Closed-Form Parametrizations

The historically first attempt to parametrize U dates back to 1950 and is due to Foldy and Wouthuysen [609]. It employs an exponential ansatz U = exp(W). Douglas and Kroll advocated the use of the so-called square-root parametriza-tion. 

But one may think of additional closed-form parametrizations for U, namely the Cayley-type form [612], 

Of course, infinitely many other unitary transformations may be constructed, such as 

At this stage it is, however, not clear which of these parametrizations should be preferred over the others for application in decoupling procedures and whether they all yield identical block-diagonal Hamiltonians. Furthermore, the four possibilities given above to parametrize unitary transformations differ obviously in their radius of convergence Rc, which is equal to unity for the square root and the McWeeny form, whereas Rc = 2 for the Cayley parametrization, and Rc = co for the exponential form. 

Within any decoupling scheme there are only a few restrictions on the choice of the transformations U. First, they have to be unitary and analytic (holomorphic) functions on a suitable domain of the one-electron Hilbert space V, since any parametrization has necessarily to be expanded in a Taylor series around W = 0 for the sake of comparability but also for later application in nested decoupling procedures (see chapter 12). Second, they have to permit a decomposition of in even terms of well-defined order in a given expansion parameter of the Hamiltonian (such as 1/c or V). It is thus possible to parametrize U without loss of generality by a power-series ansatz in terms of an antihermitean operator W, where unitarity of the resulting power series is the only constraint. In the next section this most general parametrization of U is discussed. 

---

Figure 12.2 Dependence of higher than 4th-order DKH ground-state energy of a one-eiectron atom with Z=100 on the parametrization of the unitary matrices (data adopted from Ref. [645], c=137.0359895 a.u.). To demonstrate the small effect, three closed-form parametrizations presented in chapter 11 have been chosen. Note the oscillatory convergence behavior of the...

The equations (16)-(21) have closed-form parametric solution. Multiple steady states occur. The relation between the reactor volume Da and the reactor-outlet concentration za.2, obtained for recycle free of P (zp3 = 0), is given by Eq. (22) and presented in Fig 3b. 

The equation (7.36) is too complicated to be useful in establishing closed-form parametric stability boundaries. However, some important special cases can be proven from the above Routh-Hurwitz conditions which are listed here in the form of simple lemmas. 

Every polygon is thus a parametric curve, and so is their limit, the limit curve. To every real value of parameter (and we shall use the letter t to denote the parameter) between 0 and the arity times the number of original sides of the polygon there corresponds a point of the limit curve11. It may not be possible to write a closed form for this function (except in some special cases) and the function may not be differentiable or even continuous, but it is in principle defined at every real value in the domain, as the limit of a sequence of points lying somewhere on consecutive polygons12. 

If a hyperspherical parametrization can be represented by a tree, the coordinates can be shown to form an orthogonal set. This implies that the Laplacian on the hypersphere will contain no cross terms, the corresponding Laplace equations are separable and the hyperspherical harmonics can be constructed in closed form [24]. 

It is possible to consider a stationary distribution of the process I, (if it exists) and to optimize the expected value of an objective function with respect to the stationary distribution. Typically, such a stationary distribution cannot be written in a closed form and is difficult to compute accurately. This introduces additional technical difficulties into the problem. Also, in some situations the probability distribution of the random variables D, is given in a parametric form whose parameters are decision variables. We will discuss dealing with such cases later. 

Once the conditional optimal values for b and are obtained, the value of the goodness-of-flt function can be computed by Equation (2.121) whereas the normalizing constant in the posterior PDF does not affect the parametric identification results. By maximizing the goodness-of-fit function with respect to n, the updated model parameters can be obtained. Therefore, the closed-form solution of the conditional optimal parameters reduces the dimension of the original optimization problem from - - - - 1 to N only. 

The mathematical treatment of joint analysis is to set up a series of differential equations to describe the state of stress and strain in a joint. By using stress functions or other methods, closed-form algebraic solutions may be obtained. In the simplest elastic case it should be possible to devise a solution for given boundary conditions. As non-linearities arise, such as joint rotation and material plasticity, various assumptions need to be made to give solutions. However, once obtained, these solutions may be used to great advantage in a parametric study, provided the limits of the simplifications are borne in mind. The classical early work of Volkersen(15) and of... 

The thermal evaluation of sohd-state devices and integrated circuits (ICs), and VLSI-based packaging takes two forms theoretical analysis and experimental charaaerization. Theoretical analysis utilizes various approaches from simple to complex closed-form analytical solutions and numerical analysis techniques, or a combination of both. Experimental characterization of the device/chip junction/surface temperature (s) of packaged/unpackaged structures takes both direct, infrared microradiometry, or Kquid crystals and thermographic phosphorous, or, to a lesser extent, thermocouples, and indirect (parametric) electrical measurements. 

There is, however, one very special parametrization for the transformation Ho that avoids the expansion of the one-electron operator in any way and is therefore free from convergence issues. This particular parametrization yields operators that can be converted into the closed-form free-particle Foldy-Wouthuysen expression defined by Eqs. (11.27) or (11.35), which produces the closed-form Hamiltonian /i derived in section 11.3. Its expansion... 

Closed-form equations for designing these inductors are not available. However, several parametric equations are known for wire-wound solenoids (Figure 9.29). One equation is provided in Reference 33 of the form ... 

FIGURE 1.4 Time dependent representation of the fitting curves (1.58)-(1.60) for the parametric values k j=kj= lO s kj=10 M" s , [8 =19 , and [EJ=10" M. The dashed line corresponds to equation (1.58) and involves the ff-Lambert closed form solution (1.26). The thin continuous line is the representation of linear equation (1.60) while the thick continuous line is the plot of non-linear equation (1.59) being both based on the logistic solution (1.51). Abscise and ordinate scales are given in arbitrary units (Putz et al., 2007). 

This implies that, at constant k, the line integral of the differential form s de, parametrized by time t, taken over the closed curve h) zero. This is the integrability condition for the existence of a scalar function tj/ e) such that s = d j//de (see, e.g., Courant and John [13], Vol. 2, 1.10). This holds for an elastic closed cycle at any constant values of the internal state variables k. Therefore, in general, there exists a function ij/... 

The phase-space model has been extended by Miller (1970) in order to incorporate the effect of closed channels. He made use of a parametrized form of the S matrix previously developed for compound-state resonances in atom-molecule collisions (Micha, 1967). Indicating with Sd the S-matrix for direct scattering, i.e. in the absence of coupling to closed channels, one can write (omitting the index J). 

The present paper furthers the discussion of these kinSs of circuit models in two major respects. Hie first has to do with the point-to-point variations which may be expected to occur. This has been discussed to some extent by Choudhury and Patterson (19, 24). They provide a very useful parametric method which automatically adjusts the local resistances but it was not presented in a form that is easy to visualize. Hence, a reformulation of their approach will be outlined here which ties it in more closely with the more traditionally accepted "fixed resistor" analogs. The second major point has to do with extending the equivalent circuit models in such a way as to allow for additional mobile ionic species in the electrolyte. ... 

The ACE model is stored in the form ofp pairs of yj,g yj)] and The transformation functions are not in closed or analytical form in contrast to parametric models. The transformation is obtained on the basis of an optimality criterion, according to which the variance of the error e in Eq. (6.122) is minimized with respect to the variance of the transformed variable y. 

To close the system of equations represented by Eq. (15.41) it is necessary to express the right-hand part in terms of moments. To this end, the coagulation kernel should have a special form (for example, the form of a homogeneous polynomial of degrees V and co), or it is necessary to accept that the distribution conforms to a certain class (for example, a logarithmic normal distribution or a gamma distribution). The first method is called the method of fractional moments, and the second one the parametric method. 

---