Schmidt transformation

A simple standard procedure to render functions mutually orthogonal is to apply the Schmidt orthogonal transformation (39,42,47,53). For a set of functions i, the Schmidt orthogonalization transformation is to render each function, in turn, orthogonal to the preceding one in the series. The examples discussed in this chapter, concern only the rendering of two functions, 11) and 2), mutually orthogonal, so the Schmidt transformation... 

Figure 3.2 Generation of the orthogonal Slater lithium 2s radial function by Schmidt transformation.

This conclusion is evident when we write the Schmidt transformation as the matrix equation... 

The Schmidt transformation of ketones to IH-tetrazoles (predominantly isomer A) with sodium azide-titanium tetrachloride was rmorted by Suzuki el at [93S1218]. Mechanistically, a geminal diazide undergoes loss of nitrogen to form an a-azidonitrene which undergoes a [l,2]-substituent shift to lead to preferential formation of isomer A. However, in a few cases [R = QH5 and = 4-Me(X 6H4. = CgHs and R = CH3]... 

Nonorthogonal vectors, qj,..., q are transformed into r orthogonal vectors Pj, Pj,..., Pr- The remaining vectors p are preserved as they are orthogonal to each other, but also orthogonal with respect to the new vectors, since they do not appear in the definition q. This transformation can be carried out by various different techniques, such as the Gram-Schmidt transformation (Golub and Van Loan, 1983). 

J.M. Schmidt, Transforming between discrete and continuous angle distribution models appbeation to protein 1 torsions, J. Biomol. NMR 54 (2012) 97—114. 

A potential advantage of the Schmidt reaction is illustrated by the conversion of progesterone to the 17j5-acetylamino derivative, without affecting the A-ring. A 35 % yield is obtained when 1 mole of sodium azide in polyphos-phoric acid is used. With excess azide the A-ring is transformed into an a,jS-unsaturated lactam ... 

Thus, acetylation of aniline affords acetanilide (20), an analgesic widely used in proprietary headache remedies. A similar transformation on p-aminophenol gives the analgesic, acetaminophen (21). It is of interest that the latter is also formed in vivo on administration of 21. An interesting preparation of this drug involves Schmidt rearrangement of the hydrazone (24) from p-liydroxyacetophenone. ... 

Schmidt S, P Eortnagel, R-M Wittich (1993) Biodegradation and transformation of 4,4 - and 2,4-dihalo-diphenyl ethers by Sphingomonas sp. strain SS33. Appl Environ Microbiol 58 3931-3933. 

Drzyzga O, A Schmidt, K-H Blotevogel (1996) Cometabolic transformation and cleavage of nitrodiphenylamines by three newly isolated sulfate-reducing bacterial strains. Appl Environ Microbiol 62 1710-1716. 

The obtained spiro keto esters 2-181 could be converted into 2-183, employing among other transformations an intramolecular Schmidt rearrangement [101] of the azido ketones 2-182. 

Alkyl azides have been involved in the synthesis of indolizidinone derivatives in several ways. One example (Scheme 7) is the intramolecular Schmidt reaction between alkyl azides and ketones which can be used to transform azidoketone 24 into the corresponding indolizidinones 26 through intermediate 25 <2001JOC886> or with epoxides to obtain the indolizidine 27 <2004JOC3093>. 

For the case in which the Schmidt number is equal to 1, it can be shown [7] that the conservation equations [in terms of Cl see Eq. (6.17)] can be transposed into the form used for the momentum equation for the boundary layer. Indeed, the transformations are of the same form as the incompressible boundary layer equations developed and solved by Blasius [30], The important difference... 

Trautmann A, Akdis M, Kleemann D, Altznauer F, Simon HU, Graeve T, NoU M, Brocker EB, Blaser K, Akdis CA T cell-mediated Fas-induced ker-atinocyte apoptosis plays a key pathogenetic role in eczematous dermatitis. J Clin Invest 2000 106 25-35. Schmidt-Weber CB, Blaser K Regulation and role of transforming growth factor-beta in immune tolerance induction and inflammation. 

Schmidt demonstrated that a carbon chain analog of the 110-112 transformation exists. He found that j8-chlorovinyl phenyl ketone (113) reacts in the presence of stannic chloride with phenylacetylene to afford a small yield of 2,6-diphenylpyrylium salt (115), presumably via the... 

A proposed mechanism for this transformation, provided in Scheme 42, is based on the identification of alcohol-carbene complexes by Movassaghi and Schmidt. Mesityl substituted imidazolinylidine carbene acts as a Brpnsted base as transesterification occurs to produce LXVII. Upon O N acyl transfer, the observed product is formed. The evidence provided for this mechanism includes the control experiment in which LXVII is resubjected to the reaction conditions and proceeds with amide formation. A similar mechanism has recently been reported in a theoretical study of transesterification by Hu and co-workers [139], In light of this work, it seems reasonable to suggest a similar that mechanism is operative in the transesterification reactions discussed throughout this section. 

Also propiogeninic type macrolide moieties should be accessible via this inverse type hetero-Diels-Alder methodology. This was recently exhibited starting from cis-propenylether as heterodienophile (Schmidt, R.R. Haag-Zeino, B. Hoch, M. Liebigs Ann. Chem., in press). In a highly endo-selective cycloaddition reaction and in subsequent diastereoselective transformations of the methyl substituted dihydropyran obtained a 2,4-dimethyl-... 

Stepanova, L., M. Finegold, F. DeMayo, E. V. Schmidt, and J. W. Harper. The oncoprotein kinase chaperone CDC37 functions as an oncogene in mice and collaborates with both c-myc and cyclin D1 in transformation of multiple tissues. Mol Cell Biol. 20 4462-73.2000. 

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