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the tensor of two diagrams as the composition of whiskerings ′ = (′) ′. Note that because the diagram is in generic form (i.e. each layer contains exactly one box) the definition of tensor is necessarily biased: the diagram on the left hand-side comes above the one on the right-hand side.
A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.
Tensor networks or tensor network states are a class of variational wave functions used in the study of many-body quantum systems [1] and fluids. [ 2 ] [ 3 ] Tensor networks extend one-dimensional matrix product states to higher dimensions while preserving some of their useful mathematical properties.
Vector and tensor polarization of particle/nuclei with spin S ≥ 1 are specified by the unit polarization vector and the polarization tensor P `. Additional tensors composed of products of three or more spin matrices are needed only for the exhaustive description of polarization of particles/nuclei with spin S ≥ 3 ⁄ 2 .
In mathematics, a tensor is a certain kind of geometrical entity and array concept. It generalizes the concepts of scalar , vector and linear operator , in a way that is independent of any chosen frame of reference .
In the tensor calculus formulation, the electromagnetic tensor F αβ is an antisymmetric covariant order 2 tensor; the four-potential, A α, is a covariant vector; the current, J α, is a vector; the square brackets, [ ], denote antisymmetrization of indices; ∂ α is the partial derivative with respect to the coordinate, x α.
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechanics and thermodynamics , it places a heavy emphasis on the commonalities between the topics covered.
In combination with polarizers, it can be used as a shutter or modulator. The values of K depend on the medium and are about 9.4×10 −14 m·V −2 for water, [citation needed] and 4.4×10 −12 m·V −2 for nitrobenzene. [10] For crystals, the susceptibility of the medium will in general be a tensor, and the Kerr effect produces a ...