These spaces are bigraded, with one grading denoting the length of a tensor product, and the other denoting the total degree.
where Curl denotes the curl of a tensor field with respect to the material/reference coordinate.4 For a given normal n in the lattice configuration, the vector G T n measures the net Burgers vector of dislocation lines per unit area passing through a plane of normal n.
For convenience, the coordinates are usually numbered from 1 to n, and each component of a tensor is denoted by a letter having superscripts and subscripts, each of which independently takes on the values 1 to n.
The symmetric part of a tensor is denoted by sym.
An entry of a vector a, a matrix A, or a tensor is denoted by a i, ai,j, ai,j,k, etc., depending on the number of modes.
If we need more than one quantum bit to perform any calculations, we can use a quantum register which is a system of qubits, joined by a tensor product (denoted by symbol (otimes )) in a mathematical sense.
By convention (A,B,C .Λ denotes the tensor of rank R of coordinates ∑ r = 1 R λ r A ir B jr C kr. Minimizing error (10) means finding the best rank-R approximate of and its CP decomposition.
where ∇ H α ( H ) = ∇ H + α ( H ) - ∇ H - α ( H ), both ⊙ and / denote element-wise calculations, A∘B denotes J×K×P tensor with elements a j p b k p when A and B are J×P and K×P[27], and 〈A,B〉{C},{D} denotes a contracted product [15].
Thus, CBR is widely accepted as the form of a = F·A, where F denotes a two-point deformation gradient tensor, A and a denote one lattice vector on the respective undeformed and deformed crystals.
