This equation represents a system of ntf equations with n variables (n > mr). So Ms equations need to be added to the system to ensure the uniqueness of the solution during the process of generating the ILDM. These equations are of the form [Pg.78]

So, with x(t coordinates, called natural coordinates, the system of n differential equations with n variables becomes n equations of a single variable. [Pg.1227]

Z>) Variational Equations for Systems with n Variables.—It can be shown10 that for differential systems of the form [Pg.345]

For a given system of m nonlinear equations gj(x) = 0 with n variables jc, the linearized equations fj(x) around the vector of approximate solution x in each iteration k are obtained by [Pg.450]

Assuming that all variables can be measured, how many measurements do you need for the design of a control system with N controlled variables [Pg.609]

This process is easily generalized to systems with more variables and hence higher-dimension phase spaces. In general we specify some (N — l)-dimen-sional plane in the N-dimensional phase space and perform a similar single-cycle integration. We then have N — 1 differences, which form a vector Ax which is a function of the N — 1 initial concentration xq. We wish to make all the components of Ax zero, and the appropriate Newton Raphson form is then [Pg.139]

As the initial state, take a system with N components, home to R independent transformations, characterized by p external intensive variables (/, T, etc.), the molar fractions of the different components At in each phase a () and by the quantity of matter in the different phases. [Pg.45]

In a closed system with N constituents and M elements, in which R reactions take place (R N — H), numbers of moles of the individual constituents are not independent variables, as they are linked by relationships which we call mass balance equations. There are two fundamental ways of expressing the mass balance by means of constitution coefficients or of stoichiometric coefficients. We shall show that the two ways are equivalent under the assumption R = [Pg.29]

For a nonreacting equilibrium system with n species and p phases, the number of independent phase equilibrium equations is (p — l)n. The number of phase-rule variables is 2 + (n — l)p, consisting of intensive variables of temperature pressure and (n — 1) compositions for each phase. The difference between the phase-rule variables and the number of independent phase equilibrium equations is the degrees of freedom of the system, F [Pg.35]

The calculated values y of the dependent variable are then found, for x, corresponding to the experimental observations, from the model equation (2-71). The quantity 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.279]

At the intensive level it was established that, for an adequate treatment in the quantum space of the polyatomic combinations, the electronic density p r) rather than the already historical wave function (r,...rjy) stays as the main variable for a system with N electrons. This is because, contrary to the wave fimction, the electronic density is an experimental detectable quantity, defined in the real three dimensional space, and not within a 3N Hilbert abstract one, being also directly related to the total number of electrons in the concerned system through the functional relation N=Jp. [Pg.95]

The nonstationary motion of the medium was modeled by difference analogs of the Euler s gasdynamics equation represented in a spherical coordinate system with independent variables d and n = — R /r (see Ref. 14) [Pg.235]

Eq.(2.2-4) is the phase rule of Gibbs. According to this rule a state with II phases in a system with N components is frilly determined (all intensive thermodynamic properties can be calculated) if a number of F of the variables is chosen, provided that g of all phases as function of pressure, temperature and composition is known. [Pg.20]

If at least one point ao of the manifold is known, then we can calculate the progress of the kinetic system using the following system of differential equations with N, variables [Pg.245]

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]

This relationship fully describes all of the stable equilibrium states of a simple system with n components. However, there is no single fundamental equation governing the properties of all materials. The fundamental equation is represented by a surface in (3 + n) dimensional space. Quasi-static processes can be represented by a curve on this surface. The points on this surface represent stable equilibrium states of this simple system. For the entropy and internal energy, the canonical variables consist of extensive parameters. For a simple system, the extensive properties are S, U, and V, and the fundamental equations define a fundamental surface of entropy S = S U,V) in the Gibbs space of S, U, and V. [Pg.30]

We begin by considering an assembly of N particles and marking the initial momentum and position of each molecule as the set (p (0), r (0)), wherep (O) is a vector of the initial px, Py, Pz) momentum of every particle andr (O) is a vector of the initial (x, y, z) position of every particle. Thus, for a system with N monatomic molecules, the number of variables would be 6N. The equation of motion is usually written in the Hamiltonian form by defining the Hamiltonian as the sum of the potential and the kinetic energies of the particles [Pg.110]

As the motion around the limit cycle is periodic, we can only talk of a perturbation decaying or growing if we compare successive measurements made at the same point on the cycle. Thus we impose an initial perturbation Ax0 and observe its temporal evolution at the end of each successive circuit of the limit cycle. For a system with n independent variables, the perturbation Ax is an n-component vector. If the oscillatory period is given by Tp, then the perturbation at the end of the first cycle can be represented in the form [Pg.358]

It is a fundamental postulate of quantum mechanics that the wave function contains all possible information about a system in a pure state at zero temperature, whereas at nonzero temperature this information is contained in the density matrix of quantum statistical mechanics. Normally, this is much more information that one can handle for a system with N = 100 particles the many-body wave function is an extremely complicated function of 300 spatial and 100 spin17 variables that would be impossible to manipulate algebraically or to extract any information from, even if it were possible to calculate it in the first place. For this reason one searches for less complicated objects to formulate the theory. Such objects should contain the experimentally relevant information, such as energies, densities, etc., but do not need to contain explicit information about the coordinates of every single particle. One class of such objects are Green s functions, which are described in the next subsection, and another are reduced density matrices, described in the subsection 3.5.2. Their relation to the wave function and the density is summarized in Fig. 1. [Pg.19]

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. n,vlhe clusters will be in n-dimensional space. Principal component analysis (pea) 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 A I, 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. [Pg.34]

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