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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 ...
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 ...
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.
If V is finite-dimensional then one can identify V with its double dual V ∗∗. One can then show that B 2 is the transpose of the linear map B 1 (if V is infinite-dimensional then B 2 is the transpose of B 1 restricted to the image of V in V ∗∗). Given B one can define the transpose of B to be the bilinear form given by
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 ...
Let K be the set C of all complex numbers, and let V be the set C C (R) of all continuous functions from the real line R to the complex plane C. Consider the vectors (functions) f and g defined by f(t) := e it and g(t) := e −it. (Here, e is the base of the natural logarithm, about 2.71828..., and i is the imaginary unit, a square root of −1.)
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.
The tachyon field equation of motion in the linear dilaton background requires it to take an exponential solution. The Polyakov action in this background is then identical to Liouville field theory, with the linear dilaton being responsible for the background charge term while the tachyon contributing the exponential potential.