Variables and

Equation (10a) is somewhat inconvenient first, because we prefer to use pressure rather than volume as our independent variable, and second, because little is known about third virial coefficients It is therefore more practical to substitute... 

The algorithm employed in the estimation process linearizes the constraint equations at each iterative step at current estimates of the true values for the variables and parameters. 

Measured Variables and Estimates of Their True Values for Acetone(1)/Methanol(2) System (Othmer, 1928)... 

Subroutine VLDTA2. VLDTA2 loads the binary vapor-liquid equilibrium data to be correlated. If the data are in units other than those used internally, the correct conversions are made here. This subroutine also reads the estimated standard deviations for the measured variables and the initial parameter estimates. All input data are printed for verification. 

According to this concept, a reduced property is expressed as a function of two variables, and I/, and of the acentric factor, cd ... 

In order to draw the property-yield curves for gasolines , it suffices to choose the initial point, which coilild be or 20°C, the end point being variable and situated between the end point of the heaviest gasoline cut which can be produced (200-220°C) and about 350°C. 

A Monte Carlo simulation is fast to perform on a computer, and the presentation of the results is attractive. However, one cannot guarantee that the outcome of a Monte Carlo simulation run twice with the same input variables will yield exactly the same output, making the result less auditable. The more simulation runs performed, the less of a problem this becomes. The simulation as described does not indicate which of the input variables the result is most sensitive to, but one of the routines in Crystal Ball and Risk does allow a sensitivity analysis to be performed as the simulation is run.This is done by calculating the correlation coefficient of each input variable with the outcome (for example between area and UR). The higher the coefficient, the stronger the dependence between the input variable and the outcome. 

This short-cut method could be repeated to include another variable, and could therefore be an alternative to the previous two methods introduced. This method can always be used as a last resort, but beware that the range of uncertainty narrows each time the process is repeated because the tails of the Input variables are always neglected. This can lead to a false impression of the range of uncertainty in the final result. 

The econom/c mode/for evaluation of investment (or divestment) opportunities is normally constructed on a computer, using the techniques to be introduced in this section. The uncertainties in the input data and assumptions are handled by establishing a base case (often using the best guess values of the variables) and then performing sensitivities on a limited number of key variables. 

In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle coordinates r. and velocities dr./dt at any given time. These evolve according to Newton s equations of motion. In principle, one can write down equations involving the state variables and forces acting on the particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t, provided one knows the precise state of the classical system at time t. In quantum mechanics, the state of the system at time t is instead described by a well behaved mathematical fiinction of the particle coordinates q- rather than a simple list of positions and velocities. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

In experimental work it is usually most convenient to regard temperature and pressure as die independent variables, and for this reason the tenn partial molar quantity (denoted by a bar above the quantity) is always restricted to the derivative with respect to Uj holding T, p, and all the other n.j constant. (Thus iX = [right-hand side of equation (A2.1.44) it is apparent that the chemical potential... 

Theory shows that these equations must be simple power series in the concentration (or an alternative composition variable) and experimental data can always be fitted this way.)... 

The equation of state detemiined by Z N, V, T ) is not known in the sense that it cannot be written down as a simple expression. However, the critical parameters depend on e and a, and a test of the law of corresponding states is to use the reduced variables T, and as the scaled variables in the equation of state. Figure A2.3.5 bl illustrates this for the liquid-gas coexistence curves of several substances. As first shown by Guggenlieim [19], the curvature near the critical pomt is consistent with a critical exponent (3 closer to 1/3 rather than the 1/2 predicted by van der Waals equation. This provides additional evidence that the law of corresponding states obeyed is not the fomi associated with van der Waals equation. Figure A2.3.5 (b) shows tliat PIpkT is approximately the same fiinction of the reduced variables and... 

Figure A2.3.5 (a) PIpkT as a fimction of the reduced variables and and (b) coexisting liquid and vapour densities in reduced units pp as a fimction of Jp for several substances (after [19]).

Unlike the pressure where p = 0 has physical meaning, the zero of free energy is arbitrary, so, instead of the ideal gas volume, we can use as a reference the molar volume of the real fluid at its critical point. A reduced Helmlioltz free energy in tenns of the reduced variables and F can be obtained by replacing a and b by their values m tenns of the critical constants... 

Pestak M W, Goldstein R E, Chan M H W, de Bruyn J R, Balzarini D A and Ashcroft N W 1987 Three-body interactions, scaling variables, and singular diameters in the coexistence curves of fluids Phys. Rev. B36 599-614... 

We start with a simple example the decay of concentration fluctuations in a binary mixture which is in equilibrium. Let >C(r,f)=C(r,f) - be the concentration fluctuation field in the system where is the mean concentration. C is a conserved variable and thus satisfies a conthuiity equation ... 

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. 

T is the free energy fiinctional, for which one can use equation (A3.3.52). The summation above corresponds to both the sum over the semi-macroscopic variables and an integration over the spatial variableThe mobility matrix consists of a synnnetric dissipative part and an antisyimnetric non-dissipative part. The syimnetric part corresponds to a set of generalized Onsager coefficients. 

This solution can be obtained explicitly either by matrix diagonalization or by other techniques (see chapter A3.4 and [42, 43]). In many cases the discrete quantum level labels in equation (A3.13.24) can be replaced by a continuous energy variable and the populations by a population density p(E), with replacement of the sum by appropriate integrals [Hj. This approach can be made the starting point of usefiil analytical solutions for certain simple model systems [H, 19, 44, 45 and 46]. 

What is addressed by these sources is the ontology of quantal description. Wave functions (and other related quantities, like Green functions or density matrices), far from being mere compendia or short-hand listings of observational data, obtained in the domain of real numbers, possess an actuality of tbeir own. From a knowledge of the wave functions for real values of the variables and by relying on their analytical behavior for complex values, new properties come to the open, in a way that one can perhaps view, echoing the quotations above, as miraculous. ... 

These terms are analogous to those on p. 265 of [7], It will be noted that the symbol c has been reinstated as in Section VI.F, so as to facilitate the order of magnitude estimation in the nearly nonrelativistic limit. We now proceed based on Eq. (168) as it stands, since the transformation of Eq. (168) to modulus and phase variables and functional derivation gives rather involved expressions and will not be set out here. 

Fig. 11. The Speedup of LN at increasing outer timesteps for BPTI (2712 variables), lysozyme (6090 variables), and a large water system (without nonbonded cutoffs 37179 variables). For lysozyme, the CPU distribution among the fast, medium, and slow forces is shown for LN 3, 24, and 48.

The two main ways of data pre-processing are mean-centering and scaling. Mean-centering is a procedure by which one computes the means for each column (variable), and then subtracts them from each element of the column. One can do the same with the rows (i.e., for each object). ScaUng is a a slightly more sophisticated procedure. Let us consider unit-variance scaling. First we calculate the standard deviation of each column, and then we divide each element of the column by the deviation. 

The profits from using this approach are dear. Any neural network applied as a mapping device between independent variables and responses requires more computational time and resources than PCR or PLS. Therefore, an increase in the dimensionality of the input (characteristic) vector results in a significant increase in computation time. As our observations have shown, the same is not the case with PLS. Therefore, SVD as a data transformation technique enables one to apply as many molecular descriptors as are at one s disposal, but finally to use latent variables as an input vector of much lower dimensionality for training neural networks. Again, SVD concentrates most of the relevant information (very often about 95 %) in a few initial columns of die scores matrix. 

Variable and pattern selection in a dataset can be done by genetic algorithm, simulated annealing or PCA... 

One task of data analysis is to establish a model which quantitatively describes the relationships between data variables and can then be used for prediction. 

In matrix notation PCA approximates the data matrix X, which has n objects and m variables, by two smaller matrices the scores matrix T (n objects and d variables) and the loadings matrix P (d objects and m variables), where X = TPT... 

Multiple linear regression (MLR) models a linear relationship between a dependent variable and one or more independent variables. 

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