Bilinear functionals

Fig. 4.6. Control points and Q nodes used to reconstruct A. The Qi basis function is a bilinear function equal to 1 at a Q node and zero at the surrounding Qi nodes. Four control points per cell were used. The derivatives of A were evaluated at each control point using a linear interpolation based on the neighboring data points...

If the linear relations are valid (Eq. 96), II is a bilinear function in the A s. One implication of the positive definiteness of II is that all the pure transport terms La cannot be negative. It can be shown that in... 

In this sense, electronic spin-spin contact integrals can be computed as simple bilinear functions of overlap integrals, and electron repulsion integrals are expressible as bilinear fimctions of some sort of two electron nuclear attraction integrals. 

The static reactivity can be expressed as a ratio of bilinear functionals ... 

The development and application of generalized perturbation theory (GPT) has made considerable progress since its introduction by Usachev (i(S). Usachev developed GPT for a ratio of linear flux functionals in critical systems. Gandini 39) extended GPT to the ratio of linear adjoint functionals and of bilinear functionals in critical systems. Recently, Stacey (40) further extended GPT to ratios of linear flux functionals, linear adjoint functionals, and bilinear functional in source-driven systems. A comprehensive review of GPT for the three types of ratios in systems described by the homogeneous and the inhomogeneous Boltzmann equations is given in the book by Stacey (41). In the present review we formulate GPT for composite functionals. These functionals include the three types of ratios mentioned above as special cases. The result is a unified GPT formulation for each type of system. 

The second-order perturbation expressions that Stacey derived for ratios of linear and bilinear functionals are not unique to the variational approach. In this section these expressions were obtained with a straightforward perturbation approach (50). Moreover, Seki (63) has shown that the Usachev-Gandini approach can be used to derive a GPT expression for the static reactivity for perturbations that do not retain criticality. This does not detract from the value of the variational approach. The latter provides stationary functionals for all kinds of integral parameters from which one can obtain exact expressions for these parameters [e.g., Eq. (140) for the static reactivity] as well as a consistent derivation of the equations for the distribution functions and of constraints imposed on them. 

Generalized-function formulations of GPT for homogeneous systems are the source of sensitivity functions for different integral parameters Equation (189) for reactivity worths, and Eq. (162) for ratios of linear and bilinear functionals. The first-order perturbation theory expression for reactivity [Eq. (132)] can also be used for sensitivity studies. 

The purpose in using bilinear functionals for many applications is to improve the accuracy of the calculation relative to the accuracy that can be achieved with linear functionals. Bilinear functionals are used, for example, to construct stationary functionals for some integral parameter in which first-order errors in the value of the distribution functions cause only second-order errors in the parameter. There may be problems in which second-order errors in the value of the stationary functional (due to the neglect of spectral effects), exceed first-order errors in the value of the same parameter if calculated from a linear functional formulation. In other words, spectral effects may impair the advantage of bilinear formulations. 

The paramount fundamental problem to resolve is that of the central worth discrepancy. It is important not only for its academic interest or for reassuring the reactor physicist in the adequacy of the calculational methods and nuclear data he u.ses, but also for practical reasons. Safety factors that account for the CWD must be included in the design of fast reactors an economic penalty is associated with these safety factors. Related to the question of the CWD is the more general problem of the multigroup calculation of bilinear functionals. The questions arising here are (I) for which applications, and under what conditions, are spectral fine structure effects non-negligible (2) how can these fine structure effects be taken into account in the multigroup formulation. 

In equilibriim thermodynamics the energy of a system may be considered to be a homogeneous bilinear function of pairs of intensive and extensive variables, either of which can be considered as the independent variable. For example, either pressure or volume may be considered as an independent variable depending upon the environment. The difference between the heat capacity at constant volume and at constant pressure is well known in equilibrium thermodynamics. Thus, in a single component equilibrium system where temperature. 

The resulting relationships may be described with polynominal functions (simplest case parabolic) or bilinear functions, depending on the data set and the computing efforts (Figure 3.5). 

The term a gives the slope of the left-hand ascending side of the curve and (a - b) that of the right-hand descending side. The non-linear parameter jS, which must be estimated by a stepwise iteration procedure, relates to the volume ratio of the aqueous and lipid phases in the system. Setting jS = 1 and b 2a produces the original McFarland model. Kubiny s bilinear model can be derived from kinetically controlled model systems as well as from equilibrium models, indicating that it is valid under diffusion control as well as under equilibrium or pseudo-equilibrium conditions. For many data sets, the bilinear function aptly fits the experimental observations. Difficulties in calculations may arise from unbalanced data sets, which often occur in environ-... 

Figure 3.6 Example of a non-linear log PQ -dependent model (Veith, Call and Brooke, 1983) in comparison to experimental data on fish toxicity (log I/LC50 in mmol/1). The bilinear function provides a good description of the activity data for the compounds with log < log Pow(max.) whereas the model is less well defined for the highly lipophilic substances because their data are extremely variable.

The bilinear curve resumes a linearly increasing part between log P 0 and 6, where the empirically postulated coincidence of log P and log BCF is reflected by a near-unity slope (0.99) for the first-order log P term and the intercept of about 0. Maximum log BCF values of approximately 7 are obtained for compounds with log P between 7 and 8. Compounds that are more lipophilic are expected to be less accumulating, which corresponds to the negative slope derived for the second log P term of the bilinear function. If restricted to compounds with log F w < the equation can be simplified thus ... 

There exist circumstances that sometimes facilitate the search. In fact the Hamiltonians appearing in mechanics and physics are usually quadratic (bilinear) functions. Therefore one should first examine the linear subspace of quadratic functions contained in a commutative Lie algebra V. As a rule, this examination is rather simple, and we will demonstrate it on concrete examples. 

In a bilinear transformation, the variable s in Ha (s) is replaced with a bilinear function of z to obtain H (z). Bilinear transformations for the four standard types of filters, namely, low-pass filter (LPF), high-pass filter (HPF), bandpass filter (BPF), and bandstop filter (BSF), are shown in Table 8.8. The second column in the table gives the relations between the variables s and z. The value of T can be chosen arbitrarily without affecting the resulting design. The third column shows the relations between the analog... 

If N is finite and x(t) is constant within each interval, then the associated Wiener integral of some functional ajx(t)j = a(x, .., Xj ) looks like an ordinary multiple integral, and in fact this expression can be evaluated exactly whenever a is the exponential of some linear or bilinear function of the x s. For the more general case of infinite N, Kac proved that if F(x) is a real, positive, continuous function of x, then... 

The entropy production is a bilinear function of the generalized forces and the currents. This is useful if we want to identify the correct form of the transport coefficients Ljj as we shall see in the following. Another point worth mentioning is illustrated... 

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