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With this method, we only apply the general tensor sketch method to order 2 tensors, which avoids the exponential dependency in the number of rows. It can be proved [ 15 ] that combining c {\displaystyle c} dimensionality reductions like this only increases ε {\displaystyle \varepsilon } by a factor c {\displaystyle {\sqrt {c}}} .
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces.
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T • (V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product.It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property ...
These are the coefficients of the characteristic polynomial of the deviator (() /), such that it is traceless. The separation of a tensor into a component that is a multiple of the identity and a traceless component is standard in hydrodynamics, where the former is called isotropic, providing the modified pressure, and the latter is called ...
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
In this sense, the unit dyadic ij is the function from 3-space to itself sending a 1 i + a 2 j + a 3 k to a 2 i, and jj sends this sum to a 2 j. Now it is revealed in what (precise) sense ii + jj + kk is the identity: it sends a 1 i + a 2 j + a 3 k to itself because its effect is to sum each unit vector in the standard basis scaled by the ...
In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite).
In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature.These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors formed from these by the operations of taking dual contractions and covariant differentiations.