Multiplier function

Guldi D M and Asmus K-D 1997 Photophysical properties of mono- and multiply-functionalized fullerene derivatives J. Phys. Chem. A 101 1472-81... 

Silyl ethers are among the most frequently used protective groups for the alcohol function. This stems largely from the fact that their reactivity (both formation and cleavage) can be modulated by a suitable choice of substituents on the silicon atom. Both steric and electronic effects are the basic controlling elements that regulate the ease of cleavage in multiply functionalized substrates. In plan-... 

Possible ways to multiply" functional groups a) and b) a) classically" with carbon (four times) and benzene (six times), respectively, and c) dendritically" (n times) by functionalizing the dend-rimer periphery. 

The presence of the Lagrange multiplier function o)(r) is fundamental in this variation, as it this multiplier that assumes the role of the effective Kohn-Sham potential ... 

The second term in Eq. (3) is, as stated at the end of Section IV, the relaxation mean of the integral I[St ] for the information-minimizing (generalized canonical) S, having the basic macroscopic functions p + t dpjdt)0, wa + t duJ0t)o, E + t(SE/dt)0. This ", may be calculated directly, but with the involvement of Lagrange multiplier functions (see below), and the same will follow for l[8F Then, to the first order,... 

On the other hand, we have the identity Sra = 0 throughout F so that SSra = 0 also. This can be written in terms of the Kronecker delta as Sap SSa 3 = 0. It introduces, according to the classical theory, a Lagrange multiplier function of (x), /(x) so the deduction from Eq. (29) is... 

J. Marco-Contelles, P. Ruiz, B. Sanchez, and M. L. Jimeno, A new synthetic route to chiral, multiply functionalized cyclopentane rings, Tetrahedron Lett. 33 5261 (1992). 

This secondary or intermediate level has an important multiplier function. It is firmly linked to the national standards and provides traceable calibration means (mainly certified reference materials) and test samples to the workshop level, which consists essentially of chemical testing laboratories (including medical laboratories) which are required to give evidence to their customers that their measurement results are traceable to recognized references. In the case of medical laboratories, the traceability requirement also has a legal background. 

This multiplier function is of growing importance because it will not be possible in future for the national lab-... 

The multiplier function at the intermediate level is fulfilled by the verification authorities of the federal states. The result is the deployment of a large number of verified gas analysers whose measurements are traceable to the SI units through the structure described. 

Another potential of functionalized alkoxide ligands was uncovered about ten years earlier. Tailor-made, predominantly multiply functionalized ligands attracted attention as models for binding sites of metallo-enzymes [95]. 

Krambeck (1994b) has observed that this way of describing reversibility is probably not physically meaningful, because, for instance, detailed balancing would not be satisfied. On the other hand, one could add a nonunity multiplying function to any example of reversible continuous linear kinetics, thus obtaining reversible uniform kinetics. 

Trace amounts of complexes 3 and 20 promote the synthesis of optically pure, multiply functionalized, versatile intermediates such as pyrones or lactones from activated, acid-labile siloxydienes with aldehydes. The reagents typically work under mild conditions and therefore promote the survival of valuable functionality in the dienophile, the diene, and cycloadduct [105-107]. As a consequence this procedure is applied in the total synthesis of various natural products, often requiring an intramolecular Diels-Alder approach [106]. Specific interactivity of the chiral precatalyst Eu(hfc)3 (hfc = 3-(heptafluorpropylhydroxymethylene)-D-camphorate with Danishefsky s diene bearing a chiral auxiliary resulted in cycloaddition products of high diastereofacial excess (95 % eq. (8)) [105]. 

Let us show that this effect is associated with the asymptotical behavior of the effective potential inversely proportional the square distance. The expression for the density of bound states (13) contains a multiplier, -function accounting for the loss of ions with the trajectories meeting the grain. Its argument is given by... 

A typical multiplying function for the continuously acting voltage restraint is -... 

Recently, extensive synthetic studies on multiply functionalized chiral arylphosphines by Pd-catalyzed arylation of primary and secondary phosphines with aryl halides have been carried out by Stelzer [2]. The synthetic methods can be summarized by the following general schemes. 

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