Ian G. Lisle

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In this paper we give a method which uses a nite number of di erentiations and linear operations to determine the Cartan structure of structurally transitive Lie pseudogroups from their in nitesimal de ning equations. These equations are the linearized form of the pseudogroup de ning system { the system of pdes whose solutions are the transformations(More)
Since Chevalley’s seminal work [12], the definition of Lie group has been universally agreed. Namely, a Lie group G is an analytic manifold G on which is defined an analytic group operation ∗ with analytic inverse. The historical evolution of this definition was not direct. In the 1870s when Lie began his work on “continuous groups of transformations”, the(More)
We present an algorithm COMMUTATION.RELATIONS, which can calculate the commutation relations for the Lie symmetry algebra of symmetry operators for any system of PDEs. Unlike existing methods, COMMUTATION_RELATIONS does not depend on the heuristic process of integrating the associated differential equations for the symmetry operators (i.e. integrating the(More)
Spherical harmonic (SH) lighting models require efficient and general libraries for evaluation of SH functions and of Wigner matrices for rotation. We introduce an efficient algebraic recurrence for evaluation of SH functions, and also implement SH rotation via Wigner matrices constructed for the real SH basis by a recurrence. Using these algorithms, we(More)
We describe a method which uses a finite number of differentiations and linear operations to determine the Cartan structure coefficients of a structurally transitive Lie pseudogroup from its infinitesimal defining equations. If the defining system is of first order and the pseudogroup has no scalm invariants, the structure coefficients can be simply(More)
Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G f , or equivalently of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated over-determined ‘defining system’ of differential(More)
This work is part of a sequence in which we develop and refine algorithms for computer symmetry analysis of differential equations. We show how to exploit partially integrated forms of symmetry defining systems to assist the differential elimination algorithms that uncover structure of the Lie symmetry algebras. We thus incorporate a key advantage of(More)
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