Integrating factor, denominator

If a function L does not admit of an exact differential of the form (9.1.2) it may nevertheless be possible, under conditions established below, to set up functions q x, ..., Xi,..., Xn) such that the ratio dL/q = dR does constitute an exact differential. Pfaffian forms of this genre are of special interest they are said to be holonomic or integrable. For obvious reasons q is said to be an integrating denominator and 1/, an integrating factor. 

The manner in which y is transformed to y(x) is arbitrary, but is commonly done by applying an integrating denominator , or an integrating factor. Thus, if the differential expression... 

Thus, the statistical integrals are defined with the help of functional integrals. The denominator in Equations 15 and 16 is a normalizing factor which prevents the divergence of the statistical integrals—this constitutes the first renormalization procedure. 

The first integral factor of the Eq. 20 only depends on the kinetic energy of a system and is equivalent to a system of non-interacting particles, the denominated ideal gas. Therefore this factor is called the ideal part Q of the partition... 

So far, the themiodynamic temperature T has appeared only as an integrating denominator, a fiinction of the empirical temperature 0. One now can show that T is, except for an arbitrary proportionality factor, the same as the empirical ideal-gas temperature 0jg introduced earlier. Equation (A2.1.15) can be rewritten in the fomi... 

The Caratheodory theorem establishes the existence of an integrating denominator for systems in which the Caratheodory principle identifies appropriate conditions — the existence of states inaccessible from one another by way of adiabatic paths. The uniqueness of such an integrating denominator is not established, however. In fact, one can show (but we will not) that an infinite number of such denominators exist, each leading to the existence of a different state function, and that these denominators differ by arbitrary factors of . Thus, we can make the assignment that A F (E ) = = KF(E) = 1. 

This expression may be interpreted as a ratio of two partition functions. In the denominator we have the partition function Z / of all trajectories starting in region with endpoint anywhere the integral in the numerator is the partition function Zai t) of all trajectories starting in. c/ and ending in 38 [this is the normalizing factor of (7.11)]. We can then view the ratio of partition functions as the exponential of the free energy difference between these two ensembles of trajectories... 

In equation 46-79, Twu represents the mean computed transmittance for Uniformly distributed noise and the parenthesized (1) in both the numerator and the denominator is a surrogate for the actual voltage difference between successive values represented by the A/D steps essentially a normalization factor for the actual physical voltages involved. In any case, if the actual voltage difference were used in equation 46-79, it would be factored out of both the numerator and the denominator integrals, and the two would then cancel. Since the denominator is unity in either case, equation 46-79 now simplifies to... 

The factor (z1 -f- xr2) in the numerator cancels the one in the denominator and therefore the two integrations can be performed separately. Their evaluation follows exactly the argument given in obtaining Eq. (347) the result is... 

The denominator is finite. In the numerator we may omit the factor nR (z because it is finite and non-zero and hence does not affect the convergence. The remaining triple integral can be split ... 

Here Peq, the equilibrium probability density defined earlier, is integrated (dw) over the 6N dimensional reactant zone A to obtain the normalizing factor in the denominator. In the numerator, the same density, is integrated (dor) over the 6N-1 dimensional surface S, with various weight factors which, like Peq, are functions of the coordinates q and momenta p. The factor u (ja,q) is the normal component of the veloci-... 

Due to this rapid decrease of the integral, the only part of significance is that where p and p" differ little from one another (and, hence, from Pmax as well)- This allows us to disregard ((p")2/2m) — ((p )2/2m) in the denominator compared to Ek — E0. Then, in the expression for U, the following sum is factored out ... 

Due to the excitation energies of both S and Q in the denominator the local integrations do not lead directly to monomer factors, but by invoking the Unsold approximation [97] it is possible to split the denominator into a product ... 

Partial Fractions. It is often convenient or desirable (e.g., in some difficult integrations) to break up a complicated factored polynomial expression in the denominator into partial fractions involving new denominators of order no higher than 2. For instance, it can be shown that the fraction on the left can be decomposed into a sum of the simpler fractions on the right ... 

Now consider the contributions of Eq. (3.19) from intermolecular interactions. Comparing Eq. (3.17) and Eq. (3.18), this is seen to be the logarithm of a ratio of integrals. Simple proportionality factors cancel in forming the ratio. Then the denominator of that ratio is a partition function for the uncoupled N + 1)-molecule system, i.e., without interactions between the A -molecule solution and the distinguished molecule. The numerator is similarly proportional to the partition function for the physical N +1)-molecule system. We thus write... 

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