Now we discuss some properties of this problem. Problem (2.16), (2.18) is linear. According to the theory of partial differential equations, the solution can be represented as a sum of two functions [Pg.212]

From some property of a reference set by definition (primary values). This method assumes that structural effects on the data set to be studied are a linear function of those which occur in the reference set. Secondary values of these parameters can be estimated by various methods. [Pg.556]

The 4> u, v) function arises in a number of problems involving linear equilibrium some of the properties of this function are given in Table 9.1. [Pg.680]

Some properties of the particle number operators are worth studying. One may ask what happens if N acts on a wave function which cannot be described by a single determinant. For the sake of simplicity, let us study first the case when operator Nj acts on a wave function which is a linear combination of two determinants [Pg.19]

The modules of maximum values of the radial part of T-function were correlated with the values of P arameter and the linear dependence between these values was foimd. Using some properties of wave function as applicable to P-parameter, the wave equation of P-parameter with the formal analogy with the equation of P-fiinction was obtained [9], [Pg.315]

In Section II we presented the standard general multilinear models, of which the bilinear and the PARAFAC and Tucker2 (T2) trilinear models are most important in spectroscopy. These models contain no information about the specimen except the linear dependence of spectral intensity on functions of each of the independent variables. However, some properties of the specimen are known, and a model that incorporates these known properties is preferred to one that does not. This is particularly true when the model is indeterminate without side conditions. In this section we discuss three settings for the application of knowledge about the specimen identifiable bilinear and T2 submodels, penalized general multilinear models, and submodels in which the dependence of the expected intensity from some components for some ways has a specific mathematical form. [Pg.688]

The dimensional equations are usually expansions of the dimensionless expressions in which the terms are in more convenient units and in which all numerical factors are grouped together into a single numerical constant. In some instances, the combined physical properties are represented as a linear function of temperature, and the dimension equation resolves into an equation containing only one or two variables. [Pg.559]

The development of quantum chemistry, that is, the solution of the Schrodinger equation for molecules, is almost exclusively founded on the expansion of the molecular electronic wave function as a linear combination of atom-centered functions, or atomic orbitals—the LCAO approximation. These orbitals are usually built up out of some set of basis functions. The properties of the atomic functions at large and small distances from the nucleus determines to a large extent what characteristics the basis functions must have, and for this purpose it is sufficient to exanoine the properties of the hydro-genic solutions to the Schrodinger equation. If we are to do the same for relativistic quantum chemistry, we should first examine the properties of the atomic solutions to determine what kind of basis functions would be appropriate. [Pg.100]

The process of estimating and the change in the shape (deflection from the known geometry) based on the known mechanical properties and the mechanical load closely resembles that used in the example of the twisted rod. In the case of small strains, dimensional analysis produces a correct solution with a precision up to a dimensionless constant. The maximum stress, is directly proportional to the load force, P, and to the longitudinal linear parameter, 1 it is inversely proportional to the transverse linear parameter (thickness), b, and represents some kind of a function of the beam thickness f h) (P/E)(l/b)f(h). [Pg.196]

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