Rearrangements result

Eliminating T dS and rearranging results in a differential equation for the Hugoniot... 

Consider an experiment that in N trials, evenLs A and B occur together N(A B) times, and event B occurs N(B) times. The conditional probability of A given B is equation 2.4-i or 2.4-2. Rearranging, results in equation 2.4-3. 

However, equation 2.(i-2 IS valid because A, B are commuting variables that lead to equation 2.6-3. Rearranging, results in one of the usual forms of the Bayes equation (equation 2.6-4). PiA E) is the prior probability of A given E. P(B A E is probability that is... 

With cyclic a-diazo ketones, e.g. a-diazo cyclohexanone 9, the rearrangement results in a ring contraction by one carbon " ... 

Film rearrangement resulting in the formation of oxide subgrain and grain boundaries these paths of easy ion migration promote the formation of oxide islands and result in an increase in the growth rate of the oxide. 

Scheme 9.9 a) SAE on dienols. b) Payne rearrangement, resulting in byproducts. 

Substituting specific growth rate based on the Monod rate equation into (6.8.2), the rearranging results in ... 

Above C4, aliphatic aldehydes undergo the McLafferty rearrangement, resulting in an observed m/z 44 ion, provided the a-carbon is not substituted. Substitution on the a-carbon results in a higher m/z peak. (See the proceeding text.)... 

Another compound for which degenerate Cope rearrangements result in equivalence for all the carbons is hypostrophene W1). In the case of the compound barbaralane (108) (bullvalene in which one CH=CH has been replaced by a CH2) ... 

Molecular rearrangement resulting from molecular collisions or excitation by light can be described with time-dependent many-electron density operators. The initial density operator can be constructed from the collection of initially (or asymptotically) accessible electronic states, with populations wj. In many cases these states can be chosen as single Slater determinants formed from a set of orthonormal molecular spin orbitals (MSOs) im as / =... 

The reaction of crotonaldehyde and methyl vinyl ketone with thiophenol in the presence of anhydrous hydrogen chloride effects conjugate addition of thiophenol as well as acetal formation. The resulting j3-phenylthio thioacetals are converted to 1-phenylthio-and 2-phenylthio-1,3-butadiene, respectively, upon reaction with 2 equivalents of copper(I) trifluoromethanesulfonate (Table I). The copper(I)-induced heterolysis of carbon-sulfur bonds has also been used to effect pinacol-type rearrangements of bis(phenyl-thio)methyl carbinols. Thus the addition of bis(phenyl-thio)methyllithium to ketones and aldehydes followed by copper(I)-induced rearrangement results in a one-carbon ring expansion or chain-insertion transformation which gives a-phenylthio ketones. Monothioketals of 1,4-diketones are cyclized to 2,5-disubstituted furans by the action of copper(I) trifluoromethanesulfonate. ... 

In contrast to MDA and hydroxynonenai, other aldehyde products of lipid peroxidation are hydrophobic and remain closely associated with LDL to accumulate to mil-limolar concentrations. Aldehydes at these elevated levels react with the protein portion of the LDL molecule, apolipoprotein B (apoB). Accumulated aldehydes bind the free amino groups from lysine residues in addition to other functional groups (-OH, -SH) on the apoB polypeptide. Consequently, the protein takes on a net negative charge and complete structural rearrangement results in the formation of ox-LDL. ox-LDL is no longer recognized by the LDL receptor, and has several pro-inflammatory properties (discussed below). 

The above studies show that dithionite reduction of mitosenes results in the formation of lOa-sulfite esters as well as sulfonates. The presence of the excellent sulfite-leaving group at the mitosene 1 Oa-position suggests that alkylation reactions at this position could still occur. The subsequent sulfite to sulfonate rearrangement results in loss of alkylation capability by this position. 

Substituting Eq. (13-34) into Eq. (13-32) and rearranging results in the basic equation governing the filter performance ... 

The impulse to the study of these cycloadditions came from the discovery that 5-spirocyclopropane isoxazolidines (or isoxazolines) undergo a thermal rearrangement resulting in the production of selectively substituted tetrahydro-(or dihydro) pyrid-4-ones (Scheme 42) [64], In particular, cyclic nitrones gave ultimately N-bridgehead bicyclic ketones, molecular skeleton of many alkaloid families [65]. 

The reaction of a-halosulphonc with a base to give an olefin is known as Ramberg-Backlund reaction52. A vinylogous version of this rearrangement results in the formation of 1,3-dienes (equation 21)53. Another variation of this reaction is shown in equation 2254. These rearrangements proceed with moderate stereoselectivity. 

Third, the metric tensor is determined by the variables 4>, //, A. On the other hand and v never appear in Eqs.(6)-(9) (reflecting the fact that x° and x5 constant dilatations are always possible without harming the commutator relations for the Killing motions), so these equations are of first order on 4>, / and A. However, the equations can be rearranged resulting in the following symbolic structure ... 

In one final improvement to the synthesis of the racemate, after it had been shown that even the unnatural enantiomer of the pheromone attracted bugs, the Z-isomer of diene ester 139 was used in the ester-enolate Claisen rearrangement, resulting in a slightly higher yield, and avoiding the use of HMPA as a solvent in that step [113]. 

A similar linear relationship is obtained for the large anodic values of rj. Starting from Eq. (6.49), taking the logarithm, and rearranging results in... 

Oxidative rearrangements resulting in the formation of heterocyclic compounds are relatively less explored, although some of them are quite important. A basic reaction is the conversion of a chalcone to 3,3-dimethoxy-l-phenylpropanone (Eq. 2) (85TL2961). This process may be adapted to heterocyclic synthesis. 

The stereochemical isomerization or structural rearrangement resulting in the interconversion of epimers. For example, the conversion of jS-D-glucose to jS-D-galac-tose involves epimerization at the C-4 carbon atom, and the epimerization at the C-2 of jS-D-glucose results in the synthesis of jS-D-mannose. 

A plausible mechanism is shown in Fig. 35. It seems to involve addition of the allyl group at the less hindered carbonyl group followed by a [3,3] sigmatropic rearrangement, resulting in the formation of allyl-substituted hydroquinones or their oxidation products. 

---