Defined Functions

Next, we shall consider four kinds of integrals. The first is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at that nucleus. The second is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at a different point (usually another nucleus). Then, we will consider the matrix element of a Coulomb term between two primitive basis functions at different centers. The third case is when one basis function is centered at the nucleus considered. The fourth case is when both basis functions are not centered at that nucleus. By that we mean, for two Gaussian basis functions defined in Eqs. (73) and (74), we are calculating... 

Although the elemental stiffness Equation (2.55) has a common form for all of the elements in the mesh, its utilization based on the shape functions defined in the global coordinate system is not convenient. Tliis is readily ascertained considering that shape functions defined in the global system have different coefficients in each element. For example... 

In the consistent streamline upwind Petrov-Galerkin (SUPG) scheme all of the terms in Equation (3.52) are weighted using the function defined by Equation (3.53) and hence Wjj = Wj. 

Let V be some known function defined in the domain fic- If v and the boundary dflc = F U F+ U Fj are sufficiently smooth, then we can define values of v at the boundary (the exact smoothness conditions are studied in Section 1.4). In particular, having the values u p+ and u p-, we introduce the jump of v at Fc by the formula... 

We recall some definitions which are useful in the work to follow. The smallest a-algebra containing all compact sets in r 9r is called the Borel a-algebra (Landkof, 1966). Any a-additive real-valued function defined on the Borel a-algebra which is finite for all compact sets B c r 9r is called a measure on 9r. Thus, for a measure p and a set A, the a-additivity means... 

Proof. Let us consider a linear space >V of functions defined on L, dT p ... 

Let us next show that the space Cq (L, ) is included in >V. By Cq (L, ) we denote the space of continuously differentiable functions defined on... 

This section is concerned with an extreme crack shape problem for a shallow shell (see Khludnev, 1997a). The shell is assumed to have a vertical crack the shape of which may change. From all admissible crack shapes with fixed tips we have to find an extreme one. This means that the shell displacements should be as close to the given functions as possible. To be more precise, we consider a functional defined on the set describing crack shapes, which, in particular, depends on the solution of the equilibrium problem for the shell. The purpose is to minimize this functional. We assume that the... 

In this subsection we construct a nonnegative measure characterizing the work of interacting forces. The measure is defined on the Borel subsets of I. The space of continuous functions defined on I with compact supports is denoted by Co(I). 

Fig. 11. Liaear discriminant function defines the boundary between two categories.

The dominant crystal size, is most often used as a representation of the product size, because it represents the size about which most of the mass in the distribution is clustered. If the mass density function defined in equation 33 is plotted for a set of hypothetical data as shown in Figure 10, it would typically be observed to have a maximum at the dominant crystal size. In other words, the dominant crystal size is that characteristic crystal dimension at which drajdL = 0. Also shown in Figure 10 is the theoretical result obtained when the mass density is determined for a perfectiy mixed, continuous crystallizer within which invariant crystal growth occurs. That is, mass density is found for such systems to foUow a relationship of the form m = aL exp —bL where a and b are system-dependent parameters. 

A norm on iT is a real-valued function/defined on iT with the following properties ... 

Although stratification, according to the plot in Fig. 10, occurs continuously as increases, it is accompanied by a curious structural reorganization in transverse directions (i.e., parallel to the planar substrate). A suitable measure of transverse structure is the pair correlation function defined in Eq. (62). However, for simplicity we are concerned only with the in-plane pair correlation function defined as [see Eq. (62)]... 

A random variable is a real-valued function defined over tlie sample space S of a random experiment (Note tliat tliis application of probability tlieorem to plant and equipment failures, i.e., accidents, requires tliat tlie failure occurs randomly. 

All three terms are again functionals of the electron density, and functionals defining the two components on the right side of Equation 57 are termed exchange functionals and correlation functionals, respectively. Both components can be of two distinct types local functionals depend on only the electron density p, while gradient-corrected functionals depend on both p and its gradient, Vp. ... 

Statement functions are defined before any other executable statements in the program and are called in the same way that subprogram or intrinsic functions are called (see subprogram statements later). They are one-line expressions that receive one or more parameters from the calling statement and return a single calculated value to the function name in the calling statement. For example, a statement function defined as... 

All of Newtonian mechanics is developed from the independent and absolute concepts of space, lime, and mass. These quantities cannot be exactly defined, but they may be functionally defined as follows ... 

One possible order parameter, proposed by Paris [par83], is not so much an order parameter as an order function, Define Qap to be the overlap between the states a and f3 ... 

Wave function (Section 1.2) A solution to the wave equation for defining the behavior of an electron in an atom. The square of the wave function defines the shape of an orbital. 

The transducer function defines the efficiency of the system to translate receptor stimulus into response and defines the efficacy of the agonist. Specifically, it is the fitting parameter of the hyperbolic function linking receptor... 

Hyperbola (hyperbolic), a set of functions defining nonlinear relationships between abscissae and ordinates. This term is used loosely to describe nonlinear relationships between the initial interaction of molecules and receptors and the observed response (i.e., stimulus-response cascades of cells). 

The antimode is the cut-off value separating different functionally defined groups in a bi-modal or multimodal frequency distribution. 

Generally speaking, the outcome of any digital computation is a set of numbers in machine representation. Often the problem as originally formulated mathematically is to obtain a function defined over some domain, but the computation itself can give only (approximations to) a finite number of its functional values, or a finite number of coefficients in an expansion, or some other form of finite representation. At any rate, each number y in the finite set of numbers explicitly sought can be thought of, or perhaps even explicitly represented as, some function of the input data x ... 

As a concrete illustration of these ideas, let us investigate whether the function defined by (o > 0, b > 0)... 

The conditional probability density functions defined by Eq. (3-170) are joint probability density functions for fixed values of xn... 

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