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Charles Curry

Publications and source records attributed to Charles Curry.

13 recordsLinked to original sources
arXiv2026

Finite element discretization of Yang--Mills connections

We propose a finite element method for Yang--Mills connections on nontrivial principal bundles with abelian structure group. Local connection forms satisfy internal jump conditions induced by the transition functions of the bundle. We discretize these forms in broken finite element exterior calculus spaces and enforce the jump conditions weakly by Lagrange multipliers. We prove well-posedness of the resulting saddle-point problem and derive an a priori convergence estimate. A numerical experiment for the Hopf fibration illustrates the method and exhibits linear convergence when the sphere is approximated by a piecewise linear mesh.

math.NA↗
arXiv2023

The Prytz connections

The Prytz planimeter is a simple mechanical device that historically was used to approximate areas of plane regions. In this article, we present a mathematical description and analysis of the planimeter in terms of sub-Riemannian geometry and in terms of connections and horizontal lifts -- central concepts in differential geometry.

math.DG↗
arXiv2020

Planarly branched rough paths and rough differential equations on homogeneous spaces

The central aim of this work is to understand rough differential equations on homogeneous spaces. We focus on the formal approach, by giving an explicit expansion of the solution at each point of the real line in terms of decorated planar forests. For this we develop the notion of planarly branched rough paths, following M. Gubinelli's branched rough paths. The definition is similar to the one in the flat case, the main difference being the replacement of the Butcher--Connes--Kreimer Hopf algebra of non-planar rooted forests by the Munthe-Kaas--Wright Hopf algebra of planar rooted forests. We show how the latter permits to handle rough differential equations on homogeneous spaces using planarly branched rough paths, the same way branched rough paths are used in the context of rough differential equations on finite-dimensional vector spaces. An analogue of T. Lyons' extension theorem is proven. Finally, under analyticity assumptions on the coefficients and when the Hölder index of the driving path is equal to one, we show convergence of the planar forest expansion in a small time interval.

math.CA↗
arXiv2020

The Magnus expansion and Post-Lie algebras

We relate the classical and post-Lie Magnus expansions. Intertwining algebraic and geometric arguments allows to placing the classical Magnus expansion in the context of Lie group integrators.

math.NA↗
arXiv2019

Principal symmetric space analysis

We develop a novel analogue of Euclidean PCA (principal component analysis) for data taking values on a Riemannian symmetric space, using totally geodesic submanifolds as approximating lower dimnsional submanifolds. We illustrate the technique on n-spheres, Grassmannians, n-tori and polyspheres.

math.ST↗
arXiv2018

Algebraic Structures and Stochastic Differential Equations driven by Levy processes

We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and independent of the governing vector fields. This holds provided the driving processes possess moments of all orders and the vector fields are sufficiently smooth. Moreover the efficient integrator in question is optimal within a broad class of perturbations for half-integer global root mean-square orders of convergence. We obtain these results using the quasi-shuffle algebra of multiple iterated integrals of independent Levy processes.

math.PR↗
arXiv2018

Convergence of Lie group integrators

We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local bounds. In doing so, we prove for the first time that the Lie-Butcher theory of Lie group integrators leads to global error estimates.

math.NA↗
arXiv2018

Quasi-shuffle algebras and renormalisation of rough differential equations

The objective of this work is to compare several approaches to the process of renormalisation in the context of rough differential equations using the substitution bialgebra on rooted trees known from backward error analysis of $B$-series. For this purpose, we present a so-called arborification of the Hoffman--Ihara theory of quasi-shuffle algebra automorphisms. The latter are induced by formal power series, which can be seen to be special cases of the cointeraction of two Hopf algebra structures on rooted forests. In particular, the arborification of Hoffman's exponential map, which defines a Hopf algebra isomorphism between the shuffle and quasi-shuffle Hopf algebra, leads to a canonical renormalisation that coincides with Marcus' canonical extension for semimartingale driving signals. This is contrasted with the canonical geometric rough path of Hairer and Kelly by means of a recursive formula defined in terms of the coaction of the substitution bialgebra.

math.PR↗
arXiv2013

Levy Processes and Quasi-Shuffle Algebras

We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfill the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed ones and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Levy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Levy processes form a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Levy processes.

math.PR↗
arXiv1995

On the Global Stability of Magnetized Accretion Disks. II. Vertical and Azimuthal Magnetic Fields

We investigate the global stability of a differentially rotating fluid shell threaded by vertical and azimuthal magnetic fields to linear, axisymmetric perturbations. This system, which models a thick accretion disk in the vicinity of its midplane, is susceptible to the Velikhov-Chandrasekhar (VC) instability in the absence of the azimuthal field. In most cases, the azimuthal field tends to stabilize the VC instability, although strong fields (Alfvén speed of order the characteristic rotational speed in our incompressible model) are required for complete stabilization. Stability diagrams are constructed, indicating critical values of the two fields for instability. We find an additional strong field instability that arises when the azimuthal Alfvén speed exceeds the characteristic rotational speed. This instability, in the case of a freely bounded configuration, has certain similarities to the sausage instability for interpenetrating fields in plasma physics, and may be important for very massive disks or filamentary molecular clouds. An application to the L1641 region in Orion A is briefly discussed. Finally, we find that the effect of a radially varying vertical field (without an azimuthal field) is mainly stabilizing.

astro-ph↗