Combination of Variables Method

The range of variables can be open (unbounded) or closed (bounded). If all the independent variables are closed, we classify such equations as boundary value problems. If only initial values are necessary, with no bounds specified then we speak of initial value problems. In the next section, we develop the method of combination of variables, which is strictly applicable only to initial value problems. 

We introduced the idea of change of variables in Section 10.1. The coupling of variables transformation and suitable initial conditions often lead to useful particular solutions. Consider the case of an unbounded solid material with initially constant temperature Tg in the whole domain 0 x oo. At the face of the solid, the temperature is suddenly raised to 7 (a constant). This so-called step change at the position jc = 0 causes heat to diffuse into the solid in a wavelike fashion. For an element of this solid of cross-sectional area A, density p, heat capacity C, and conductivity k, the transient heat balance for an element Ax thick is 

For some position within the solid matrix (say x,), we certainly expect the temperature to rise to Z, eventually, that is, when t approaches infinity. Moreover, for some finite time (say f,) we certainly expect, far from the face at X = 0, that temperature is unchanged that is, we expect at f, that T = Tg as x approaches infinity. We can write these expectations as follows 

There appears to be some symmetry for conditions at zero and infinity, and this is the first prerequisite for a combination of variables approach to be applicable. Now, the combination of variables method is often called a similarity transform. This nomenclature arises from the manner we use to select a combined variable. Let us next write a rough approximation to Eq. 10.74, to an order of magnitude 

This suggests roughly at which we interpret to mean the change in at is similar in size to a change in. Or, equivalently, a change in is similar 

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One of the procedures found useful in solving partial differential equations is the so-called combination of variables method, or similarity transform. The strategy here is to reduce a partial differential equation to an ordinary one by judicious combination of independent variables. Considerable care must be given to changing independent variables. 

Combination of Variables Method 413 The conditions required to find the arbitrary constants C, B are simply... 

Combination of Variables Method 417 since, for half the symmetrical reactor, we have ... 

The method of combination of variables requires that a suitable combination ofy and t can be found. Dimensionally, equation 10.19 can be written as... 

A general alternative to stepwise-type searching methods for variable selection would be methods that attempt to explore as much of the possible solution space as possible. An exhaustive search of all possible combinations of variables is possible only for problems that involve relatively few x variables. However, it... 

Various approaches can be taken for constructing the U matrix. With PCR, a principal components analysis is used because PCA is an efficient method for finding linear combinations of variables that describe variation in the row space of R (See Section 4.2.2). With analytical chemistry data, it is usually possible to describe the variation in R using significantly fewer PCs than the number of original variables. This small number of columns effectively eliminates the matrix inversion problem. 

In the field of chemometrics, PCR and PLS are the most widely used of the inverse calibration methods. Tliese methods solve the matrix inversion problem inherent to the inverse methods by using a linear combination of variables in... 

Simplex Optimization Results. Of the 56 combinations of variables for use with the simplex in Table IV, only the 4 combinations highlighted with boxes have been utilized. Gearly the optimization of SFC separations via the simplex algorithm is still in its infancy, as are all other systematic methods of optimization for SFC. Nevertheless, as described below, the results provided by the simplex approach were quite good. The results of the 2 and 3-parameter simplexes are especially informative for the novice because their movement can be visualized in 3-dimensional space, in contrast to simplexes of four parameters and higher which cannot be depicted graphically. 

Future work. As mentioned earlier, use of the simplex algorithm for the systematic optimization of SFC separations is still in its early stages. The success already achieved, however, merits continued research along these lines. Research opportunities include (i) extension of the simplex method to less ideal variables and/or greater than 4 variables (ii) investigation of the benefits of the simplex method to packed columns and modified mobile phases and (iii) development of the capability to predict, for a given type of sample, the best combination of variables to optimize. 

First of all, it is necessary to continue and improve the monitoring of the principal parameters of the Black Sea and the Sea of Azov environment, which is subjected to a strong variability. In so doing, it is very important to use a combination of different methods of research such as traditional, satellite, drifter, numerical and laboratory modelling. Precisely this kind of approach should allow us to obtain reliable results by comparing the data acquired with different techniques. 

One of the simplest methods is to create a single, long, data matrix from the original three-way tensor. In the case of Table 5.17, we have four samples, which could be arranged as a 4 x 5 x 6 tensor (or box ). The three dimensions will be denoted /, J and A-. It is possible to change the shape so that any binary combination of variables is converted to a new variable, for example, the intensity of the variable at J = 2 and K = 3, and the data can now be represented by 5 x 6 = 30 variables and is the unfolded form of the original data matrix. This operation is illustrated in Figure 5.16. 

Comparisons between the different intra-arterial thrombolysis trials and between intraarterial thrombolysis and intravenous thrombolysis is hampered by differences in methodology and type of thrombolytic therapy. In addition, within the intra-arterial thrombolysis trials, thrombolytic deUvery has varied between regional into a parent vessel of the thrombosed vessel, local into the affected artery and into the thrombus itself, or combinations of these methods. In addition, the infusion process has been variable, ranging from continuous to pulsed infusion. Some studies have allowed physical clot dispersion using the tip of the microcatheter while this was prohibited in others, for instance in the PROACT trials. 

The ILDM technique proposed by Maas and Pope overcomes this problem by describing geometrically the optimum slow manifold of a system. The criterion for reduction is based on the time-scales of linear combinations of variables and not on species themselves. The main advantage of the technique is that it requires no information concerning which reactions are to be assumed in equilibrium or which species in quasi-steady-state. The only inputs to the system are the detailed chemical mechanism and the number of degrees of freedom required for the simplified scheme. The ILDM method then tabulates quantities such as rates of production on the lower-dimensional manifold. For this reason, it is necessarily better suited to numerical problems since it does not result in sets of rate... 

Once identified all possible combination of the variables could of course be tested, however, this is typically prohibitively resource intensive to be practical. DoE is a class of statistically ba.sed methods to select combinations of variables that may be tested to yield the same information using a reduced number of experiments. DoE is often used in a totally empirical manner however, the knowledge of potentially critical variables can greatly simplify the process. DoEs may take the form of simple experiments which bracket the extremes of variable combinations, the so called extreme vertices method, or may be more intricate. Commonly, a partial matrix type screening DoE will be executed to zero in on the ranges in variables that show the maximum impact and/or that most closely bracket the desired responses from the process. This is followed by a more focused matrix DoE to determine optimum ranges of operations. 

If the same variable parameter is used for two independent methods of analysis for the purpose of obtaining different information, in general a combination of the two techniques will provide a method which is more powerful than cither one taken alone. It is our purpose to show how a combination of thermal methods of analysis with mass spectrometry results in a single technique of exceptionally high significance (13). 

The questions ofhowto construct a chemical mechanism withouthaving in-process concentration measurements and how to excite the dynamics during model identification while having few manipulated variables can be addressed by a combination of three methods ... 

Linear discriminant analysis (LDA) is aimed at finding a linear combination of descriptors that best separate two or more classes of objects [100]. The resulting transformation (combination) may be used as a classifier to separate the classes. LDA is closely related to principal component analysis and partial least square discriminant analysis (PLS-DA) in that all three methods are aimed at identifying linear combinations of variables that best explain the data under investigation. However, LDA and PLS-DA, on one hand, explicitly attempt to model the difference between the classes of data whereas PCA, on the other hand, tries to extract common information for the problem at hand. The difference between LDA and PLS-DA is that LDA is a linear regression-like method whereas PLS-DA is a projection technique... 

We use the method of combination of variables, with the combined variable f = z/yf. ... 

To develope a model for this mathematical problem we can either simplify the differential species mass balance equation (1.39) appropriately or combine the transient shell species mass balance written for the thin layer Az with Pick s law for binary diffusion. The resulting partial differential equation is called Pick s second law. A simple way to obtain a solution for this differential equation is to adopt the method of combination of variables. It is then necessary to define a new independent variable that enable us to transform the partial differential equation into an ordinary differential equation. 

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