Mathematical properties

Like the geometry of Euclid and the mechanics of Newton, quantum mechanics is an axiomatic subject. By making several assertions, or postulates, about the mathematical properties of and physical interpretation associated with solutions to the Scluodinger equation, the subject of quantum mechanics can be applied to understand behaviour in atomic and molecular systems. The fust of these postulates is ... 

The mathematical properties discussed above are central to the next postulate ... 

In essence, a BE-matrix li.sts all the valence clcctron.s of the atoms in a molecule, both the ones involved in bonds and those associated as free electrons with an atom. A BE-matrix has a scries of interesting mathematical properties that directly... 

Here (0 is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irro-tational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects m be neglected. Such flows without viscous effec ts are called in viscid flows. 

The mathematical properties of P(A), tlie probability of event A, are deduced from tlie following postulates governing tlie assigmiient of probabilities to tlie elements of a sample space, S. 

The mathematical properties of the set of equations describing chemical equilibrium in the synthesis gas system indicate that the carbon-producing regions are defined solely by pressure, temperature, and elemental analysis. Once a safe blend of reactants is determined from the ternary, the same set of equations which was used to derive the ternary may be used to determine the gas composition. 

Sums of Independent Random Variables.—Sums of statistically independent random variables play a very important role in the theory of random processes. The reason for this is twofold sums of statistically independent random variables turn out to have some rather remarkable mathematical properties and, moreover, many physical quantities, such as thermal noise voltages or measurement fluctuations, can be usefully thought of as being sums of a large number of small, presumably independent quantities. Accordingly, this section will be devoted to a brief discussion of some of the more important properties of sums of independent random variables. 

The research for finding these conditions, has been intense and fruitful [5-13]. Thus, although an exact procedure for determining directly an -representable 2-RDM has not been found, many mathematical properties of these matrices are now known and several methods for approximating RDM s and for employing them have been developed [14-19],... 

Because of this a study of mathematical properties of function U led to understanding geometrical and mechanical features of level surfaces. Also, with a help of potential it was proved that external surface of earth with an accuracy of flattening of the first order has to be spheroid. The next step in developing the theory of the... 

We first state the postulates succinctly and then elaborate on each of them with particular regard to the mathematical properties of linear operators. The postulates are as follows. 

The next step is to evaluate the numerical constants axi and bxi- In order to accomplish these evaluations, we must first investigate some mathematical properties of the eigenfunctions Sxi(p). 

The radial functions Sni p) and R i(r) may be expressed in terms of the associated Laguerre polynomials L p), whose definition and mathematical properties are discussed in Appendix F. One method for establishing the relationship between Sniip) and L p) is to relate Sni p) in equation (6.50) to the polynomial L p) in equation (F.15). That process, however, is long and tedious. Instead, we show that both quantities are solutions of the same differential equation. 

Generally, we can write the transfer function as the ratio of two polynomials in 5.1 When we talk about the mathematical properties, the polynomials are denoted as Q s) and P(s), but the same polynomials are denoted as Y(s) and X(s) when the focus is on control problems or transfer functions. The orders of the polynomials are such that n > m for physical realistic processes.2... 

A group is a set of abstract elements (members) that has specific mathematical properties. In general it is not necessary to specify the nature of the members of the group or the way in which they are related. However, in the applications of group theory of interest to physicists and chemists, the key word is symmetry. 

Thanks to tensors mathematic properties, it is possible to add the same constant to each one of the diagonal terms, which allows the elimination of one of the axial qaudrupoles. 

We examine these models not only as mathematical entities but also as a means of determining what the mathematical properties of schemes tell us regarding programming problems and languages. In studying alternative models an important point to consider is their relative power. 

To solve this equation, an appropriate basis set ( >.,( / ) is required for the nuclear functions. These could be a set of harmonic oscillator functions if the motion to be described takes place in a potential well. For general problems, a discrete variable representation (DVR) [100,101] is more suited. These functions have mathematical properties that allow both the kinetic and potential energy... 

The extension of vector methods to more dimensions suggests the definition of related hypercomplex numbers. When the multiplication of two three-dimensional vectors is performed without defining the mathematical properties of the unit vectors i, j, k, the formal result is... 

Maxwell used the mathematical properties of state functions to derive a set of useful relationships. These are often referred to as the Maxwell relations. Recall the first law of thermodynamics, which may be written as... 

Before turning to the transport equation for Po t), let us add some remarks about the mathematical properties of the basic operators of the theory. 

M. A. Savageau, Biochemical systems analysis. I. Some mathematical properties of the rate law for the component enzymatic reactions. J. Theor. Biol. 25, 365 369 (1969). 

The PCA scores have a very powerful mathematical property. They are orthogonal to each other, and since the scores are usually centered, any two score vectors are uncorrelated, resulting in a zero correlation coefficient. No other rotation of the coordinate system except PCA has this property. 

Omitting all discussion of the mathematical properties and the subtleties with regard to the continuum of final state energies, we will hop to the perturbation expansion in three short equations. The initial state describing the unperturbed reactant (DA) system describes the solution to the zeroth order Schrodinger equation ... 

In the MPC theory, the problem is not even posed. One starts defining the purely mathematical concept of dynamical system without any reference to a representation of reality. (The baker s transformation or the Bernoulli shift are obvious examples.) From here on, one proves mathematically the existence of a class of abstract dynamical systems (K-flows) that are intrinsically stochastic —that is, that possess precise mathematical properties (including a temporal symmetry breaking that can be revealed by a change of representation). 

Fields as property values in 3-D space can either be evaluated and encoded on a regular (often cubic) grid [71], or approximated by certain distribution functions. Most often, Gaussians [83] have been employed here, as they have some desirable mathematical properties and can usually approximate the original field reasonably well with not too many parameters. 

The mathematical properties of the above function r 4 ) indicate that, when 0 < 0.2, t is close to unity. Under such conditions, there would be no diffusional resistance to reaction. On the other hand, when 0 > 5, 7 = 1/0 is a good approximation and for such conditions, diffusion is the rate-limiting process. 

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