In this paper we define a set of radii called thickness for simple closed curves denoted by K, which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these curves. One can think of these radii as representing the thickness of a rope in space and of K as the core of the rope. Great care is taken to define our radii in order to gain freedom from small pieces with large curvature in the curve. Intuitively, this means that we tend to allow the surface of the ropes that represent the knots to deform into a non smooth surface. But as long as the radius of the rope is less than the thickness so defined, the surface of the rope will remain a two manifold and the rope (as a solid torus) can be deformed onto K via strong deformation retract. In this paper we explore basic properties of these thicknesses and discuss the relationship amongst them.

In this paper we construct a symmetric monoidal smash product of Γ-spaces modelling the smash product of connective spectra. For the corresponding theory of ring-spectra, we refer the reader to [Sch].

We obtain a result compatible with Bloch's conjecture for Mumford's fake plane. In the process some new surfaces associated with Mumford's surface are found.

One of the main problems in the theory of inductive limits of Banach spaces is the projective description problem, finding a reasonable representation for the continuous seminorms. The problem is nontrivial even in the simplest cases. Recall that given, for example, an increasing sequence of Banach spaces (Yk)∞k=1 with continuous embeddings Yk[rarrhk ]Yk+1 the inductive limit is the space Y=∪kYk endowed with the finest locally convex topology τ such that every embedding Yk[rarrhk ](Y, τ) becomes continuous. It is possible to give abstract definitions for families of continuous seminorms generating the topology τ, but the connection with the norms of the step spaces Yk is not necessarily very close. For example, if the spaces Yk are Banach spaces of continuous functions endowed with weighted sup-norms, it is not clear if the continuous seminorms of the inductive limit are of the same type.

We mention that inductive limits of spaces of continuous and holomorphic functions occur in many areas of analysis like linear partial differential operators, convolution equations [BD1], [E], complex and Fourier analysis and distribution theory. The projective description problem in these spaces has been thoroughly studied in [BMS1, BB1, BB2, BB3, BT, BM1, BM2], to mention some examples. We refer to the survey articles [BM1,BMS2, BB3]. The present work is also connected with the factorization problems which are treated in the book [Ju].

Let n≥5. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball in (n+2)-space.) Furthermore, the projection is not the projection of any unknotted sphere in (n+2)-space. There are two inequivalent embeddings of an n-manifold in (n+2)-space such that the singular point set of the projection of each embedding consists of double points and that the number of components of the singular point set of each embedding is one.

In this paper we advertise the category of Γ-spaces as a convenient framework for doing ‘algebra’ over ‘rings’ in stable homotopy theory. Γ-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (−1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for Γ-spaces. The study of ‘rings, modules and algebras’ based on Γ-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg–MacLane spectra counterparts.

We study when c0 is isomorphic to a quotient of Lp(μ, X) (1<p<∞). The results are extended to order continuous Köthe function spaces.

If G is an abelian group, then G# denotes G equipped with the weakest topology that makes every character of G continuous. This is the Bohr topology of G. If G=ℤ, the additive group of the integers and A is a Hadamard set in ℤ, it is shown that: (i) A−A has 0 as its only limit point in ℤ#; (ii) no Sidon subset of A−A has a limit point in ℤ#; (iii) A−A is a Λ(p) set for all p<∞. This leads to an explicit example of a set which is Λ(p) for all p<∞ and is dense in ℤ#. If f(x) is a quadratic or cubic polynomial with integer coefficients, then the closure of f(ℤ) in the Bohr compactification of ℤ is shown to have Haar measure 0. Every infinite abelian group G contains an I0 set A of the same cardinality as G such that 0 is the only limit point of A−A in G#.

For a class of (n+1)-dimensional manifolds of hyperbolic type, the spectral measure is described in terms of Poisson kernels. This implies that the spectrum is absolutely continuous and it is contained in [n2/4; +∞).

If G is a compact Lie group and M a Mackey functor, then Lewis, May and McClure [4] define an ordinary cohomology theory H*G(−; M) on G-spaces, graded by representations. In this article, we compute the ℤ/p-rank of the algebra of integer-degree stable operations [Ascr ]M, in the case where G=ℤ/p and M is constant at ℤ/p. We also examine the relationship between [Ascr ]M and the ordinary mod-p Steenrod algebra [Ascr ]p.

The main result implies that while [Ascr ]M is quite large, its image in [Ascr ]p consists of only the identity and the Bockstein. This is in sharp contrast to the case with M constant at ℤ/p for q≠p; there [Ascr ]M≅[Ascr ]q.

In this paper we study the cohomology groups Hn(I, I*) and Hn([Uscr ], [Uscr ]*) where [Uscr ] is a Banach algebra with a bounded approximate identity and I is a codimension one closed two-sided ideal of [Uscr ]. This is applied to the case when [Uscr ] is the group algebra L1(G) of a locally compact group G and I={f∈L1(G)[mid ] ∫Gf=0}, the augmentation ideal of G. We show that if G is inner amenable, then I is always weakly amenable, i.e. [Hscr ]1(I, I*)={0}.

We introduce an analogue of Calderón's first commutator along a parabola and establish its L2 boundedness under essentially sharp hypotheses.

We study the action of fractional differentiation and integration on weighted Bergman spaces and also the Taylor coeffficients of functions in certain subclasses of these spaces. We then derive several criteria for the multipliers between such spaces, complementing and extending various recent results. Univalent Bergman functions are also considered.

We present a class of approximately finite dimensional C*-algebras whose local multiplier algebras are simple with real rank zero and stable rank one.

For any topological space X let C(X) be the realization of the singular cubical set of X; let [midast ] be the topological space consisting of one point. In [1] Antolini proves, as a corollary to a general theorem about cubical sets, that C(X) and X×C([midast ]) are homotopy equivalent, provided X is a CW-complex. In this note we give a short geometric proof that for any topological space X there is a natural weak homotopy equivalence between C(X) and X×C([midast ]).

Let f and h be transcendental entire functions and let g be a continuous and open map of the complex plane into itself with g∘f=h∘g. We show that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g−1(J(h))=J(f). Here J(·) denotes the Julia set of a function. We conclude that if f has no wandering domains, then h has no wandering domains. Further, we show that for given transcendental entire functions f and h, there are only countably many entire functions g such that g∘f=h∘g.

We investigate the best order of smoothness of Lp(Lq). We prove in particular that there exists a [Cscr ]∞-smooth bump function on Lp(Lq) if and only if p and q are even integers and p is a multiple of q.

Stokes and Darwin drift, for irrotational waves and flow past rigid bodies, are analysed in a general framework and shown to be fundamentally the same thing.

The notion of cycle-free partial order (CFPO) was defined in [9] and the class of sufficiently transitive CFPOs was investigated, and in many cases, classified. In [10] a complete classification was given for the countable 3- or 4-CS-transitive CFPOs having a finite chain and embedding an infinite ‘alternating chain’ ALT. Here, the major case is that of the so-called ‘skeletal’ CFPOs, in which the CFPO itself forms a mere skeleton of the structure of the whole picture, which is more accurately provided by its Dedekind–MacNeille completion. It was asserted that a similar classification should also be possible in the infinite chain case (still with the countability restriction), and it is the object of the present paper to finish this task.

The overall plan of the work is similar in style to [10]. The CFPOs we study can all be construed as arising from a chain, suitable ‘adorned’ with instructions as to how to branch (and repeatedly). An obvious difference from the finite chain case is that, this time, many of the points in the chain will actually survive as elements of the structure; whereas in [10], since the chains of the structure had to have length 2, only the endpoints would. On the whole, the treatment is parallel, with just a small number of additional cases.

We recall the main ideas from [10], quote results that are needed, and concentrate on aspects of the structures which are new or specific to the infinite chain case. We conclude by remarking on some direct connections between the finite and infinite chain cases.