It is well known that an n-dimensional crystallographic group can be reconstructed from its point group, the integral representation of the point group which arises from its action on the translation lattice, and the 2-cocycle which glues the point group to the lattice ([2]). In practice, this constitutes a complicated list of invariants. When confronted with the classification of objects possessing a rich structure, the algebraic geometer first attempts to find more coarse birational invariants. We begin such a programme for torsion-free crystallographic groups. More precisely, if Γ is a torsion-free crystallographic group and k is a field then the group algebra k[Γ] is a non-commutative domain (see [6], chapter 13). It can be localized at its centre to yield a division algebra k(Γ) which is a crossed product; the Galois group is the point group and it acts on the rational function field generated by k and the lattice (regarded multiplicatively), which is a maximal subfield ([3]). What are thecommon invariants of Γ1 and Γ2 when k(Γ1) and k(Γ2) are isomorphic k-algebras?

An embedding of the Jacobian variety of a curve of genus 2 is given, together with an explicit set of defining equations. A pair of local parameters is chosen, for which the induced formal group is defined over the same ring as the coefficients of . It is not assumed that has a rational Weierstrass point, and the theory presented applies over an arbitrary ground field (of characteristic ╪ 2, 3, or 5).

This paper is devoted to the study of unions of ideals in commutative rings. The starting point is the prime avoidance lemma and an accompanying but diverse body of results on coverings of ideals by unions of ideals, which is described in Section 1. In Sections 4 and 5 these known facts, about finite and infinite unions, are combined and generalized, two distinct but overlapping cases emerging. All that is proved in Sections 4 and 5 turns on the crucial Lemma 2·1 in Section 2, which shows that a cover of an arbitrary ideal by cosets can be lifted to a cover of the entire ring. In Section 3 we introduce and define α-sieves, which provide a concise framework for the expression of applications. In Sections 6 to 9 various applications of Sections 4 and 5 are investigated.

Let G be an infinite connected graph. A ray (from ν) in G is a 1-way infinite path in G (with initial vertex ν). An infinite connected subgraph of a ray R ⊂ G is called a tail of R. If X ⊂ G is finite, the infinite component of R\X will be called the tail of R in G\X.

Following the techniques of ordinary homotopy theory, a theoretical treatment of proper homotopy theory, including the known proper homotopy groups, is provided within Baues's theory of cofibration categories.

Let K be an oriented knot in the 3-sphere S3. An exterior of K is the closure of the complement of a regular neighbourhood of K, and is denoted by E(K). A Seifert surface for K is an oriented surface S( ⊂ S3) without closed components such that ∂S = K. We denote S ∩ E(K) by Ŝ, and we regard S as obtained from Ŝ by a radial extension. S is incompressible if Ŝ is incompressible in E(K). A tunnel for K is an embedded arc τ in S3 such that τ ∪ K = ∂τ. We denote τ ∪ E(K) by τ, and we regard τ as obtained from τ by a radial extension. Let τ1, τ2 be tunnels for K. We say that τ1 and τ2 are homeomorphic if there is a self-homeomorphism f of E(K) such that f(τ1) = τ2. The tunnels τ1 and τ2 are isotopic if τ1 is ambient isotopic to τ2 in E(K). Then the main result of this paper is as follows: Theorem. Let K be a knot in S3, and let τ1, τ2 be tunnels for K. Suppose that there are incompressible Seifert surfaces S1 S2 for K such that S1 ∪ S2 = K, and τi ⊂ Si (i = 1, 2). If τ1 and τ2 are isotopic, then there is an ambient isotopyhτ (0 ≤ t ≤ 1) of S3 such that ht(K) = K, and h1(τ1) = τ2.

Some results are presented which establish connections between shape theory and the theory of multivalued maps. It is shown how to associate an upper-semi-continuous multivalued map F: X → Y to every approximative map f = {fk, X → Y} in the sense of K. Borsuk and it is proved that, in certain circumstances, if F is ‘small’ and admits a selection, then the shape morphism S(f) is generated by a map, and if F admits a coselection then S(f) is a shape domination.

In supersymmetric field theories, the use of Berezin-like integrals for curved superspace presupposes the existence of a smooth manifold structure on the underlying curved space (the body). The aim of this paper is to address the following question: when does a Rogers supermanifold have a body manifold?

Elementary techniques from the theory of foliations are used to derive some necessary and sufficient conditions. The main result is the necessity of regularity of the soul foliation in the sense of Catenacci, Reina and Teofilatto. Regularity is not sufficient (for example, punctured superspace is regular with non-Hausdorff body). A sufficient condition (albeit highly sensitive to the choice of charts) is the transitivity of the body relation ˜, a property enjoyed by De Witt supermanifolds.

We show that the body of a quotient supermanifold S/Γ is the quotient of the body (under the induced action). Several examples in the text illustrate that this is extremely useful.

New characterizations ρ-supermanifolds and their saturations are given and the saturation construction is extended to all bodied supermanifolds.

The aim of this paper is to give a representation for the types in some stable Banach sequence spaces, namely in the Orlicz, Lorentz and dual of Lorentz sequence spaces. We also find a characterization for the Lorentz sequence spaces whose class of weakly-null types is locally compact for the topology of uniform convergence on bounded subsets.

A strictly real unital Banach algebra is one in which every element has real spectrum. An antilattice partial order and its associated open-interval topology are defined on by taking as positive cone the principal component of the maximal group of the algebra, and their properties are studied. The topology coincides with the semimetric topology of the spectral radius, which is a seminorm, making into a locally convex partially ordered topological algebra with continuous inversion and normal cone. Every positive element has a unique positive square root, and logarithm, and these functions and exp are continuous and monotone. The cone has the same closure in both open-interval and normal topologies, the closure being a wedge for an associated preorder on , this preorder coinciding with the preorder of Kelley and Vaught. The partial order and the associated preorder have the same dual cone. There is a modified Gelfand representation for (possibly non-commutative) , and a boundary integral representation for functionals in the order dual. The compact convex subset in the dual is metrizable if the cone of the partial order in is separable.

Cohen's Factorization Theorem says, in its basic form, that if A is a Banach algebra with a bounded left approximate identity, then every element x ∈ A may be written as a product x = ay for some a, y ∈ A. Such is the beauty and importance of this result that much interest attaches to the question of whether the hypothesis of a bounded left approximate identity can be weakened, or whether a converse result exists. This paper contributes to the study of that question.

This paper presents a uniqueness result for solutions to the general nonlinear coagulation-fragmentation equation

where

Equation (1·1) has many applications in the applied sciences (cf. [1, 3, 8, 13, 15]) and a brief physical interpretation can be found in Melzak [12] or the survey article by Drake[7]. c(x, t), for x ≥ 0, t ≥ 0, denotes the number of particles of size x at time t and the non-negative kernels K and F describe, respectively, the rates at which particles of size x coalesce with those of size y and particles of size (x + y) break-up into those of sizes x and y.

Using a non-linear variation of constants formula, sufficient conditions are derived for the global asymptotic stability of the trivial solution of

A purely algebraic result. We begin by stating the following theorem. Theorem. Let E be a finite set, and letdenote the set of real E × E matrices with non-negative off-diagonal elements and with non-positive row sums. Let A be a symmetric element of, and let V be a diagonal real E × E matrix. Then there exists a unique pair (H+, H−) of elements ofsuch that

I denoting the identity E × E matrix, and the superscript T signifying transpose. It is an immediate consequence that

Let A1,A2,…,An be events on a given probability space, and let mn be the number of those Aj which occur. Put S0 = S0,n = 1, and

where the summation is over all subscripts satisfying 1 ≤ i1 < i2 < … < ik ≤ n. For convenience in some formulae we adopt the convention Sk, n = 0 if k > n. By turning to indicator variables one immediately finds that

Inequalities of the form

where ck = ck(r, n) and dk = dk(r, n) are constants (not dependent on the events Aj, 1 ≤ j ≤ n), possibly zero, are called Bonferroni-type inequalities. This same name applies if P(mn = r) is replaced by P(mn ≥ r) in the middle. The best known such inequalities are the method of inclusion and exclusion

where j ≥ 0 is an arbitrary integer. An extension of (4) to arbitrary r, called Jordan's inequalities (see Takács [16]), is as follows: for 0 ≤ r ≤ n and for any integer j ≥ 0,

It is observed by Galambos and Mucci[9] that (5) follows from (4), and indeed, one can always generate inequalities of the form (3) for arbitrary r from the special case r = 0 if one utilizes only Sr,Sr+1… in the case of P(mn = r). As a matter of fact, from the instructions of Galambos and Mucci one has that the inequalities

hold for an arbitrary sequence A1,A2,…,An of events if, and only if, for an arbitrary sequence A1,A2,…,An−r of events,

where