The Z/2\mathbb{Z}/2 ordinary cohomology of BGU(1)B_G U(1)

With $G = \mathbb{Z}/2$, we calculate the ordinary $G$-cohomology (with Burnside ring coefficients) of $\mathbb{C}P_G^\infty = B_GU(1)$, the complex projective space, a model for the classifying space for $G$-equivariant complex line bundles. The $RO(G)$-graded ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators, including the Euler class of the canonical bundle.Comment: 68 pages; complete rewrite with much better method of proof and more information on the equivariant cohomology of a point; some material from arXiv:1708.06009v1, which will ultimately be rewritten for the odd-order case onl

The Z/2\mathbb{Z}/2-equivariant cohomology of complex projective spaces

In this article we compute the cohomology of complex projective spaces associated to finite dimensional representations of $\mathbb{Z}/2$ graded on virtual representations of its fundamental groupoid. This fully graded theory, unlike the classical $RO(G)$-graded theory, allows for the definition of push-forward maps between projective spaces, which we also compute. In the computation we use relations and generators coming from the fully graded cohomology of the projective space of $\mathscr {U}$, the complete complex $\mathbb{Z}/2$-universe, as carried out by the first author. This work is the first step in a program for developing $\mathbb{Z}/2$-equivariant Schubert calculus.Comment: 42 pages, this is a minor update. The proof of the multiplicative additive structure has been uniformized with the rest of the paper. Typos have been correcte

Unoriented bordism for odd-order groups

AbstractWe consider the action of the Burnside ring on equivariant unoriented bordism for an odd-order group. Known splittings of the bordism rings are shown to correspond to idempotents of the Burnside ring

Parameter estimation in kinetic reaction models using nonlinear observers facilitated by model extensions

An essential part of mathematical modelling is the accurate and reliable estimation of model parameters. In biology, the required parameters are particularly difficult to measure due to either shortcomings of the measurement technology or a lack of direct measurements. In both cases, parameters must be estimated from indirect measurements, usually in the form of time-series data. Here, we present a novel approach for parameter estimation that is particularly tailored to biological models consisting of nonlinear ordinary differential equations. By assuming specific types of nonlinearities common in biology, resulting from generalised mass action, Hill kinetics and products thereof, we can take a three step approach: (1) transform the identification into an observer problem using a suitable model extension that decouples the estimation of non-measured states from the parameters; (2) reconstruct all extended states using suitable nonlinear observers; (3) estimate the parameters using the reconstructed states. The actual estimation of the parameters is based on the intrinsic dependencies of the extended states arising from the definitions of the extended variables. An important advantage of the proposed method is that it allows to identify suitable measurements and/or model structures for which the parameters can be estimated. Furthermore, the proposed identification approach is generally applicable to models of metabolic networks, signal transduction and gene regulation

Continuous functors as a model for the equivariant stable homotopy category

In this paper, we investigate the properties of the category of equivariant diagram spectra indexed on the category W_G of based G-spaces homeomorphic to finite G-CW-complexes for a compact Lie group G. Using the machinery of Mandell, May, Schwede, and Shipley, we show that there is a "stable model structure" on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal G-spectra. We construct a second "absolute stable model structure" which is Quillen equivalent to the "stable model structure". Our main result is a concrete identification of the fibrant objects in the absolute stable model structure. There is a model-theoretic identification of the fibrant continuous functors in the absolute stable model structure as functors Z such that for A in W_G the collection {Z(A smash S^W)} form an Omega-G-prespectrum as W varies over the universe U. We show that a functor is fibrant if and only if it takes G-homotopy pushouts to G-homotopy pullbacks and is suitably compatible with equivariant Atiyah duality for orbit spaces G/H_+ which embed in U. Our motivation for this work is the development of a recognition principle for equivariant infinite loop spaces.Comment: This is the version published by Algebraic & Geometric Topology on 8 December 200

Units of equivariant ring spectra

It is well known that very special $\Gamma$-spaces and grouplike \E_\infty spaces both model connective spectra. Both these models have equivariant analogues. Shimakawa defined the category of equivariant $\Gamma$-spaces and showed that special equivariant $\Gamma$-spaces determine positive equivariant spectra. Costenoble and Waner showed that grouplike equivariant \E_\infty-spaces determine connective equivariant spectra. We show that with suitable model category structures the category of equivariant $\Gamma$-spaces is Quillen equivalent to the category of equivariant \E_\infty spaces. We define the units of equivariant ring spectra in terms of equivariant $\Gamma$-spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.Comment: More detailed introduction, added appendix