Mathematical apparatus

Theory of the fictitious temperature field allows us to analyze the problems of residual stresses in glass using the mathematical apparatus of thermoelasticity. In this part we formulate the boundary-value problem for determining the internal stresses. We will Lheretore start from the Duhamel-Neuinan relations... 

Most of the propositions in Chapters 2-4 are of independent value although, for the present book they are used only as part of the auxiliary mathematical apparatus. Some of them were known earlier and the rest were discovered and proven in recent years in connection with the rapid development of the theory of difference schemes. 

The instantaneous composition of a copolymer X formed at a monomer mixture composition x coincides, provided the ideal model is applicable, with stationary vector ji of matrix Q with the elements (8). The mathematical apparatus of the theory of Markov chains permits immediately one to wright out of the expression for the probability of any sequence P Uk in macromolecules formed at given x. This provides an exhaustive solution to the problem of sequence distribution for copolymers synthesized at initial conversions p l when the monomer mixture composition x has had no time to deviate noticeably from its initial value x°. As for the high-conversion copolymerization products they evidently represent a mixture of Markovian copolymers prepared at different times, i.e. under different concentrations of monomers in the reaction system. Consequently, in order to calculate the probability of a certain sequence Uk, it is necessary to average its instantaneous value P Uk over all conversions p preceding the conversion p up to which the synthesis was conducted. 

This is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

The mathematical apparatus for treating combinations of symmetry operations lies in the branch of mathematics known as group theory. A mathematical group behaves according to the following set of rules. A group is a set of elements and the operations that obey these rules. 

The theory of surface tension, in other words, the problem how certain known facts and certain assumptions about the liquid state can be made to account for the existence of a surface tension, has been treated exhaustively by Laplace, by Gauss, and more recently by Van der Waals. The mathematical apparatus employed is very considerable, and we must confine ourselves to a statement of the... 

For the second example, let us consider the random sphere model (RSM), which can be referred to as an intermediate deterministic-stochastic approach. This model and an appropriate mathematical apparatus were originally offered by Kolmogorov in 1937 for the description of metal crystallization [254], Later, this model became widely applicable for the description of phase transformations and other processes in PS, and usually without references to the pioneer work by Kolmogorov [134,149-152,228,255,256],... 

A model is a mathematical representation of the modelers s reality, a way of capturing some aspects of a particular reality within the framework of a mathematical apparatus that provides us with a means for exploring the properties of the reality mirrored in the model. 

The mathematical apparatus presented in 2.3.3 is restricted to irreversible reactions. Hydrolysis, due to the large excess of one reactant, water, is for practical purposes irreversible. In a transesterification the concentration of the leaving alcohol (from... 

According to the ground rules laid down at the beginning of this book, multiple scattering is excluded from consideration. But it is not always prudent to pretend that multiple scattering does not exist. Fortunately, it is almost trivial—the mathematical apparatus of radiative transfer theory is unnecessary—to extend our treatment of scattering and circular polarization to multiple scattering media, and in this instance it is worth the small amount of effort required to do so. 

In order to utilize the mathematical apparatus of the angular momentum theory, we have to write all the operators in. /-representation, i.e., to express them in terms of the quantities which transform themselves like the eigenfunctions of operators j2 and jz. For example, the explicit form of the spherical components of the spin operator in. /-representation, according to [28], is as follows ... 

We shall obtain a new type of a relativistic wave function if we use in Eq. (2.15) expression (2.18) instead of (2.16). However, usually in this case / is chosen to equal —1/2. Such a form leads to simpler phase multipliers for some quantities of the corresponding mathematical apparatus, but does not affect the matrix elements of the operators corresponding to physical quantities. 

Moreover, for atoms with open shells, the difficulties in calculating the angular parts of the PT expansion grow very rapidly with the increase in the order of expansion terms. Even the methods of their calculations are not developed sufficiently, unlike the usual mathematical apparatus of the theory of atomic spectra. Therefore, in order to successfully apply the PT to complex atoms, further development of the many-body theory... 

The mathematical apparatus of the angular momentum theory can be applied to describe the tensorial properties of electron creation and annihilation operators in the space of occupation numbers of a certain definite one-particle state a). It follows from (13.29) and (13.30) that the operators... 

A. P. Jucys, I. B. Levinson and V. V. Vanagas. Mathematical Apparatus of the Angular Momentum Theory, Vilnius, 1960 (in Russian). English translations Israel Program for Scientific Translations, Jerusalem 1962 Gordon and Breach, New York, 1963. 

Part 2 is devoted to the foundations of the mathematical apparatus of the angular momentum and graphical methods, which, as it has turned out, are very efficient in the theory of complex atoms. Part 3 considers the non-relativistic and relativistic cases of complex electronic configurations (one and several open shells of equivalent electrons, coefficients of fractional parentage and optimization of coupling schemes). Part 4 deals with the second-quantization in a coupled tensorial form, quasispin and isospin techniques in atomic spectroscopy, leading to new very efficient versions of the Racah algebra. 

---