we see in this simple case that the closeness of the approximation depends upon the size the second term in Eq. (35) whether it is really a small perturbation upon the system. With these matrices the approximation would be good only if the two diagonal elements of H are close in value. The 2x2 case is rather special, however, and we give further more complicated examples. [Pg.31]

For naphthalene we examine the H and S matrices based upon the both the HLSP functions and the standard tableaux functions for the system. In both cases we include the non-ionic structures, only. This will give a picture of how the situation compares for the two sorts of basis functions. In both cases, of course, the dimensions of the matrices are 42x42, the number of non-ionic Rumer diagrams for a naphthalene structure. Some statistics concerning the commutator are shown in Table 6. It is clear that, [Pg.31]

When an ST03G AO basis full VB calculation of CH4 is carried out, there are 1716 singlet standard tableaux functions all together. When these are combined into functions of symmetry Mi the number of independent linear combinations is reduced to 164. Thus the symmetry factored H and S matrices are 164x164. We show the statistics for the HS—SH matrix for standard tableaux functions in Table 7. The statistics for HLSP functions are not available in this case. It is immediately obvious that the numbers [Pg.32]

V is the boil-up rate. Design a 2x2 MIMO system with PI controllers and decouplers as in Fig. 10.14. [Pg.210]

It can be demonstrated that in the case of a 2x2 system the elements in the steady state RGA can be calculated by the following relation [Pg.488]

Consider the two-dimensional system x = A - r x, where r = x and A is a 2x2 constant real matrix with complex eigenvalues a itu. Prove that there exists at least one limit cycle for a > 0 and that there are none for a < 0. [Pg.232]

One exception to this general behavior is the system p(2x2) 0.25 0/Pt( 111), where, due to the high tendency of coadsorbed CO and O for C02 formation on Pt(lll), the value of S0 for CO adsorption is the same to the one for the clean surface, although on other substrates oxygen behaves as a typical electronegative modifier of CO adsorption. [Pg.61]

Avoid very interactive loops, as RGA element close to 0.5 in a 2x2 system. [Pg.493]

In this section, we will generalize the simple 2x2 case and, at the same time, take a closer look at a different type of system, i.e., a quantum system in contact with its environment. Our theoretical formulation includes random-, thermal- and quantum-fluctuations together. This is a fundamentally difficult problem since we are dealing with systems out of equilibrium. We will demonstrate that it is possible to incorporate the temperature in a quasiequilibrium context and at the same time show that constructive interaction may exist between the environment and the open quantum system. [Pg.98]

HREELS and TFD have played a unique role In characterizing the surface chemistry of systems which contain hydrogen since many surface techniques are not sensitive to hydrogen. We have used these techniques to characterize H2S adsorption and decomposition on the clean and (2x2)-S covered Ft(111) surface (5). Complete dissociation of H,S was observed on the clean Ft(lll) surface even at IlOK to yield a mixed overlayer of H, S, SH and H2S. Decomposition Is primarily limited by the availability of hydrogen adsorption sites on the surface. However on the (2x2)-S modified Ft(lll) surface no complete dissociation occurs at IlOK, Instead a monolayer of adsorbed SH Intermediate Is formed (5) [Pg.200]

If we pick an arbitrary element we can see that it is represented by the xy-coordinates of the four nodal points, as depicted in Fig. 9.16. The figure also shows a -coordinate system embedded within the element. In the r/, or local, coordinate system, we have a perfectly square element of area 2x2, where the element spreads between —1 > < 1 and — 1 > rj < 1. This attribute allows us to easily allows us to use Gauss quadrature as a numerical integration scheme, where the limits vary between -1 and 1. The isoparametric element described in the //-coordinate system is presented in Fig. 9.17. [Pg.475]

All applications of the lattice-gas model to liquid-liquid interfaces have been based upon a three-dimensional, typically simple cubic lattice. Each lattice site is occupied by one of a variety of particles. In the simplest case the system contains two kinds of solvent molecules, and the interactions are restricted to nearest neighbors. If we label the two types of solvents molecules S and Sj, the interaction is specified by a symmetrical 2x2 matrix w, where each element specifies the interaction between two neighboring molecules of type 5, and Sj. Whether the system separates into two phases or forms a homogeneous mixture, depends on the relative strength of the cross-interaction W]2 with respect to the self-inter-action terms w, and W22, which can be expressed through the combination [Pg.166]

The states of a dynamic system are simply the variables that appear in the time differential. The time-domain differential equation description of multivariable systems can be used instead of Laplace-domain transfer functions. Naturally, the two are related, and we derive these relationships below. State variables are very popular in electrical and mechanical engineering control problems, which tend to be of lower order (fewer differential equations) than chemical engineering control problems. Transfer function representation is more useful in practical process control problems because the matrices are of lower order than would be required by a state variable representation. For example, a distillation column can be represented by a 2X2 transfer function matrix. The number of state variables of the column might be 200. [Pg.435]

Even if H and S are functionally independent, one still might argue that the commutator is likely to be small, and, thus, the idea could be a useful approximation. The difficulty here is with the subtleties of the concept of smallness in this context. We will not attempt to address this question quantitatively, but satisfy ourselves by examining the commutators of H and S for three systems. The first of these is a simple 2x2 system for which we may obtain an algebraic answer. The other two are matrices from real VB calculations of CH4 and the 7r-system of naphthalene. [Pg.30]

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