Small parameters

By assumption, the mass ratio = m/M is a small parameter. Thus, rescaling the Schrbdinger equation properly in time and potential transforms it into the singularly perturbed equation... 

Given a small parameter c > 0, we write down the penalized equation... 

For a small parameter 5, the family of problems defined in the domain fig is considered. We want to find a function such that... 

Banichuk N.V. (1970) The small parameter method in finding a curvilinear crack shape. Izvestiya USSR Acad. Sci., Mechanics of Solid (2), 130-137 (in Russian). 

The operator U shifts the qj oscillator coordinate to its equilibrium through the distance QoCj/cOj, the sign depending on the state of the TLS. All the coupling now is put into the term proportional to the tunneling matrix element and the small parameter of the theory is zIq rather... 

In comparison to the constant of propagation of the a-helix formation (kp — 1010s 1) and the double-helix formation (kp — 107s-1), a comparatively small parameter concerning the formation of triple helix has been found (fcp = 8 x 10 3s1). A higher entropy of activation is assumed as the main cause of this occurrence which means a lower frequence factor in the Arrhenius equation. 

Only at the end of the 19th century did the first attempt to approach this subject systematically appear. In fact, Poincar6 became interested in certain problems in celestial mechanics,1 and this resulted in the famous small parameters method of which we shall speak in Part II of this chapter. In another earlier work2 Poincar6 investigated also certain properties of integral curves defined by ike differential equations of the nearly-linear class. 

As we wish to stay within the scope of the small parameters method, we assume that a, b, and c are small numbers. 

It would be erroneous to think that present-day mathematical methods are adequate to explain or predict all that exists in the enormous volume of experimental material that physicists and engineers have accumulated. This is due partially to the fact that we possess fairly uniform theoretical approaches only in the domain of small parameters, and also because conditions in some oases are so complicated that in spite of a knowledge of the theory, one is handicapped by computational difficulties. 

Sink (in graph theory), 258 "Slack variables, 294 Slightly-ionized gases, 46 "Slow time, 362 Small parameter methods, 350 S-matrix, 599,649,692 Smirnova, T. S., 726 Smoluchowski, R., 745 Sokolov, A. V., 768 Sommerfeld, C. M., 722 Sonine polynomials, 25 Source (in graph theory), 258 Space group... 

The perturbation theory presented in Chapter 2 implies that orientational relaxation is slower than rotational relaxation and considers the angular displacement during a free rotation to be a small parameter. Considering J(t) as a random time-dependent perturbation, it describes the orientational relaxation as a molecular response to it. Frequent and small chaotic turns constitute the rotational diffusion which is shown to be an equivalent representation of the process. The turns may proceed via free paths or via sudden jumps from one orientation to another. The phenomenological picture of rotational diffusion is compatible with both... 

Andreev, V. and Savin, I. (1995) On the uniform convergence in a small parameter of Samarskii scheme and its modification. Zh. Vychisl. Mat. i Mat. Fiz., 35, 739-752 (in Russian) English transl. in USSR Comput. Mathem. and Mathem. Physics. 

In perturbation theory we assume that H may be expanded in terms of a small parameter A... 

A. Deprit, Canonical transformations depending on a small parameter, Cel. Mech. 1,12 (1969). 

In this section, we take an approach that is characteristic of conventional perturbation theories, which involves an expansion of a desired quantity in a series with respect to a small parameter. To see how this works, we start with (2.8). The problem of expressing ln(exp (—tX)) as a power series is well known in probability theory and statistics. Here, we will not provide the detailed derivation of this expression, which relies on the expansions of the exponential and logarithmic functions in Taylor series. Instead, the reader is referred to the seminal paper of Zwanzig [3], or one of many books on probability theory - see for instance [7], The basic idea of the derivation consists of inserting... 

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

At low-conversion copolymerization in classical systems, the composition of macromolecules X whose value enters in expression (Eq. 69) does not depend on their length l, and thus the weight composition distribution / ( ) (Eq. 1) equals 5(f -X°) where X° = jt(x°). Hence, according to the theory, copolymers prepared in classical systems will be in asymptotic limit (/) -> oo monodisperse in composition. In the next approximation in small parameter 1/(1), where (/) denotes the average chemical size of macromolecules, the weight composition distribution will have a finite width. However, its dispersion specified by formula (Eq. 13) upon the replacement in it of l by (l) will be substantially less than the dispersion of distribution (Eq. 69)... 

Here, A is a differentially small parameter which quantifies the strength of the external perturbation, but which will not appear in the final equations. It serves only to separate terms of different magnitudes of the resulting expressions. For instance, a sum of a term which contains a part linear in A and one quadratic in A can be simplified - because A is supposed to be differentially small, the quadratic term will be infinitely smaller than the linear term, so it can be neglected. If, however, the linear term happens to be zero, the quadratic term must be taken into account. 

Equation (273) is still extraordinarily difficult to solve. As we are, however, interested in a limiting law which corresponds to a weak coupling result (see Eq. (171)) we may formally consider the charge e as a smallness parameter we thus have ... 

Finally, making use of eqs. (18) and (20) the evolution matrix can be expanded in a power series of small parameter e ... 

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