Hydrolysis of alpha-alkyl-alpha-(methylthio)methylene Meldrum's acids. A kinetic and computational investigation of steric effects

The rates of hydrolysis of alpha-R-alpha-(methylthio)methylene Meldrum's acids (8-R with R = H, Me, Et, s-Bu, and t-Bu) were determined in basic and acidic solution in 50% DMSO-50% water (v/v) at 20 degrees C. In basic solution (KOH), nucleophilic attack to form a tetrahedral intermediate (T(OH)-) is rate limiting for all substrates (k1(OH)). In acidic solution (HCl) and at intermediate pH values (acetate buffers), water attack (k1(H2O) is rate limiting for 8-Me, 8-Et, and 8-s-Bu; the same is presumably the case for 8-t-Bu, but rates were too slow for accurate measurements at low pH. For 8-H, water attack is rate limiting at intermediate pH but at pH < 4.5 MeS- departure from the tetrahedral intermediate becomes rate limiting. Our interpretation of these results is based on a reaction scheme that involves three pathways for the conversion of T(OH)- to products, two of which being unique to hydrolysis reactions and taking advantage of the acidic nature of the OH group in T(OH)-. This scheme provides an explanation why even at high [KOH] T(OH)- does not accumulate to detectable levels even though the equilibrium for OH- addition to 8-R is expected to favor T(OH)-, and why at low pH water attack is rate limiting for R = Me, Et, s-Bu, and t-Bu but leaving group departure becomes rate limiting with the sterically small R = H. The trend in the k1(OH) and k1(H2O) indicates increasing steric crowding at the transition state with increasing size of R, but this effect is partially offset by a sterically induced twisting of the C=C double bond in 8-R which leads to its elongation and makes the substrate less stable and hence more reactive. Our computational results suggest that this effect becomes particularly pronounced for R = t-Bu and explains why k1(OH) for 8-t-Bu is somewhat higher than for the less crowded 8-s-Bu.     

Substituted 2-methylene-1,3-oxazolidines, -1,3-thiazolidines, -1,3-benzothiazines, -1,3-oxazines, and substituted imidazopyrimidinediones from Cl(CH2)nNCO and Cl(CH2)nNCS and active methylene compounds

The reaction of omega-chloroalkyl isocyanates Cl(CH2)nNCO (n = 2 (2), 3 (4)) and isothiocyanate Cl(CH2)2NCS (3) with active methylene compounds CH2YY' 1 in the presence of Et3N or Na give 2-YY'-methylene-1,3-oxazolidines, (E,Z)-1,3-thiazolidines, and 1,3-oxazines from 2, 3, and 4, respectively. 2-(Chloromethyl)phenyl isocyanate 8 gives with 1 the corresponding benzo-oxazines. Ethyl 2-isothiocyanatobenzoate 10 gives the corresponding benzothiazolinone, whereas the analogous isocyanate 12 gives noncyclic enols. Ethoxycarbonyl isothiocyanate 14 gives an open-chain thioenol or an enol-thioamide. The cyanoamides CH2(CN)CONHR, R = H, Me, CHPh2, give with Et3N and 2 the bicyclic imidazopyrimidinediones 16, derived from two molecules of 2, but with their preformed Na salt they give the 1,3-oxazolidines. Reaction of cyanoacetamide with 3 in the presence of Na gave a tricyclic triaza(thia)indacene, derived from two molecules of 3. A reaction mechanism involving an initial attack of the anion 1- on the N=C=X (X = O, S) moiety gives an anion 18, which cyclizes intramolecularly and after tautomerization gives the mono-ring heterocycle. With the cyanoamides, the N- site of the ambident ion 18 attacks another molecule of 2 giving the anion 20, which by intramolecular attack on the CN, followed by expulsion of the Cl- gives the bicyclic 16 after tautomerization.     

On non-solvable Camina pairs

In this paper we study non-solvable and non-Frobenius Camina pairs (G,N). It is known [D. Chillag, A. Mann, C. Scoppola, Generalized Frobenius groups II, Israel J. Math. 62 (1988) 269–282] that in this case N is a p-group. Our first result (Theorem 1.3) shows that the solvable residual of G/O_p(G) is isomorphic either to SL(2,p^e),p is a prime or to SL(2,5), SL(2,13) with p=3, or to SL(2,5) with p7.Our second result provides an example of a non-solvable and non-Frobenius Camina pair (G,N) with |O_p(G)|=5⁵ and G/O_p(G)SL(2,5). Note that G has a character which is zero everywhere except on two conjugacy classes. Groups of this type were studies by S.M. Gagola [S.M. Gagola, Characters vanishing on all but two conjugacy classes, Pacific J. Math. 109 (1983) 363–385]. To our knowledge this group is the first example of a Gagola group which is non-solvable and non-Frobenius.     

Spin Entropy

Two types of randomness are associated with a mixed quantum state: the uncertainty in the probability coefficients of the constituent pure states and the uncertainty in the value of each observable captured by the Born’s rule probabilities. Entropy is a quantification of randomness, and we propose a spin-entropy for the observables of spin pure states based on the phase space of a spin as described by the geometric quantization method, and we also expand it to mixed quantum states. This proposed entropy overcomes the limitations of previously-proposed entropies such as von Neumann entropy which only quantifies the randomness of specifying the quantum state. As an example of a limitation, previously-proposed entropies are higher for Bell entangled spin states than for disentangled spin states, even though the spin observables are less constrained for a disentangled pair of spins than for an entangled pair. The proposed spin-entropy accurately quantifies the randomness of a quantum state, it never reaches zero value, and it is lower for entangled states than for disentangled states.     

Competitive facility location under attrition

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed point and saddle point problems, and noncooperative games as particular cases. In this paper sufficient conditions for the existence of solutions of an equilibrium problem are given by weakening the assumption of quasiconvexity of the involved equilibrium bifunction. The existence of solutions is established both in presence of compactness of the feasible set as well with a coercivity assumption. The results are obtained in an infinite dimensional setting, and they are based on the so called finite solvability property which is weaker than the recently introduced finite intersection property and in turn, weaker than most common cyclic and proper quasimonotonicity. Some examples are presented to illustrate the various cases in which other existence results for equilibrium problems do not apply. Finally, applications to the solution of quasiequilibrium problems, quasioptimization problems and generalized quasivariational inequalities are discussed.