A Special Case

For example, when the kernel is [2] of a binary scheme, the kernel of its quaternary square is [4], giving the matrix 

The process of integrating back up, for those submatrices which do not contain the 4 , gives schemes with only polynomial eigenvectors. There is no discontinuity at all. This fits exactly what we know about B-splines, which are indeed fully continuous at all points in the interior of the spans. 

We have derived aD of the general projjerties of the multidimensional distributions that have been noted in Sections 3.2.4 and 3.2.5 (for the case N = 2), except that of the special case 

If the matrix (4.93) is a unit matrix, the multidimensional GF Gt factors into a simple product of N one-dimensional GF, each of which depends on variables and parameters of one vibrational component only. In addition, if the parameters P /pj, = P, far ad components i are equal, the corresponding distribution generated by (4.95) coincides with formula (4.35). 

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Separation of mixtures of condensable and non-condensable components. If a fluid mixture contains both condensable and noncondensable components, then a partial condensation followed by a simple phase separator often can give a food separation. This is essentially a single-stage distillation operation. It is a special case that deserves attention in some detail later. 

This type of analysis requires several chromatographic columns and detectors. Hydrocarbons are measured with the aid of a flame ionization detector FID, while the other gases are analyzed using a katharometer. A large number of combinations of columns is possible considering the commutations between columns and, potentially, backflushing of the carrier gas. As an example, the hydrocarbons can be separated by a column packed with silicone or alumina while O2, N2 and CO will require a molecular sieve column. H2S is a special case because this gas is fixed irreversibly on a number of chromatographic supports. Its separation can be achieved on certain kinds of supports such as Porapak which are styrene-divinylbenzene copolymers. This type of phase is also used to analyze CO2 and water. 

The volume fraction of water (S J and the saturation exponent n can be considered as expressing the increased difficulty experienced by an electrical current passing through a partially oil filled sample. (Note is only a special case of C, when a reservoir... 

The measuren.cnts are performed on a sample made from Inconell IN 718 placed at our disposal by Me oren- und Turbinen-Union Miinchen GmbH (MTU). The sample is of disk shape and has a curved side and a flat side, see also fig. 6. On the curved side MTU had detected a segregation reaching to the surface but with unknown depth. It had been detected by segregation etching and its existence was confirmed by eddy current testing. Both methods are successful only in such a special case where the segregation extends to the surface. 

Equation 11-3 is a special case of a more general relationship that is the basic equation of capillarity and was given in 1805 by Young [1] and by Laplace [2]. In general, it is necessary to invoke two radii of curvature to describe a curved surface these are equal for a sphere, but not necessarily otherwise. A small section of an arbitrarily curved surface is shown in Fig. II-3. The two radii of curvature, R and / 2[Pg.6]

A monolayer can be regarded as a special case in which the potential is a square well however, the potential well may take other forms. Of particular interest now is the case of multilayer adsorption, and a reasonable assumption is that the principal interaction between the solid and the adsorbate is of the dispersion type, so that for a plane solid surface the potential should decrease with the inverse cube of the distance (see Section VI-3A). To avoid having an infinite potential at the surface, the potential function may be written... 

A special case of adsorption in cavities is that of clatherate compounds. Here, cages are present, but without access windows, so for adsorption to occur the solid usually must be crystallized in the presence of the adsorbate. Thus quinol crystallizes in such a manner that holes several angstroms in diameter occur and, if crystallization takes place in the presence of solvent or gas... 

Rigid linear molecules are a special case in which an extended MS group, rather than the MS group, is isomorphic to the point group of the equilibrium structure see chapter 17 of [1]. 

Equation (A2.1.21) includes, as a special case, the statement dS > 0 for adiabatic processes (for which Dq = 0) and, a fortiori, the same statement about processes that may occur in an isolated system (Dq = T)w = 0). If the universe is an isolated system (an assumption that, however plausible, is not yet subject to experimental verification), the first and second laws lead to the famous statement of Clausius The energy of the universe is constant the entropy of the universe tends always toward a maximum. ... 

These two equations are a special case of the corresponding tliree-dimensional definition, connnon in XRD, with the surface nonnal n replacing the third lattice vector c. 

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

To see that this phase has no relation to the number of ci s encircled (if this statement is not already obvious), we note that this last result is true no matter what the values of the coefficients k, X, and so on are provided only that the latter is nonzero. In contrast, the number of ci s depends on their values for example, for some values of the parameters the vanishing of the off-diagonal matrix elements occurs for complex values of q, and these do not represent physical ci s. The model used in [270] represents a special case, in which it was possible to derive a relation between the number of ci s and the Berry phase acquired upon circling about them. We are concerned with more general situations. For these it is not warranted, for example, to count up the total number of ci s by circling with a large radius. 

Note that the equation for metal-metal ion systems is a special case of this general equation since the reduced state is the metal itself and the concentration of a solid is a constant and omitted from the equation.)... 

An instability of the impulse MTS method for At slightly less than half the period of a normal mode is confirmed by an analytical study of a linear model problem [7]. For another analysis, see [2]. A special case of this model problem, which gives a more transparent description of the phenomenon, is as follows Consider a two-degree-of-freedom system with Hamiltonian p + 5P2 + + 4( 2 This models a system of two springs con-... 

This approximation is a special case of the Baker-Campbell-Hausdorff lemma for additional discu.ssion and extensions to more general classes of methods. 

Charged species are not a special case in RAMSES. As ean be seen from Figure 2-57, rr-electron systems can accommodate any number of electrons between zero and two times the number of atoms involved. 

Note. Formaldehyde is a special case. It reacts similarly, but a primary alcohol is necessarily produced. This alcohol, however, will always contain one carbon atom more than that obtained in Reaction (2). 

The acetoacetic ester condensation (involving the acylation of an ester by an ester) is a special case of a more general reaction term the Claisen condensation. The latter is the condensation between a carboxylic ester and an ester (or ketone or nitrile) containing an a-hydrogen atom in the presence of a base (sodium, sodium alkoxide, sodamide, sodium triphenylmethide, etc.). If R—H is the compound containing the a- or active hydrogen atom, the Claisen condensation may be written ... 

Before we start with a systematic discussion of the syntheses of difunctional molecules, we have to point out a formal difficulty. A carbonmultiple bond is, of course, considered as one functional group. With these groups, however, it is not clear, which of the two carbon atoms has to be named as the functional one. A 1,3-diene, for example, could be considered as a 1,2-, 1,3-, or 1,4-difunctional compound. An a, -unsaturated ketone has a 1.2- as well as a 1,3-difunctional structure. We adhere to useful, although arbitrary conventions. Dienes and polyenes are separated out as a special case. a, -Unsaturated alcohols, ketones, etc. are considered as 1,3-difunctional. We call a carbon compound 1,2-difunctional only, if two neighbouring carbon atoms bear hetero atoms. 

The original DR equation is thus a special case of the Dubinin-Astakhov equation, with m = 2 parameter of Equation (4.18) for m = 2 is related to the structural constant B of the DR treatment through the simple expression... 

In applying the BET procedure to Type III isotherms, c = 1 constitutes a special case insertion of c = 1 into the standard BET equation (2.12) leads to the simplified equation... 

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