Multiple solutions

Multiple solution s tgj and tij are possible for this last eq nation. The wave functions for individual electrons, g/j, are called molecular orbilals, and the energy, Gp of an electron in orbital is called the orbital... 

The adsorbed layer at G—L or S—L surfaces ia practical surfactant systems may have a complex composition. The adsorbed molecules or ions may be close-packed forming almost a condensed film with solvent molecules virtually excluded from the surface, or widely spaced and behave somewhat like a two-dimensional gas. The adsorbed film may be multilayer rather than monolayer. Counterions are sometimes present with the surfactant ia the adsorbed layer. Mixed moaolayers are known that iavolve molecular complexes, eg, oae-to-oae complexes of fatty alcohol sulfates with fatty alcohols (10), as well as complexes betweea fatty acids and fatty acid soaps (11). Competitive or preferential adsorption between multiple solutes at G—L and L—L iaterfaces is an important effect ia foaming, foam stabiLizatioa, and defoaming (see Defoamers). 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

Adsorption and ion exchange share so many common features in regard to apphcation in batch and fixed-bed processes that they can be grouped together as sorption for a unified treatment. These processes involve the transfer and resulting equilibrium distribution of one or more solutes between a fluid phase and particles. The partitioning of a single solute between fluid and sorbed phases or the selectivity of a sorbent towards multiple solutes makes it possible to separate solutes from a bulk fluid phase or from one another. 

This may again have multiple solutions, but by choosing the lowest A value the minimization step is selected. The maximum step size R may be taken as a fixed value, or allowed to change dynamically during the optimization. If for example the actual energy change between two steps agrees well witlr that predicted from the second-order Taylor expansion, the trust radius for the next step may be increased, and vice versa. 

In Fig. 28, the abscissa kt is the product of the reaction rate constant and the reactor residence time, which is proportional to the reciprocal of the space velocity. The parameter k co is the product of the CO inhibition parameter and inlet concentration. Since k is approximately 5 at 600°F these three curves represent c = 1, 2, and 4%. The conversion for a first-order kinetics is independent of the inlet concentration, but the conversion for the kinetics of Eq. (48) is highly dependent on inlet concentration. As the space velocity increases, kt decreases in a reciprocal manner and the conversion for a first-order reaction gradually declines. For the kinetics of Eq. (48), the conversion is 100% at low space velocities, and does not vary as the space velocity is increased until a threshold is reached with precipitous conversion decline. The conversion for the same kinetics in a stirred tank reactor is shown in Fig. 29. For the kinetics of Eq. (48), multiple solutions may be encountered when the inlet concentration is sufficiently high. Given two reactors of the same volume, and given the same kinetics and inlet concentrations, the conversions are compared in Fig. 30. The piston flow reactor has an advantage over the stirred tank... 

I) I / y)dy < 0.75. For problems without multiple solutions, the proposed equation shows... 

Hence there are multiple solutions for the final set of 10000 compounds. The final selection can be diversity driven using for example cluster analysis based on multiple fingerprints [63], hole filling strategies by using scaffold/ring analysis (LeadScope [66], SARVision [66]) or pharmacophore analysis [67, 68]. For a review of computational approaches to diversity and similarity-based selections, see the paper of Mason and Hermsmeier [69] and the references therein. 

We would be remiss in our obligations if we did not point out that the regions of multiple solutions are seldom encountered in industrial practice, because of the large values of / and y required to enter this regime. The conditions under which a unique steady state will occur have been described in a number of publications, and the interested student should consult the literature for additional details. It should also be stressed that it is possible to obtain effectiveness factors greatly exceeding unity at relatively low values of the Thiele modulus. An analysis that presumed isothermal operation would indicate that the effectiveness factor would be close to unity at the low moduli involved. Consequently, failure to allow for temperature gradients within the catalyst pellet could lead to major errors. 

For integral values of n i 4, theoretically this polynomial always can be solved for r in terms of C, although some questions may arise about the propen choice from among multiple solutions. 

Nf = 0 The problem is exactly determined. If NF = 0, then the number of independent equations is equal to the number of process variables and the set of equations may have a unique solution, in which case the problem is not an optimization problem. For a set of linear independent equations, a unique solution exists. If the equations are nonlinear, there may be no real solution or there may be multiple solutions. 

Some equations such as/(x)=0 cannot be explicitly solved for x. If multiple solutions are not expected in a narrow range, Newton s method is often simple to implement and has faster convergence than the natural method of interval splitting. The method is recursive and uses the first-order expansion off (x) in the vicinity of the fcth guess... 

Eq (16) can be derived in several different ways. The original derivation of eq (16), presented in ref 9, has been based on the analysis of the mathematical relationships between multiple solutions of nonlinear equations representing different CC approximations (CCSD, CCSDT, etc.). An elementary derivation of eq (16), based on applying the resolution of identity to an asymmetric energy expression. 

Bui, P.-A., D. G. Vlachos, and P. R. Westmoreland. 1999. On the local stability of multiple solutions and oscillatory dynamics of spatially distributed flames. Combustion Flame 117 307-22. 

The energy and spectral optimization problems are convex programs so when there are multiple solutions the solution sets form a convex set. The following corollary characterizes how these convex sets of solutions relate to solutions of the Euler equation. In the formulation of this corollary we use the notion of optimal gap Ao—the gap achieved by optimal P and S. The optimal gap is a characteristic of the energy problem, depending only on H and S. 

P. Piecuch and K. Kowalski, In search of the relationship between multiple solutions characterizing coupled-cluster theories, in Computational Chemistry Reviews of Current Trends, Vol. 5 (J. Leszczynski, ed.), World Scientific, New York, 2000, p. 1. 

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