Elementary tensors

Elementary tensors are also known as decomposable tensors. 

Now recall the definition of an elementary tensor (Definition 2.15). It tons out that elementary tensors always correspond to unentangled states, and vice versa. 

Proof. First, consider an elementary tensor, i.e., a vector of the form x =... 

We may assume, without loss of generality, that every one of the vector spaces Vk is finite dimensional, due to the following argument. Because x is an element of the tensor product, it must be a finite sum of elementary tensors ... 

DC left-hand plots) and pure explosive right-hand plots) sources contaminated by random noise and displayed in three different source-type plots, (a, b) The diamond CLVD-ISO plot, (c, d) the Hudson s skewed diamond plot, and (e, f) the Riedesel-Jordan plot. The dots show 1,000 DC sources defined by the elementary tensor Edc (see Eq. (12)) and contaminated by noise with a uniform distribution from —0.25 to 4-0.25. The noise is superimposed to all tensor components... 

To provide an elementary treatment, in this seetion the theory is eon-strueted in terms of the elassieal small strain tensor s defined as the symmetrie... 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

A little elementary acoustics components of the strain tensor are... 

C ). (Some readers may already know that spin-1 spin states are described by vectors in C others might see Section 10.4.) We will use tensor products in Proposition 7.7, our mathematical description of the elementary states of the... 

Next we suppose that a state [x] is not entangled and show that it is elementary. We proceed by induction on n, the number of factors in the tensor product. The base case is trivial if = 1, then any x e Vi is elementary. 

We will exploit the vector space isomorphism between the scalar product space Vo and the scalar product space Hom(V,, Vjf), introduced in Proposition 5.14 and Exercise 5.22. (Note that (V ) = V hy Exercise 2.15.) Instead of working directly with x 7 0, we will work with the corresponding linear transformation X 0. We will show that X Vq Etc has rank one and that its image is generated by an elementary element of the tensor product Ei 0 0 E . Then we will deduce that x itself is elementary in the tensor product Eo El 0 0 E . 

Because the rank of X is one, Exercise 5.14 implies that x is elementary in the tensor product Vb 0 Vx- In other words, there are vectors xq e Vb and xk e Vk such that x = xq 0 xk- It remains to show that xk is elementary. 

The spacing of the different energy levels studied by NQR is due to the interaction of the nuclear quadrupole moment and the electric field gradient at the site of the nucleus considered. Usually the electric quadrupole moment of the nucleus is written eQ, where e is the elementary charge Q has the dimension of an area and is of the order of 10 24 cm2. More exactly, the electric quadrupole moment of the nucleus is described by a second order tensor. However, because of its symmetry and the validity of the Laplace equation, the scalar quantity eQ is sufficient to describe this tensor. 

For application, Eq. (II.4) has to be considerably simplified. The simplest model is the assumption of point charges. We take the crystal lattice as composed from the point charges nte, where e is the elementary charge. The index i, o < i k, distinguishes the different kinds of charged points (particles) within the lattice. Assuming the system of crystal axes already transformed to the principal axes system of the tensor, we calculate the coupling constant from... 

Nuclear quadrupole moments are conventionally defined in a different way from molecular quadrupole moments. Q is an area and e is the elementary charge. eQ is taken to be the maximum expectation value of the zz tensor element. The values of Q for some nuclei are listed in table 6.3. 

Pq, is expressed in Cartesian coordinates. These polar tensors T), can be derived from experimental intensities by elementary coordinate transformation. If the axes x, y, and z are chosen such that the bonds are oriented along one of the axes, then the derivatives can be used to interpret the changes of the electron clouds during a vibration. Besides, considering the definitions of the axes, it is possible to transfer atomic polar tensors between similar molecules and to estimate their intensities (Person and Newton, 1974 Person and Overend, 1977). 

The starting approach will be the elasticity theory (3). In the elementary theory of beams, the only component of the stress tensor differing from zero is Gxx = Ey/R, which, according to the theory developed for the elastic case, can be written as... 

The electromagnetic Green s tensors Ge, G// are introduced as the fields of an elementary electric source (Zhdanov, 1988 Felsen and Marcuvitz, 1994). They satisfy the Maxwell s equations... 

According to the linearity of the wave equation, the vector field of an arbitrary source can be represented as the sum of elementary fields generated by the point pulse sources. However, the polarization (i.e., direction) of the vector field does not coincide with the polarization of the source, F . For instance, the elastic displacement field generated by an external force directed along axis x may have nonzero components along all three coordinate axes. That is why in the vector case not just one scalar but three vector functions are required. The combination of those vector functions forms a tensor object G" (r, t), which we call the Green s tensor of the vector wave equation. 

The complexity of these formulae, especially of the second, is the justification of the remark made in section 3A that they could hardly be obtained by elementary manipulations and they give a good demonstration of the great power of the irreducible tensor method. 

The Cauchy stress principle arises through consideration of the equilibrium of body forces and surface tractions in the special case of the infinitesimal tetrahedral volume shown in fig. 2.7. Three faces of the tetrahedron are perpendicular to the Cartesian axes while the fourth face is characterized by a normal n. The idea is to insist on the equilibrium of this elementary volume, which results in the observation that the traction vector on an arbitrary plane with normal n (such as is shown in the figure) can be determined once the traction vectors on the Cartesian planes are known. In particular, it is found that = crn, where a is known as the stress tensor, and carries the information about the traction vectors associated with the Cartesian planes. The simple outcome of this argument is the claim that... 

---