Alternate tensor product

For any natural number n, define the alternate tensor product... 

Common examples of pseudo-vectors that will be relevant later include the angular velocity vector f2, the torque T, the vorticity vector co (or the curl of any true vector), and the cross product of two vectors. The inner scaler product of a vector and a pseudo-tensor or a pseudo-vector and a regular tensor will both produce a pseudo-vector. It will also be useful to extend the notion of a pseudo-vector to scalers that are formed as the product of a vector and a pseudo-vector. The third-order, alternating tensor e is a pseudo-tensor of third order as may be verified by reviewing its definition... 

Quantum mechanics can also be formulated in terms of an alternate Hilbert space whose elements are operators, with the density operator p and the typical measurables F among them. A variety of complete sets of basis operators in the space may be constructed, for example, as the tensor product of a basis set of the Hilbert space vectors > and its dual, yielding elements of the form... 

Using the tensor products notation (see Chapter 2) an alternative expression of the PARAFAC model is ... 

In our previous work [96], we have developed an alternative multireference method, block correlated coupled cluster (BCCC) approach, for the electronic structure calculation of systems with noticeable multireference character. In this approach, the reference function is expressed as the tensor product of the most important many-electron states in each block, assuming that all orbitals in a system... 

Alternatively the can be defined on a grid of points. Now let us introduce a second layer, L2, by considering the four logical modes Q21 = Qi,Q2, Qn = 23,24, 223 = 2s,26, 224=127,2s - Taking as an example the 221 particle, we may combine linearly the nine elements of the tensor product of the L functions for the modes 2i and 2z and contract them to, say, three single-particle functions. 

During the last two decades a number of new versions of the Racah algebra or its improvements have been suggested [27]. So, the exploitation of the community of transformation properties of irreducible tensors and wave functions allows one to adopt the notion of irreducible tensorial sets, to deduce new relationships between the quantities considered, to simplify further on the operators already expressed in terms of irreducible tensors, or, in general, to offer a new method of calculating the matrix elements, as an alternative to the standard Racah way. It is based on the utilization of tensorial products of the irreducible operators and wave functions, also considered as irreducible tensors. 

An alternative method for computing the dot product uses the direct metric tensor. The direct metric tensor is a 3 x 3 symmetric matrix written in terms of the six lattice... 

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