Optimization series

Predescu, C. Doll, J. D., Optimal series representations for numerical path integral simulations, J. Chem. Phys. 2003,117, 7448-7463... 

The optimal series CSTR configuration to minimize cost for a given disturbance attenuation requirement and instantly reacting reagent comprises equal-sized tanks for typical treatment systems. 

Triantaphyllou, E. (2000). Multicriteria decision making methods A comparative study. Applied Optimization Series, Kluwer Academic Publishers. 

The first two methods are heavily constrained on the available structural data. The assignment of, at least, protein backbone resonances is clearly the prerequisite to be achieved prior to a chemical optimization series. Lacking an assignment, the information obtained by NMR-based fragment identification and by their corresponding binding affinities can be transferred to X-ray crystallography to reduce the number of trials. 

FD, FFD and other related methods have in common that they suggest ideal experiments that, in a later stage, have to be associated with real molecules. A complete different approach starts from a set of real molecules, candidates to be included into the series. Provided that we can define an objective quality criterion for the series, it is possible to extract from the universe of candidates a few compounds yielding the best results, and therefore called optimal series. As a criterion of quality it has been proposed the value of... 

According to the Karhunen-Loeve (K-L) theorem, a stochastic process on a bounded interval can be represented as an infinite linear combination of orthogonal functions, the coefficients of which constitute uncorrelated random variables. The basis functions in K-L expansions are obtained by eigendecomposition of the autocovariance function of the stochastic process and are shown to be its most optimal series representation. The deterministic basis functions, which are orthonormal, are the eigenfunctions of the autocovariance function and their magnitudes are the eigenvalues. The Karhunen-Loeve expansion converges in the mean-square sense for any distribution of the stochastic process (Papoulis and Pillai 2002). A K-L representation of a zero-mean stochastic process f(t, 6) can be represented in the form... 

Figure S.19 The approach based on optimization of a reducible structure starts with the most general configuration and simplifies. (From Eliceche and Sargent, IChemE Symp. Series No. 61 1, 1981 reproduced by permission of the Institution of Chemical Engineers...

The general constrained optimization problem can be considered as minimizing a function of n variables F(x), subject to a series of m constraints of the fomi C.(x) = 0. In the penalty fiinction method, additional temis of the fomi. (x), a.> 0, are fomially added to the original fiinction, thus... 

G. Ramachandran and T. Schlick. Beyond optimization Simulating the dynamics of supercoiled DNA by a macroscopic model. In P. M. Pardalos, D. Shal-loway, and G. Xue, editors. Global Minimization of Nonconvex Energy Functions Molecular Conformation and Protein Folding, volume 23 of DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, pages 215-231, Providence, Rhode Island, 1996. American Mathematical Society. 

We use the sine series since the end points are set to satisfy exactly the three-point expansion [7]. The Fourier series with the pre-specified boundary conditions is complete. Therefore, the above expansion provides a trajectory that can be made exact. In addition to the parameters a, b and c (which are determined by Xq, Xi and X2) we also need to calculate an infinite number of Fourier coefficients - d, . In principle, the way to proceed is to plug the expression for X t) (equation (17)) into the expression for the action S as defined in equation (13), to compute the integral, and optimize the Onsager-Machlup action with respect to all of the path parameters. 

Ferrites aHowing for operation at frequencies well above 1 MH2 have also become available, eg, 3F4 and 4F1 (Table 6). Other newer industrial power ferrites are the Siemens-Matsushita N-series (28,97) the TDK PC-series (28,100), and the Thomson B-series (28,103). While moving to higher frequencies, the ferrites have been optimized for different loss contributions, eg, hysteresis losses, eddy current losses, and resonance losses. Loss levels are specified at 100°C because ambient temperature in power appHcations is about 60°C plus an increase caused by internal heat dissipation of about 40°C. 

The design of the sludge-blanket clarifiers used primarily in the water industry is based on the jar test and a simple measurement of the blanket expansion and settling rate (12). Different versions of the jar test exist, but essentially it consists of a bank of stirred beakers used as a series flocculator to optimize the flocculant addition that produces the maximum floc-setfling rate. Visual floc-size evaluation is usually included. 

This method of optimization is known as the generalized reduced-gradient (GRG) method. The objective function and the constraints are linearized ia a piecewise fashioa so that a series of straight-line segments are used to approximate them. Many computer codes are available for these methods. Two widely used ones are GRGA code (49) and GRG2 code (50). 

Westerberg, A. W. Optimization in A. K. Sunol, D. W. T. Rippin, G. V. Reklaitis, O. Hortacsu (eds.). Batch Pr ocessing Systems Engineering Current Status and Future Dir ections. vol. 143, NATO ASI Series F, Springer, Berlin (1995). 

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