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Browsing by Subject "Moving frames"

Now showing 1 - 3 of 3
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    Applications of moving frames to group foliation of differential equations
    (2013-10) Thompson, Robert
    The classical group foliation algorithm uses the continuous symmetries of a differential equation to aid in its integration. This is accomplished by transforming the differential equation into two alternative systems, called the resolving and automorphic systems. Incorporating the theory of equivariant moving frames for Lie pseudogroups, a completely symbolic and systematic version of the group foliation algorithm is introduced. In this version of the algorithm, the resolving system is derived using only knowledge of the structure of the differential invariant algebra, requiring no explicit formulae for differential invariants. Additionally, the automorphic system is replaced by an equivalent reconstruction system, again requiring only symbolic computation. The efficacy of this approach is illustrated through several examples. Further applications of aspects of group foliation are given, including the construction of Backlund transformations using resolving systems and a reconstruction process for an invariant submanifold flow corresponding to a given invariant signature evolution.
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    Integrable planar curve flows and the vortex membrane flow in Euclidean 4-Space using moving frames and the variational bicomplex
    (2014-10) Benson, Joseph Jonathan
    Evolution equations with infinite hierarchies of symmetries have been shown to nat- urally arise within the context of geometric, arc-length preserving flows of curves in the plane and in three dimensions. In the following work, a systematic investigation into this phenomenon is conducted for the case of group actions on planar curves. Equivariant moving frames and the variational bicomplex are used. A catalog of results is produced, connecting many invariant curve flows with integrable equations such as Burgers', KdV, mKdV, and Sawada-Kotera. In the last chapter, the techniques are extended to an investigation of the evolution of curvature of 2-dimensional surfaces in 4-dimensional Euclidean space under the Skew-Mean-Curvature flow.
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    Involutive Moving Frames
    (2017-06) Arnaldsson, Orn
    This thesis combines two major equivalence methods, Cartan's equivalence method and the method of the equivariant moving frame for pseudo-groups, to obtain a more efficient tool with which to tackle a large class of equivalence and symmetry problems. These include, for example, all equivalence problems for sections of tensor bundles under change of variable and those arising in the calculus of variations. Furthermore, once the connection between the two original equivalence methods has become clear, we provide a proof of termination of Cartan's method in these cases. To obtain this termination result, we develop a novel algorithm for the computation of Pommaret bases for polynomial modules.