Expansion in powers

This statement is not exactly true - the slightly different system of ODEs is defined by an asymptotic expansion in powers of At which is generally divergent. 

An alternative form of the virial equation expresses Z as an expansion in powers of pressure about the real-gas state at zero pressure (zero density) ... 

In addition to a near-shock and an acoustic region, Deshaies and Clavin (1979) distinguished a third—a near-piston region—where nonlinear effects play a role as well. As already pointed out by Taylor (1946), the near-piston flow regime may be well approximated by the assumption of incompressibility. For each of these regions, Deshaies and Clavin (1979) developed solutions in the form of asymptotic expansions in powers of small piston Mach number. These solutions are supposed to hold for piston Mach numbers lower than 0.35. 

The approxmations reviewed so far were all developed for the low-piston Mach number regime. Cambray and Deshaies (1978), on the other hand, developed a solution of the similarity equations by asymptotic expansions in powers of high-piston Mach numbers. These solutions are supposed to hold for piston Mach numbers higher than 0.7. Finally, Cambray et al. (1979) suggested an interpolation formula to cover the intermediate-piston Mach number range. 

If A = 0, then H = Hq, 4/ = o md W = Eq. As the perturbation is increased from zero to a finite value, the new energy and wave function must also change continuously, and they can be written as a Taylor expansion in powers of the perturbation parameter A. 

An expansion in powers of 1 /c is a standard approach for deriving relativistic correction terms. Taking into account electron (s) and nuclear spins (1), and indicating explicitly an external electric potential by means of the field (F = —V0, or —— dAjdt if time dependent), an expansion up to order 1/c of the Dirac Hamiltonian including the... 

In digital computation, numbers are represented by expansion in powers of a base (or radix) b, which is usually either 2 or 10, with coefficients that are non-negative integers <6. Thus... 

Unfortunately, it is not known whether solutions of these equations exist. It is known that the above set is consistent when the solution is obtained as a perturbation expansion in power of e, but very little can be said about the convergence of such a series. 

It was found on expansion in powers of -r), neglecting all terms beyond... 

G is then a generating function for these integrals, which occur as coefficients in its expansion in powers of u and and it can he evaluated with the use of the generating function for the associated Laguerre polynomials, given in equation (19). Thus we have... 

If the upper integration limit in (C.5a) is y = nh rather than oo, i.e. for finite n, a simple closed expression is not obtained. However, one can estimate the leading term in an expansion in powers of n, such that... 

From this asymptotic expansion in powers of n no conclusions on the radius of convergence of ed h) are possible, but there are some hints that the radius of convergence is that ofcosech anh ), i.e.the series (A.4) probably converges for... 

Here the point p belongs to the spherical surface A of radius R. In order to find the upper limit on the left hand side of this equality, let us recall that T is the disturbing potential. In other words, it is caused by the irregular distribution of masses whose sum is equal to zero. This means that its expansion in power series with Legendre s functions does not contain a zero term. The next term is also equal to zero, because the origin coincides with the center of mass. Therefore, the series describing the function T starts from the term, which decreases as r. This means that the product r T O if oo and... 

Here, the final three terms are a Ginzburg-Landau expansion in powers of i j. The coefficient t varies as a function of temperature and other control variables. When it decreases below a critical threshold, the system undergoes a chiral symmetry-breaking transition at which i becomes nonzero. The membrane then generates effective chiral coefficients kHp = k n>i f and kLS = which favor membrane curvature and tilt modulations, respec-... 

A great many of the difficulties (and sometimes the misunderstandings) arise from point (c). It is however important to notice that the APM describes the properties of solutions as finite differences between suitable composition-dependent averages and the properties of the pure components. Series expansions in powers of 6, p, 6, and a were introduced afterwards for the purpose of qualitative discussion and comparison with other treatments, e.g., the theory of conformal solutions.34>85>36 They introduce artificial difficulties due to their slow convergencef which have nothing to do with the physical ideas of the APM. Therefore expansions of this type should be proscribed for all quantitative applications one should instead use the compact expressions of the excess functions. 

The resulting EoS is expressed as an expansion in powers of k/, and the value of A 0.65 GeV is adjusted to the empirical binding energy per nucleon. In its present form the validity of this approach is clearly confined to relatively small values of the Fermi momentum, i.e. rather low densities. Remarkably for SNM the calculation appears to be able to reproduce the microscopic EoS up to p 0.5 fm-3. As for the SE the value obtained in this approach for 4 = 33 MeV is in reasonable agreement with the empirical one however, at higher densities (p > 0.2 fm-3) a downward bending is predicted (see Fig. 4) which is not present in other approaches. 

Dunham obtained these eigenvalues using the semiclassical approximation for the potential (1.8) which is an expansion in powers of (r - re)/re. The results for the Morse potential [Eq. (1.14)] can also be written in this form, as can results for other potentials. One therefore often uses Eq. (1.71) as a convenient empirical form. A slightly different form of (1.71) is... 

Shell corrections can also be evaluated without recourse to an expansion in powers of v, but existing calculations such as Refs. [13,14] are based on specific models for the target atom and, unlike equation (19), do not end up in expressions that would allow to identify the physical origin of various contributions. It is clear, however, that orbital motion cannot be the sole cause of shell corrections The fact that the Bethe logarithm turns negative at 2mv /I< 1 cannot be due to the neglect of orbital motion but must be of a purely mathematical nature. Unfortunately, the uncertainty principle makes it impossible to eliminate orbital motion in an atom from the beginning. 

Traditionally the effect is incorporated into stopping theory via expansion in powers of Zj. As pointed out by Lindhard [21], there are two independent dimensionless parameters containing Zj, namely, the Bohr parameter k (equation (17)) and the Barkas parameter... 

While the Bloch correction represents a series expansion in powers of k, the leading term in the Barkas-Andersen correction was found [22] to be oc B. 

We want to mention here that the application of the near-nuclear corrections could have been performed with the complete relativistic functional, and we have utilized the semi-relativistic expressions just for simplicity and for testing them. For not large Z, the remaining errors above mentioned should be addressed to limitations of the semiclassical approach and of the procedure utilized for the near-nuclear corrections, rather than to the truncation of the expansion in powers of... 

Based on a perturbation expansion using the KS Hamiltonian [26,27], recently a new systematic scheme for the derivation of orbital-dependent Ec has been proposed [12]. While this representation is exact in principle, an explicit evaluation requires the solution of a highly nonlinear equation, coupling Exc and the corresponding x>xc [19]. For a rigorous treatment of this Exc one thus has to resort to an expansion in powers of e, which allows to establish a recursive procedure for the evaluation of Exc and the accompanying Vxc-... 

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