Functions implicit

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

Thus, one specifies either T or P and either the liquid-phase or the vapor-phase composition, fixing 1 + (N - 1) or N phase-rule variables, exactly the number required by the phase rule for vapor/liquid equilibrium. All of these calculations require iterative schemes because of the complex functionality implicit in Eqs. (12.1) and (12.2). In particular, we have the following functional relationships for low-pressure VLE ... 

The expressions in Eqs. (13-39) and (13-40) define the linear and quadratic response functions implicitly. 

In the context of total synthesis, the reduction in the level of functionalization implicit in these processes is not always at variance with synthetic objectives, a simple example being the aldol condensation-deoxygenation sequence of Scheme 1, which could replace a (frequently more difficult) direct enolate alkylation. The conversion of readily available, polyfunctionalized materials such as carbohydrates into specifically deoxygenated or deaminated derivatives provides a variety of chiral synthons for the assembly of more complex substances. 

Excessive calcium entry into depolarized neurons contributes significantly to neuronal injury. Voltage-sensitive calcium channels (VSCCs) regulate, among other functions, cellular excitability and neurosecretory activity, functions implicit in epileptogenic events... 

The radial distribution function implicitly takes account of the enhancement of the number density in the neighborhood of the collision sphere due to the excluded volume of the other molecules making up the system. 

When we have a complete understanding of the physics of a problem and have no difficulty with the formulation but are mathematically stuck on the solution, we refer to dimensional analysis for a functional (implicit) form of the solution. Three distinct methods exist for dimensional analysis ... 

Since dimensional analysis can provide only a functional (implicit) relationship between dimensionless numbers, Eqs. (5.132) and (5.133) are synonymous dimensionless results. That is, by suitable transformations, a dimensionless result can be made identical to... 

Having learned the functional (implicit) relation among the dimensionless numbers of forced convection and of natural convection, we proceed to Chapter 6 for explicit relations among these numbers... 

In saturated molecules such spin correlation or antisymmetry effects, combined with nuclear geometry, localize electrons into lone pairs and electron pair bonds. Thus an H.F. MO wave function implicitly contains the localization properties. ... 

Somewhat surprisingly, the calculated spectrum shows an almost perfect agreement with the experiment. This seems to indicate that either the density functional implicitly accounts for the missing double excitation (see also Ref. 142) or that its influence on the shape of the excited-state potential energy surface is small or similar to that of the HOMO LUMO - -1 and HOMO 1 LUMO excitations. Based upon the intensity inducing b mode with a frequency of 94cm , which leads to mixing of the Si and Si states, several excitations of totally symmetric modes with low quantum numbers are observable. 

The probability density functions of the observations are generally unknown, but the Gauss-Markov theorem ensures that least-squares is always an acceptable estimator. However, the results of least squares are strongly influenced by discordant observations, so-called outliers. The robust-resistant techniques use weight-modification functions of O—Cy which progressively down-weight outliers. Tnese functions implicitly define probability functions p. They may alternatively be interpreted as an appreciation of the reliability of certain measurements. This approaches the frequently used option to simply omit discordant observations because they are judged to be unreliable. 

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