Coupled systems

These equations are the coupled system of discrete equations that define the rigorous forward problem. Note that we can take advantage of the convolution form for indices (i — I) and (j — J). Then, by exciting the conductive media with a number N/ oi frequencies, one can obtain the multifrequency model. The kernels of the integral equations are described in [13] and [3j. 

By analogy with the correlation function for the ftilly coupled system, the pair correlation ftmction g(r A) for an intennediate values of A is given by... 

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. 

Anderson S M, Zink J I and Neuhauser D 1998 A simple and accurate approximation for a coupled system-bath locally propagating Gaussians Chem. Phys. Lett. 291 387... 

Master equation methods are not tire only option for calculating tire kinetics of energy transfer and analytic approaches in general have certain drawbacks in not reflecting, for example, certain statistical aspects of coupled systems. Alternative approaches to tire calculation of energy migration dynamics in molecular ensembles are Monte Carlo calculations [18,19 and 20] and probability matrix iteration [21, 22], amongst otliers. 

In this paper, we consider the symplectic integration of the so-called Quantum-Classical Molecular Dynamics (QCMD) model. In the QCMD model (see [11, 9, 2, 3, 6] and references therein), most atoms are described by classical mechanics, but an important small portion of the system by quantum mechanics. This leads to a coupled system of Newtonian and Schrddinger equations. 

Summing (3.99) and (3.100), we clearly obtain exactly (3.98). Therefore, to prove the solvability of variational inequality (3.98), it suffices to establish solvability for the coupled system of variational inequalities (3.99) and (3.100). 

Multiple and Multisite, Coreceptor- and Coupled-System Substrate Recognition... 

FIG. 13-6 Thermally coupled systems for separation into three products, (a) Fractionator with vapor sidestream and side-cut rectifier, (h) Petlyuk towers. 

This derivation indicates a strong coupling between the momentum equation and the energy equation, which implies that the momentum and energy balance equations should be solved as a coupled system. In particular, the dis-... 

There is also a general failure to recognize that cooling water quality can be very dynamic. Do not, for example, make the mistake of installing a new tower, placing it into operation, and ignoring the water treatment for a few days. Some closely coupled systems with small water volumes (evaporative condensers and fluid coolers lending the best examples) can be scaled in a matter of hours. 

Giddings pointed out (32) that separated compounds must remain resolved throughout the whole process. This situation is illustrated in Figure 1.5, where two secondary columns are coupled to a primary column, and each secondary column is fed a fraction of duration Ar from the eluent from the first column. The peak capacity of the coupled system then depends on the plate number of each individual separation and on At. The primary column eliminates sample components that would otherwise interfere with the resolution of the components of interest in the secondary columns. An efficient primary separation may be wasted, however, if At is greater than the average peak width produced by the primary column, because of the recombination of resolved peaks after transfer into a secondary column. As At increases, the system approaches that of a tandem arrangement, and the resolution gained in one column may be nullified by the elution order in a subsequent column. 

LC-LC coupling systems are also employed to perform separations requiring very large plate numbers. However, it has been demonstrated (see equation (5.20) that for coupled columns peak capacity increases linearly with the square root of n... 

Although SFE and SFC share several common features, including the use of a superaitical fluid as the solvent and similar instrumentation, their goals are quite distinct. While SFE is used mainly for the sample preparation step (extraction), SFC is employed to isolate (chr-omatography) individual compounds present in complex samples (11 -15). Both techniques can be used in two different approaches off-line, in which the analytes and the solvent are either vented after analysis (SFC) or collected (SFE), or on-line coupled with a second technique, thus providing a multidimensional approach. Off-line methods are slow and susceptible to solute losses and contamination the on-line coupled system makes possible a deaease in the detection limits, with an improvement in quantification, while the use of valves for automation results in faster and more reproducible analyses (16). The off-line... 

Figure 6.7 shows a schematic diagram of an on-line SFE-SFC coupled system (42), with details of the interface being shown in Figure 6.8 (42). 

The first bioanalytical application of LC-GC was presented by Grob et al. (119). These authors proposed this coupled system for the determination of diethylstilbe-strol in urine as a replacement for GC-MS. After hydrolysis, clean-up by solid-phase extraction and derivatization by pentafluorobenzyl bromide, the extract was separated with normal-phase LC by using cyclohexane/1 % tetrahydrofuran (THE) at a flow-rate of 260 p.l/min as the mobile phase. The result of LC-UV analysis of a urine sample and GC with electron-capture detection (ECD) of the LC fraction are shown in Ligures 11.8(a) and (b), respectively. The practical detection limits varied between about 0.1 and 0.3 ppb, depending on the urine being analysed. By use of... 

One of the possibilities is to study experimentally the coupled system as a whole, at a time when all the reactions concerned are taking place. On the basis of the data obtained it is possible to solve the system of differential equations (1) simultaneously and to determine numerical values of all the parameters unknown (constants). This approach can be refined in that the equations for the stoichiometrically simple reactions can be specified in view of the presumed mechanism and the elementary steps so that one obtains a very complex set of different reaction paths with many unidentifiable intermediates. A number of procedures have been suggested to solve such complicated systems. Some of them start from the assumption of steady-state rates of the individual steps and they were worked out also for stoichiometrically not simple reactions [see, e.g. (8, 9, 5a)]. A concise treatment of the properties of the systems of consecutive processes has been written by Noyes (10). The simplification of the treatment of some complex systems can be achieved by using isotopically labeled compounds (8, 11, 12, 12a, 12b). Even very complicated systems which involve non-... 

The simultaneous determination of a great number of constants is a serious disadvantage of this procedure, since it considerably reduces the reliability of the solution. Experimental results can in some, not too complex cases be described well by means of several different sets of equations or of constants. An example would be the study of Wajc et al. (14) who worked up the data of Germain and Blanchard (15) on the isomerization of cyclohexene to methylcyclopentenes under the assumption of a very simple mechanism, or the simulation of the course of the simplest consecutive catalytic reaction A — B —> C, performed by Thomas et al. (16) (Fig. 1). If one studies the kinetics of the coupled system as a whole, one cannot, as a rule, follow and express quantitatively mutually influencing single reactions. Furthermore, a reaction path which at first sight is less probable and has not been therefore considered in the original reaction network can be easily overlooked. 

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