Numerical Schubert calculus

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gr\"obner basis for the Pl\"ucker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.Comment: 24 pages, LaTeX 2e with 2 figures, used epsf.st

The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings

In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum ?1+...+? r of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding ?(?1,...,? r ). In this paper we extend this correspondence in a natural way to cover also non-coherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos

Has management accounting research been critical?

This paper examines the contributions Management Accounting Research (MAR) has (and has not) made to social and critical analyses of management accounting in the twenty-five years since its launch. It commences with a personalised account of the first named author’s experiences of behavioural, social and critical accounting in the twenty-five years before MAR appeared. This covers events in the UK, especially the Management Control Workshop, Management Accounting Research conferences at Aston, the Inter-disciplinary Perspectives on Accounting Conferences; key departments and professors; and elsewhere the formation of pan-European networks, and reflections on a years’ visit to the USA. Papers published by MAR are analysed according to year of publication, country of author and research site, research method, research subject (type of organization or subject studied), data analysis method, topic, and theory. This revealed, after initial domination by UK academics, increasing Continental European influence; increasing use of qualitative methods over a wide range of topics, especially new costing methods, control system design, change and implementation, public sector transformation, and more recently risk management and creativity. Theoretical approaches have been diverse, often multi-disciplinary, and have employed surprisingly few economic theories relative to behavioural and social theories. The research spans mainly large public and private sector organisations especially in Europe. Seven themes perceived as of interest to a social and critical theory analysis are evaluated, namely: the search for ‘Relevance Lost’ and new costing; management control, the environment and the search for ‘fits’; reconstituting the public sector; change and institutional theory; post-structural, constructivist and critical contributions; social and environmental accounting; and the changing geography of time and space between European and American research. The paper concludes by assessing the contributions of MAR against the aspirations of groups identified in the opening personal historiography, which have been largely met. MAR has made substantial contributions to social and critical accounting (broadly defined) but not in critical areas endeavouring to give greater voice and influence to marginalised sectors of society worldwide. Third Sector organisations, politics, civil society involvement, development and developing countries, labour, the public interest, political economy, and until recently social and environmental accounting have been neglected

A Family of Sparse Polynomial Systems Arising in Chemical Reaction Systems

A class of sparse polynomial systems is investigated which is dened by a weighted directed graph and a weighted bipartite graph. They arise in the model of mass action kinetics for chemical reaction systems. In this application the number of real positive solutions within a certain affine subspace is of particular interest. We show that the simplest cases are equivalent to binomial systems while in general the solution structure is highly determined by the properties of the two graphs. First we recall results by Feinberg and give rigorous proofs. Secondly, we explain how the graphs determine the Newton polytopes of the system of sparse polynomials and thus determine the solution structure. The results on positive solutions from real algebraic geometry are applied to this particular situation. Examples illustrate the theoretical results

Polyhedral End Games for Polynomial Continuation

Bernshtein's theorem provides a generically exact upper bound on the number of isolated solutions a sparse polynomial system can have in (C ) n , with C = C n f0g. When a sparse polynomial system has fewer than this number of isolated solutions some face system must have solutions in (C ) n . In this paper we address the process of recovering a certificate of deficiency from a diverging solution path. This certificate takes the form of a face system along with approximations of its solutions. We apply extrapolation to estimate the cycle number and the face normal. Applications illustrate the practical usefulness of our approach. keywords : homotopy continuation, polynomial systems, Newton polytopes, Bernshtein bound, cycle number. AMS(MOS) Classification : 14Q99, 52A39, 52B20, 65H10. 1 Introduction All isolated complex solutions to polynomial systems can be approximated numerically by homotopy continuation methods. The strategy is to set up a collection of implicitly d..

SPIN: A Software for Computing State Polytopes of Toric Ideals

this paper we present a software package for computing the state polytope of a toric ideal based on theory and algorithms presented in [10]. Given an integer matrix A 2

Fractionally Spaced CMA Equalizers under Periodic and Correlated Inputs

CMA Fractionally Spaced Equalizers (CMA-FSEs) have been shown, under certain conditions, to be globally asymptotically convergent to a setting which provides perfect equalization. Such a result relies heavily on the assumptions of a white source and no channel noise (as is the case in much of the literature's analysis of CMA). Herein, we relax the white source assumption and examine the e#ect of source correlation on CMA. Analytic results are meshed with examples showing CMA-FSE source correlation e#ects. Techniques for finding all stationary and saddle points on the CMA-FSE error surface are presented using recent developments in the Algebraic-Geometry community. 1. INTRODUCTION "There are some aspects of blind deconvolution that do not fit easily into the theory presented in this chapter. The most important one is probably what happens if the input sequence is not white. Little work has been done on this matter." -Bellini[4] In the equalization problem for digital communication, a..