Perturbed /-values

The air spring effect results from adiabatic expansion and compression of the air in the actuator casing, Niirnericallv, the small perturbation value for air spring stiffness in Newtons/rneter is given bv Eq, (8-107),... 

Moreover, if the wave function + Xxp P is used as a trial function 0, then the quantity W from equation (9.2) is equal to the second-order energy determined by perturbation theory. Any trial function 0 with parameters which reduces to -h 20o for some set of parameter values yields an approximate energy W from equation (9.2) which is no less accurate than the second-order perturbation value. 

After all steady state variables are determined, each variable is represented by its steady state value plus a "perturbation" value — for example M = M88 + M. All non-linear terms resulting from this substitution are expanded in a Taylor series about the steady state values and only linear terms are retained. The result of this operation may be summarized by the following matrix equationst... 

One at a time, perturb the elements of the tear variable xTi. Calculate the dependent variables, and evaluate the tear equations. Calculate the gradients of/, g, h, and h with respect to each xT i by a forward difference equation in which the xD are the perturbed values and x7 are the unperturbed values. 

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 scalar mean (4>) unchanged. 

Maximum perturbed value of time delays Weighting parameters White plant disturbance intensity White sensor noise intensity Initial conditions... 

The RIPS conformational search starts by perturbing all of the atoms X, Y, and Z coordinates by plus or minus the perturbation value (usually 2 A or less) the sign ( ) is randomly assigned. Next the molecule is energy minimized to a user-defined RMS gradient requirement. After each molecule is minimized, a check is preformed for a duplicate conformation existing in the conformer list. If the current structure is a new conformation, it is added to the list and the failure count (number of times a conformation that exists in the conformer list is repeated) is set to zero. A duplicate conformer is discarded and the failure count is increased by one. The search is considered complete once the number of failures has reached the user-defined values. 

The superscript 0 indicates the steady-state value and the prime indicates the perturbed value. The perturbation of F /a yields... 

This means that the change in volume caused by the reaction is of the same sign as the initial perturbation thus there is no volume moderation at constant T and p. In other words, if the pressure and volume of a system are perturbed and the pressure is then maintained constant at the perturbed value, the reaction will proceed in a direction such that the volume will continue to change in the same direction as the initial perturbation. If the initial perturbation is an increase in pressure and decrease in volume, then if the pressure is maintained at the new value, the reaction will proceed in the direction which leads to a continued decrease in volume until a new equilibrium position is reached. This result may be inferred from the fact that pressure moderation is observed. For following the initial increase in pressure the reaction will tend to proceed in the direction leading to a decrease in the pressure but if the pressure is maintained at the higher value the reaction continues in the same direction, and there must be a further decrease in volume before equilibrium is reached. 

The pair functions can be further approximated by the AHS modeP or the more exact y (r) perturbation values. Usually simple sum rules are used to obtain the intermolecular potential between different species. However, the extrapolation of pure fluid theories directly to mixtures should be undertaken with caution as some features, such the unimportance of different well depths in the liquid, are not so certain in mixtures. Hence any deviations from the above model, for example, excess nonlinear concentration dependence of should not automatically be attributed to three-body effects. Only in isotopic or isopotential mixtures, which can be described very nearly as the pure fluid, could an unambiguous detection of ternary contributions to the concentration dependence be feasible. 

When perturbed from equilibrium in the z direction, the system returns to equilibrium in a first-order rate process with a time constant T the spin-lattice, or longitudinal, relaxation time. Consequently, the rate of return from a perturbed value to the equilibrium value Mq is given by... 

The temperature dependence of the reference /(-values in the reduced model is expressed by equation set 12.41. Parameters 7, As, B, B, are determined by solving these equations at the base and perturbed values above, giving the following values ... 

This equation determines the perturbed values of a and E. They restricted their attention to the ground state (n = 1, Z = 0) only, and for large values of r0. 

Figure 2 Orientational dependence of SCF and correlation components to interaction energy in the water dimer as a function of deviation of H-bond from linearity. A refers to second-order Mpller-Plesset correlation contribution, corrected by the counterpoise method Eji p is a shorthand notation for the perturbation value of Edisp -...

Margenau has discussed the fact that the true distribution function contains two parts (a) the statistical distribution giving rise to modified frequencies corresponding to perturbation values which the configuration of atoms has produced, and (b) additional frequencies which result when kinetic energy is lost or gained during an optical process. The formulation developed implicitly includes both effects. 

To examine the convergence of nonlinear estimation routine, the converged values of parameters (see Table III) were perturbed by 10% of the converged values. These perturbed values were used as initial estimates of the parameters for the nonlinear least squares. The least-squares routine terminated at the exactly same parameter values as those presented in Table III. This indicates that the estimation routine converges at least at a local minimum on the sum of squares surface. 

Important is that equilibrium states passed during such reversible processes are stable. Such a reversible process must be slow because the time scale (for volume) in the usual (irreversible) processes is much less than the relaxation time— practically finite time of achieving an equilibrium state during a change of V from the old (perturbed) value to the new equilibrium value during the process in B. These results... 

The difference between the results obtained by the parallel and the sequential procedure is that for the latter the sum of the trajectories is constant but depends on the perturbation value A (Eq. 11.88). For the parallel procedure, such a dependence is absent (Eq. 11.84). Furthermore, the dependence of the sum of the concentration trajectories on experiment A can be eliminated by running two sequential procedures with opposite values of A ( back-to-back ). Writing c t) and c °(t) for the concentrations during the second sequential procedure we obtain... 

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