Equation scaled

Keywords Polyelectrolytes Poisson-Boltzmann theory Density functional theory Integral equations Scaling theory Annealed polyelectrolytes Computer Simulations... 

As the rate constant is in the units of cm molecule s all the concentrations should be converted into molecule cm (as per PSS example). The answer should be converted into ppb h by dividing the answer in cm molecule s into ppbv and turning the seconds into hours. For NO2 = 2 ppbv, NO = 1 ppbv, P(03) = 7.1 ppbv h The equation scales as per the absolute values of NO2/NO used. 

Every term in the coupled cluster amplitude equations that is nonlinear in T may be factored into linear components. As a result, each step of the iterative solution of the CCSD equations scales at worst as ca. 0(X ) (where X is the number of molecular orbitals). The full CCSDT method in which all Tycon-taining terms are included requires an iterative 0(X ) algorithm, whereas the CCSD(T) method, which is designed to approximate CCSDT, requires a noniterative O(X ) algorithm. The inclusion of all T4 clusters in the CCSDTQ method scales as... 

In total [176], the solution of the CCSD equation scales as where is the... 

Parameter to be kept constant Function Scale-up equation Scale-up exponent x... 

The elimination procedure described in the last sections forms a process, commonly called Gauss elimination. It is the backbone of the direct methods, and is the most useful in solving linear equations. Scaling and pivoting are essential in the Gauss elimination process. 

Dimensionless number Definition Equation Scaling Value ... 

Note as well that the second term of the van der Waals equation scales as length-. The interaction energies of mutually induced dipoles scale in a like fashion. In the limit of large V and high T, Equations (3.50) and (3.51) operate more like the ideal... 

We define the Schrodinger equation scale (at which the decaying scalet solutions become very close to the Schrodinger equation solution they are converging to), asch(b), to correspond to the scale at which the lowest order asymptotic correction is dominant. This definition is more sensitive to the slower convergence rate of the aforementioned, kinetic energy related, scaling transform derivatives. 

The average entropy for the subsets S, .., S produced by dividing the data set S on the basis of the states of the /tth attribute, is given by the sum of entropy for each subset, given by the previous equation, scaled by the subset s proportion to S. 

Going from CCS to CCSD, the main additional cost arises from the last two terms in the doubles right-hand side (14.6.33), which scale as - see Sections 13.7.8 and 13.7.9. The remaining terms in the CCSD equations scale as the fifth or lower powers of the size of the system. Likewise, going from CCSD to CCSDT, the extra cost arises mainly from the last two terms in (14.6.36), which scale as H. As we shall show shortly, all other terms in the CCSDT equations scale as the seventh or lower powers of the size of the system. We also make the observation that, at each level, the terms of highest cost - fire last two terms in (14.6.33) and the last two terms in (14.6.36) - all describe higher-order adjustments of the top-level excitations - the doubles in CCSD theory and the triples in CCSDT theory. As such, they are not required if we wish to describe the top-level excitations in an approximate manner. 

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