Number of variables

Radiography provides the only means of reliably detecting voids in pre-stressed cable ducts or of detecting loss of section or fracture of eables inside the duets. The maximum thiekness of eonerete whieh ean be radiographed for confident loeation of voids inside ducts is of course dependant on a number of variables, e g. amount of reinforcing bars, size of void in duet etc... 

To define the thennodynamic state of a system one must specify fhe values of a minimum number of variables, enough to reproduce the system with all its macroscopic properties. If special forces (surface effecls, external fields—electric, magnetic, gravitational, etc) are absent, or if the bulk properties are insensitive to these forces, e.g. the weak terrestrial magnetic field, it ordinarily suffices—for a one-component system—to specify fliree variables, e.g. fhe femperature T, the pressure p and the number of moles n, or an equivalent set. For example, if the volume of a surface layer is negligible in comparison with the total volume, surface effects usually contribute negligibly to bulk thennodynamic properties. 

For a very large number of variables, the question of storing the approximate Hessian or inverse Hessian F becomes important. Wavefunction optimization problems can have a very large number of variables, a million or more. Geometry optimization at the force field level can also have thousands of degrees of freedom. In these cases, the initial inverse Hessian is always taken to be diagonal or sparse, and it is best to store the... 

Due to the large number of variables in wavefiinction optimization problems, it may appear that fiill second-order methods are impractical. For example, the storage of the Hessian for a modest closed-shell wavefiinction with 500... 

In Eq. (12), SE is the standard error, c is the number of selected variables, p is the total number of variables (which can differ from c), and d is a smoothing parameter to be set by the user. As was mentioned above, there is a certain threshold beyond which an increase in the number of variables results in some decrease in the quality of modeling. In fact, the smoothing parameter reflects the user s guess of how much detail is to be modeled in the training set. 

Multivariate statistics is the discipline to analyze data, to elucidate the intrinsic structure within the data, and to reduce the number of variables needed to describe the data. 

HyperChem allows the visualization of two-dimensional contour plots for a certain number of variables, fh esc contour plots show the values of a spatial variable (a property f(x,y,z) in normal th rce-dimensional Cartesian space ) on a plane that is parallel to the screen. To obtain these contour plots the user needs to specify ... 

The dimensionality of a data set is the number of variables that are used to describe eac object. For example, a conformation of a cyclohexane ring might be described in terms c the six torsion angles in the ring. However, it is often found that there are significai correlations between these variables. Under such circumstances, a cluster analysis is ofte facilitated by reducing the dimensionality of a data set to eliminate these correlation Principal components analysis (PCA) is a commonly used method for reducing the dimensior ality of a data set. 

At present it is not possible to determine which of these mechanisms or their variations most accurately represents the behavior of Ziegler-Natta catalysts. In view of the number of variables in these catalyzed polymerizations, both mechanisms may be valid, each for different specific systems. In the following example the termination step of coordination polymerizations is considered. 

Ethylene glycol can be produced by an electrohydrodimerization of formaldehyde (16). The process has a number of variables necessary for optimum current efficiency including pH, electrolyte, temperature, methanol concentration, electrode materials, and cell design. Other methods include production of valuable oxidized materials at the electrochemical cell s anode simultaneous with formation of glycol at the cathode (17). The compound formed at the anode maybe used for commercial value direcdy, or coupled as an oxidant in a separate process. 

The hberated iodine is measured spectrometricaHy or titrated with Standard sodium thiosulfate solution (I2 +28203 — 2 1 VS Og following acidification with sulfuric acid buffers are sometimes employed. The method requires measurement of the total gas volume used in the procedure. The presence of other oxidants, such as H2O2 and NO, can interfere with the analysis. The analysis is also technique-sensitive, since it can be affected by a number of variables, including temperature, time, pH, iodide concentration, sampling techniques, etc (140). A detailed procedure is given in Reference 141. 

Process Systems. Because of the large number of variables required to characterize the state, a process is often conceptually broken down into a number of subsystems which may or may not be based on the physical boundaries of equipment. Generally, the definition of a system requires both definition of the system s boundaries, ie, what is part of the system and what is part of the system s surroundings and knowledge of the interactions between the system and its environment, including other systems and subsystems. The system s state is governed by a set of appHcable laws supplemented by empirical relationships. These laws and relationships characterize how the system s state is affected by external and internal conditions. Because conditions vary with time, the control of a process system involves the consideration of the system s transient behavior. 

The degrees of freedom, F, is by definition the difference between the number of variables and the number of independent equations ... 

The ordered set of measurements made on each sample is called a data vector. The group of data vectors, identically ordered, for all of the samples is called the data matrix. If the data matrix is arranged such that successive rows of the matrix correspond to the different samples, then the columns correspond to the variables as in Figure 1. Each variable, or aspect of the sample that is measured, defines an axis in space the samples thus possess a data stmcture when plotted as points in that / -dimensional vector space, where n is the number of variables. 

Eig. 1. Eormat of the data matrix, where Ai is the number of samples and iC is the number of variables. 

Rules of matrix algebra can be appHed to the manipulation and interpretation of data in this type of matrix format. One of the most basic operations that can be performed is to plot the samples in variable-by-variable plots. When the number of variables is as small as two then it is a simple and familiar matter to constmct and analyze the plot. But if the number of variables exceeds two or three, it is obviously impractical to try to interpret the data using simple bivariate plots. Pattern recognition provides computer tools far superior to bivariate plots for understanding the data stmcture in the //-dimensional vector space. 

Finding the best solution when a large number of variables are involved is a fundamental engineering activity. The optimal solution is with respect to some critical resource, most often the cost (or profit) measured in doUars. For some problems, the optimum may be defined as, eg, minimum solvent recovery. The calculated variable that is maximized or minimized is called the objective or the objective function. 

Theorem 1. The number of products in a complete set of B-numbers associated with a physical phenomenon is equal to n — r, where n is the number of variables that are involved in the phenomenon and ris the rank of the associated dimensional matrix. 

Let Dhe the dimensional matrix of order mhy n associated with a set of variables of a physical phenomenon, where m is the number of chosen reference dimensions and n the number of variables of the set. Without loss of generaUty, it may be assumed that n > m. Consider the augmented matrix (eq. 21) ... 

The least-squares technique can be extended to any number of variables as long as the equation is linear in its coefficients. The linear correlation ofj vs X can be extended to the correlation ofj vs multiple independent variables generating an equation of the form ... 

Venous Nomogra.phs, The alignment chart is restricted neither to addition operations, nor to three-variable problems. Alignment charts can be used to solve most mathematical problems, from linear ones having any number of variables, to ratiometric, exponential, or any combination of problems. A very useful property of these alignment diagrams is the fact that they can be combined to evaluate a more complex formula. Nomographs for complex arithmetical expressions have been developed (108). 

The general problem is posed as finding the minimum number of variables necessary to define the relationship between n variables. Let (( i) represent a set of fundamental units, hke length, time, force, and so on. Let [Pj represent the dimensions of a physical quantity Pj there are n physical quantities. Then form the matrix Ot) ... 

The intensive state of a PVT system is established when its temperature and pressure and the compositions of all phases are fixed. However, for equihbrium states these variables are not aU independent, and fixing a hmited number of them automaticaUy estabhshes the others. This number of independent variables is given by the phase rule, and is called the number of degrees of freedom of the system. It is the number of variables which may be arbitrarily specified and which must be so specified in order to fix the intensive state of a system at equihbrium. This number is the difference between the number of variables needed to characterize the system and the number of equations that may be written connecting these variables. 

The total number of independent equations is therefore (tt — )N + r In their fundamental forms these equations relate chemical potentials, which are functions of temperature, pressure, and composition, the phase-rule variables. Since the degrees of freedom of the system F is the difference between the number of variables and the number of equations. 

Variables It is possible to identify a large number of variables that influence the design and performance of a chemical reactor with heat transfer, from the vessel size and type catalyst distribution among the beds catalyst type, size, and porosity to the geometry of the heat-transfer surface, such as tube diameter, length, pitch, and so on. Experience has shown, however, that the reactor temperature, and often also the pressure, are the primary variables feed compositions and velocities are of secondary importance and the geometric characteristics of the catalyst and heat-exchange provisions are tertiary factors. Tertiary factors are usually set by standard plant practice. Many of the major optimization studies cited by Westerterp et al. (1984), for instance, are devoted to reactor temperature as a means of optimization. 

Construction Time The duration of construction is difficult to estimate owing to the large number of variables involved. In general, estimates of construction time tend to be overoptimistic, especially for... 

System, Equipment, and Refrigerant Selection There is no universal rule which can be used to decide which system, equipment type, or refrigerant is the most appropriate for a given application. A number of variables influence the finm-design decision ... 

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