Special formulae

There exist several sources where special formulae for the n/-symbols are presented. One reference compilation is represented by the Landolt-Bornstein series [1]. Of great advantage is the computer system Mathematica which is capable of deriving 3/ - and 6y -symbols in both symbolic and numerical forms [2], Another source using the FORTRAN computer program is available [3]. 

The 3/-symbol is invariant under the even permutations of the columns in odd permutations of the columns the sign factor is (— l), where s is the sum of the upper symbols for example 

It is invariant by switching the upper and lower members of any two columns 

The 67-symbol I/1 /2 /3 1 is non-zero only when the elements of each 

The 9/ -symbol is invariant in even permutations of rows and columns it has to be multiplied by the factor of (—1/ in odd permutations of rows and columns, where s is equal to the sum of all nine symbols. An example is 

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In 1962 special formula reading machines [41] were constructed and used at BASF Lridwigshafcn. They scanned formulas drawn on transparent grid sheets... 

This is the general result for a second-order batch reaction. The mathematical form of the equation presents a problem when the initial stoichiometry is perfect, ao =bo- Such problems are common with analytical solutions to ODEs. Special formulas are needed for special cases. 

An alternative way to find a special formula for a special case is to apply L Hospital s rule to the general case. When bo ao. Equation (1.34) has an indeterminate form of the 0/0 type. Differentiating the numerator and denominator with respect to bo and then taking the limit gives... 

Effective treatment of maple syrup urine disease involves the restriction of dietary branched-chain amino acids. Long-term treatment entails the dietary restriction of the BCAAs. This is accomplished by administration of a special formula from which these amino acids are removed. The outlook for intellectual development is favorable in youngsters in whom diagnosis is made early and who do not suffer recurrent, severe episodes of metabolic decompensation [17]. 

Fortunately, pest-control operators and others are now proceeding more cautiously, and fogging is recognized for what it is—a specialized control procedure for certain specific insect problems, not a cure-all for any problem that has presented unusual difficulties. As a specialized procedure it works best when special formulas and special equipment are used—formulas and equipment designed for the various special fogging problems, after research has first revealed the difficulties and how to overcome or avoid them. 

For the computation of the master tables given below, the necessary. elds and TVs were first computed8 for the desired values of p and t, ana then combined in accordance with the following Eqs. (l8)-(50) to give the S values.9 These formulas were obtained by the use of Eqs. (15)-(17) in connection with Eqs. (14) and (7). It will be noted that a separate formula is given in each case for /—0, for reasons that will be understood on looking at Eqs. (16). Certain additional special formulas will now be described. 

To solve permutation questions, you must use a special formula ... 

The preceding discussions have focused on an analysis of the most common operating modes for solid-liquid filtration. The analysis however is general, and does not allow practical calculations without further development of working expressions and more careful consideration of proper boundary conditions and assumptions. We now focus on a series of special formulas developed specifically for assessing optimum conditions of filtration. These offer simplified calculation methods that can be readily performed on a PC, and in some cases with a hand-held calculator. 

It is sometimes of interest to invert the transformations, computing the pair of (0. I ) coordinates from a given pair (R, Z). All four transformations can be inverted trigonometrically. The inversion for AF has been presented by Svir and Oleinick [540] (with a small typographical problem), and that for OAS by those authors themselves [428]. For some inversions, there are alternative expressions, and there are cases where special formulae must be applied, as is described below. 

The sample value of by using the special formula is [1114x 18- 13x551- 100 200... 

However, better results are obtained if specific values Kpq are taken, i.e. different proportionality factors for different pairs of atoms and orbitals, and a special formula is used for one-center off-diagonal elements (for details, see 58>). The K factors are not fitted on empirical data but obtained by trial and error after ab initio SCF calculations on model compounds (for ethylene, KCc — 1.0 for valence-shell orbitals, Xhh = 1.18, Ka,n = 1.05, K p,H = 0.98). 

The procedures described above are only valid when a size-consistent method is used to calculate the various energies. Special formulae have therefore been developed to deal with BSSE in CI(SD) calculations, since these lack size consistency. However, it seems preferable to use Eq. (20) as it stands, and to apply size-consistency corrections (cf. Section III.C) to all energies. 

For non-totally symmetric modes in any point group, there are no trace contributions. Therefore G° = 0 and Pi is given by the special formula... 

Inborn errors of metabolism PN use may be necessary in children who are unable to tolerate certain specialized formulas enterally because of... 

As will be seen in this discussion, the evolution of cleaners developed from simple soap powders to liquid formulations to products that are more specialized. All-purpose cleaners are the backbone of this development. Along with specialized formulas for specific cleaning problems, in some cases these products are augmented by specialized packaging. Generally, the packaging contributes more to the convenience of the product than the efficacy. 

Aerosol packaging has played a smaller role in this area than in others like furniture polishes, air fresheners, and hair care. It has survived most often when a thick foam meant to stick to vertical surfaces is needed as in the case of oven cleaning or bathtub enclosures. In future it can be expected that as packaging innovation continues, specialized formulas will be matched to them to create more convenient and targeted cleaning systems for consumers to use. 

BMW LT71141 Honda ZL Mitsubishi Diamond SP-II SP-I11 Nissan J-Matic Toyota Type T, T-III, and T-IV BMW special formulation ATF for Honda automatics Special formula for Mitsubishi Special formula for Nissan Toyota, ATF, and Lexus transmissions... 

The evaluation of Clebsch-Gordan coefficients, 3/-, 67- and 97-symbols can be performed on the basis of special formulae (Appendix 3) or a computer program (Appendix 5). 

The latter can be rearranged and expressed through a special formula as... 

This -symbol can be generated by utilizing the special formula [16]... 

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