Dependent geometric interpretation

Singularity of the matrix A occurs when one or more of the eigenvalues are zero, such as occurs if linear dependences exist between the p rows or columns of A. From the geometrical interpretation it can be readily seen that the determinant of a singular matrix must be zero and that under this condition, the volume of the pattern P" has collapsed along one or more dimensions of SP. Applications of eigenvalue decomposition of dispersion matrices are discussed in more detail in Chapter 31 from the perspective of data analysis. 

As can be seen from Eqs. (3.55) to (3.58), depends only on composition, and geometric interpretation for different order models are to be found in the reference literature [12]. Lack of fit of models is checked in each control point by means of Students- test ... 

The geometrical interpretation of the time dependence of a reaction proceeding can be described using the idea of trajectories relaxing onto a series of increasingly lower-dimensional manifolds and eventually to equilibrium. Roussel and Fraser present an interesting picture of the relax-... 

Figure 5.1 The shape domains of local convexity of a MIDCO surface G(a) are shown. A geometrical interpretation of the classification of points r of G(a) into locally concave Dp. locally saddle-type D, and locally convex D2 domains is given when comparing local neighborhoods of the surface to a tangent plane T. Each point r of G(a) is classified into domains Dp. D, and D2 depending on whether at point r a local neighborhood of point r on the tangent plane (r not included) falls within the interior of the surface G(a), or it cuts into the surface G(a) within any small neighborhood of point r, or it falls on the outside of G(a).

The more general case of b >t0 corresponds to a generalization of the concept of convexity [156,199]. This concept of relative local convexity has a u.seful geometrical interpretation. Eor a fixed value of parameter b, relative local convexity classifies the points r of G(a) into domains D x(b), depending whether at point r the surface G(a) is curved more in all directions, more in. some and less in some other directions, or less in all directions, than a test sphere T of radius 1/b. The corresponding three types of domains are denoted by Do(b), D (b), and D2(b), where the molecular contour surface G(a) is locally concave, of the saddle-type, and convex, respectively, relative to curvature b. 

In this chapter we discuss the concept of a mathematical function and its relationship to the behavior of physical variables. We define the derivative of a function of one independent variable and discuss its geometric interpretation. We discuss the use of derivatives in approximate calculations of changes in dependent variables and describe their use in finding minimum and maximum values of functions. 

Having this geometric interpretation in mind it is easy to see that the interconversion between the rotor machine and arbitrary reference-frame variables is obtained using the matrix Afr = M(p(t) - 9 t)) and its inverse, as given in (8.4), where the projection depends on the angle (p — 6t) ... 

In each case, we first studied the laser driven dynamics of the system in the framework of the Floquet formalism, described in Sect. 6.5 of Chap. 6, which provides a geometrical interpretation of the laser driven dynamics and its dependence on the frequency and amplitude of the laser field, through the analysis of the eigenvalues of the Floquet operator, called quasienergies. Various effective models were used for that purpose. This analysis allowed us to explain the shape of the relevant quasienergy curves as a function of the laser parameters, and to obtain the parameters of the laser field that induce the CDT. We then used the MCTDH method to solve the TDSE for the molecule in interaction with the laser field and compare these results with those obtained from the effective Hamiltonian described in Sect. 8.2.3 above. 

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