Let K⊂C be a field finitely generated over Q, K(a)⊂C the algebraic closure of K and G(K)=Gal (K(a)/K) its Galois group. For each positive integer m we write K(μm) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime [lscr ] we write K([lscr ]) for the subfield of K(a) obtained by adjoining to K all [lscr ]-power roots of unity. We write K(c) for the subfield of K(a) obtained by adjoining to K all roots of unity in K(a). Let K(ab)⊂K(a) be the maximal abelian extension of K. The field K(ab) contains K(c); if K=Q then Q(ab)=Q(c) (the Kronecker-Weber theorem). We write χ[lscr ][ratio ]G(K)→Z*[lscr ] for the cyclotomic character defining the Galois action on all [lscr ]-power roots of unity. We write χ[lscr ]= χ[lscr ] mod [lscr ][ratio ]G(K) →Z*[lscr ]→(Z/[lscr ]Z)* for the cyclotomic character defining the Galois action on the [lscr ]th roots of unity. The character χ[lscr ] identifies Gal (K([lscr ])/K) with a subgroup of Z*[lscr ]=Gal (Q([lscr ])/Q). Let μ(Z[lscr ]) be the finite cyclic group μ(Z[lscr ]) of all roots of unity in Z*[lscr ]. Its order is equal to [lscr ]−1 if [lscr ] is odd and 2 if [lscr ]=2. Let Q([lscr ])′ be the subfield of μ(Z[lscr ])-invariants in Q([lscr ]). Clearly, Gal (Q([lscr ])/Q([lscr ])′)=μ(Z[lscr ]) and Gal (Q([lscr ])′/Q)= Z*[lscr ]/μ(Z[lscr ]) is isomorphic to Z[lscr ].

For a non-zero integer n, let d(n) denote the number of positive divisors of n. Let a, b and c be integers with a>0, and set Δ=b2−4ac. If the quadratic polynomial ax2+bx+c is irreducible over the rational numbers Q (that is, if Δ is not the square of an integer), then one has

formula here

as X→∞, for some λ depending on a, b and c (see [7]). In this paper we discuss the way in which λ depends on a, b and c, giving a precise, compact expression in terms of class numbers. This extends previous work for the case a=1, Δ<0 (see [4]).

For the case a=1, b=0, a much better description of the error is given in [2], with the following expression for λ:

formula here

Here ρ is a multiplicative function, defined below, and (p/q) is the Legendre/Jacobi symbol.

In the tangle model for DNA site-specific recombination, one is required to solve simultaneous equations for unknown tangles which are summands of observed DNA knots and links. For 0[les ]i[les ]3, given fixed 4-plats Ki where the set {K1, K2, K3} contains at least two distinct 4-plats, let O and R denote unknown 2-string tangles such that {O, R} are the variables in the system of four tangle equations N(O+iR)=Ki, where N is the numerator construction, and nR denotes the tangle sum of n copies of R. Then there is at most one simultaneous solution {O, R} and this solution must be of the form R an integral tangle and O either a rational tangle or the sum of two rational tangles. In addition, if there exists a solution, then at least one of the 4-plats is chiral. We exhibit an algorithm for solving simultaneous tangle equations of this form.

The metric projection mapping πX plays an important role in nonlinear approximation theory. Usually X is a closed subset of a Banach space [] and, for each e∈[], πX(e) is the set, perhaps empty, of all points in X which are nearest to e. From a classical theorem due to Stečkin [7] it is known that, when [] is uniformly convex, the metric projection πX(e) is single valued at each typical point e of [] (in the sense of the Baire categories), i.e. at each point e of a residual subset of []. More recently Zamfirescu [8] has proven that, if X is a typical compact set in ℝn (in the sense of Baire categories) and n[ges ]2, then the metric projection πX(e) has cardinality at least 2 at each point e of a dense subset of ℝn. This result has been extended in several directions by Zhivkov [9, 10], who has also considered the case of the metric antiprojection mapping νX (which associates with each e∈[] the set νX(e), perhaps empty, of all ∈X which are farthest from e). For this mapping De Blasi [2] has shown that, if [] is a real separable Hilbert space with dim[]=+∞ and n is an arbitrary natural number not less than 2, then, for a typical compact convex set X⊂[], the metric antiprojection νX(e) has cardinality at least n at each point e of a dense subset of []. A systematic discussion of the properties of the maps πX and νX, and additional bibliography, can be found in Singer [5, 6] and Dontchev and Zolezzi [3].

In the present paper we consider some further properties of the metric projection mapping πX, with X a compact set in a real separable Hilbert space []. If dim[]=n and 2[ges ]n<+∞, it is proven that for a typical compact set X⊂[], the metric projection πX(e) has cardinality exactly n+1 at each point e of a dense subset of [], while the set of those points e∈[] where πX(e) has cardinality at least n+2 is empty. Furthermore it is shown that, if dim[]=+∞, then for a typical compact set X⊂[] the metric projection πX(e) has cardinality at least n (for arbitrary n[ges ]2) at each point e of a dense subset of []. Incidentally we obtain a characterization of the dimension of the space [] by means of a typical property holding in the space of the compact subsets of [].

The main result of this paper is to show that if G∗ is a simplicial group of finite length, then HnG∗ also has finite length. Here the length of a simplicial group means the length of the corresponding Moore normalization and HnG∗ is the simplicial abelian group given by [k][map ]HnGk. A similar fact is true if we replace G∗ by a simplicial ring and we take the algebraic K-functors instead of group homology.

The origin of such results goes back to the classical paper of Dold and Puppe (see Hilfsatz 4·23 of [7]), where the following was proved: let

formula here

be a functor between abelian categories of degree d, meaning that the (d+1)st cross-effect functor in the sense of Eilenberg and MacLane [9] vanishes. Then l(TX∗)[les ]dl(X∗) for any simplicial object X∗ in A. Here l(X∗) denotes the length of X∗. Actually, this property characterizes the degree of functors in the abelian case (see Lemma 3·6).

One can modify the notion of the cross-effects in the nonabelian framework (see [2]) and define the notion of degree of functors. One can also use the above inequality to define the simplicial degree in the nonabelian set-up. However in this way we get generally different invariants and the aim of this paper is to establish the relationship between the different notions. We show that many classical functors arising in homological algebra and K-theory have finite simplicial degree. Our Conjecture 4·7 claims that this should be always the case.

We remark that if the Moore normalization of a simplicial group G∗ is zero in dimensions >k then πiG∗=0 for i>k as well. Therefore, for a functor T of simplicial degree d, one has

formula here

Our work can therefore be considered as a new general method proving vanishing results. Here is a sample application of the main result.

We give a construction of Gusarov's groups [Gscr]n of knots based on pure braid commutators, and show that any element of [Gscr]n is represented by an infinite number of prime alternating knots of braid index less than or equal to n+1. We also study [Vscr]n, the torsion-free part of [Gscr]n, which is the group of equivalence classes of knots which cannot be distinguished by any rational Vassiliev invariant of order less than or equal to n. Generalizing the Gusarov–Ohyama definition of n-triviality, we give a characterization of the elements of the nth group of the lower central series of an arbitrary group.

Let V⊂S3 be a solid, knotted torus. Through the work of Birman and Menasco [2], the observation has been made that a satellite link L=C[midast ]P⊂V (with companion C, pattern P and essential torus T=∂V) falls into one of two broad categories: reverse string and non-reverse string. These categories are borne of the three embedding types identified in [2] and are distinguished by the existence of, or lack of, a meridional disc D⊂V with an orientation, whose point-intersections with the oriented satellite C[midast ]P are all similarly oriented. If no such D exists, then L is said to be a reverse string satellite.

If C[midast ]P is a non-reverse string satellite, it is known that the braid index b(C[midast ]P) is dependent only on b(C) and certain specified properties of the pattern and not on the choice f∈Z of framing parameter. In [2] it is conjectured that, if C[midast ]fP is a reverse string satellite (in which the framing parameter is f), the braid index b(C[midast ]fP) depends on the arc index α(C), properties of the pattern and also the framing. We study this dependence further via established upper and lower bounds for braid index. The upper bound comes from explicit closed braid diagrams of L, for which constructions are shown. The lower bound comes from the Homfly polynomial, via the Morton–Franks–Williams inequality [5, 8]. We will prove the following theorem.

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].

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#.

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}.

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 ]).

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.