Localization for nonabelian group actions

Suppose $X$ is a compact symplectic manifold acted on by a compact Lie group $K$ (which may be nonabelian) in a Hamiltonian fashion, with moment map $\mu: X \to {\rm Lie}(K)^*$ and Marsden-Weinstein reduction \xred = \mu^{-1}(0)/K. There is then a natural surjective map $\kappa_0$ from the equivariant cohomology $H^*_K(X)$ of $X$ to the cohomology H^*(\xred). In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of \xred of any \eta_0 \in H^*(\xred) whose degree is the dimension of \xred, provided that $0$ is a regular value of the moment map $\mu$ on $X$. This formula is given in terms of any class $\eta \in H^*_K(X)$ for which $\kappa_0(\eta ) = \eta_0$, and involves the restriction of $\eta$ to $K$-orbits $KF$ of components $F \subset X$ of the fixed point set of a chosen maximal torus $T \subset K$. Since $\kappa_0$ isComment: 42 pages, LaTex version no. 2.09, Introduction and Section 8 have been rewritten in revised versio

The Hodge--Poincar\'e polynomial of the moduli spaces of stable vector bundles over an algebraic curve

Let X be a nonsingular complex projective variety that is acted on by a reductive group $G$ and such that $X^{ss} \neq X_{(0)}^{s}\neq \emptyset$. We give formulae for the Hodge--Poincar\'e series of the quotient $X_{(0)}^s/G$. We use these computations to obtain the corresponding formulae for the Hodge--Poincar\'e polynomial of the moduli space of properly stable vector bundles when the rank and the degree are not coprime. We compute explicitly the case in which the rank equals 2 and the degree is even.Comment: Final published version. arXiv admin note: text overlap with arXiv:math/0305346, arXiv:math/0305347 by other author

ESRI Demand Responsiveness Enquiry. ESRI Memorandum Series No. 115 1975

Irish manufacturing industry suffered a fall in its sales volume in 1975 over it's 1974 level. In an effort to clarify the relative importance of price competitiveness vis a vis other factors, the authors conducted a survey amongst firms in Irish manufacturing industry in December 1975. The results indicate what managers of firms perceived as the reasons for their poor sales performance. The questions put, of their very nature require subjective answer. Thus, managers, when faced with a leftward shift in their firms demand curve, were asked to distinguish the separate effects of a fall in consumer demand, and of any loss of price competitiveness. An effort was also made to assess the degree of price responsiveness of demand amongst sectors in both domestic and export markets. The normal caveats about this type of subjective enquiry of course apply. The survey covered those firms which participate in the monthly CII/ESRI Business Opinion Survey. Of a total of 320 questionnaires despatched, 218 usable replies were received, a response rate of just over 68%. For the purposes of the survey the firms were classified in accordance with the ten sector classification used by the CSO in the Quarterly Industrial Enquiry. The actual processing and calculation of the results was carried out by computer, each firm's replies being weighed by that firm's turnover weight as used in the CII/ESRI survey. Sectoral output weights were derived from the finer sectoral classification of the same survey

How many invariant polynomials are needed to decide local unitary equivalence of qubit states?

Given L-qubit states with the fixed spectra of reduced one-qubit density matrices, we find a formula for the minimal number of invariant polynomials needed for solving local unitary (LU) equivalence problem, that is, problem of deciding if two states can be connected by local unitary operations. Interestingly, this number is not the same for every collection of the spectra. Some spectra require less polynomials to solve LU equivalence problem than others. The result is obtained using geometric methods, i.e. by calculating the dimensions of reduced spaces, stemming from the symplectic reduction procedure.Comment: 22 page

The C.1.1.- E.S.R.I. Quarterly and Monthly Surveys of Business Attitudes: Methods and Uses. Quarterly Economic Commentary Special Article, March 1975

Beginning with the fourth quarter of 1961, the Confederation of Irish Industry (C.1.1.) and the Economic and Social Research Institute (E.S.R.I.) have been jointly administering a quarterly survey of businessmen's attitudes in the Republic of Ireland. The results of this survey were circulated to C.I.I. members and since 1968 were published in the E.S.R.I.'s Quarterly Economic Commentary. Similar monthly surveys have been carried out on a coordinated basis by the member countries of the European Economic Community since 1961. On Ireland's accession to the E.E.C. it was decided to revise the C.1.1.-E.S.R.I. survey so as to ensure comparability with those in other member countries. The process of revision has now been completed: the last quarterly survey referred to the fist quarter of 1974, while the first monthly survey was carried out in March 1974, and has been continued on a regular basis since then. The purpose of this note is to describe the methods and coverage of the two surveys, to outline the uses to which the results may be put, and to suggest some directions for further research into both the methods and application of the surveys

Stratifying quotient stacks and moduli stacks

Recent results in geometric invariant theory (GIT) for non-reductive linear algebraic group actions allow us to stratify quotient stacks of the form [X/H], where X is a projective scheme and H is a linear algebraic group with internally graded unipotent radical acting linearly on X, in such a way that each stratum [S/H] has a geometric quotient S/H. This leads to stratifications of moduli stacks (for example, sheaves over a projective scheme) such that each stratum has a coarse moduli space.Comment: 25 pages, submitted to the Proceedings of the Abel Symposium 201

Matrix String Theory and its Moduli Space

The correspondence between Matrix String Theory in the strong coupling limit and IIA superstring theory can be shown by means of the instanton solutions of the former. We construct the general instanton solutions of Matrix String Theory which interpolate between given initial and final string configurations. Each instanton is characterized by a Riemann surface of genus h with n punctures, which is realized as a plane curve. We study the moduli space of such plane curves and find out that, at finite N, it is a discretized version of the moduli space of Riemann surfaces: instead of 3h-3+n its complex dimensions are 2h-3+n, the remaining h dimensions being discrete. It turns out that as $N$ tends to infinity, these discrete dimensions become continuous, and one recovers the full moduli space of string interaction theory.Comment: 30 pages, LaTeX, JHEP.cls class file, minor correction

Three Applications of Instanton Numbers

We use instanton numbers to: (i) stratify moduli of vector bundles, (ii) calculate relative homology of moduli spaces and (iii) distinguish curve singularities.Comment: To appear in Communications in Mathematical Physic

Applying model checking to agent-based learning systems

In this thesis we present a comprehensive approach for applying model checking to Agent-Based Learning (ABL) systems. Model checking faces a unique challenge with ABL systems, as the modelling of learning is thought to be outwith its scope. The practical work performed to model these systems is presented in the incremental stages by which it was carried out. This allows for a clearer understanding of the problems faced and of the progress made on traditional ABL system analysis. Our focus is on applying model checking to a specific type of system. It involves a biologically-inspired robot that uses Input Correlation learning to help it navigate environments. We present a highly detailed PROMELA model of this system, using embedded C code to avoid losing accuracy when modelling it. We also propose an abstraction method for this type of system: Agent-centric abstraction. Our abstraction is the main contribution of this thesis. It is defined in detail, and we provide a proof of its soundness in the form of a simulation relation. In addition to this, we use it to generate an abstract model of the system. We give a comparison between our models and traditional system analysis, specifically simulation. A strong case for using model checking to aid ABL system analysis is made by our comparison and the verification results we obtain from our models. Overall, we present a framework for analysing ABL systems that differs from the more common approach of simulation. We define this framework in detail, and provide results from practical work coupled with a discussion about drawbacks and future enhancements