In this paper, we look at the symmetric outer product decomposition (SOPD), a summation of rank-one fully (partially) symmetric tensors.
The outer product of the vectors x, y, z [member of] [R.
The outer product of three nonzero vectors is a third-order rank-one tensor; the outer product of k nonzero vectors is a kth-order rank-one tensor.
A kth-order tensor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called rank-one if it can be written as an outer product of k vectors, i.
A kth-order rank-one tensor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is partially symmetric if it can be written as an outer product of k vectors and if there exist modes l and m such that [a.
xI] is symmetric if it can be written as an outer product of k identical vectors, i.
Their outer product
R(4) describes Minkowskian space-time, whose inner products form bivectors, pseudo-vectors, and pseudo-scalars, describing spin, momentum-energy, and action, respectively, which automatically satisfy the Heisenberg Uncertainty Principle.
Using the standard modules of the library of CAS REDUCE (for example, a matrices product subprogram, a matrices determinant subprogram, a matrix inversion subprogram) and several nonstandard algorithms as the matrices elements cofactor, a matrices determinant cross-expansion and a matrices outer product we deduced the symbolic expressions of coefficients of the geometric properties, material matrices and global stiffness matrices of the following types of the finite elements:
The original approach in all procedures was that the outer product of a shape functions matrices was used on the element stiffness matrix as well as the full anisotropy and polynomial representation of elasticity constants in the material matrix.
5 [micro]L outer product
and NP- 1 as 1 primer and NP1 s2 (5'-CTCACCTGCGAGCTCTGTAAGTA) primer was performed at an annealing temperature of 55[degrees]C.
The coextruder is a single-screw machine, which, in accordance with the requirements of each application, plastifies the virgin PET necessary for the outer product
The outer product
inside the cans is over-processed due to slow heating of the center, causing poor quality and taste.