimorphisms on the p-primary component of πn+2p-3Sn, p > 2, is shown in work of J. F. Adams [1]. It is the purpose of this note to document the above computation in the case n = 3 for the use of others.

L. Smith, in a recent paper [11], studied a class of maps X →Y which he called ramified coverings. Roughly speaking, these are maps with a multiple-valued inverse Y → SPdX; cf. 1·1. He showed that X → X/G is a ramified covering whenever a finite group G acts on X. Using results of [4] on infinite symmetric powers SP∞X of CW-complexes X he obtained transfer homomorphisms .

In his study of the structure of periodic homeomorphisms of surfaces, Jakob Nielsen [7] asked, in effect, how much about an effective group action on a surface is determined by its induced action on homology. (It is well known that the induced action on homology is faithful, provided the surface has negative Euler characteristic. See Farkas and Kra[3], v·3, for example.) Two periodic maps T1 and T2 on an oriented surface M are conjugate if there is an orientation preserving homeomorphism h: M → M such that hT1h–1 = T2; the two maps T1 and T2 are symplectically equivalent if there is an orientation preserving homeomorphism h: M → M so that hT1h–1 and T2 induce the same automorphisms on H1(M). By standard results this is the same as T1* and T2* being conjugate by an automorphism of H1(M) which preserves intersection numbers.

0. If Fg is a closed, orientable surface of genus g, then the mapping class group of Fg is the group whose elements are orientation preserving self homeomorphisms of Fg modulo isotopy. We shall denote this group by Mg. Recall that a group is said to be linear if it admits a faithful representation as a group of matrices (where the entries for this purpose will be in some field).

The aim of this paper is to prove some new results towards answering the question: When does Dehn surgery on a knot give a non-prime manifold? This question has been raised on several occasions (see for instance [5] or [4]; concerning the latter see below). Recall that a 3-manifold is prime if, in any connected sum decomposition

one of M1, M2 is S3. (For standard definitions of low-dimensional topology see [2] or [16].)

We construct examples of compact 3-manifolds which have finite cyclic first homology groups but which are not realizable by Dehn surgery on a knot in a homotopy 3-sphere. A necessary condition for a 3-manifold M to be the result of such a surgery is that π1( M) be normally generated by a single element (corresponding to a meridian of the knot), that is π1(M) should have weight 1. As we do not expect this to hold in general for the free product of non-trivial groups we are naturally led to consider connected sums. For instance, if G denotes the binary icosahedral group, it is known that ℤ* G has weight greater than 1 (see [2]). Thus, if H is the Poincaré sphere and M is any 3-manifold with first homology ℤ, M # H is not the result of surgery on a knot in a homotopy 3-sphere. Unfortunately this purely group-theoretic approach is too difficult in general owing to the intractability of the weight 1 condition (see § 9 of [4]). Thus we adopt a strategy involving an interplay between group theory and geometry. Our method is based on the following:

In this paper I shall show how certain bounds on the possible diagrams presenting a given oriented knot or link K can be found from its two-variable polynomial PK defined in [3]. The inequalities regarding exponent sum and braid index of possible representations of K by a closed braid which are proved in [5] and [2] follow as a special case.

Let X be a non-empty compact Hausdorff space and let C(X) denote either the space of all continuous real-valued functions or the space of all complex-valued functions on X, endowed with the norm , where for each non-empty closed F ⊆ X and each f ∈ C(X) we write

Let p(X1, …, Xn) be a polynomial in the commuting indeterminates X1, …, Xn.

Define

Let 1 ≤ p < ∞, let be the open unit ball of lp and let Δ be the open unit disc in ℂ. We prove that there is a holomorphic map such that F(Δ) is dense in and such that whenever .

Let Λ ⊂ Ω ⊂ ℝnbe open subsets. We construct a natural space of test functions on Λ such that the dual consists exactly of those distributions on Λ which can be extended to distributions on Ω. As an application of this representation we calculate the space of multiplication operators on as well as the space of absolutely regular extendable distributions.

We shall be concerned here with two classical families of summability methods, and with links between them, together with applications in probability theory and elsewhere.