Asymptotic dependence

For sufficiently slow addition (t/ > tmt) the selectivity becomes independent on the feed time. The position of the asymptote depends on whether the reaction is controlled by mixing on the molecular scale or by chemical reaction it is higher for the former case. 

Fig. 4 Curves Xsp versus copolymer composition X = X for macromolecules of length / = 103 at values of parameters (Eq. 20) equal to b- 0.01 and h-2 (2) 5 (3) 15 (4). Asymptotic dependences at h 0 and h oo are depicted by thin lines (Eq. 1) and (Eq. 5), respectively. Volume fraction of monomeric units in a globule = 0.9...

Figure 16 Representative Asymptotic Dependency of the Relative Retention of Peptides on the Concentration of an Ion-Pairing Reagent in RPCt231ab...

In this formula, p. stands for gTd This is certainly an eigensystem equation. However, we must add a small correction to H, and moreover, this correction is not known in advance. It turns out that this correction can be left out, unless it is important that the linear equation is exactly solved. This is not necessary if the object is to find a good step for a macroiteration. Moreover, it turns out that, in such a context, the discrepancy introduced between this method and the exact NR-steps has the same asymptotic dependence as the error. Therefore, the method is still a second-order method with this modification, and there is no way to say a priori that this method is better or worse than the exact NR-iterations. This method is called the augmented Hessian (AH-)method. It is seen to be equivalent to a Newton-Raphson using a shifted hessian. This can be very advantageous, since this shift tends to keep the step down, and to keep the shifted hessian positive definite, when one is far from a solution. The size of... 

In case the distortion producted by the stress of potential barriers is insignificant (nk k 1) the following asymptotic dependencies (Tk = 0kx) were obtained under stationary flow ... 

Furthermore, at very large separations the arguments of the modified Bessel functions are exponentially small, it therefore becomes justified to consider the small argument expansion of the /o s. Multiplying out the two 7o terms and expanding the natural logarithm leads to the following asymptotic dependence on separation,... 

The catalytic reaction is simply a bimolecular reaction between B and R, with boundary conditions given by ce,m lz=0+ = cxBm, cj >m z=0+ = c] m. The yield of S increases monotonically as the Damkohler number of the catalytic reaction, Das, increases, and finally attains an asymptotic value when the catalytic reaction reaches its mass transfer limited asymptote. This feature is illustrated in Fig. 19, where the variation of Ys with Das is shown. It is interesting to note from Fig. 19, that the value of the mass transfer limited asymptote depends on the micromixing limitation of the homogeneous reaction. Larger is the micromixing limitation (rj) of the homogeneous reaction, more is the local... 

This asymptotic dependence of F on 0 is well known. Freed also proposed that F could well be approximated by the following anal)4ic form [3] ... 

Since equation (5-27) shows that the temperature T depends on both

evaluated numerically. However, equation (19) shows that the asymptotic dependence of t (as ( - oo) is... 

FIGU RE 11.3. The asymptotic dependence of the combustion efficiency on the dimensionless chamber length Z2 for various values of the exponent 2 the ordinary Rosin-Rammler distribution (/2 = - 2 " hh /C2 = 1-... 

On the other hand, by using Eq. (1.2.10) it is possible to obtain the expression of the asymptotic dependence I jP for low viscosity solvents at T/n i.e. for so high values of TfT/n at which, even for the highest values of Tj, the condition TflTj > 1 is obeyed. In this case, it is possible to obtain an asymptotic expansion for F... 

Figure 6. An asymptotic dependence of spin label intensity proportional to suggests diffusion of the label into the coal.

It can be argued, however, that (15) and (17) are really not independent of each other, because of the inverse variation of I with atomic size, which can be seen from the asymptotic dependence of p(r) upon radial distance from the nucleus [29,165-167] ... 

The possibility of the factorization of all diagrams involving quadruple excitation denominators is essential since this permits the elimination of all quadruple excitation denominators from fifth-order energy diagrams. This reduces the computational dependence of such terms from n9 (n is the number of basis functions) down to n6 (n7 when triple excitation denominators are present). [In all asymptotic dependences it should be recognized that the actual dependence is always far less since (1) the actual dependence is n ntin, not np+q and (2) space and spin symmetry greatly reduce the number of distinct terms. When we say np+[Pg.327]

Now we can see that (E.22) is very analogous to (E.20) the form function of a random chain of dimension D, embedded in a d-dimensional space, has the same asymptotic dependence on to q, as a solid body of dimension D, oriented at random in a d-dimensional space (with d > D). 

The method which has been implemented to generate the eigenvalues and eigenvectors of a real symmetric matrix is not, in fact, the fastest method there are methods which have asymptotic dependence on floating point operations, while the Jacobi method depends asymptotically on m. f77 implementations of these methods (the Givens and Householder methods) are available for most computers and calls to eigen may be simply replaced by corresponding calls to the other routine. 

The two kinds of Bessel functions, thus, have the asymptotic dependence of slowly damped cosines and sines. In analogy with Euler s formula = cos X i sinx, we define Hankel functions of the first and second kinds ... 

An alternative measure of chain extension is the persistent length, P , which behaves in a similar asymptotic dependence on n as observed for C in Figure 12.2.11. It is meant as the capacity of the chain to preserve the direction of the first residue (vector) of the chain. Therefore, as the directional persistence dissipates with increasing chain length and the direction of the terminal residue vector L loses correlation with that of the initial vector L, P approaches an asymptotic limit for sufficiently long chains. Both the and P functions in Figure 12.2.11 reveal the [a-D-(l-3)-gle] chain to be much more extended flian that of [P-D-(l-3)-gle]n. 

Tanguy, C. (2008a). Asymptotic dependence of average failure rate and MTTF for a recursive, meshed network architecture. In K. Ko owrocki, J. Soszynska, and E. Zio (Eds.), Proceedings of the 2nd Summer Safety Reliability Seminars, Volume 2, ul. G. Naru-towicza 11/12, 80-952 Gdansk, Poland, pp. 347-353. Polish Safety and... 

It has been proved that each of the Hartree-Fock orbitals has the same asymptotic dependence on the distance from the molecule (N.C. Handy, M.T. Matron, HJ. Silverstone,PM x. Rev. 180 (1969) 45), i.e. const - exp( —V—2Ema. r), where Cma. is the orbital energy of HOMO. Earlier, people thought the orbitals decay as exp(—2c,-r), where c,- is the orbital energy expressed in atomic units. The last formula, as is easy to prove, holds for the atomic orbitals of hydrogen atoms (see p. 178). R. Ahlrichs,... 

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