Variance, 269 of a distribution, 120 significance of, 123 of a Poisson distribution, 122 Variational equations of dynamical systems, 344 of singular points, 344 of systems with n variables, 345 Vector norm, 53 Vector operators, 394 Vector relations in particle collisions, 8 Vectors, characteristic, 67 Vertex, degree of, 258 Vertex, isolated, 256 Vidale, M. L., 265 Villars, P.,488 Von Neumann, J., 424 Von Neumann projection operators, 461 [Pg.785]

The calculated values y of the dependent variable are then found, for jc, corresponding to the experimental observations, from the model equation (2-71). The quantity ct, the variance of the observations y is calculated with Eq. (2-90), where the denominator is the degrees of freedom of a system with n observations and four parameters. [Pg.47]

Amolecular dynamics simulation usually starts with amolecular structure refined by geometry optimization, but without atomic velocities. To completely describe the dynamics of a classical system containing N atoms, you must define 6N variables. These correspond to 3N geometric coordinates (x, y, and z) and 3N variables for the velocities of each atom in the x, y, and z directions. [Pg.73]

For a system containing N chemical species distributed at equihbrium among 7C phases, the phase-rule variables are temperature and pressure, presumed uniform throughout the system, and N — mole fraciions in each phase. The number of these variables is 2 -t- (V — 1)7T. The masses of the phases are not phase-rule variables, because they have nothing to do with the intensive state of the system. [Pg.534]

The fundamental idea of this procedure is as follows For a system of two fluid phases containing N components, we are concerned with N 1 independent mole fractions in each phase, as well as with two other intensive variables, temperature T and total pressure P. Let us suppose that the two phases (vapor and liquid) are at equilibrium, and that we are given the total pressure P and the mole fractions of the liquid phase x, x2,. .., xN. We wish to find the equilibrium temperature T and the mole fractions of the vapor phase yu y2,. .., yN-i- The total number of unknowns is N + 2 there are N 1 unknown mole fractions, one unknown temperature, and two unknown densities corresponding to the two limits of integration in Eq. (6), one for the liquid phase and the other for the vapor phase. To solve for these N +2 unknowns, we require N + 2 equations of equilibrium. For each component i we have an equation of the form [Pg.171]

Principal component analysis (PCA) is a statistical method having as its main purpose the representation in an economic way the location of the objects in a reduced coordinate system where only p axes instead of n axes corresponding to n variables (p

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. [Pg.105]

Our discussion of solids and alloys is mainly confined to the Ising model and to systems that are isomorphic to it. This model considers a periodic lattice of N sites of any given symmetry in which a spin variable. S j = 1 is associated with each site and interactions between sites are confined only to those between nearest neighbours. The total potential energy of interaction [Pg.519]

The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). [Pg.536]

Using Table 52 the variables are El(FL ), L(L), d(L), (d - d,)(L), T(FL), and P(F). Note that this I is moment-area which is in the units of ft (not to be confused with I given in Table 52 which is moment of inertia, see Chapter 2, Strength of Materials, for clarification). The number of FI ratios that will describe the problem is equal to the number of variables (6) minus the number of fundamental dimensions (F and L, or 2). Thus, there will be four FI ratios (i.e., 6-2 = 4), FI, flj, fl, and FI. The selection of the combination of variables to be included in each n ratio must be carefully done in order not to create a complicated system of ratios. This is done by recognizing which variables will have the fundamental dimensions needed to cancel with the fundamental dimensions in the other included variables to have a truly dimensionless ratio. With this in mind, FI, is [Pg.374]

Hamiltonian Systems A Hamiltonian system is characterized by an even number of dimensions N = 2n = number of degrees of freedom), with variables conventionally labeled as (representing canonical positions) and (representing [Pg.171]

We shall now consider the properties of systems the state of which is determined by the values of the absolute temperature T, and n other independent variables x , 2, x3i. . . xn. If the latter are chosen in such a way that no external work is done when the temperature changes provided all the s are maintained constant, they, along with T, are called the normal variables, and the state so defined is said to be normally defined (Duhem Mecanique chimiqne, I., 83). [Pg.107]

A surface of separation between two phases is called a specific surface of separation, and in considering the statesof such systems it is evident that every specific surface constitutes a new independent variable. If there are n components in r phases with x specific surfaces, the Phase Rule will therefore read [Pg.446]

The change of n, with time was calculated according to first-order kinetics. It is given by a system of r linear differential equations and 0 r(r - 1) variables [Pg.138]

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