Separated variables approximation

Note that the function 103 has the form of the zeroth-order separated variables approximation (SVA). ... 

In such a case, the over2dl rotation variables, which may be expressed as rotational angles around some orthogonal axes, and the internal motion variables, which may be written as internal coordinates, are completely separable. Because of this separability, this approximate full Hamiltonian operator may be regarded as local... 

The stationary motions in a weak electric field are, however, essentially different from those in a spherically symmetrical field differing only slightly from a Coulomb field. In the latter (for which the separation variables are polar co-ordinates) the path is plane it is an ellipse with a slow rotation of the perihelion. In the former (separable in parabolic co-ordinates) it is likewise approximately an ellipse, but this ellipse performs a complicated motion in space. If then, in the limiting case of a pure Coulomb field, k or nf be introduced as second quantum number, altogether different motions would be obtained in the two cases. The degenerate action variable has therefore no significance for the quantisation. 

In multiple shooting, the integration horizon is divided into time intervals, with the control variables approximated by polynomials in each control interval and differential variables assigned initial values at the beginning of each interval. The DAE system is solved separately within each control interval, as shown in Fignre 14.2b. Profiles for partial derivatives with respect to the optimization variables, as well as the initial conditions of the state variables in each time interval, are obtained through integration of the sensitivity eqnations. These state and sensitivity profiles are solved independently over each time interval and can even be computed in parallel. Additional equations are inclnded in the NLP to enforce continuity of state variables at the time interval boundaries. 

The variable approximation parameter values (0d/ ei/ e2/ e3/ arid as functions of the molar volumes Vrf3 calculated from the X-ray data (ICDD Database, 1993) form fairly accurate linear dependences (see Figure 7) for two series of RF3 (R = La-Pm, and R = Sm-Lu). The separation into two series is caused by the formation of different polymorphs by these compounds. Under standard conditions, lanthanide trifluorides from lanthanum to promethium inclusive have LaFs-type structures, in which the CN of R is 6. All the other RF3 are characterized p-YF3-type crystal structures with the CN = 9. 

For small temperature ranges or approximate calculations, we can assume A/i independent of temperature. If we separate variables in (Equation 9.18), we get ... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. 

Power Supplie.s Iligh-voltage ac and dc power supplies for electrostatic separators are iisiiallv of solid-state construction and feature variable outputs ranging from 0 to 30,()()() for ac wiper transformers to 0 to 60,000 for the dc supply The maximum current requirement is approximately 1,0 to 1,5 rnA/rn of electrode length. Powder supplies for industrial separators are typically oil-insulated, but smaller diw-epoxv-insulated supplies are also available. 

The choice of variables remaining with the operator, as stated before, is restricted and is usually confined to the selection of the phase system. Preliminary experiments must be carried out to identify the best phase system to be used for the particular analysis under consideration. The best phase system will be that which provides the greatest separation ratio for the critical pair of solutes and, at the same time, ensures a minimum value for the capacity factor of the last eluted solute. Unfortunately, at this time, theories that predict the optimum solvent system that will effect a particular separation are largely empirical and those that are available can be very approximate, to say the least. Nevertheless, there are commercially available experimental routines that help in the selection of the best phase system for LC analyses, the results from which can be evaluated by supporting computer software. The program may then suggest further routines based on the initial results and, by an iterative procedure, eventually provides an optimum phase system as defined by the computer software. 

A prominent part of many of the techniques is separation of variables. In that method, the deflection variables, or the variation In deflection variables, are arbitrarily separated into functions of plate coordinate x alone times functions of y alone. Wang [5-8] determined that separation of variables leads to exact solutions for some classes of plate problems, but does not for others, I.e., the deflections are not always separable. A specific example of an approximate use of separation of variables due to Ashton [5-9] will be discussed in Section 5.3.2. Other exact uses of the method abound throughout Section 5.3 through 5.5. 

Lagrangian as a functional -(v,v). Note however that unlike functionals used in the Time-dependent Hartree Fock approximation (14), this Lagrangian is not complex analytic in the variables (v,v) separately. 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

To a first approximation the three terms in equation (1.46) and (1.47) can be treated as independent variables. For a fixed value of n Figure 1.8 Indicates the influence of the separation factor and capacity factor on the observed resolution, when the separation factor equals 1.0 there is no possibility of any separation. The separation factor is a function of the distribution coefficients of the solutes, that is the thermodynamic properties of the system, and without some... 

Hence, a series of measurements with several Tcp values will provide a data set with variable decays due to both diffusion and relaxation. Numerical inversion can be applied to such data set to obtain the diffusion-relaxation correlation spectrum [44— 46]. However, this type of experiment is different from the 2D experiments, such as T,-T2. For example, the diffusion and relaxation effects are mixed and not separated as in the PFG-CPMG experiment Eq. (2.7.6). Furthermore, as the diffusion decay of CPMG is not a single exponential in a constant field gradient [41, 42], the above kernel is only an approximation. It is possible that the diffusion resolution may be compromised. 

---