The relationships, in many cases equivalences, between lattice distributivity, adjunction and continuity have been studied by many authors, for example [1, 3–8, 12, 13, 15, 17–20, 22, 23]. Very roughly, we refer to the following circle of ideas. Let L be an ordered set, and L a class of subsets of L, and suppose that L has a supremum for each element in L. We might say that L has -sups. The ‘distributivity’ we refer to is that of infs over -sups. The ‘adjunction’ is that given by a left adjoint to the map V: L→L. Now the latter has a left adjoint if and only if it preserves infs, and this means roughly that the -sup of an intersection is an inf of -sups. When one does succeed in identifying the -sup of an intersection as a -sup of infs, one has an instance of distributivity.

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.

We shall prove below part of a conjecture made by Shigefumi Mori, David Morrison and Ian Morrison in the course of their investigations into the properties of isolated terminal cyclic quotient singularities of prime Gorenstein index in dimension four [1]. The reader of the present paper need have no knowledge of algebraic geometry, because we quickly reduce the problem to one about the geometry of numbers that can be solved by elementary calculations. The calculations are very lengthy and not quite routine, so what the reader does need is either patience, if he intends to check them, or faith, if he does not. We give only part of the calculations below. Full details may be obtained from the author.*

It is claimed in [3], proposition 4 that if a Morita context (R, V, W, S) is such that WVW = W and S/WV is flat as a right S-module, then Vω = 0, ω ∈ W, implies ω = 0. The proof given for this uses proposition 5 of Azumaya[l] in which rings have unity and so is only applicable when the ring S has a unity. Thus proposition 4 in [3] should be amended so that the additional hypothesis that S has a unity is imposed for statement (4) to be equivalent to the first three statements. It is the purpose of this note to give an example to show that the claim in [3] is false in general.

An S-system over a monoid S is a set A together with a function ø: A × S → A such that (a, 1) ø = a and (a, st) ø = ((a, s) ø, t) ø for all a ∈ A and for all s, t ∈ S. So an S-system is a set on which S acts unitarily on the right. For (a, s) ø we write simply as: the notions of S-homomorphism, S-subsystem etc. are defined in the obvious manner. Many papers have been written characterizing monoids by properties of their S-systems; here we are concerned with injectivity and a related concept, absolute purity. Further details of the terms we use are given in Section 2.

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.

In [1] Marc Feighn proves the following result: a proper, C2, codimension-1 immersion in a manifold with no first homology separates the ambient space. In [4] Michelangelo Vaccaro proves a related result: the C1-immersed image of a compact n-manifold with image a (curved) polyhedron has non-zero Hn with ℤ2 coefficients. (In Vaccaro's terminology f(M) is a curved polyhedron if f is smooth and f(M) is the (non-PL) embedded image of a simplicial complex.) Using ideas similar to Feighn's we prove here the following result.

In this paper we study the classes of almost f-algebras and d-algebras. Apart from a survey of known properties we present some ‘new’ properties of almost f-algebras and d-algebras and we consider their connection with f-algebras. To be more precise, we show, among other things, in an elementary intrinsic manner, that every Archimedean almost f-algebra is commutative. This result is a considerable improvement upon the fact that every Archimedean f-algebra is commutative. Furthermore, we give a description of the set of nilpotent elements in both an Archimedean d-algebra and an Archimedean almost f-algebra.

The difference between the maximal C*-norm ‖ ‖max and the Haagerup norm ‖ ‖h on the tensor product space of C*-aIgebras is studied. Let A and B be C*-algebras. It is shown that ‖ ‖max is equivalent to ‖ ‖h on A ⊗ B if and only if A or B is finite-dimensional.

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.

A metric d on a set has maximal symmetry provided its isometry group is not properly contained in the isometry group of any metric equivalent to d. This concept was introduced by Janos [7] and subsequently Williamson and Janos [17] proved that the standard euclidean metric on ℝn has maximal symmetry. In Bowers [2], an elementary proof that every convex, complete, two-point homogeneous metric for which small spheres are connected has maximal symmetry is presented. This result in turn implies that the standard metrics on the classical spaces of geometry – hyperbolic, euclidean, spherical and elliptic – are maximally symmetric. In this paper we study homogeneous metrics that possess maximal symmetry and, in particular, address the problem of the existence of such metrics and, to a lesser extent, their uniqueness.

In p-primary stable homotopy theory, recent developments have shown the importance of the Morava K-theory spectra K(n) for positive integers n. A current major problem concerns the behaviour of the K(n)-cohomologies on the classifying spaces of finite groups and on related spaces. In this paper we show how to compute the Morava K-theory of extended power constructions Here Xp is the p-fold product of some space X and Cp is the cyclic group of order p. In particular, if we take X as the classifying space BG for some group G, then Dp(X) forms the classifying space for , the wreath product of G by Cp.

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.

The set of critical values of a map of the plane (of corank 1) can be regarded as the apparent contour of a smooth surface. It is a classical result of Whitney[13] that generically the apparent contour is a smooth fold curve with isolated cusps and transverse fold crossings. More recent classifications of smooth map-germs of the plane of low codimension (occurring in generic 2- or 3-parameter families), e.g. in [1, 5, 7], were motivated by a question in differential geometry: given any smooth surface which is generically embedded in ℝ3, produce a list of all possible (orthogonal or central) projections of such a surface.

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.

Throughout, ‘link’ means ‘oriented link in S3’, and ‘polynomial’ means ‘Laurent polynomial’. Let L be a link; then denotes the oriented polynomial of L (see [2, 8]), written in Morton's variables (see [5]), and denotes the semi-oriented polynomial of L (see [3]). Let L have components K1, …, Kn. The total linking of L, denoted t(L), is the sum of the linking numbers lk (Ki, Kj) with 1 ≤ i < j ≤ n. Let f be a framing of L (that is, f assigns an integer to each Ki); then f(L) denotes the total framing of L (that is, the sum of these integers), and A(L, f) denotes the naturally associated union of annuli embedded in S3 Define the framed polynomial {L, f} (v, z) to be (–1)n plus (v−1–v)z−1 times the sum, over all non-empty sublinks K of L, of (where K has k components with 1 ≤ k ≤ n).

A class of representations on Fock space is associated to a representation of the *-algebra structure of a cocommutative graded bialgebra with an involution. We prove that the Gelfand–Naimark–Segal (GNS) representation given by the convolution exponential of a conditionally positive linear functional can be embedded into a representation of this class. Our theory generalizes a well-known construction for infinitely divisible positive definite functions on a group. Applying our general result, we obtain a complete characterization of the GNS representations given by infinitely divisible states on involutive Lie superalgebras.

Cannon's recognition problem [10] asks for a short list of topological properties that is reasonably easy to check and that characterizes topological manifolds. In dimensions below three the answer has been known for a long time: see [6, 24]. In dimensions above four it is now known, due to the work of J. W. Cannon [11], R. D. Edwards [14] (see also [12] and [18]), and F. S. Quinn [21], that topological n-manifolds (n ≥ 5) are precisely ENR ℤ-homology n-manifolds with Cannon's disjoint disc property (DDP) [11] and with a vanishing Quinn's local surgery obstruction [23]. In dimension four there is a resolution theorem of Quinn [22] (with the same obstruction as in dimensions ≥ 5) and a 1-LCC shrinking theorem of M. Bestvina and J. J. Walsh [5]. However, it is still an open problem to find an effective analogue of Cannon's DDP for this dimension, one which would yield a shrinking theorem along the lines of that of Edwards [14]. For more on the history of the recognition problem see the survey [24].

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