The geometry of orthogonal groups over finite fields
... forms over finite fields. The emphasis is placed on geometric and combinatorial objects, rather than the orthogonal group itself. Our goal is to introduce dual polar spaces as distance-transitive graphs in a self contained way. Prerequisites are linear algebra, and finite fields. In the later part o ...
... forms over finite fields. The emphasis is placed on geometric and combinatorial objects, rather than the orthogonal group itself. Our goal is to introduce dual polar spaces as distance-transitive graphs in a self contained way. Prerequisites are linear algebra, and finite fields. In the later part o ...
18.03 Differential Equations, Lecture Note 33
... A matrix times a column vector is the linear combination of the columns of the matrix weighted by the entries in the column vector. When is this product zero? One way is for x = 0 = y. If [a ; c] and [b ; d] point in different directions, this is the ONLY way. But if they lie along a single line, we ...
... A matrix times a column vector is the linear combination of the columns of the matrix weighted by the entries in the column vector. When is this product zero? One way is for x = 0 = y. If [a ; c] and [b ; d] point in different directions, this is the ONLY way. But if they lie along a single line, we ...
Positivity for toric vectorbundles
... line bundles L1 ⊕ · · · ⊕ Lr . Moreover, each such Li is equivariantly isomorphic to some Lui |Uσ , where the class of ui is uniquely determined in M/M ∩σ ⊥ . If σ is a top-dimensional cone, then in fact the multiset {u1 , . . . , ur } is uniquely determined by E and σ. A vector bundle E on X is nef ...
... line bundles L1 ⊕ · · · ⊕ Lr . Moreover, each such Li is equivariantly isomorphic to some Lui |Uσ , where the class of ui is uniquely determined in M/M ∩σ ⊥ . If σ is a top-dimensional cone, then in fact the multiset {u1 , . . . , ur } is uniquely determined by E and σ. A vector bundle E on X is nef ...
![arXiv:math/0105237v3 [math.DG] 8 Nov 2002](http://s1.studyres.com/store/data/017056594_1-99fcf3e85def2a6659feb2cb8a9b60c3-300x300.png)
Cartesian tensor
In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is through an orthogonal transformation.The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product.Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity. While orthonormal bases may be found for some such coordinate systems (e.g. tangent to spherical coordinates), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a passive transformation, since the coordinates are changed and not the physical system.