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Each lattice Λ in Rd determines a sequence of Brillouin zones Bn, fundamental regions for Λ bounded by Bragg hyperplanes; for example B1 is the Dirichlet region. Basic geometric and topological properties of these zones are established, and we obtain asymptotic estimates (valid for almost all Λ)...
Let L and L¯ be orthogonal complementary rational linear subspaces of En, and let Λ = L ∩ Zn and Λ¯ = L¯ ∩ Zn be the sublattices of the usual integer lattice Zn induced by L and L¯. Then the determinants of Λ and Λ are equal. The same relationship holds between the determinants of the lattices ∏...
Let Z and Z¯ be the images of a regular unit n‐cube in En under orthogonal projection on to orthogonal complementary subspaces of dimensions d and n — d respectively. It is shown that the d‐volume of Z and the (n — d)‐volume of Z¯ are equal. This generalizes to a connexion between the volumes of...
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