Vector equality - pseudo-vectors

We shall subsequently see that the second and third requirements play a critical role in the 

Because det L can have only two values for an orthogonal transformation, +1 or —1, it follows that the difference between pseudo-vectors or pseudo-tensors and normal vectors and 

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 

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Symmetry properties of gap states and Ginzburg-Landau theory Triplet pair states are characterised by the vector gap function d(k) defined in sect. 2. Due to the constraint of equal (pseudo-)spin pairing (i.e., = 0) the d-vector is confined to the... 

The unique properties of liquid crystals have also provided opportunity for study of novel nonlinear optical processes. An example involves the ability to modify the pitch of cholesteric liquid crystals. Because a pseudo-wave vector may be associated with the period of pitch, a number of interesting Umklapp type phasematching processes (processes in which wave vector conservation is relaxed to allow the vector addition to equal some combination of the material pseudo-wave vectors rather than zero) are possible in these pseudo-one-dimensional media. Shen and coworkers have investigated these employing optical third harmonic generation (5.) and four-wavemixing (6). 

The dye molecules are positioned at sites along the linear channels. The length of a site is equal to a number ns times the length of c, so that one dye molecule fits into one site. Thus ns is the number of unit cells that form a site we name the ns-site. The parameter ns depends on the size of the dye molecules and on the length of the primitive unit cell. As an example, a dye with a length of 1.5 nm in zeolite L requires two primitive unit cells, therefore ns = 2 and the sites are called 2-site. The sites form a new (pseudo) Bravais lattice with the primitive vectors a, b, and ns c in favorable cases. 

Having obtained the solution vector x of the pseudo-primal problem, we relax the set of equality constraints of the primal problem h(x) = 0 to the form... 

These properties follow from the fact that and IJ are true vectors, whereas jf and to are pseudovectors. And the entities appearing on opposite sides of an equality sign must be of the same tensorial type, either pseudo or true. 

Let us now consider Gibbs triple product, which is a scalar equal to the signed volume of the parallelopiped spanned by the three vectors. Using the fact that pseudo-scalars commute with everything in the algebra ... 

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