Direct linear transformation

All contributions to the molecular derivatives involving the higher electronic derivatives [Eqs. (59) and (60)] may be treated as direct linear transformations and calculated in terms of inactive Fock matrices containing multiply one-index transformed integrals. For example, the first anharmonic-ity contains the term... 

Abdel-Aziz, Y. I., and Karara, H. M. (1971). Direct linear transformation from comparator coordinates into object-space coordinates, Close-range photogrammetry. American Society of Photogrammetry, Falls Church, Va. 

Instead of a theoretical model that requires careful measurements of distances, angles, and so on, an experimental calibration approach is preferred. The experimental calibration estimates the model parameters based on the images of a calibration target as recorded by each camera. A linear imaging model that works well for most cases, the pinhole camera model, is based on geometrical optics. This leads to the following direct linear transform equations, where x,y are image coordinates, and X,Y,Z are object coordinates. This physics-based model cannot describe nonlinear phenomena such as lens distortions. 

Tbe linear transformation expressed by Eq. (104) has the same form if the vector is rotated in the clockwise direction by the angle

coordinate axes remain fixed. 

The direct linear plot (O Figure 4-6) requires no transformation of data, and this approach is arguably the most appropriate for determining and values. Individual values of [S] are plotted as their negatives on the x-axis and measured values of v are plotted on the axis. Corresponding pairs of ( — [S], 0)... 

Because the successive application of two linear transformations is itself a linear transformation, it must also correspond to a matrix. The matrix corresponding to a compound transformation can be computed directly from the matrices corresponding to the individual transformations. Let us derive the formulas for simphcity we shall work in two dimensions. Given two hnear transformations ... 

Diffuse reflectance R is a function of the ratio K/S and proportional to the addition of the absorbing species in the reflecting sample medium. In NIR practice, absolute reflectance R is replaced by the ratio of the intensity of radiation reflected from the sample and the intensity of that reflected from a reference material, that is, a ceramic disk. Thus, R depends on the analyte concentration. The assumption that the diffuse reflectance of an incident beam of radiation is directly proportional to the quantity of absorbing species interacting with the incident beam is based on these relationships. Like Beer s law, the Kubelka-Munk equation is limited to weak absorptions, such as those observed in the NIR range. However, in practice there is no need to assume a linear relationship between NIRS data and the constituent concentration, as data transformations or pretreatments are used to linearize the reflectance data. The most used linear transforms include log HR and Kubelka-Munk as mathemati-... 

It is necessary, for some further discussion of this model presented in Section VI E, that the inequality signs be directed as shown and that Qi,. . . , Qm be greater than or equal to zero. This requirement can always be met by making linear transformations of the qualities if needed. 

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 . 

According to the dressed oscillator model, the normal modes describing the dissociative state are assumed to be part of the set of normal modes for the initial bound state. However, the initial and final states (G and D for direct photodissociation, or Q and D for indirect photodissociation) are each characterized by their own set of normal modes, that are related to each other by a linear transformation (2,40). 

Since a CASSCF calculation is faster than a direct SC calculation, owing to the advantages associated with orbital orthogonality in CASSCF, it is practical to extract an approximate SC wave function (or another type of VB function, e.g., a multiconfigurational one) from a CASSCF wave function. The conversion from one wave function to the other relies on the fact that a CASSCF wave function is invariant under linear transformations of the active orbitals. Based on this invariance principle, two different procedures were developed and both share the same name CASVB . Thus, CASVB is not a straightforward VB method, but rather a projection method that bridges between CASSCF and VB wave functions. 

Let us assume that the coulombic branch is resolved, for K = 0 and for a given direction K, by the diagonalization of (1.70). This means that we know, for each direction K, the eigenenergies a>e(K) for each excitonic mode, as well as the eigenvectors e>, which are linear functions by the transformation (1.32), (1.51) of the creation and annihilation operators of molecular states. Furthermore, let us define the excitonic dipolar moment by the same linear transformation on the molecular dipoles,... 

The solution of Eq. (23) is thus carried out by forming the linear transformations of trial vectors directly without constructing or explicitly,... 

The third term in Eq. (69) should be calculated carrying out the linear transformation G(1)A(1). [The reason this and similar terms discussed later should be calculated using linear transformations is that we only need to know the projection of G(1) onto the solution vectors A(1), and not G(1) itself. Constructing G(1) explicitly constitutes a much heavier task than carrying out linear transformations in a direct fashion.] The transformation Gll,/U) is identical to GW)b described in Eqs. (94)—(97), except that the /<0) integrals are replaced by /(1). The linear transformation requires /(1) and S(1), /<0) with two general and two active indices. The requirements on the integrals are therefore the same as in the calculation of E° ... 

The last contribution to fF(3), 3G<1)P<1)P<1), is best calculated by first carrying out the linear transformations GU)P(I) in a direct fashion, requiring first derivative integrals in the MO basis. 

The other type of linear transformation, to which equations (4) and (5) belong, is characterised because those transformations neutralise or mask the energetic heterogeneity of the adsorption data when C takes high values. Under these conditions, the regression is linear and can be used to obtain the value of the constant. Of both equations, (4) and (5), the second one permits a simpler and more precise determination of the constant since it is calculated directly from the slope. The energetic heterogeneity of adsorption is reflected in the value of the C parameter. Its values can be obtained directly from the BET equation once the value of Hm is known. 

A more general approach would be to directly consider the elastic energy associated by deforming cell A into cell B so as to achieve coherency. If this deformation is designated by the 2x2 linear transformation matrix T... 

It was supposed, that each of the phases is characterized by two parameters the ohmic conductivity <7o(r) and the Hall factor p(r). However each of properties CTo(r) and p(r) from the conductivity tensor (285) admit of only two values Co = cii and p [5, in the first phase, cto = ct2 and p = p2 in the second phase. The essence of ideas described in [118,119] consists in linear transformations from the old fields (j,E) to new fields (j,E ) such that the macroscopic properties of the new system are equivalent to those of the original system. These transformations can be applied only to a two-dimensional system, since they do not then change the laws governing a direct current ... 

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