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Differential equations, Spencer cohomology, and computing resolutions
, 2002
"... Abstract. We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a ..."
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Abstract. We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a Cotor. We discuss concrete methods for its construction based on homological perturbation theory and resolutions. 1.
A Discrete Vector Calculus in Tensor Grids
, 2003
"... The key to the success of mimetic discretization methods is that they discretize some description of continuum mechanics, e.g. vector calculus or differential forms. For a ..."
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The key to the success of mimetic discretization methods is that they discretize some description of continuum mechanics, e.g. vector calculus or differential forms. For a
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 092, 41 pages Existence and Construction of Vessiot Connections ⋆
, 909
"... doi:10.3842/SIGMA.2009.092 Abstract. A rigorous formulation of Vessiot’s vector field approach to the analysis of general systems of partial differential equations is provided. It is shown that this approach is equivalent to the formal theory of differential equations and that it can be carried thro ..."
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doi:10.3842/SIGMA.2009.092 Abstract. A rigorous formulation of Vessiot’s vector field approach to the analysis of general systems of partial differential equations is provided. It is shown that this approach is equivalent to the formal theory of differential equations and that it can be carried through if, and only if, the given system is involutive. As a byproduct, we provide a novel characterisation of transversal integral elements via the contact map. Key words: formal integrability; integral element; involution; partial differential equation; Vessiot connection; Vessiot distribution
Existence and Construction of Vessiot Connections ⋆
"... doi:10.3842/SIGMA.2009.092 Abstract. A rigorous formulation of Vessiot’s vector field approach to the analysis of general systems of partial differential equations is provided. It is shown that this approach is equivalent to the formal theory of differential equations and that it can be carried thro ..."
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doi:10.3842/SIGMA.2009.092 Abstract. A rigorous formulation of Vessiot’s vector field approach to the analysis of general systems of partial differential equations is provided. It is shown that this approach is equivalent to the formal theory of differential equations and that it can be carried through if, and only if, the given system is involutive. As a byproduct, we provide a novel characterisation of transversal integral elements via the contact map. Key words: formal integrability; integral element; involution; partial differential equation;
Involution Analysis of Field Theories
"... We describe the application of the theory of involutive systems to eld theories in mathematical physics and engineering sciences. Emphasis is put on eective constraint algorithms, on counting degrees of freedom and on the initial value problem. ..."
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We describe the application of the theory of involutive systems to eld theories in mathematical physics and engineering sciences. Emphasis is put on eective constraint algorithms, on counting degrees of freedom and on the initial value problem.