47 research outputs found
Formal Solutions of a Class of Pfaffian Systems in Two Variables
In this paper, we present an algorithm which computes a fundamental matrix of
formal solutions of completely integrable Pfaffian systems with normal
crossings in two variables, based on (Barkatou, 1997). A first step was set in
(Barkatou-LeRoux, 2006) where the problem of rank reduction was tackled via the
approach of (Levelt, 1991). We give instead a Moser-based approach. And, as a
complementary step, we associate to our problem a system of ordinary linear
singular differential equations from which the formal invariants can be
efficiently derived via the package ISOLDE, implemented in the computer algebra
system Maple.Comment: Keywords: Linear systems of partial differential equations, Pfaffian
systems, Formal solutions, Moser-based reduction, Hukuhara- Turritin normal
for
Formal Solutions of Completely Integrable {Pfaffian} Systems With Normal Crossings
In this paper, we present an algorithm for computing a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in several variables. This algorithm is a generalization of a method developed for the bivariate case based on a combination of several reduction techniques and is implemented in the computer algebra system Maple
Elliptic singularities on log symplectic manifolds and Feigin--Odesskii Poisson brackets
A log symplectic manifold is a complex manifold equipped with a complex
symplectic form that has simple poles on a hypersurface. The possible
singularities of such a hypersurface are heavily constrained. We introduce the
notion of an elliptic point of a log symplectic structure, which is a singular
point at which a natural transversality condition involving the modular vector
field is satisfied, and we prove a local normal form for such points that
involves the simple elliptic surface singularities
and . Our main application is to the classification of Poisson
brackets on Fano fourfolds. For example, we show that Feigin and Odesskii's
Poisson structures of type are the only log symplectic structures on
projective four-space whose singular points are all elliptic.Comment: 33 pages, comments welcom
Trajectory generation for the N-trailer problem using Goursat normal form
Develops the machinery of exterior differential forms, more particularly the Goursat normal form for a Pfaffian system, for solving nonholonomic motion planning problems, i.e., motion planning for systems with nonintegrable velocity constraints. The authors use this technique to solve the problem of steering a mobile robot with n trailers. The authors present an algorithm for finding a family of transformations which will convert the system of rolling constraints on the wheels of the robot with n trailers into the Goursat canonical form. Two of these transformations are studied in detail. The Goursat normal form for exterior differential systems is dual to the so-called chained-form for vector fields that has been studied previously. Consequently, the authors are able to give the state feedback law and change of coordinates to convert the N-trailer system into chained-form. Three methods for planning trajectories for chained-form systems using sinusoids, piecewise constants, and polynomials as inputs are presented. The motion planning strategy is therefore to first convert the N-trailer system into Goursat form, use this to find the chained-form coordinates, plan a path for the corresponding chained-form system, and then transform the resulting trajectory back into the original coordinates. Simulations and frames of movie animations of the N-trailer system for parallel parking and backing into a loading dock using this strategy are included
Realizing non-Abelian statistics
We construct a series of 2+1-dimensional models whose quasiparticles obey
non-Abelian statistics. The adiabatic transport of quasiparticles is described
by using a correspondence between the braid matrix of the particles and the
scattering matrix of 1+1-dimensional field theories. We discuss in depth
lattice and continuum models whose braiding is that of SO(3) Chern-Simons gauge
theory, including the simplest type of non-Abelian statistics, involving just
one type of quasiparticle. The ground-state wave function of an SO(3) model is
related to a loop description of the classical two-dimensional Potts model. We
discuss the transition from a topological phase to a conventionally-ordered
phase, showing in some cases there is a quantum critical point.Comment: 20 pages in two-column format. v2: fixed typos and added reference