Asymptote right-hand

Thus, when p(0) and p(L) both become sufficiently Large, the right hand side of equation (10.10) should asymptote at a value which is independent of the particular substances employed or the composition of the mixture, and... 

Formula ( ) is the base for asymptotic computations of and and (1") lends itself to generalizations. (Indeed, in the general term of the series on the right hand side of (1") we recognize the cycle index of the symmetric group of n elements.)... 

It is a consequence of the assumptions of the lemma that the second term of the right-hand side of (4.36) is either regular or at least "less seriously" singular than the first term on the right the second term is, in particular, regular if / 0. The proposition states that the coefficient behaves asymptotically like the coefficient of x" in the power expansion of... 

If the second term on the right-hand side of the equation is omitted, the latter is transformed into Eq. (2.76). As the omission is possible only for t < tj, Fourier transformation of the reduced equation holds for co-tj 1 only. Consequently, the equality (2.75) is of asymptotic character, and may not be utilized to find full g(co) or its Fourier-transform Kj(t) at any times. When it was nevertheless used in [117], the rotational correlation function turned out to be alternating in sign. The oscillatory behaviour of Kj(t) occured not only in a compressed gas, but also at normal pressure, when Kj(t) should vanish monotonically, if not exponentially. The origin of these non-physical oscillations is easily... 

Plotting the overpotential against the decadic logarithm of the absolute value of the current density yields the Tafel plot (see Fig. 5.3). Both branches of the resultant curve approach the asymptotes for r RT/F. When this condition is fulfilled, either the first or second exponential term on the right-hand side of Eq. (5.2.28) can be neglected. The electrode reaction then becomes irreversible (cf. page 257) and the polarization curve is given by the Tafel equation... 

The second term on the right-hand side of Eq. (146) vanishes in the continuum limit when use is made of electrical neutrality. For the defects in the impure crystal the term is again zero. In the intrinsic case it is not identically zero but is much smaller than the other terms (details can be found in Ref. 4). The final term to be evaluated in Eq. (146) is found, by substituting the asymptotic value of mu, to be... 

The solutions lie on the part of the hyperbola situated on the right-hand side of the asymptote... 

Remark. Equation (7.14) is strictly correct in the sense that it is the first term of a mathematically well-defined asymptotic expansion in 1/y. To estimate whether it is also a good approximation we require that y 1 P(1 differential operator on the left-hand side of (7.9) has eigenvalues of order 1, the zero eigenvalue having been extracted by the solubility condition. Thus P(1 is comparable with the right-hand side, i.e., of order dP(0 /dx — /P(0. Our requirement amounts therefore to... 

Consequently, in this case, the surface is practically free. Let us consider the asymptotes at high (6g - oo) and low (6g -> 0) pressures. If 6g - oo, the equation for cA will have a small parameter l/6g but not for all summands and on the right-hand side of the equation for 9Z there appears a large parameter 6g. Let us write... 

For linear sets of differential equations having an cu-invariant limited polyhedron, an eigenvalue for the matrix of the right-hand side can be either zero or have a negative real part, i.e. after eliminating linear laws of conservation, a steady-state point of these systems becomes asymptotically stable. 

Figure 4.31 Data for the characterization of an electrostatic lens, (a) Positions of the focal and principal planes (left-hand and right-hand sides are indicated by the subscripts Y and r respectively) and their distances (optical sign conventions are disregarded, i.e., the distances are described only by their lengths). (b) Geometrical construction applied to image the arrow ye by means of characteristic asymptotic trajectories, (c) Geometrical construction for an asymptotic ray with a pencil angle a,e. The shaded areas are needed for the derivation of the linear and angular magnification factors of the lens. For details see main text.

The linewidth for the 0 = 1 states shows quite the opposite dependence on J. It first rises with J and reaches a maximum around J = 3. In addition, it is significantly smaller than the linewidths for the 0=0 resonances. Unlike the latter, the 0=1 quasi-bound states cannot directly decay to the j = 0 asymptotic channel which has the helicity quantum number 0 = 0. Transitions between different helicity states can be promoted only through Coriolis coupling, i.e., the last two terms on the right-hand side of (11.7). Coriolis coupling, on the other hand, is diagonal in the rotational quantum number j. Therefore, the j = 1,0 = 1 resonances can only decay via a two-step process of the form... 

Finally, by inserting this result in the left-hand side of Eq. (86), we confirm that the right-hand side of this equation decays asymptotically as stated in theorem 1. 

Since the KLI equation must be satisfied in the asymptotic region, the expression in square brackets on the right-hand side of Eq. (Ill) must vanish identically for r —> oo. The term involving >(Af -i)[Pg.46]

The dominant term in the Boltzmann Equation (84) is assumed to be the collision term, i.e., the second term on the right-hand side of (84). This then implies that, as t —> 00, solutions to Equation (82), denoted 11l0 (r. v p. u, e) represent a good approximation to the asymptotic solution to (84). Consequently, we choose... 

This matrix is nnitary, b( canse all the matrices on the right hand side of E(p(115) are unitary. It should be noted that E(i.(115) does not describe the formal transformation from the body-fixed to space-fixed frame by coordinate transformation [1], but just gives the transformation of electronic basis sets in the asymptotic region. 

Th( analysis nnderlying the ( valuation of the S matrix elements was fominlated for the J = 0 (and / = 0) case [56] and did not take proper account of the correct asymptotic phases of the spherical Bessel functions [124]. This phase should have been exp — , R. — / tt/2) rather than the phase given in Eep (45) above. To correct for this omission in both the reactant and product duuinels we must multiply by a phase of exp(— / 7t/2) = ir for the products and i for the reactants. These factors are included on the right hand side of Ecj. (45). 

Convergence of these kind of formal expansions is questionable, however, it is believed today that at least asymptotic convergence behavior is to expected. Collecting terms in equations (21) and (22) that are proportional to the same power of X and p on the left and right hand sides we obtain recursion formulas... 

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