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Differential Invariants of Conformal and Projective Surfaces
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2007
"... We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in threedimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames. ..."
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Cited by 16 (13 self)
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We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in threedimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.
Scaling Invariants and Symmetry Reduction of Dynamical Systems
, 2012
"... Scalings form a class of group actions that have theoretical and practical importance. A scaling is accurately described by a matrix of integers. Tools from linear algebra over the integers are exploited to compute their invariants and offer a scheme for the symmetry reduction of dynamical systems. ..."
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Cited by 5 (3 self)
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Scalings form a class of group actions that have theoretical and practical importance. A scaling is accurately described by a matrix of integers. Tools from linear algebra over the integers are exploited to compute their invariants and offer a scheme for the symmetry reduction of dynamical systems. A special case of the symmetry reduction algorithm applies to reduce the number of parameters in physical, chemical or biological models.
Discrete moving frames and discrete integrable systems
 Foundations of Computational Mathematics, Volume 13, Issue 4 (2013), Page 545582
"... Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving frame, which is essentially a sequence of moving frames with ..."
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Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer–Cartan invariants, for which there are recursion formulae. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differentialdifference systems. In particular, we show that in the centroaffine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices respectively under Miura transformations. We also show that a specified invariant map of polygons in the centroaffine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere. Dedicated to Peter Olver in celebration of his 60th birthday 1
Rational Invariants of Finite Abelian Groups
, 2014
"... We investigate the field of rational invariants of the linear action of a finite abelian group in the non modular case. By diagonalization, the group is accurately described by an integer matrix of exponents. We make use of linear algebra to compute a minimal generating set of invariants and the sub ..."
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We investigate the field of rational invariants of the linear action of a finite abelian group in the non modular case. By diagonalization, the group is accurately described by an integer matrix of exponents. We make use of linear algebra to compute a minimal generating set of invariants and the substitution to rewrite any invariant in terms of this generating set. We show that the generating set can be chosen to consist of polynomial invariants. As an application, we provide a symmetry reduction scheme for polynomial systems the solution set of which are invariant by the group action. We furthermore provide an algorithm to find such symmetries given a polynomial system.
Algebraic and rational differential invariants
"... These notes start with an introduction to differential invariants. They continue with an algebraic treatment of the theory. The algebraic, differential algebraic and differential geometric tools that are necessary to the development of the theory are explained in detail. We expose the recent result ..."
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These notes start with an introduction to differential invariants. They continue with an algebraic treatment of the theory. The algebraic, differential algebraic and differential geometric tools that are necessary to the development of the theory are explained in detail. We expose the recent results on the topic of rational and algebraic differential invariants. Finally we give a new algebraic version of the finiteness theorem of Lie–Tresse for the case of finite dimensional algebraic groups.
Invariants of objects and their images under surjective maps
, 2015
"... Abstract: We examine the relationships between the differential invariants of objects and of their images under a surjective maps. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the image, and the case when only a proper subgroup of th ..."
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Abstract: We examine the relationships between the differential invariants of objects and of their images under a surjective maps. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the image, and the case when only a proper subgroup of the entire group acts projectably. In the former case, we establish a constructible isomorphism between the algebra of differential invariants of the images and the algebra of fiberwise constant (gauge) differential invariants of the objects. In the latter case, we describe residual effects of the full transformation group on the image invariants. Our motivation comes from the problem of reconstruction of an object from multipleview images, with central and parallel projections of curves from threedimensional space to the twodimensional plane serving as our main examples. 1 Introduction. The subject of this paper is the behavior of invariants and, particularly, differential invariants under surjective maps. While our theoretical results are valid for manifolds of arbitrary dimension, the motivating examples are central and parallel projections from threedimensional space onto the twodimensional plane, as prescribed by simple cameras. We concentrate on