4,615 research outputs found
Multihomogeneous resultant formulae by means of complexes
We provide conditions and algorithmic tools so as to classify and construct
the smallest possible determinantal formulae for multihomogeneous resultants
arising from Weyman complexes associated to line bundles in products of
projective spaces. We also examine the smallest Sylvester-type matrices,
generically of full rank, which yield a multiple of the resultant. We
characterize the systems that admit a purely B\'ezout-type matrix and show a
bijection of such matrices with the permutations of the variable groups. We
conclude with examples showing the hybrid matrices that may be encountered, and
illustrations of our Maple implementation. Our approach makes heavy use of the
combinatorics of multihomogeneous systems, inspired by and generalizing results
by Sturmfels-Zelevinsky, and Weyman-Zelevinsky.Comment: 30 pages. To appear: Journal of Symbolic Computatio
Geometric Finite Element Discretization of Maxwell Equations in Primal and Dual Spaces
Based on a geometric discretization scheme for Maxwell equations, we unveil a
mathematical\textit{\}transformation between the electric field intensity
and the magnetic field intensity , denoted as Galerkin duality. Using
Galerkin duality and discrete Hodge operators, we construct two system
matrices, (primal formulation) and (dual
formulation) respectively, that discretize the second-order vector wave
equations. We show that the primal formulation recovers the conventional
(edge-element) finite element method (FEM) and suggests a geometric foundation
for it. On the other hand, the dual formulation suggests a new (dual) type of
FEM. Although both formulations give identical dynamical physical solutions,
the dimensions of the null spaces are different.Comment: 22 pages and 4 figure
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