Continuous variables examples

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

Figure 19.8.5 displays the graph of tliis cdf. Anodier example of a cdf of a continuous random time variable is shown in Figure 19.8.6. A cdf of a continuous variable (a normal distribudon - to be reviewed in the next chapter) is provided in Figure 19.8.7.

The amount of energy a molecule contains is not continuously variable but is quantized. That is, a molecule can stretch or bend only at specific frequencies corresponding to specific energy levels. Take bond-stretching, for example. Although we usually7 speak of bond lengths as if they were fixed, the numbers... 

We shall in this chapter be most concerned with the following example of Hilbert space. Each element /> is a complex-valued numerical function f(x) of one or more continuous variables represented collectively by the symbol x, such that the integral of its square modulus exists ... 

A useful expression for evaluating expectation values is known as the Hell-mann-Feynman theorem. This theorem is based on the observation that the Hamiltonian operator for a system depends on at least one parameter X, which can be considered for mathematical purposes to be a continuous variable. For example, depending on the particular system, this parameter X may be the mass of an electron or a nucleus, the electronic charge, the nuclear charge parameter Z, a constant in the potential energy, a quantum number, or even Planck s constant. The eigenfunctions and eigenvalues of H X) also depend on this... 

On occasion you need to obtain correlation coefficients between two variables. Correlation coefficients are a way of measuring linear relationships between two variables. A correlation coefficient of 1 or -1 indicates a perfect linear relationship, and a coefficient of 0 indicates no strong linear relationship. Pearson correlation coefficients are useful for continuous variables, while Spearman correlation coefficients are useful for ordinal variables. For example, look at the following SAS code ... 

Also, it is possible to combine stochastic and deterministic methods as hybrid methods. For example, a stochastic method can be used to control the structural changes and a deterministic method to control the changes in the continuous variables. This can be useful if the problem involves a large number of integer variables, as for such problems, the tree required for branch and bound methods explodes in size. 

The results for this scenario were obtained using GAMS 2.5/CPLEX. The overall mathematical formulation entails 385 constraints, 175 continuous variables and 36 binary/discrete variables. Only 4 nodes were explored in the branch and bound algorithm leading to an optimal value of 215 t (fresh- and waste-water) in 0.17 CPU seconds. Figure 4.5 shows the water reuse/recycle network corresponding to fixed outlet concentration and variable water quantity for the literature example. It is worth noting that the quantity of water to processes 1 and 3 has been reduced by 5 and 12.5 t, respectively, from the specified quantity in order to maintain the outlet concentration at the maximum level. The overall water requirement has been reduced by almost 35% from the initial amount of 165 t. 

Continuous variables can assume any value within an interval discrete variables can take only distinct values. An example of a discrete variable is one that assumes integer values only. Often in chemical engineering discrete variables and continuous variables occur simultaneously in a problem. If you wish to optimize a compressor system, for example, you must select the number of compressor stages (an integer) in addition to the suction and production pressure of each stage (positive continuous variables). Optimization problems without discrete variables are far easier to solve than those with even one discrete variable. Refer to Chapter 9 for more information about the effect of discrete variables in optimization. 

Objective functions that allow only discrete values of the independent variable ) occur frequently in process design because the process variables assume only specific values rather than continuous ones. Examples are the cost per unit diameter of pipe, the cost per unit area for heat exchanger surface, or the insulation cost considered in Example 1.1. For a pipe, we might represent the installed cost as a function of the pipe diameter as shown in Figure 4.2 [see also Noltie (1978)]. For... 

Relations (fl)-(g) define the MINLP problem. It is important to note that the relations between the binary and continuous variables in Equation (/) are linear. It is possible to impose the desired relations nonlinearly. For example, one could replace Cl by Cl 71 everywhere Cl appears. Then if 71 = 0, Cl does not appear, and if 71 = 1, Cl does appear. Alternatively, one could replace Cl by the conditional expression (if 71 = 1 then Cl else 0). Both these alternatives create nonlinear models that are very difficult to solve and should be avoided if possible. 

Limitations on our ability to measure constrain the extent to which the real-world situation approaches the theoretical, but many of the variables studied in toxicology are in fact continuous. Examples of these are lengths, weights, concentrations, temperatures, periods of time, and percentages. For these continuous variables, we may describe the character of a sample with measures of central tendency and dispersion that we are most familiar with the mean, denoted by the symbol x and also called the arithmetic average, and the standard deviation SD, denoted by the symbol [Pg.870]

Examples of novel engine and transmission technologies are homogeneous combustion compression ignition (HCCI), combined combustion system (CCS), combined autoignition (CAI) and continuously variable transmission (CVT). 

The effect size of a continuous variable is frequently expressed as the difference between the mean of the experimental minus the mean of the control group divided by the pooled standard deviation. For example, in Chapter 5, data from the National Institute of Mental Health collaborative study demonstrated that antipsychotic-treated patients averaged a 4.2-point increase on a 6-point improvement scale, whereas the placebo patients averaged only a 2.2-point increase (i.e., an average difference of 2 points). The standard deviation of these data was approximately 1.7, so in effect size units, the improvement was approximately 1.2 (i.e., 2.0 1.7) SD units. For discontinuous data, the effect size for a drug-placebo comparison is usually expressed as the difference between the percent improvement with the experimental drug and the percent improvement with placebo. 

It is this large and continuous variability in bulk composition coupled with the fact that crystal structures may be different for the same anhydrous or hydrous bulk composition which makes zeolite identification so difficult (see Breck, 1970, for example and Deer, et al., Vol. 4, 1962). The factors determining which species of zeolite will crystallize are undoubtedly complex, involving such variables as the chemical activity of dissolved ionic species, crystal growth rate and ease of nucleation however, certain patterns of mineral paragensis can be discerned through a survey of the literature. 

Box Spatial subunit to approximate the continuous spatial variation of state variables. A box can be characterized by the spatially averaged concentration of one or several state variables Example Well-mixed reactor... 

Titanium dioxide, TiOa (rutile), In the structures so far considered, all the atoms have occupied very special positions in the unit cell there were no continuously variable parameters to be determined. The structure of rutile, now to be considered, is a simple example of a structure in which there is one parameter. This structure has been described on p. 226, where it was introduced in connexion with the calculation of structure amplitudes. The general arrangement of the atoms was assumed, and the effect of the variation of the oxygen parameter on the structure amplitudes of the reflections was demonstrated (Fig. 123). Here we shall consider the evidence which leads to a knowledge of the general arrangement of the atoms. 

Formulation of the mathematical model here adopts the usual assumptions of equimolar overflow, constant relative volatility, total condenser, and partial reboiler. Binary variables denote the existence of trays in the column, and their sum is the number of trays N. Continuous variables represent the liquid flow rates Li and compositions xj, vapor flow rates Vi and compositions yi, the reflux Ri and vapor boilup VBi, and the column diameter Di. The equations governing the model include material and component balances around each tray, thermodynamic relations between vapor and liquid phase compositions, and the column diameter calculation based on vapor flow rate. Additional logical constraints ensure that reflux and vapor boilup enter only on one tray and that the trays are arranged sequentially (so trays cannot be skipped). Also included are the product specifications. Under the assumptions made in this example, neither the temperature nor the pressure is an explicit variable, although they could easily be included if energy balances are required. A minimum and maximum number of trays can also be imposed on the problem. 

The multicommodity capacity facility location-allocation problem is of primary importance in transportation of shipments from the original facilities to intermediate stations and then to the destinations. In this illustrative example we will consider such a problem which involves I plants, J distribution centers, K customers, and P products. The commodity flow of product p which is shipped from plant i, through distribution center j to customer k will be denoted by the continuous variable z tp. It is assumed that each customer k is served by only one distribution center j. Data are provided for the total demand by customer k for commodity p, Dkp, the supply of commodity p at plant i denoted as Sip, as well as the lower and upper bounds on the available throughput in a distribution center j denoted by V " and Vf7, respectively. 

This MILP example features three binary variables, one continuous variable and is of the form ... 

Illustration 7.4.7 This example is taken from Glover (1975) and is as follows We have a continuous variable x, and a 0-1 variable y ... 

Remark 1 The resulting optimization model is an MINLP problem. The objective function is linear for this illustrative example (note that it can be nonlinear in the general case) and does not involve any binary variables. Constraints (i), (v), and (vi) are linear in the continuous variables and the binary variables participate separably and linearly in (vi). Constraints (ii), (iii), and (iv) are nonlinear and take the form of bilinear equalities for (ii) and (iii), while (iv) can take any nonlinear form dictated by the reaction rates. If we have first-order reaction, then (iv) has bilinear terms. Trilinear terms will appear for second-order kinetics. Due to this type of nonlinear equality constraints, the feasible domain is nonconvex, and hence the solution of the above formulation will be regarded as a local optimum. 

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