Variables interdependent

Correlation analysis reveals the interdependence between variables. The statistical measure for the interdependence is the correlation coefficient. 

Two variables of primary importance, which are interdependent, are reaction temperature and ch1orine propy1ene ratio. Propylene is typically used ia excess to act as a diluent and heat sink, thus minimising by-products (eqs.2 and 3). Since higher temperatures favor the desired reaction, standard practice generally involves preheat of the reactor feeds to at least 200°C prior to combination. The heat of reaction is then responsible for further increases in the reaction temperature toward 510°C. The chlorine propylene ratio is adjusted so that, for given preheat temperatures, the desired ultimate reaction temperature is maintained. For example, at a chlorine propylene molar ratio of 0.315, feed temperatures of 200°C (propylene) and 50°C (chlorine) produce an ultimate reaction temperature of approximately 500°C (10). Increases in preheat temperature toward the ultimate reactor temperature, eg, in attempts to decrease yield of equation 1, must be compensated for in reduced chlorine propylene ratio, which reduces the fraction of propylene converted and, thus aHyl chloride quantity produced. A suitable economic optimum combination of preheat temperature and chlorine propylene ratio can be readily deterrnined for individual cases. 

Develop (e.g., write) the hyperbolic equation in terms of the dimensionless variables. This breaks the interdependence of exponential and pre-exponential terms. 

MW and MWD are very significant parameters in determining the end use performance of polymers. However, difficulty arises in ascertaining the structural properties relationship, especially for the crystalline polymers, due to the interdependent variables, i.e., crystallinity, orientation, crystal structure, processing conditions, etc., which are influenced by MW and MWD of the material. The presence of chain branches and their distribution in PE cause further complications in establishing this correlation. 

These equations describe the full oxidation of a conducting polymer Submitted to a potential step under electrochemically stimulated confer-mational relaxation control as a function of electrochemical and structural variables. The initial term of /(f) includes the evolution of the current consumed to relax the structure. The second term indicates an interdependence between counter-ion diffusion and conformational changes, which are responsible for the overall oxidation and swelling of the polymer under diffusion control. 

The classification of methods for studying electrode kinetics is based on the criterion of whether the electrical potential or the current density is controlled. The other variable, which is then a function of time, is determined by the electrode process. Obviously, for a steady-state process, these two quantities are interdependent and further classification is unnecessary. Techniques employing a small periodic perturbation of the system by current or potential oscillations with a small amplitude will be classified separately. 

Correlation analysis investigates stochastic relationships between random variables on the basis of samples. The interdependence of two variables x... 

Because the Y-matrix and X-matrix are interdependently decomposed the B-matrix fits better and more robust than in PCR the calibration. The evaluation is carried out by Eq. (6.88) according to X = YB. The application of PLS to only one y-variable is denoted as PLS 1. When several y-variables are considered in the form of a matrix the procedure is denoted PLS 2 (Manne [1987] H0skuldsson [1988] Martens and ISLes [1989] Faber and Kowalski [1997a, b]). 

Unlike conventional experimental designs which have independent variables, mixture designs possess variables which are interdependent in that the summation of the q component proportions must be unity. Typically, the individual component levels are restricted by lower (a ) and upper (bj) constraints imposed on the system by physical or chemical limitations of the formulation or by the selection of the level values by the formulator. These constraints are represented as ... 

In general, the effects of the process variables on electrocodeposition are often interdependent and therefore, are ill understood. Often a slight change of one variable can sometimes lead to a dramatic change in the amount of particle incorporation. For specific systems, the current density at which maximum incorporation occurs seems to be related to a change in the slope of the current-potential relation-... 

Copper production is quite a complex process to plan and to schedule due to the many process interdependencies (shared continuous casters and cranes, emission level restrictions, limited material availability, to name a few). This makes it very difficult to foresee the overall consequences of a local decision. The variability of the raw material has alone a significant impact on the process, various disturbances and equipment breakdowns are common, daily maintenance operations are needed and material bottlenecks occur from time to time. The solution that is presented here considers simultaneously, and in a rigorous and optimal way, the above mentioned aspects that affect the copper production process. As a consequence, this scheduling solution supports reducing the impact of various disturbance factors. It enables a more efficient production, better overall coordination and visualization of the process, faster recovery from disturbances and supports optimal... 

The constraints of a two-stage stochastic linear program can be classified into constraints on the first-stage variables only (9.3.2) and constraints on the first and on the second-stage variables (9.3.3). The latter represent the interdependency of the stages. All constraints are represented as linear inequalities with the matrices Aer>x">, Tb6 R 2xn Wm Rm2x 2, and the vectors b Rm2 and K, e R 2. 

Analytical methods often contain many different variables which need to be optimized to attain best performance. The different variables are not always independent. For example, pH and polarity in a solution may be interdependent. Optimization by changing one variable at a time, while... 

A sample may be characterized by the determination of a number of different analytes. For example, a hydrocarbon mixture can be analysed by use of a series of UV absorption peaks. Alternatively, in a sediment sample a range of trace metals may be determined. Collectively, these data represent patterns characteristic of the samples, and similar samples will have similar patterns. Results may be compared by vectorial presentation of the variables, when the variables for similar samples will form clusters. Hence the term cluster analysis. Where only two variables are studied, clusters are readily recognized in a two-dimensional graphical presentation. For more complex systems with more variables, i.e. //, the clusters will be in -dimensional space. Principal component analysis (PCA) explores the interdependence of pairs of variables in order to reduce the number to certain principal components. A practical example could be drawn from the sediment analysis mentioned above. Trace metals are often attached to sediment particles by sorption on to the hydrous oxides of Al, Fe and Mn that are present. The Al content could be a principal component to which the other metal contents are related. Factor analysis is a more sophisticated form of principal component analysis. 

We return to the complex formation equilibria described in Chapter 2 (Eqs. 2.1 -2.10). The equilibrium constants as given in these equations are essentially intrinsic constants valid for a (hypothetically) uncharged surface. In many cases we can use these constants as apparent constants (in a similar way as non-activity corrected constants are being used) to illustrate some of the principal features of the interdependent variables that affect adsorption. Although it is impossible to separate the chemical and electrical contribution to the total energy of interaction with a surface without making non-thermodynamic assumptions, it is useful to operationally break down the interaction energy into a chemical and a Coulombic part ... 

Variations in ferritin protein coats coincide with variations in iron metabolism and gene expression, suggesting an Interdependence. Iron core formation from protein coats requires Fe(Il), at least experimentally, which follows a complex path of oxidation and hydrolytic polymerization the roles of the protein and the electron acceptor are only partly understood. It is known that mononuclear and small polynuclear Fe clusters bind to the protein early in core formation. However, variability in the stoichiometry of Fe/oxidant and the apparent sequestration and stabilization of Fe(II) in the protein for long periods of time indicate a complex microenvironment maintained by the protein coats. Full understanding of the relation of the protein to core formation, particularly at intermediate stages, requires a systematic analysis using defined or engineered protein coats. 

Obvious goals in bioprocessing are to minimise raw material costs and reaction times, and to maximise product concentrations, yields and purities. However the effect of varying any one variable on the overall process must be accurately assessed. This can be difficult because of the interdependence of many of these variables, making computer models a very useful tool. Thus the use of a cheaper raw material may appear to be attractive, but if this material is of significantly lower purity and so requires costly purification prior to use, or if the product derived from it needs more extensive... 

Inflation factor in regression analysis, a term which evaluates the interdependence of the independent variables A low inflation factor is desirable for use of the regression equation for prediction. 

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