Topological Equivalence of Linear Representations for Cyclic Groups, I & II

In the two parts of this paper we solve a problem of De Rham, proving that Reidemeister torsion invariants determine topological equivalence of linear G-representations, for G a finite cyclic group. Methods in controlled K-theory and surgery theory are developed to establish, and effectively calculate, a necessary and sufficient condition for non-linear similarity in terms of the vanishing of certain non-compact transfer maps. For cyclic groups of 2-power order, we obtain a complete classification of non-linear similarities.Comment: The first version of this paper appeared as MPI Preprint 1997-58, Max Planck Institut fuer Mathematik, Bonn. The final version includes many improvements in exposition and new results. It is now divided into two parts. Part I (36 pages) will appear in Annals of Mathematics, and Part II (43 pages) will appear in Forum Mat

Jensen's Operator Inequality

We establish what we consider to be the definitive versions of Jensen's operator inequality and Jensen's trace inequality for functions defined on an interval. This is accomplished by the introduction of genuine non-commutative convex combinations of operators, as opposed to the contractions used in earlier versions of the theory. As a consequence, we no longer need to impose conditions on the interval of definition. We show how this relates to the pinching inequality of Davis, and how Jensen's trace inequlity generalizes to C*-algebras..Comment: 12 p

Stability in controlled L-theory

We prove a squeezing/stability theorem for delta-epsilon controlled L-groups when the control map is a fibration on a finite polyhedron. A relation with boundedly-controlled L-groups is also discussed.Comment: This is the version published by Geometry & Topology Monographs on 22 April 200

More examples of discrete co-compact group actions

We survey some results and questions about free actions of infinite groups on products of spheres and euclidean spaces, and give some new co-compact examples

Combined production of broilers and fruits

Combined production of broilers and fruit trees is a subject often discussed in organic fruit production in Denmark. Very little research has been carried out on this type of production system. In organic production in Denmark, nearly no pesticides are allowed, so the need for alternative pest control is large. Apple sawfly (Hoplocampa testudinea) and pear midge (Contarinia pyrivora) cause big crop losses in apples and pears respectively, in unsprayed organic fruit production. Both insects infest fruitlets and cause these to drop prematurely after which the pests pupate in the topsoil. In the present experiment a research orchard with the varieties ‘Discovery’ and ‘Conference’ were used as outdoor area for broilers to minimise the population of sawflies and pear midges, and to reduce the need for weeding and manuring. The trees were kept unsprayed. Fruit yield and fruit quality were assessed at harvest. White sticky traps were placed in the test area in order to measure the occurrence of sawfly over time. The infestation of pear midge was investigated counting the infested fruitlets in clusters on trees at the centre of the plots. The catch of apple sawflies was reduced in the combined apple and broiler production, but no significant effect on the yield or the fruit quality was seen. Experiences from on-farm research show that combining fruit and egg-production is one way to reduce the problem with apple sawfly, but poultry alone is not a sufficient way of controlling sawflies. The welfare and health of the broilers were excellent under fruit trees

Convex Multivariable Trace Functions

For any densely defined, lower semi-continuous trace \tau on a C*-algebra A with mutually commuting C*-subalgebras A_1, A_2, ... A_n, and a convex function f of n variables, we give a short proof of the fact that the function (x_1, x_2, ..., x_n) --> \tau (f(x_1, x_2, ..., x_n)) is convex on the space \bigoplus_{i=1}^n (A_i)_{self-adjoint}. If furthermore the function f is log-convex or root-convex, so is the corresponding trace function. We also introduce a generalization of log-convexity and root-convexity called \ell-convexity, show how it applies to traces, and give a few examples. In particular we show that the trace of an operator mean is always dominated by the corresponding mean of the trace values.Comment: 13 pages, AMS TeX, Some remarks and results adde