The Multiplier

Multiplier, a common term in the consulting business, is one measure of a firm s efficiency. Refer again to Table 10.7, which is a hypothetical year-to-date income statement for a consulting engineering firm. The multiplier (M), a dimensionless parameter, is defined as net revenues divided by the direct cost of labor used to produce the revenues. In other words, the hours of labor that cost the firm 400,000 must generate total net revenues of 1,200,000 so that M = 3.0. Therefore, the multiplier is a factor that the salary chargeable to projects must be marked up to cover the raw salary itself, non-reimbursables, overhead, taxes, and profit. 

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Equations (169) and (171), together with Eqs. (170), fomi the basic equations that enable the calculation of the non-adiabatic coupling matrix. As is noticed, this set of equations creates a hierarchy of approximations starting with the assumption that the cross-products on the right-hand side of Eq. (171) have small values because at any point in configuration space at least one of the multipliers in the product is small [115]. 

Given the modified Huekel matrix and the orthogonal transform in Exereise 7-1, earry/ out the multiplieation... 

Complex numbers ean be added, subtraeted, multiplied and divided like real numbers. For example, the multiplieation of z by z gives ... 

A similar effect is obtained by using the spin-constrained UHF method (SUHF). In this method, the spin contamination error in a UHF wave function is constrained by the use of a Lagrangian multiplier. This removes the spin contamination completely as the multiplier goes to infinity. In practice, small positive values remove most of the spin contamination. 

Both remaining CgHi4 isomers have two methyl groups as substituents on a four carbon chain Thus the parent chain is butane When the same substituent appears more than once use the multiplying prefixes di tri tetra and so on A separate locant is used for each substituent and the locants are separated from each other by commas and from the words by hyphens... 

The multiplying affixes di-, tri-, tetra-, penta-, hexa-, hepta-, octa-, nona-, deca-, undeca-, and so on are used to indicate a set of identical unsubstituted radicals or parent compounds. The forms bis-, tris-, tetrakis-, pentakis-, and so on are used to indicate a set of identical radicals or parent compounds each substituted in the same way. The affixes hi-, ter-, quater-, quinque-, sexi-, septi-, octi-, novi-, deci-, and so on are used to indicate the number of identical rings joined together by a single or double bond. 

In the case of complex entities such as organic ligands (particularly if they are substituted) the multiplying prefixes bis-, tris-, tetrakis-, pentakis-,. . . are used, i.e.,-kis is added starting from tetra-. The modified entity is often placed within parentheses to avoid ambiguity. 

Since the electrophilic reagent attacks the multiply-bonded nitrogen atom, as shown for (68) and (69), the orientation of the reaction product is related to the tautomeric structure of the starting material. However, any conclusion regarding tautomeric equilibria from chemical reactivity can be misleading since a minor component can react preferentially and then be continually replenished by isomerization of the major component. 

Substituents are expected to alter the electron density at the multiply-bonded nitrogen atom, and therefore the basicity, in a manner similar to that found in the pyridine series. The rather limited data available appear to bear out these assumptions. The additional ring nitrogen atoms in triazoles, oxadiazoles, etc. are quite strongly base-weakening this is as... 

Azoles containing a free NH group react comparatively readily with acyl halides. N-Acyl-pyrazoles, -imidazoles, etc. can be prepared by reaction sequences of either type (66) -> (67) or type (70)->(71) or (72). Such reactions have been carried out with benzoyl halides, sulfonyl halides, isocyanates, isothiocyanates and chloroformates. Reactions occur under Schotten-Baumann conditions or in inert solvents. When two isomeric products could result, only the thermodynamically stable one is usually obtained because the acylation reactions are reversible and the products interconvert readily. Thus benzotriazole forms 1-acyl derivatives (99) which preserve the Kekule resonance of the benzene ring and are therefore more stable than the isomeric 2-acyl derivatives. Acylation of pyrazoles also usually gives the more stable isomer as the sole product (66AHCi6)347). The imidazole-catalyzed hydrolysis of esters can be classified as an electrophilic attack on the multiply bonded imidazole nitrogen. 

The multiplier tells how the optimal value of the objective func tion changes for this small change in the value of a constraint while holding all the other constraints at zero. It is for this reason that they are often called shadow prices. 

The above ratings are for single bars. When multiple bars are used, apply the multiplying factors, as recommended in Table 50.5. These factors will account for the restricted heat dissipation and additional skin effect due to the larger number of bars. 

Most produets have a number of eomponents, subassemblies and assemblies whieh all must funetion in order that the produet system funetions. Eaeh eomponent eontri-butes to the overall system performanee and reliability. A eommon eonfiguration is the series system, where the multiplieation of the individual eomponent reliabilities in the system, Rj, gives the overall system reliability, as shown by equation 4.67. It applies to system reliability when the individual reliabilities are statistieally independent (Leiteh, 1995) ... 

The minimum speeified maximum linear limit of the input to the multiplier (pin3) is 2.5 V. This level should be the peak value of the divided reetified input waveform at the highest expeeted ae input voltage at the erest of the sinusoid (370 V). If a sense eurrent of 200jUA is seleeted at this point the resistor divider beeomes... 

Step 4. Find the required bottle volume from the displacement and the multiplier. 

For an applied stress of 1 MN/m and letting X be the multiplier on this stress, we can determine the value of X to make the Tsai-Hill equation become equal to 1. 

Hence once again a stress of Ojt = 92 MN/m is the limiting condition, (iii) Tsai-Hill Criterion Letting X be the multiplier for... 

The contribution to the stress from electromechanical coupling is readily estimated from the constitutive relation [Eq. (4.2)]. Under conditions of uniaxial strain and field, and for an open circuit, we find that the elastic stiffness is increased by the multiplying factor (1 -i- K ) where the square of the electromechanical coupling factor for uniaxial strain, is a measure of the stiffening effect of the electric field. Values of for various materials are for x-cut quartz, 0.0008, for z-cut lithium niobate, 0.055 for y-cut lithium niobate, 0.074 for barium titanate ceramic, 0.5 and for PZT-5H ceramic, 0.75. These examples show that electromechanical coupling effects can be expected to vary from barely detectable to quite substantial. 

Minimization of the ErrF subject to the normalization constraint is handled by the Lagrange method (Chapter 14), and leads to the following set of linear equations, where A is the multiplier associated with the normalization. 

If there is more than one constraint, one additional multiplier term is added for each constraint. The optimization is then performed on the Lagrange function by requiring that the gradient with respect to the x- and A-variable(s) is equal to zero. In many cases the multipliers A can be given a physical interpretation at the end. In the variational treatment of an HF wave function (Section 3.3), the MO orthogonality constraints turn out to be MO energies, and the multiplier associated with normalization of the total Cl wave function (Section 4.2) becomes the total energy. 

SA(A)1011]. Static dipole polarizabilities were eomputed up to the MP4(SDQ) level [94MP557]. A study of the eleetronie strueture of the and S" " states of 192 showed that inelusion of dynamie eleetron eorrelation effeets is very important [98JPC(A)8021]. The multiplieity of the 1,3,5-triazine dieation is predieted to be a high-spin triplet while the trieation is most likely a doublet. In hexahydro-... 

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