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In this paper we develop an effective way to access the topology of the gauge group for a smooth K-principal bundle P = (K, π, P, M) with possibly infinite-dimensional structure group K over a compact manifold with corners M. For this purpose we introduce the concept of a not necessarily finite-dimensional manifold with corners and show that C ∞ (M, K) is a Lie group if M is a compact manifold with corners. This enables us in the second section to consider the gauge group Gau(P), with a natural topology on it, as an infinite-dimensional Lie group if M is compact and K is locally exponential. In the last section we discuss some applications. We show that the inclusion Gau(P) → Gauc(P) of smooth into continuous gauge transformations is a weak homotopy equivalence, apply this result to the calculation of πn(Gau(P)).

We study a variant of System F that integrates and generalizes several existing proposals for calculi with structural typing rules. To the usual type constructors (!, , All, Some, Rec) we add anumber of type destructors, each internalizing a useful fact about the subtyping relation. For example, in F with products every closed subtype of a product S T must itself be a product S 0 T 0 with S 0 <:S and T 0 <:T. We internalise this observation by introducing type destructors.1 and.2 and postulating an equivalence T = T.1 T.2 whenever T<:U V (including, for example, when T is a variable). In other words, every subtype of a product type literally is a product type, modulo-conversion. Adding type destructors provides a clean solution to the problem of polymorphic update without introducing new term formers, new forms of polymorphism, or quanti cation over type operators. We illustrate this by giving elementary presentations of two well-known encodings of objects, one based on recursive record types and the other based on existential packages. The formulation of type destructors poses some tricky meta-theoretic problems. We discuss two di erent variants: an \ideal &quot; system where both constructors and destructors appear in general forms, and a more modest system, F TD, which imposes some restrictions in order to achieve a tractable metatheory. The properties of the latter system are developed in detail. 1

Abstract. Let G be the group of rational points of a semisimple algebraic group of rank 1 over a nonarchimedean local field. We improve upon Lubotzky’s analysis of graphs of groups describing the action of lattices in G on its Bruhat–Tits tree assuming a condition on unipotents in G. The condition holds for all but a few types of rank 1 groups. A fairly straightforward simplification of Lubotzky’s definition of a cusp of a lattice is the key step to our results. We take the opportunity to reprove Lubotzky’s part in the analysis from this foundation.

The last ten years have seen large progress in the investigation of surfaces of constant mean curvature. In particular, for CMC-surfaces without umbilics many important new results were obtained. Starting with the work of Wente [20], CMC-tori were first constructed and then classified in terms of algebraic geometric data [1, 18, 2, 10, 14].

We linearize the Artin representation of the braid group given by (right) automorphisms of a free group providing a linear faithful representation of the braid group. This result is generalized to obtain linear representations for the coloured braid groupoid and pure braid group too. Applications to some areas of two-dimensional physics are discussed. 1 1

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Abstract—Coproducts of monads on Set have arisen in both the study of computational effects and universal algebra. We describe coproducts of consistent monads on Set by an initial algebra formula, and prove also the converse: if the coproduct exists, so do the required initial algebras. That formula was, in the case of ideal monads, also used by Ghani and Uustalu. We deduce that coproduct embeddings of consistent monads are injective; and that a coproduct of injective monad morphisms is injective. Two consistent monads have a coproduct iff either they have arbitrarily large common fixpoints, or one is an exception monad, possibly modified to preserve the empty set. Hence a consistent monad has a coproduct with every monad iff it is an exception monad, possibly modified to preserve the empty set. We also show other fixpoint results, including that a functor (not constant on nonempty sets) is finitary iff every sufficiently large cardinal is a fixpoint. Index Terms—monads, coproducts, bialgebras, computational effects, fixpoints

ABSTRACT. The purpose of this note is to establish a connection between the notion of (n-2)-tightness in the sense of N.H. Kuiper and T.F. Banchoff and the total absolute curvature of compact submanifolds-with-boundary of even dimension in Euclidean space. The argument used is a certain geometric in-equality similar to that of S.S. Chern and R.K. Lashof where equality charac-terizes (n-2)-tightness.