Compact matrices

Figure 8.7 Numerical simulation of the stretching of four randomly placed compact matrices of particles into a square channel for the free helix geometry operated in extrusion mode a) initial position of the matrices atz = 0, b) end view of the particles at 100 s of rotation, and c) three-dimensional view of the particles after 100 s of rotation...

The integral-driven procedure indicated above is practicable only if the elements of the two-particle density matrix can be rapidly accessed. In the closed-shell Hartree-Fock case, the two-particle density matrix can be easily constructed from the one-particle density. The situation is similar for open-shell and small multiconfigurational SCF wavefunctions the two-particle density matrix can be built up from a few compact matrices. In most open-shell Hartree Fock theories (Roothaan, 1960), the energy expression (Eq. (23))... 

The unrestricted Hartree-Fock (UHF) case is completely analogous to the closed-shell one. New terms do appear in the open-shell SCF and the few-configuration case. Nevertheless, the preferred technique is quite similar to the closed-shell case. In particular, the two-particle density matrix can be constructed from compact matrices, and the solution of the derivative Cl equations is very simple, due to the small dimension. 

Inserting the fc-frame expansit)us (G.8) of the correlation functions into the RSOZ Eqs. (7.41a)-(7.42), the angular integral buried in the products /j > fs can be easily performed by using the orthogonality of the spherical harmonics [258] [cf. Eq. (F.39)]. Introducing, for compactness, matrices with elements... 

Greater bit design freedom is generally available with matrix body bits because they are cast in a moldlike natural diamond bits. Thus, matrix body bits typically have more complex profiles and incorporate cast nozzles and waterways. In addition to the advantages of bit face configuration and erosion resistance with matrix body bits, diamond compact matrix bits often utilize natural... 

The identity of several rows and columns in the matrix in Eq. (4.3) shows that the same information is contained in a more compact matrix. This matrix can be written with the four distinguishable states on the 2nnd lattice indexing the rows and columns, in the order in which they are listed in Table 4.1 [153] ... 

SPME/GC/MS is an efficient technique to reveal the presence of resinic substances in archaeological samples. Indeed, volatile terpenes are still present in very old archaeological samples (4000 years old), particularly in the case of compact matrixes, and can be trapped by the SPME fibre. In comparison with methylene chloride extraction, SPME is very specific and allows the direct analysis of the volatile terpenes content in complex mixtures including oils, fats or waxes. For this reason, headspace SPME is the first method to use when analysing an archaeological sample it will either allow the identification of the resin or indicate further sample treatment in order to detect characteristic triterpenes. The method is not really nondestructive because it uses a little of the sample but the same sample can be used for several SPME extractions and then for other chemical treatments. 

Let us define /ad(R>,) as an -dimensional nuclear motion column vector, whose components are xad(R>,) through x d(Rx)- The w-electronic-state nuclear motion Schrodinger equation satisfied by xad(R>,) can be obtained by inserting Eqs. (12) and (14) into Eq. (13) and using Eqs. (7)-(10). The resulting Schrodinger equation can be expressed in compact matrix form as [26]... 

Geometrically, this means sectioning the sensor space with straight lines, each bisecting the space. The result is a partitioning of the space into volumes, each defining one class. Considering a set P of experimental data, the previous set of equations can be written in a compact matrix form as ... 

The four expressions are usually combined into a more compact matrix equation... 

The approximation of Eq. (Bl) allows one to reduce Eqs. (A10) and (A11) to a linearized boundary-value problem (183,184,186). The latter can then be solved analytically and yields a compact matrix-form solution for the concentration profiles in the film region [58], Such a solution gives simple analytical expressions for the component fluxes with regard to the homogeneous reaction in the fluid films (see Ref. 135), which can be of particular value when large industrial reactive separation units are considered and designed. 

The expansion (1.12) of H may be written in a compact matrix form using a scalar product in a fictitious space where Bx and Bfa are components of vectors, with the notation... 

It is possible to interconnect the matrices J and Qk through a compact matrix equation [48]. This follows from the tridiagonal structure of the matrix in Eq. (60), which allows one to deduce Eq. (78) directly from the product JQt at the set of the eigenvalues uk. By the same reasoning, this conclusion can also be extended to encompass Eq. (84) at values of u other than uk. Thus, the polynomials Q ( ) satisfy the matrix equation ... 

This completes the derivation of the derivatives needed for the gradient of the energy functional. The compact matrix forms of these results can be manipulated using matrix algebra and are readily implemented using optimized computer subroutine libraries. 

Optimal linear coefficients can be computed using the compact matrix form of equation (4.4), supposing both sides equal ... 

Kirkwood and Buff [15] obtained expressions for those quantities in compact matrix forms. For binary mixtures, Kirkwood and Buff provided the results listed in Appendix A. Starting from the matrix form and employing the algebraic software Mathematica [16], analytical expressions for the partial molar volumes, the isothermal compressibility and the derivatives of the chemical potentials for ternary mixtures were obtained by us. They are listed in Appendix B together with the expressions at infinite dilution for the partial molar volumes and isothermal compressibility. 

Expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility were derived by Kirkwood and BufP in compact matrix forms (see Appendix 1). The derivation of explicit expressions for the above quantities in multicomponent mixtures required an enormous number of algebraic transformations, which could be carried out by using a special algebraic software (Maple 8 was used in the present paper). A full set of expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibilities in a quaternary mixture were derived. However, our main interest in this paper is related to the derivatives of the activity coefficient with respect to the mole fractions (all of the expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility can be obtained from the authors at request), namely, the derivatives of the form (9 In where Xg, Xg,... 

Kirkwood and BufP expressed the concentration derivatives of the chemical potentials, the partial molar volumes, and the isothermal compressibility in compact matrix forms as follows ... 

In order to obtain the residual in a compact matrix form, we define the following integral matrices ... 

Is is convenient to use a more compact matrix notation in place of Eq. (IV. 14). For this purpose we introduce a column matrix 0 corresponding to a generalized velocity vector, 0. In the fourth, fifth and sixth position of 0 we still use the angular velocities n>a, mb, me and we will continue to use them as long as possible. For later reference the relation between 0 and the corresponding column matrix... 

Arranging these moments into column vectors n and nq leads to the more compact matrix notation given by [for printing purposes, Eqs. (IV.24) and (IV.24 ) show the transposed vectors] ... 

This chapter provides a tutorial on the fundamental concept of Parallel factor (PARAFAC) analysis and a practical example of its application. PARAFAC, which attains clarity and simplicity in sorting out convoluted information of highly complex chemical systems, is a powerful and versatile tool for the detailed analysis of multi-way data, which is a dataset represented as a multidimensional array. Its intriguing idea to condense the essence of the information present in the multi-way data into a very compact matrix representation referred to as scores and loadings has gained considerable popularity among scientists in many different areas of research activities. 

Figure 4.2-6 shows the calculated temperature coefficient of reactivity for the BOC-IC condition. Curve A is the fuel prompt doppler coefficient due to heatup of the fuel compact matrix as a function of the assumed fuel temperature. Curve B is the active core isothermal temperature coefficient and is the Siam of the doppler coefficient and the moderator temperature coefficient of reactivity which is also strongly negative, due in large measure to the presence of LBP in the BOC condition. The moderator coefficient, not shown in Figure 4.2-6, would be the difference between Curve B and Curve A and would be -4.0 x 10" / C at 800 C (1472 F), for example. Curve C is the total reactor isothermal coefficient and includes the positive contribution of the reflector heatup to the estimated inner and outer reflector temperatures that would result when the fuel reaches the indicated temperature. 

Let us consider this linear equation written in compact matrix notation... 

The eigenproblem of Section B.7 is useful in the solution of coupled linear differential equations. Let these equations be represented by the following set written in compact matrix notation... 

In this new molecular new reference frame, the MOZ equation becomes a compact matrix equation (Kusalik and Patey 1988),... 

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