Rationally Isomorphic Hermitian Forms and Torsors of Some Non-Reductive Groups

Let $R$ be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an $R$-algebra with involution, which are rationally ismorphic and have isomorphic semisimple coradicals, are in fact isomorphic. The same result is also obtained for quadratic forms equipped with an action of a finite group. The results have cohomological restatements that resemble the Grothendieck--Serre conjecture, except the group schemes involved are not reductive. We show that these group schemes are closely related to group schemes arising in Bruhat--Tits theory.Comment: 27 pages. Changes from previous version: Section 5 was split into two sections, several proofs have been simplified, other mild modification

On the non-existence of an R-labeling

We present a family of Eulerian posets which does not have any R-labeling. The result uses a structure theorem for R-labelings of the butterfly poset.Comment: 6 pages, 1 figure. To appear in the journal Orde

Tax Compliance and Firms’ StrategicInterdependence

We focus on a relatively neglected area of the tax-compliance literature ineconomics, the behaviour of firms. We examine the impact of alternativeaudit rules on receipts from a tax on profits in the context of strategicinter-dependence of firms. In the market firms may compete in terms ofeither output or price. The enforcement policy can have an effect onfirms' behaviour in two dimensions - their market decisions as well astheir compliance behaviour. An appropriate design of the enforcementpolicy can thus have a "double dividend" by manipulating firms in bothdimensions.Tax compliance, evasion, oligopoly

Tax Compliance by Firms and Audit Policy

Firms are usually better informed than tax authorities about market conditions and the potential profits of competitors. They may try to exploit this situation by underreporting their own taxable profits. The tax authority could offset firms' informational advantage by adopting "smarter" audit policies .that take into account the relationship between a firm's reported profits and reports for the industry as a whole. Such an audit policy will create an externality for the decision makers in the industry and this externality can be expected to affect not only firms' reporting policies but also their market decisions. If public policy takes into account wider economic issues than just revenue raising what is the appropriate way for a tax authority to run such an audit policy? We develop some clear policy rules in a standard model of an industry and show the effect of these rules using simulations.ca3Tax compliance, evasion, oligopoly

Grid infrastructures for secure access to and use of bioinformatics data: experiences from the BRIDGES project

The BRIDGES project was funded by the UK Department of Trade and Industry (DTI) to address the needs of cardiovascular research scientists investigating the genetic causes of hypertension as part of the Wellcome Trust funded (£4.34M) cardiovascular functional genomics (CFG) project. Security was at the heart of the BRIDGES project and an advanced data and compute grid infrastructure incorporating latest grid authorisation technologies was developed and delivered to the scientists. We outline these grid infrastructures and describe the perceived security requirements at the project start including data classifications and how these evolved throughout the lifetime of the project. The uptake and adoption of the project results are also presented along with the challenges that must be overcome to support the secure exchange of life science data sets. We also present how we will use the BRIDGES experiences in future projects at the National e-Science Centre

Final-State Constrained Optimal Control via a Projection Operator Approach

In this paper we develop a numerical method to solve nonlinear optimal control problems with final-state constraints. Specifically, we extend the PRojection Operator based Netwon's method for Trajectory Optimization (PRONTO), which was proposed by Hauser for unconstrained optimal control problems. While in the standard method final-state constraints can be only approximately handled by means of a terminal penalty, in this work we propose a methodology to meet the constraints exactly. Moreover, our method guarantees recursive feasibility of the final-state constraint. This is an appealing property especially in realtime applications in which one would like to be able to stop the computation even if the desired tolerance has not been reached, but still satisfy the constraints. Following the same conceptual idea of PRONTO, the proposed strategy is based on two main steps which (differently from the standard scheme) preserve the feasibility of the final-state constraints: (i) solve a quadratic approximation of the nonlinear problem to find a descent direction, and (ii) get a (feasible) trajectory by means of a feedback law (which turns out to be a nonlinear projection operator). To find the (feasible) descent direction we take advantage of final-state constrained Linear Quadratic optimal control methods, while the second step is performed by suitably designing a constrained version of the trajectory tracking projection operator. The effectiveness of the proposed strategy is tested on the optimal state transfer of an inverted pendulum

Signatures of hermitian forms and the Knebusch Trace Formula

Signatures of quadratic forms have been generalized to hermitian forms over algebras with involution. In the literature this is done via Morita theory, which causes sign ambiguities in certain cases. In this paper, a hermitian version of the Knebusch Trace Formula is established and used as a main tool to resolve these ambiguities. The last page is an erratum for the published version. We inadvertently (I) gave an incorrect definition of adjoint involutions; (II) omitted dealing with the case $(H\times H, \widehat{\phantom{m}}\,)$. As $W(H\times H, \widehat{\phantom{m}}\,)= W(R\times R, \widehat{\phantom{m}}\,)=0$, the omission does not affect our reasoning or our results. For the sake of completeness we point out where some small changes should be made in the published version.Comment: This is the final version before publication. The last page is an updated erratum for the published versio

Semisimple quantum cohomology, deformations of stability conditions and the derived category

The introduction discusses various motivations for the following chapters of the thesis, and their relation to questions around mirror symmetry. The main theorem of chapter 2 says that if the quantum cohomology of a smooth projective variety V yields a generically semisimple product, then the same holds true for its blow-up at any number of points (theorem 3.1.1). This is a positive test for a conjecture by Dubrovin, which claims that quantum cohomology of V is generically semisimple if and only if its bounded derived category Db(V) has a complete exceptional collection. Chapter 3 generalizes Bridgeland's notion ofstability condition on a triangulated category. The generalization, a polynomial stability condtion (definition 2.1.4), allows the central charge to have values in complex polynomials instead of complex numbers. We show that polynomial stability conditions have the same deformation properties as Bridgeland's stability conditions (theorem 3.2.5). In section 4, it is shown that every projective variety has a canonical family of polynomial stability conditions. In chapter 4, we define and study the notion of a weighted stable map (definition 2.1.2), depending on a set of weights. We show the existence of moduli spaces of weighted stable maps as proper Deligne-Mumford stacks of finite type (theorem 2.1.4), and study in detail their birational behaviour under changes of weights (section 4). We introduce a category of weighted marked graphs in section 6, and show that it keeps track of the boundary components of the moduli spaces, and natural morphisms between them. We introduce weighted Gromov-Witten invariants by defining virtual fundamental classes, and prove that these satisfy all properties to be expected (sections 5 and 7). In particular, we show that Gromov-Witten invariants without gravitational descendants do not depend on the choice of weights.Halbeinfache Quanten-Kohomologie, Deformation von Stabilitätsbedingungen und die Derivierte Kategorie Die Einleitung erläutert verschiedene Ausgangspunkte für die nachfolgenden Kapitel, und ihre Verbindungen zu Fragen rund um Spiegelsymmetrie. Hauptaussage von Kapitel 2 ist Satz 3.1.1: wenn das Produkt der Quantenkohomologie einer glatten projektiven Varietät V generisch halbeinfach ist, dann gilt dasgleiche für die Aufblasung von V an beliebig vielen Punkten. Dies ist ein erfolgreicher Test für eine Vermutung von Dubrovin, die besagt, dass die Quantenkohomologie von V genau dann generisch halbeinfach ist, wenn die beschränkte derivierte Kategorie Db(V) ein vollständiges exzeptionelles System besitzt. Kapitel 3 verallgemeinert Bridgelands Begriff einer Stabilitätsbedingung in einer triangulierten Kategorie. Diese Verallgemeinerung, eine polynomiale Stabilitätsbedingung (Definition 2.1.4), lässt eine zentrale Ladung mit Werten in komplexen Polynomen statt komplexen Zahlen zu. Es wird gezeigt, dass polynomiale Stabilitätsbedingungen dieselben Deformationseigenschaften wie Bridgelands Stabilitätsbedingungen haben (Satz 3.2.5). Abschnitt 4 zeigt, dass es für jede projektive Varietät V eine kanonische Familie von polynomialen Stabilitätsbedingungen in Db(V) gibt. Kapitel 4 führt den Begriff einer gewichteten stabilen Abbildung ein (Definition 2.1.2), in Abhängigkeit einer Menge von Gewichten. Satz 2.1.4 zeigt die Existenz der Modulräume gewichter stabiler Abbildung als eigentliche Deligne-Mumford-Stacks endlichen Typs, und Abschnitt 4 beschäftigt sich im Detail mit dem birationalen Verhalten der Modulräume bei Änderungen der Gewichte. In Abschnitt 6 führen wir eine Kategorie gewichteter markierter Graphen ein, und zeigen, dass sie natürlicherweise Randkomponenten der Modulräume und die natürliche Morphismen zwischen ihnen indiziert. Gewichtete Gromov-Witten-Invarianten werden durch die Definition von virtuellen Fundamentalklassen eingefürt, und wir zeigen, dass diese alle zu erwartenden Eigenschaften erfüllen (Abschnitte 5 und 7). Insbesondere zeigen wir, dass Gromov-Witten-Invarianten ohne Kopplung an Gravitation nicht von der Wahl der Gewichte abhängen