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We give necessary and sufficient conditions for an orthogonal group defined over a global field of characteristic not equal 2 to contain a maximal torus of a given type.
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In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose).
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic to the standard torus Tn). A maximal torus is one which is maximal among such subgroups. That is, T is a maximal torus if for any torus T′ containing T we have T = T′. Every torus is contained in a maximal torus simply by dimensional considerations.
In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = (V,Q) on the associated projective space P(V). Explicitly, the projective orthogonal group is the quotient group PO(V) = O(V)/ZO(V) = O(V)/{±I} where O(V) is the orthogonal group of (V) and ZO(V)={±I} is the subgroup of all orthogonal scalar transformations of V – these consist of the identity and reflection through the origin.
In the collective imagination the villa is a manifesto of 'the good life’, often representing for architects a laboratory of experimentation and style and an exception in their portfolio. The fate of the villa in contemporary architecture and research cult ...
In the first chapter of this thesis, the macrocyclization of a new type of bifunctional substrates, omega-isocyanoaldehyde derivatives, is described. Ten different omega-isocyanoaldehydes in terms of different ring sizes and functional groups were prepared ...
Concepts of type and typology are not specific to architecture. Rather they represent an interdisciplinary approach to ordering knowledge and gaining insight. In the field of architecture, the study of types and typology offers a didactic perspective that ...