The Reidemeister theorem describes the equivalence of links in terms of diagrams (this theorem was proven in detail by Alexander [1]). A diagram for a link in S3 can be found by projecting the link to any disc recording the over or under crossings. It was shown that two links are equivalent if and only if their diagrams are related by a sequence of so called Reidemeister moves and isotopy in the disc.

Let be an oriented knot in S3 and a solid torus endowed with a preferred framing which contains in its interior. By ρ and λ we denote the wrapping and winding numbers of in respectively. That is, they are the geometric and algebraic intersection numbers of and a meridian disk of . For an integer μ, let τμ be an orientation-preserving homeomorphism of satisfying τμ(m) = m and τμ(l) = l + μm in H1(∂), where (m, l) is a meridian-longitude pair of . We call τμ(), denoted by μ, the knot obtained from by μ-twisting along .

The natural distance function on the vertices of a tree is a kernel of negative type. As a corollary, for any group G acting on a tree X, the length function |g| = d(υ, gυ) is a negative definite function on G for any given vertex υ of X.

It is proved that if G is a compact connected Hausdorff group of uncountable weight, w(G), then G contains a homeomorphic copy of [0, 1]w(G). From this it is deduced that such a group, G, contains a homeomorphic copy of every compact Hausdorff group with weight w(G) or less. It is also deduced that every infinite compact Hausdorff group G contains a Cantor cube of weight w(G), and hence has [0, 1]w(G) as a quotient space.

Let us assume throughout that (pn) denotes a sequence of reals which satisfies

For real sequences (sn) with increments an = sn – sn−1 for n ≥ 0,(where s−1 = 0), we consider the power seriesmethod of summability (P), where we say

The power series methods (P) containthe so-called (Jp)-methods (R = 1)and the Borel-type methods (Bp)(R = ∞). We consider only regular (P)-methods, i.e. sn → s implies sn → s(P). By theorem 5 in [5], p.49, we have regularity if and only if

Here we are interested in the converse conclusion, namely sn → s(P) implies sn → s, which can only be validiffurther conditions, so-called Tauberian conditions are satisfied by (sn). These so-called Tauberian theorems for power series methods have a long history; see e.g. the books [5, 14, 23], and they found new attentionrecently in the papers [6, 18, 19, 20] and [8, 9, 10, 11, 12]. The latter papers contain certain o- Tauberian theorems for all power series methods in question and O-Tauberian theorems, if the weight sequence (pn) can be interpolated by alogarithmico-exponential function g(·)(see e.g. [4]), i.e.

Continuous one-parameter semigroups {μt}t≥0 of probability measures on a locally compact group which are semistable with respect to some automorphism τ of the group, namely such that τ(μt) = μct for all t ≥ 0, for a fixed c ∈ (0, 1), have attracted considerable attention of various researchers in recent years (cf. [3], [5] and other references cited therein). A detailed study of semistable measures on (real) Lie groups is carried out in [5]. In this context it is of interest to study semistable measures on the class of p-adic Lie groups, which is another significant class of locally compact groups.

How must a sphere be covered by n equal circles so that the angular radius of the circles will be as small as possible? In this paper, conjectured solutions of this problem for n = 15 to 20 are given and some sporadic results for n > 20 (n = 22, 26, 38, 42, 50) are presented. The local optima are obtained by using a ‘cooling technique’ based on the theory of bar-and-joint structures. Thus the graph of the coverings by circles is considered as a spherical cable net in which the edge lengths are uniformly decreased, e.g. due to a uniform decrease in the temperature, until the graph becomes rigid and tensile stresses appear in the cables.

We give a new characterization of amenability for dynamical systems, in cohomological terms, which generalizes the classical characterization of amenable locally compact groups stated by Johnson.

Throughout this paper q = 2h with h ≥ 4, and PG(r, q) stands for the r-dimensional projective space over the finite field GF(q) with q elements.

We generalize the classical Bernstein's and Markov's Inequalities for polynomials on any real Banach space. We also give estimates for the derivatives of homogeneous polynomials on real Banach spaces.

The aim of this paper is to show one more facet of the role of crossed complexes as generalisations of both groups (or groupoids) and of chain complexes. We do this by defining and establishing the main properties of a classifying space functor B: from the category of crossed complexes to the category of spaces.

We give examples which (i) disprove a conjecture of Sheil-Small regarding the valence of harmonic mappings of the unit disc to bounded convex domains, and (ii) answer negatively a question of the author regarding the valence of harmonic mappings with polynomial analytic and co-analytic parts.

In this paper we will be concerned with studying operators T: C(K, X) → Y defined on Banach spaces of continuous functions. We will be particularly interested in studying the classes of strictly singular and strictly cosingular operators. In the process, we obtain answers to certain questions recently raised by Bombal and Porras in [5]. Specifically, we study Banach space X and Y for which an operator T: C(K, X) → Y with representing measure m is strictly singular (strictly cosingular) whenever m is strongly bounded and m(A) is strictly singular (strictly cosingular) for each Borel subset A of K. Along the way we establish several results dealing with non-compact operators on continuous function spaces, and we consolidate numerous results concerning extension theorems for operators defined on these same spaces. Also, we join Saab and Saab [25] in demonstrating that if l1 does not embed in X* then the adjoint T* of a strongly bounded map must be weakly precompact, thereby presenting an alternative solution to a question raised in [2].

If X is a topological space with base point, then a based map m: X × X → X is called a multiplication if m restricted to each factor of X × X is homotopic to the identity map of X. The pair (X, m) is then called an H-space. If A is a based topological space and (X, m) is an H-space, then the multiplication m induces a binary operation on the set [A, X] of based homotopy classes of maps of A into X. A classical result due to James [6, theorem 1·1 asserts that if A is a CW-complex and (X, m) is an H-space, then the binary operation gives [A, X] the structure of an algebraic loop. That is, [A, X] has a two-sided identity element and if a, b ∈[A, X], then the equations ax = b and ya = b have unique solutions x, y ∈ [A, X]. Thus it is meaningful to consider loop-theoretic properties of H-spaces. In this paper we make a detailed study of the following loop-theoretic notions applied to H-spaces: inversivity, power-associativity, quasi-commutativity and the Moufang property – see Section 2 for the definitions. If an H-space is Moufang, then it has the other three properties. Moreover, an associative loop is Moufang, and so a homotopy-associative H-space has all four properties. Since many of the standard H-spaces are homotopy-associative, we are particularly interested in determining when an H-space, in particular a finite CW H-space, does not have one of these properties.

The Hausdorff and box dimensions for measures associated with recurrent self-similar sets generated by similitudes is explicitly given. The box dimension of the attractor associated with a class of two-dimensional affine maps is also computed.

In [1], Asimov and Franks give conditions under which a collection of periodic orbits of a diffeomorphism f:M→M of a compact manifold persists under arbitrary isotopy of f. Together with the Nielsen–Thurston theory, their result has been of pivotal importance in recent work on the periodic orbit structure of surface automorphisms (for example [3, 4, 7, 8, 9, 12, 13]). However, their proof uses bifurcation theory and as such depends crucially upon the differentiability of f. The periodic orbit results which make use of the Asimov–Franks theorem are therefore applicable only in the differentiable case, a limitation which belies their topological character. In this paper we shall use classical Nielsen-theoretic methods to prove the analogue of the Asimov–Franks result for homeomorphisms.

It is well known that if h is harmonic in the open unit ball B of the Euclidean space N (where N ≥ 2), then there exist homogeneous harmonic polynomials Hn of degree n in N such that converges absolutely and locally uniformly to h in B (see e.g. Brelot[2], appendice). Further, it is easy to show that this series is unique and that each polynomial Hn is the sum of all the monomial terms of degree n in the multiple Taylor series of h about the origin 0. We call the polynomial expansion of h. Our aim is to obtain sharp upper and lower bounds for the individual terms Hn and for the partial sums of this expansion in the case where h > 0 in B.

Quantitative estimates for finite mean values

of multiplicative functions are highly applicable tools in analytic and probabilistic number theory. Extending a result of Hall [4], Halberstam and Richert[3] proved a useful inequality valid for real, non-negative g satisfying for instance a Wirsing type condition, viz for all primes p, with constants λ1 ≥ 0, 0 ≤ λ2 < 2. Their upper bound is sharp to within a factor (l + o(l)), but even a weaker and easier to prove estimate, such as

(where the implied constants depend on λ1 and λ2), may become a surprisingly strong device. For instance, setting g(p) = l ± ε, where ε is an arbitrarily small positive number, provides immediately a proof of the famous Hardy–Ramanujan theorem on the normal order of the number of prime factors of an integer. This example, and many others, are discussed in detail in our book [5] where we make extensive use of (2) for various problems connected with the structure of the set of divisors of a normal number.

Let G be a locally compact group, and let P(G) be the convex set of all continuous, positive definite functions ø on G normalized by ø(e) = 1, where e denotes the group unit of G. For ø∈P(G) the spectrum spø of ø is defined as the set of all indecomposable ψ∈P(G) which are limits, for the topology of uniform convergence on compact subsets of G, of functions of the form

(see [5], p. 43). Denoting by πø the cyclic unitary representation of G associated with ø, it is clear that sp ø consists of all ψ∈P(G) for which πψ is irreducible and weakly contained in πø (see [3], chapter 18).

This paper discusses the Wiener–Itô chaos decomposition of an L2 function φ over Wiener space, and is concerned in particular with the identification of the integrands ƒn in the chaos decomposition

First these are identified as Radon–Nikodým derivatives. Two elementary non-standard proofs of the Wiener–Itô chaos decomposition are given, based on Anderson's construction of Brownian motion and Itô integration.