Defining the Linear System

is the absorbing potential operator for all chemical arrangements and Et)) is a reference scattering state with incoming-wave boundary conditions in channel (u,j, /, J, M Et). The reaction probabihties are independent of the quantum number M, to be defined below. In what follows, we omit the ABC superscript with the understanding that we are using the ABC formulation. 

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Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

Thus, given the weights and abscissas, the micromixing term for the moments is closed. Applying DQMOM, the micromixing source terms (which are added to the right-hand sides of Eqs. (133)—(135)) can be shown to obey for each n — 1,..., N the linear system defined by... 

For the IEM model, it is well known that for a homogeneous system (i.e., when pn and .) remain constant and the locations ((). ) move according to the rates rn. Using the matrices defined above, we can rewrite the linear system as... 

In general, if all (n = l,. .., A7e) are distinct, then A will be full rank, and thus a = A 1 /3 as shown in (B.32). However, if any two (or more) (< />) are the same, then two (or more) columns of Ai, A2, and A3 will be linearly dependent. In this case, the rank of A and the rank of W will usually not be the same and the linear system has no consistent solutions. This case occurs most often due to initial conditions (e.g., binary mixing with initially only two non-zero probability peaks in composition space). The example given above, (B.31), illustrates what can happen for Ne = 2. When ((f)) = ()2, the right-hand sides of the ODEs in (B.33) will be singular nevertheless, the ODEs yield well defined solutions, (B.34). This example also points to a simple method to overcome the problem of the singularity of A due to repeated (< />) it suffices simply to add small perturbations to the non-distinct perturbed values need only be used in the definition of A, and that the perturbations should leave the scalar mean (4>) unchanged. 

From Figure 7 it is deduced that the number of the equilibrium states depends on the number of points where the straight line yo = constant intersects with the curve defined by Eq.(13). With a value of yo 0.025, there are three equilibrium points Pi, P2, P3, being P stable, P2 unstable and P3 can be stable or unstable depending on the real part of the eigenvalues of the linearized system at this point. When the line yo = constant is tangent to the curve yo = fiy ) (be. point M) a new behavior of the reactor appears, which can be characterized from dyo/dy = 0 in Eq.(13) as follows ... 

Exercise 6. Show that the equilibrium point of the model defined by Eq.(34) and the simplified model R given by Eq.(35), i.e. when the dynamics of the jacket is considered negligible, are the same. Deduce the Jacobian of the system (35) at the corresponding equilibrium point. Write a computer program to determine the eigenvalues of the linearized model R at the equilibrium point as a function of the dimensionless inlet flow 4 50. Values of the dimensionless parameters of the PI controller can be fixed at Ktd = 1-52 T2d = 5. The set point dimensionless temperature and the inlet coolant flow rate temperature are Xg = 0.0398, X40 = 0.0351 respectively. An appropriate value of dimensionless reference concentration is C g = 0.245. Does it exist some value of 2 50 for which the eigenvalues of the linearized system R at the equilibrium point are complex with zero real part Note that it is necessary to vary 2 50 from small to great values. Check the possibility to obtain similar results for the R model. 

It is assumed that the original system of equations has already been partitioned into the smallest possible blocks and Eq. (l) defines one of these blocks. The linear system given in Eq. (l) is permuted so as to arrive at the following equivalent system ... 

As for the solution of the linear system, the standard approach based on the inversion of D matrix (see equation (48)) becomes unmanageable for very large solutes due to both the computational time and the disk memory occupation it requires. To deal with these cases an iterative procedure has been developed, [112] which is able to solve equation (48) without defining and inverting the full D matrix. A specific two-step extrapolation technique proved very effective in the solution of this problem, especially for the PCM variant based on the normal... 

Different sets of LOVIs can be obtained by different choices of matrices and vectors defining the linear equation system several combinations were studied on linear alkanes (Table M-6). 

It is natural to define the outputs, of the linearized system to correspond to those of the original system ... 

Equations (24.22) and (24.24) define a linear system by which the state differences are driven by parameter variations about their optimal value, and we name this the companion model . The properties of this companion model will clearly depend on Jx and J,. A controllability test will show whether all the elements of the 0 vector can be influenced by variations in the a vector. Provided that the system is controllable, then it should be possible to drive 0 to any transient desired, in the absence of limits on the amplitude or frequency of variation of a. 

In this appendix, we derive the solution to the linear system of equations (9) that define a fluctuating-charge model, namely ... 

Some systems can be decomposed into a harmonic (quadratic) part Ho(q,p) = p M p/2 + q Aq/2 and an anharmonic part H q,p) = U q). Then a splitting method can be formulated based on exact solution of the linear system together with kicks representing impulses defined by the anharmonic part. For a simple system consisting of a harmonic oscillator Ho q,p) = -I- such a method can... 

This defines the stability region for Euler s method as a disk in the complex hX-plane centered around —1 of radius 1. This region is shown in Fig. 4.1. If hX, where A is an eigenvalue of A lies in the indicated stability region, then Euler s method will be stable for the linear system z = Az. 

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