Derived Tensor Product

Let 7 be a topological space, O a sheaf of commutative rings, and A the abelian category of (sheaves of) O-modules. Recall from (1.5.4) the definition of the tensor product (over O) of two complexes in K(.A), and its A-functorial properties. The standard theory of the derived tensor product, via resolutions by complexes of flat modules, applies to complexes in D (.4), see e.g., [H, p.93], Following Spaltenstein [Sp] we can use direct limits to extend the theory to arbitrary complexes in D(.A). Before defining, in (2.5.7), the derived tensor product, we need to develop an appropriate acyclicity notion, q-flatness.  

Example 2.5.2. P K(.A) is q-flat iff for each point x U, the stalk P is q-flat in K(.Aa ), where Ax is the category of modules over the ring Ox- (In verifying this statement, note that an exact O j-complex Qx is the stalk at x of the exact O-complex Q which associates Qx to those open subsets of U which contain x, and 0 to those which don t.) 

For instance, a complex P which vanishes in all degrees but one (say n) is q-flat if and only if tensoring with the degree n component P is an exact functor in the category of O-modules, i.e., P is a flat O-module, i.e., for each X G P, PF is a flat C j -module. 

Example 2.5.3. Tensoring with a fixed complex Q is a A-functor, and so the exact homology sequence (1.4.5) of a triangle yields that the q-fiat complexes are the objects of a A-subcategory of K( ). 

Example 2.5.4. Since (filtered) direct limits commute with both tensor product and homology, therefore any such limit of q-flat complexes is again q-flat. 

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For fixed A and variable B, Pa B is a A-functor from K to D which takes quasi-isomorphisms to isomorphisms, so by (1.5.1) there results a A-functor from D to D. Hence there is a functor of two variables, called a derived tensor product,... 

A very convenient indirect procedure for the derivation of shape functions in rectangular elements is to use the tensor products of one-dimensional interpolation functions. This can be readily explained considering the four-node rectangular element shown in Figure 2.8. 

The extra terms in the bottom row are a result of nonvanishing unit-vector derivatives. The tensor products of unit vectors (e.g., ezer) are called unit dyads. In matrix form, where the unit vectors (unit dyads) are implied but usually not shown, the velocity-gradient tensor is written as... 

Note that the many-body response functions are in general non-local, implying that the response (i.e. the polarization) at point r depends on the electric field at other locations. This makes sense In a system of interacting molecules the response of molecules at location r arises not only from the field at that location, but also from molecules located elsewhere that were polarized by the field, then affected other molecules by their mutual interactions. Also note that by not stressing the vector forms of and P we have sacrificed notational rigor for relative notational simplicity. In reality the response function is a tensor whose components are derived from the components of the polarization vector, and the tensor product ... / is the corresponding sum over vector components of and tensor components of /. 

To conclude this section on tensor-product QMOM, it is important to highlight that, although it is not necessary, the formulation of the problem in terms of translated (i.e. centered on the mean) and rotated (i.e. with diagonal covariance matrix) internal coordinates can be advantageous. In fact, if a change of variables is implemented so that the distribution is rewritten with respect to its principal coordinates, the calculations for the derivation of the quadrature approximation are simplified. These concepts will be illustrated in Exercise 3.8. A Matlab script implementing a tensor-product QMOM can be found in Section A.3.2 of Appendix A. 

The special significance of this equation derives from the fact that some of the interaction operators considered here contain double tensors, while the general formulae used in the following refer to tensor products of the form X. ... 

In order to achieve continuity of the first derivatives of the approximate solution we use hermite cubics [8] and thus, have to consider two types of basis functions. By tensor product we construct d-dimensional basis functions (2 types). Now, we restrict ourselves on the two dimensional case and get four types of hermite bicubic basis functions as shown in Fig.3.1. Furthermore, we choose the hierarchical approach [2, 9]. Then, on every hierarchy level k = ki- -k2 subspaces Ski,k2 re spanned by the basis functions with supports as indicated in Fig.3.2. Here, ki denotes the hierarchy level in direction x,. Notice that one rectangle depicts the above mentioned four different types of basis functions. Now, the usual full grid space Vs is spanned by the whole set of subspaces that are shown in Fig.3.2. However, the sparse grid space Vs is constructed only by the subspaces below the dotted line in Fig.3.2, i.e. Vn is given by the direct sum... 

Note that we have merely switched the order of the tensor product from that given in the Finger tensor, but as we will see, in general, this switch gives us different results. By a similar derivation, (see Exercise 1.10.4) as given for p, the relative area change, we can... 

Guevara-Garcia A et al. (2011) Pointing the way to the products Comparison of the stress tensor and the second-derivative tensor of the electron draisity. J Chem Phys 134 234106... 

Section II deals with the general formalism of Prigogine and his co-workers. Starting from the Liouville equation, we derive an exact transport equation for the one-particle distribution function of an arbitrary fluid subject to a weak external field. This equation is valid in the so-called "thermodynamic limit , i.e. when the number of particles N —> oo, the volume of the system 2-> oo, with Nj 2 = C finite. As a by-product, we obtain very easily a formulation for the equilibrium pair distribution function of the fluid as well as a general expression for the conductivity tensor. 

When a body undergoes vibrations, the displacements vary with time, so time averages must be taken to derive the mean-square displacements, as we did to obtain the lattice-dynamical expression of Eq. (2.58). If the librational and translational motions are independent, the cross products between the two terms in Eq. (2.69) average to zero, and the elements of the mean-square displacement tensor of atom n, U"j, are given by... 

Thus, the double tensor, defined by (23.52), is a convenient standard quantity for studies of mixed configurations. Let us turn now to the properties of the tensorial products of two-shell operators (23.52). Proceeding in the same way as in derivation of (23.11), we arrive at... 

Equations (23.56)-(23.58) hold for any 71 and j2, but for some specific values of 71 and j2 these relationships are simplified, thereby enabling us to obtain analytical expressions for individual tensorial products of double tensors. It is worth noting that such expressions can only be derived at 71 = 1/2 (j2 is arbitrary). 

Ordinary Raman scattering is determined by derivatives of the electric dipole-electric dipole tensor ae, and ROA by derivatives of cross-products of this tensor with the imaginary part G,e of the electric dipole-magnetic dipole tensor (the optical activity tensor) and the tensor Ae which results from the double contraction of the third rank electric dipole-electric quadrupole tensor Ae with the third rank antisymmetric unit tensor s of Levi-Civita. The electronic property tensors have the form ... 

To derive Eq. (9-44), a decoupling approximation was used so that a fourth rank tensor could be replaced by a product of second rank tensors. 

This contains an TCP of the TpaL tensor, which is derived from the electron spin and dipole-dipole interaction tensor(See equation (11)). Hence, the first question we confront is whether those tensors are correlated or not. In case they are not the total TCP can be decomposed into a product of auto correlations for the the electron spin and dipole-dipole interaction tensor, respectively. In case they are, however, it is necessary to consider the whole TCP and the electron spin has to be correlated with the dipole-dipole interaction tensor. The time dependence in the electron spin tensor can be obtained by integrating the time dependent Schrbdinger equation for the electron spin under the electron spin Hamiltonian. The electron spin is just like the nuclear spin precessing around the external magnetic field and influenced by molecular dynamics. 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Re3molds stress tensor The first and second terms on the RHS denote the production... 

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