Example of the function

Part a) of this lemma corresponds to Lemma 8.3 for the discrete-logarithm case and has the same consequence The result z gives no (Shannon) information about the first parameter, b, if the second parameter, y, is chosen uniformly at random from D. This is a hiding property again. 

Remark 8.7. One can generalize the constructions from this section and their computational aspects treated in Section 8.5.4 to larger tuples of permutations than pairs, similar to the construction of hash functions in [Damg88]. (Some performance analysis of the hash functions can be found in [Fox91].) However, 

3 Factoring Case Iterated Squaring and Doubling (Or A Useful Homomorphism on an Ugly Group) 

All the functions presented in this section are based on the squaring function modulo n, which is a permutation on QR if n is a generalized Blum integer (see Section 8.1.3). Before these functions are defined, two generalizations are made  

the domains and pairs of functions,/o and/j , on them are defined. 

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Only one example of the functionalization of a bis(oxazoline) ligand with fluorinated chains has been described to date [44], but the use of these functionalized hgands does not meet our description of a supported catalyst... 

The combination of the electrogenerated chloro cation [Cl] with diphenyldise-lenide is a typical example of the functionalization of an olefin. The electrochemical oxyselenation-deselenation of (1) to (2) proceeds in an MeOH-NaCl-(Pt) system (Scheme 1) [32]. The bromide salt-promoted oxyselenation of olefins is discussed in Sect. 15.2.2. Penicillin (3) can be converted into the oxazoline-azetidinone (4) by a chloride salt-promoted paired reaction in an MeOH-t-BuOH(5 l)-MgCl2-(Pt) system at —40°C in 74 93% yield (Scheme 2) [33, 34]. This conversion probably involves an initial attack of an... 

For another illustrativ example of the functional group olerance of the Stille coupling see Kumamoto, H. Tanaka, H. J. Org. Chem. 2002, 67, 3541. 

Further examples of the functionalization of sulfur compounds with iodine(III)-nitrogen ylides include copper-catalyzed imidations of phenyl benzyl sulfide with [(sulfonylimino)iodo]benzenes possessing imidazole and pyridine rings in the sulfonyl moiety [34], and uncatalyzed tosylimidations of diaminothiocarboxylate inner salts 26 (Scheme 14) with Phi = NTs to give 27 [35]. 

Table 9.5 Examples of the functional groups used to link carriers with drugs...

Among the numerous applications of this reaction reported over four decades, some typical examples of the functionalization are outlined in Schemes 3 to 10. Thus, Scheme 3 outlines the well-known Barton transformation of corticosterone acetate nitrite into aldosterone acetate oxime which, with nitrous acid, gives aldosterone 12-acetate [4]. The yield of 18-oxime is, however, rather low (21 %), since attack by the 11/3-alkoxyl radical on C-19 competes with the desired attack on C-18. Incorporation of the 1,2-double bond into the nitrite prevented undesired C-19 attack by the radical and a far better yield (47 %) of 18-oxime was achieved [5] (Scheme 4). This example may indicate that the Barton reaction is sensitive to... 

The supramolecular assembly of (3-cyclodextrin-capped AuNPs on ITO previously coated with a monolayer of ferrocene molecules was shown to exhibit interesting redox properties, as presented in Figure 8.5.25 This is a fine example of the functionalization of a nanoparticle with a functionalized recognition element to create a unique supramolecular scaffold with tailor-made properties. Specifically, the electrocatalytic determination of ascorbic acid was demonstrated by this surface assembly with excellent discrimination over dopamine. 

Further examples of Mannich reactions aimed at the modification of materials range from the use of suitable aldehyde mixtures in the crosslinking of proteins for contact Icnscs-- to the modification of rcsols with arylamines - for obtaining rapidly curing resins employed as wood adhesives. Some further examples of the functionalization of macromolecular derivatives for modifying the final properties of the material are found in Chap. Ill, D. 

As an example of the functions described above, the pre-crash function is described in more detail here. 

Figure 3. Example of the function of RPS in predicting retention times of three alkylbenzenes. Input information is underlined.

An impressive example of the functional group compatibility of the Clauson-Kaas reaction was reported by Frontata-Uribe and co-workers in the... 

Methods based on determination of specific elements In certain cases, the polymers contain functional groups of some specific element. For these polymers, it is sufficient to analyze the polymers for that particular element, and the degree of functionalization can be determined from the percentage of that element. In this case, it is assumed that no low-molecular-weight substances are adsorbed or entrapped in the polymer. Table 3-1 lists typical examples of the functional groups in polymers that can be determined by elemental analysis. 

This system is organized in vascular plants as a 30 nm diameter plasma membrane complex, known as the rosette TC, that was first described by Mueller and Brown more than 25 years ago [42]. The rosette is composed of an estimated 36 subunits comprising at least three types of cellulose synthase A proteins. All three components have been shown to be essential for correct assembly and enzymatic activity of the complex in Arabidopsis [43, 44]. The physical interaction of these three components has recently been demonstrated by bimolecular fluorescence complementation [45]. The system also offers a beautiful example of the functional aspects of this organization, as cortical microtubules guide the movement of cellulose synthase complexes through the plasma membrane during cell wall biosynthesis [46,47],... 

The homo- and copolymerization of methacrylates and acrylates and the synthesis of block copolymers is also reported [108], Recently, it is reported that new organometallic compounds (late metals) also produce ethylene/MMA copolymers [54, 55]. The development of polymerization catalysts incorporating late transition metals is a promising area of research, since late metals are typically less oxophilic, and thus more functional group tolerant, than early metals. Examples of the functional group tolerance of late metals in insertion-type reactions include reports on Ru, Rh, Ni, and Pd catalysts. 

The lateral loops of lampbrush chromosomes are active chromosomal regions, similar to puffs. These chromosomes have been studied less than polytene chromosomes from thp point of view of the differential activity of homologous chromosomes. However, there are examples of the functional heteromorphism in the behavior of homologous loops (Callan, 1963, 1965). 

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