Results 1  10
of
24
Smooth and Algebraic Invariants of a Group Action: Local and Global Constructions
 THE JOURNAL OF THE SOCIETY FOR THE FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
, 2007
"... We provide an algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group. This formulation gives rise to algorithms for constructing rational and replacement invariants. The latter are algebraic over the field of rational i ..."
Abstract

Cited by 31 (15 self)
 Add to MetaCart
(Show Context)
We provide an algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group. This formulation gives rise to algorithms for constructing rational and replacement invariants. The latter are algebraic over the field of rational invariants and play a role analogous to Cartan’s normalized invariants in the smooth theory. The algebraic algorithms can be used for computing fundamental sets of differential invariants.
Discretization of partial differential equations preserving their physical symmetries
, 2005
"... A procedure for obtaining a “minimal” discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. “Minimal” in this case means that the differential equation is replaced by a partial difference scheme involving N difference equations, where N is the nu ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
A procedure for obtaining a “minimal” discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. “Minimal” in this case means that the differential equation is replaced by a partial difference scheme involving N difference equations, where N is the number of independent and dependent variable. We restrict to one scalar function of two independent variables. As examples, invariant discretizations of the heat, Burgers and Kortewegde Vries equations are presented. Some exact solutions of the discrete schemes are obtained.
Summation invariant and its application to shape recognition
 In Proc. of ICASSP
, 2005
"... ..."
(Show Context)
Moving frames and differential invariants in centro–affine geometry
, 2009
"... Abstract. Explicit formulas for the generating differential invariants and invariant differential operators for curves in two and threedimensional centroequiaffine and centroaffine geometry and surfaces in threedimensional centroequiaffine geometry are constructed using the equivariant method ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
Abstract. Explicit formulas for the generating differential invariants and invariant differential operators for curves in two and threedimensional centroequiaffine and centroaffine geometry and surfaces in threedimensional centroequiaffine geometry are constructed using the equivariant method of moving frames. In particular, the algebra of centroequiaffine surface differential invariants is shown to be generated by a single second order invariant.
Numerical invariantization for morphological PDE schemes, in: Scale Space and Variational Methods in Computer Vision
 Department of Mathematics, Oregon State University
, 2007
"... Abstract. Based on a new, general formulation of the geometric method of moving frames, invariantization of numerical schemes has been established during the last years as a powerful tool to guarantee symmetries for numerical solutions while simultaneously reducing the numerical errors. In this pape ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
(Show Context)
Abstract. Based on a new, general formulation of the geometric method of moving frames, invariantization of numerical schemes has been established during the last years as a powerful tool to guarantee symmetries for numerical solutions while simultaneously reducing the numerical errors. In this paper, we make the first step to apply this framework to the differential equations of image processing. We focus on the Hamilton–Jacobi equation governing dilation and erosion processes which displays morphological symmetry, i.e. is invariant under strictly monotonically increasing transformations of grayvalues. Results demonstrate that invariantization is able to handle the specific needs of differential equations applied in image processing, and thus encourage further research in this direction. 1
On postLie algebras, Lie–Butcher series and moving frames. ArXiv preprint
, 1203
"... trees. PreLie (or Vinberg) algebras arise from flat and torsionfree connections on differential manifolds. They have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differenti ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
trees. PreLie (or Vinberg) algebras arise from flat and torsionfree connections on differential manifolds. They have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on preLie algebras, used in numerical analysis to study geometric properties of flows on euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of postLie algebras, a generalization of preLie algebras. Whereas preLie algebras are intimately associated with euclidean geometry, postLie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan’s method of moving frames. Lie–Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie–Butcher series are founded on postLie algebras. The functorial relations between postLie algebras and their enveloping algebras, called Dalgebras, are explored. Furthermore, we develop new formulas for computations in free postLie algebras and Dalgebras, based on recursions in a magma, and we show that Lie–Butcher series are related to invariants of curves described by moving frames. 1
Differential Invariant Algebras
"... Abstract. The equivariant method of moving frames provides an algorithmic procedure for determining and analyzing the structure of algebras of differential invariants for both finitedimensional Lie groups and infinitedimensional Lie pseudogroups. This paper surveys recent developments, including ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract. The equivariant method of moving frames provides an algorithmic procedure for determining and analyzing the structure of algebras of differential invariants for both finitedimensional Lie groups and infinitedimensional Lie pseudogroups. This paper surveys recent developments, including a few surprises and several open questions. 1. Introduction. Differential invariants are the fundamental building blocks for constructing invariant differential equations and invariant variational problems, as well as determining their explicit solutions and conservation laws. The equivalence, symmetry and rigidity properties of submanifolds are all governed by their differential invariants. Additional applications
Noncommutative Riquier Theory in Moving Frames of Differential Operators
, 2003
"... Moving frames chosen to be invariant under a known Lie group G provide a powerful generalization of the idea of choosing Ginvariant coordinates to cases where Ginvariant coordinates do not exist. Such Ginvariant formulations are of great current interest in areas such as Geometric Integration whe ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Moving frames chosen to be invariant under a known Lie group G provide a powerful generalization of the idea of choosing Ginvariant coordinates to cases where Ginvariant coordinates do not exist. Such Ginvariant formulations are of great current interest in areas such as Geometric Integration where Ginvariant integrators (e.g. symplectic integrators), can often substantially outperform noninvariant integrators. They are also of substantial interest in applications where one would like to factor out a known group. One form of classical existence and uniqueness theory for analytic PDE referred to (standard) commuting partial derivatives is that of Riquier, which was formulated and generalized by Rust using a Gröbner style development. We extend the RustRiquier existence and uniqueness theory to analytic PDE written in terms of moving frames of noncommuting Partial Dierential Operators (PDO). The main idea for the theoretical development is to use the commutation relations between the PDO to place them in a standard order. This normalization is exploited to generalize the corresponding steps of the commuting RustRiquier Theory to the noncommuting case. Given an equivalence group G Lisle has given a Ginvariant method for determining the structure of Lie symmetry groups of classes of PDE. Lisle's method for such group classication problems was illustrated on a number of challenging examples, which lead to unmanageable expression explosion for computer algebra programs using the standard (commuting) frame. He obtained new results, which for want of an existence and uniqueness theorem for PDE in noncommuting frames, had to be individually checked. We provide an existence and uniqueness theorem making rigorous the output from Lisle's method. For the finite parameter group case, the output is reformulated in terms of the integration of a system of Frobenius type, which can be numerically integrated by integrating an ODE system along a curve.