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A two-vector or bivector [1] is a tensor of type () and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector.
The Mackey–Arens theorem, named after George Mackey and Richard Arens, characterizes all possible dual topologies on a locally convex space.. The theorem shows that the coarsest dual topology is the weak topology, the topology of uniform convergence on all finite subsets of ′, and the finest topology is the Mackey topology, the topology of uniform convergence on all absolutely convex ...
The notion of a reductive dual pair makes sense over any field F, which we assume to be fixed throughout.Thus W is a symplectic vector space over F.. If W 1 and W 2 are two symplectic vector spaces and (G 1, G′ 1), (G 2, G′ 2) are two reductive dual pairs in the corresponding symplectic groups, then we may form a new symplectic vector space W = W 1 ⊕ W 2 and a pair of groups G = G 1 × G ...
A dyad is a tensor of order two and rank one, and is the dyadic product of two vectors (complex vectors in general), whereas a dyadic is a general tensor of order two (which may be full rank or not). There are several equivalent terms and notations for this product:
This ensures that a two-dimensional convolution will be able to be performed by a one-dimensional convolution operator as the 2D filter has been unwound to a 1D filter with gaps of zeroes separating the filter coefficients. One-Dimensional Filtering Strip after being Unwound. Assuming that some-low pass two-dimensional filter was used, such as:
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual.In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression.
Three pairs of dual points and lines: one red pair, one yellow pair, and one blue pair. We shall describe this polarity algebraically by following the above construction in the case that C is the unit circle (i.e., r = 1) centered at the origin.
Since ε 2 = 0 for dual numbers, exp(aε) = 1 + aε, all other terms of the exponential series vanishing. Let F = {1 + εr : r ∈ H}, ε 2 = 0. Note that F is stable under the rotation q → p −1 qp and under the translation (1 + εr)(1 + εs) = 1 + ε(r + s) for any vector quaternions r and s. F is a 3-flat in the eight-dimensional space of ...