The index of a critical point for densely defined operators of type in Banach spaces

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type acting from a real, reflexive and separable Banach space into This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in 2,$"> illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type Applications to nonlinear Dirichlet problems have appeared elsewhere.

     

Mechanistic aspects of the dehydration and dehydrohalogenation of halo-hydroxyformaldoxime conformers. A quantum chemical model study

A detailed exploration of the configurational and conformational space of chloro- and bromo-hydroxyformaldoximes, Xhfaox (X = Cl, Br) has been carried out with the aid of the B3LYP level of density functional theory, using the 6-31G(d,p) basis set. The most stable configuration in each series of the Clhfaox and Brhfaox conformers corresponds to the Z-s-cis, s-trans configuration, while the highest energy Z-(s-trans, s-cis) conformers were found at 7.0(7.6) and 6.0(6.6) kcal mol(-1), respectively, at the B3LYP(QCISD(T)) levels of theory. Saddle points were also located on the PES of the Clhfaox and Brhfaox compounds corresponding to Z-(s-cis, s-cis) conformers at 13.8(14.9) and 13.6(14.6) kcal mol(-1), respectively, at the B3LYP(QCISD(T)) levels. Upon dehydration Xhfaox could afford a number of isomeric CXNO species. The dehydration processes of Xhfaox are predicted to be endothermic, the computed heats of reactions found in the range of 20.5 to 86.2 kcal mol(-1) and 15.9 to 100.4 kcal mol(-1) at the B3LYP and QCISD(T) levels, respectively. The reaction pathways for the addition of water to halo-fulminates yielding the most stable Xhfaox conformers was predicted to be concerted with a single transition structure, but are asynchronous with activation barriers of 32.8 and 43.0 kcal mol(-1) for the chloro- and bromo-derivatives, respectively. The PES governing the isomerization reactions of the CXNO isomers have also been calculated, and possible isomerization pathways have been delineated. Upon dehydrohalogenation the Xhfaox conformers yield hydroxy-isocyanate or hydroxy-fulminate, the former being more stable by 31.8(18.8) kcal mol(-1) at the B3LYP(QCISD(T)) levels of theory. The reaction pathways for the addition of HX to hydroxy-isocyanate were predicted to be slightly exothermic, the heats of reactions being -3.2 and -5.5 kcal mol(-1), respectively, and have to surmount high activation barriers of 39.7 and 35.0 kcal mol(-1), respectively. Similarly, the addition of HX to hydroxy-fulminate was predicted to be much more exothermic, the heats of reactions being -34.7 and -37.3 kcal mol(-1), respectively, and have to surmount much lower activation barriers of only 10.5 and 7.5 kcal mol(-1) respectively, at the B3LYP level. Finally, calculated structures, relative stability, and bonding properties of all stationary points located on the PES of the systems and reactions studied are thoroughly discussed with respect to computed electronic properties.     

Universality class of criticality in the restricted primitive model electrolyte

The 1:1 equisized hard-sphere electrolyte or restricted primitive model has been simulated via grand-canonical fine-discretization Monte Carlo. Newly devised unbiased finite-size extrapolation methods using loci in the temperature-density or (T,rho) plane of isothermal rho(2-k) vs pressure inflections, of Q identical with(2)/ maxima, and of canonical and C(V) criticality, yield estimates of (T(c),rho(c)) to +/-(0.04,3)%. Extrapolated exponents and Q ratio are (gamma,nu,Q(c)) = [1.24(3), 0.63(3); 0.624(2)], which support Ising (n = 1) behavior with (1.23(9), 0.630(3); 0.623(6)), but exclude classical, XY (n = 2), self-avoiding walk (n = 0), and n = 1 criticality with potentials varphi(r)>Phi/r(4.9) when r-->infinity.     

Monte carlo study of coulombic criticality in polyelectrolytes

The role of charges in determining the water solubility of polyelectrolytes, a question of considerable relevance to biology, is currently unresolved. We use computer simulations to study the purely Coulombic phase separation of flexible polyelectrolytes with monovalent counterions in an athermal solvent. In agreement with recent theories we find that the critical temperature for this transition increases with chain length, but that the critical density remains unchanged. We therefore stress that the phase behavior of polyelectrolytes is qualitatively different from uncharged polymers, where the critical density decreases towards zero for long chains.