Variables, dependent choice

Whether ozone or oxidant measurements can be correlated with other variables depends heavily on the choice of sampling site and the manner of sample transfer. The importance of these criteria is discussed in Chapter 5, and the desirable specifications are described in this chapter. 

Numerical variables can either be continuous or discrete. Continuous variables are measured on a continuous, uninterrupted scale and can take any value on that scale. For example, height, weight, blood pressure, and heart rate are continuous variables. Depending on how accurately we want (or are able) to measure these variables, values containing one or more decimal points are certainly possible. In contrast, discrete variables can only take certain values, which are usually integers (whole numbers). The number of visits to an emergency room made by a person in one year is measured in whole numbers and is therefore a discrete variable. A subject s response to a questionnaire item that requires the choice of one of several specified levels (e.g., l=mild pain, 2=medium pain, 3=severe pain) yields a discrete variable. 

Reactor design is often discussed in terms of independent and dependent variables. Independent variables are choices such as reactor type and internals, catalyst type, inlet temperature, pressure, and fresH feed composition. Dependent variables result from independent variable selection. They may be constrained or unconstrained. Con-... 

Variables other than pressure and volume can be used equally well to construct different sets of empirical temperatures. The selection of such variables depends on the characteristics of the system that is being investigated. Clearly, for each different choice one can anticipate a distinct temperature scale this then presents a problem of unifying all different possible temperature scales—a matter that we will resolve below. 

Highly nonideal solutions are characterized by the fact that the activity coefficients and the partial molar enthalpies are strongly dependent upon composition. In order to compute the partial derivatives of these quantities which are needed in the application of the Newton-Raphson method, it is convenient to choose compositions or component-flow rates as members of the set of independent variables. Numerous choices of the independent variables have been made.6, lf 8 13,15 17 19-20 To demonstrate the formulation of the Newton-Raphson method, the choice of independent variables proposed by Naphtali and Sandholm17 is used. The Almost Band Algorithm may be formulated for other choices of independent variables as shown by Gallun and Holland.7,8 9... 

Since the velocity u is a random quantity, the concentration c that results from the solution of (18.13) is also random. What we want to do is to solve (18.13) for a particular choice of the velocity u and find the concentration c corresponding to that choice of u. For simplicity, let us assume that u is independent of x and depends only on time t. Thus the velocity u(t) is a random variable depending on time. Since u t) is a random variable, we need to specify the probability density for u(t). A reasonable assumption for the probability distribution of u(t) is that it is Gaussian... 

Write out all possibilities for boundary values of the dependent variables the choice among these will be made in conjunction with the solution method selected for the defining (differential) equation. 

With the exception of those variables having zero variance (which pick themselves), the decision about which variables to eliminate/include and the method by which this is done depends on several factors. The two most important factors are whether the dataset consists of two blocks of variables, a response block (Y) and a descriptor/predictor block (X), and whether the purpose of the analysis is to predict/describe values for one or more of the response variables from a model relating the variables in the two blocks. If this result is indeed the aim of the analysis, then it seems reasonable that the choice of variables to be included should depend, to some extent, on the response variable or variables being modeled. This approach is referred to as supervised variable selection. On the other hand, if the variable set consists of only one block of variables, the choice of variables in any analysis will be done with what are referred to as unsupervised variable selection. 

This diagram may be constructed in more than one way because the introduction of action-angle type variables depends on the choice of the basis of cycles on the torus in the neighbourhood of which these variables are introduced. Now let us discuss the character of this ambiguity. Let... 

Place spinner vessel on magnetic stirrer at 37 °C and set at 100-200 r.p.m. (this is variable depending upon the vessel size, geometry, fluid volume, and individual cell. Set a speed which visually shows complete homogeneous mixing of the cells throughout the medium—150 r.p.m. is usually a safe choice). 

Thus, the synthesis of highly active NNM catalysts toward ORR requires materials with well-developed pore structures and high density of Fe-Nx active ensembles, whose formation depends on many different variables the choice of the starting precursors (macrocycles containing transition metal, carbon, and nitrogen), the conditions of the pyrolyzation process (temperature, time, inert or reactive atmosphere). During precursors decomposition because of the pyrolyzation, many different reactions occur, with several intermediate decomposition steps. 

The choice of the variables depends on the physical process that is being studied. For the simulation of each sequence of runs, the following sea state random variables are selected ... 

Subsequently, we take Pe, ko, and r as independent variables, and calcnlate the phase behavior in the three-dimensional space spanned by them. This choice is made for the sake of simplicity. In experimental charge-stabilized colloidal suspensions, all these variables depend on each other. Experimental parameters can be mapped on our phase diagrams by estimating the effective screening length Ka and contact value pe. Note that, in our phase diagrams, two phases in coexistence have equal pressure, chemical potential, xa, and pe, but different q. 

In drawing the solid line curve of Figure 7.1 the horizontal axis has been identified as the independent variable and the vertical axis as the dependent axis. One generally likes to think of one variable being dependent on the value of some other variable. In the real physical world this is usually the case and one typically knos which variable depends on which. However, in terms of the pure analysis of the data, either variable can be considered as being the independent one and the other the dependent one. The analysis mattiematics depends little on a choice of dependent or independent variable. In some cases it is convenient to reverse the role of the variables in curve fitting techniques. 

We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

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