Denominators

RM Sum given by denominator in right-hand side of Equation... 

A similar form of indicator is the Profitability Index (PI), where the denominator is the maximum exposure of the project, and is applicable where the company is sensitive to the maximum exposure e.g. 

Two main types of harmonised standards can be distinguished the product standards and the horizontal standards. The latter, as their denomination indicates, contain issues of a horizontal nature, e.g. for materials or provisions relating to welding or welder approvals. These standards should be formulated in such a way as to be self-sufficient. That means it should be possible to apply them without having to use individual product standards. 

The apparent activation energy is then less than the actual one for the surface reaction per se by the heat of adsorption. Most of the algebraic forms cited are complicated by having a composite denominator, itself temperature dependent, which must be allowed for in obtaining k from the experimental data. However, Eq. XVIII-47 would apply directly to the low-pressure limiting form of Eq. XVIII-38. Another limiting form of interest results if one product dominates the adsorption so that the rate law becomes... 

The denominator is included in equation (A 1.1.115) because it is impossible to ensure tliat the trial fiinction is nomialized for arbitrarily chosen coefficients c. In order to minimize the value of e for the trial fiinction... 

Equation (A 1.6.94) is called the KHD expression for the polarizability, a. Inspection of the denominators indicates that the first temi is the resonant temi and the second temi is tire non-resonant temi. Note the product of Franck-Condon factors in the numerator one corresponding to the amplitude for excitation and the other to the amplitude for emission. The KHD fonnula is sometimes called the siim-over-states fonnula, since fonnally it requires a sum over all intennediate states j, each intennediate state participating according to how far it is from resonance and the size of the matrix elements that coimect it to the states i. and The KHD fonnula is fiilly equivalent to the time domain fonnula, equation (Al.6.92). and can be derived from the latter in a straightforward way. However, the time domain fonnula can be much more convenient, particularly as one detunes from resonance, since one can exploit the fact that the effective dynamic becomes shorter and shorter as the detuning is increased. 

One must now examine the integrability of the differentials hi equation (A2.1.121 and equation (A2.1.13), which are examples of what matliematicians call Pfoff differential equations. If the equation is integrable, one can find an integrating denominator k, a fiiiictioii of the variables of state, such that = d( ) where d( ) is... 

If a Pfaff differential expression DF = Xdx + Tdy+Zdz has the property that every arbitrary neighbourhood of a point P(x, y, z) contains points that are inaccessible along a path corresponding to a solution of the equation DF = 0, then an integrating denominator exists. Physically this means that there are two mutually exclusive possibilities either a) a hierarchy of non-intersecting surfaces (x,y, z) = C, each with a different value of the constant C, represents the solutions DF = 0, in which case a point on one surface is inaccessible... 

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... 

There are two important feaUires of this result. The energy difference between states 1 and 2 is A = E2 E. Wlien AE hv, the denominator of the second tenn becomes very small, and this temi dominates. This is the well known... 

In turn, an expression for is obtained, which, in the frequency domain, consists of a numerator containing a product of (.s + 1) transition moment matrix elements and a denominator of. s complex energy... 

Flere, the linear polarizability, a (oip 2), corresponds to the doorway stage of the 4WM process while to the window stage. We also see the (complex) Raman resonant energy denominator exposed. Of the tliree energy denominator factors required at third order, the remaining two appear, one each, m the two Imear polarizability tensor elements. 

Similar reasoning shows that were one to view along tiieX, Yand Z axes and polarization analyse the signal each time, whether excited by linearly or by naturally polarized light, the total intensity should be given by + 2/. Given eauation (Bl.3.23). if we add its denominator to twice the numerator we find that A... 

As for the Imear response, the transitions occur tlnough the electric-dipole operator and are characterized by the matrix elements hr equation Bl.5.30, the energy denominators involve the energy differences... 

The frequency denominators in the eight tenns of equation B 1.5.30 mtroduce a resonant enliancement in the nonlmearity when any of the tlnee frequencies 3) comcides with a transition from the ground state... 

The denominator in Eq. (13) can be interpreted as an average value over the momentum distribution from the initial wavepacket, that is,... 

Usually, the denominator, if present in a similarity measure, is just a normalizet it is the numerator that is indicative of whether similarity or dissimilarity is being estimated, or both. The characteristics chosen for the description of the objects being compared are interchangeably called descriptors, properties, features, attributes, qualities, observations, measurements, calculations, etc. In the formiilations above, the terms matches and mismatches" refer to qualitative characteristics, e.g., binary ones (those which take one of two values 1 (present) or 0 (absent)), while the terms overlap and difference" refer to quantitative characteristics, e.g., those whose values can be arranged in order of magnitude along a one-dimensional axis. 

For each combination of atoms i.j, k, and I, c is defined by Eq. (29), where X , y,. and Zj are the coordinates of atom j in Cartesian space defined in such a way that atom i is at position (0, 0, 0), atomj lies on the positive side of the x-axis, and atom k lies on the xy-plaiic and has a positive y-coordinate. On the right-hand side of Eq. (29), the numerator represents the volume of a rectangular prism with edges % , y ., and Zi, while the denominator is proportional to the surface of the same solid. If X . y ., or 2 has a very small absolute value, the set of four atoms is deviating only slightly from an achiral situation. This is reflected in c, which would then take a small absolute value the value of c is conformation-dependent because it is a function of the 3D atomic coordinates. 

In this model of electrostatic in teraction s, two atoms (i and j) have poin t charges tq and qj. The magnitude of the electrostatic energy (V[. , [ ) varies inversely with the distance between the atoms, Rjj. fh e effective dielectric constant is . For in vacuo simulations or simulation s with explicit water rn olecules, the den om in a tor equals uRjj, In some force fields, a distance-dependent dielectric, where the denominator is uRjj Rjj, represen is solvent implicitly. 

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a, b and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal latlice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2tt. of the X-ray crystallographic reciprocal lattice vectors ... 

The denominator in this expression is the molecular partition function ... 

---