The six-vertex model eigenvectors as critical limit of the eight-vertex model bethe ansatz

The critical limit of the eight-vertex model eigenvectors obtained by means of the generalized Bethe Ansatz is shown to give the six-vertex eigenvectors as constructed in a previous paper by two of the authors. Furthermore, an explicit mapping is established between these eigenvectors and the usual Bethe Ansatz eigenvectors of the six-vertex model. This allows one to show that the indexv labeling the eight-vertex eigenstates becomes exactly the third component of the total spin in the critical limit.     

Limit cycles and Lie symmetries

Given a planar vector field U which generates the Lie symmetry of some other vector field X, we prove a new criterion to control the stability of the periodic orbits of U. The problem is linked to a classical problem proposed by A.T. Winfree in the seventies about the existence of isochrons of limit cycles (the question suggested by the study of biological clocks), already answered by Guckenheimer using a different terminology. We apply our criterion to give upper bounds of the number of limit cycles for some families of vector fields as well as to provide a class of vector fields with a prescribed number of hyperbolic limit cycles. Finally we show how this procedure solves the problem of the hyperbolicity of periodic orbits in problems where other criteria, like the classical one of the divergence, fail.     

Can the pi-facial selectivity of solvation be predicted by atomistic simulation?

This work is concerned with the rationalization and prediction of solvent and temperature effects in nucleophilic addition to alpha-chiral carbonyl compounds leading to facial diastereoselectivity. We study, using molecular dynamics simulations, the facial solvation of (R)-2-phenyl-propionaldehyde in n-pentane and n-octane at a number of temperatures and compare it with experimental selectivity data for the nBuLi addition leading to syn- and anti-(2R)-2-phenyl-3-heptanol, which give nonlinear Eyring plots with the presence of inversion temperatures. We have found from simulations that the facial solvation changes with temperature and alkane. Moreover, by introducing a suitable molecular chirality index we have been able to predict break temperatures (T(CI)) for the two solvents within less than 20 degrees of the inversion temperatures experimentally observed in the diastereoselective nBuLi addition. We believe this could lead to a viable approach for predicting inversion temperatures and other subtle solvent effects in a number of stereoselective reactions.     

Chemo‐ and Enzyme‐Catalyzed Reactions Revealing a Common Temperature‐Dependent Dynamic Solvent Effect on Enantioselectivity

The enantiomeric ratio E of enzyme‐catalyzed (Candida antarctica lipase and lipase PS) and chemo‐catalyzed (L ‐proline‐based diamines) acylation reactions of 1‐(naphthalen‐2‐yl)ethanol, 2‐phenylpropanol, and 2‐benzylpropane‐1,3‐diol is dependent on solvent and temperature. Plots of ln E vs. 1/T showed the presence of inversion temperatures (T_inv). The T_inv values for the bio‐catalyzed and the chemo‐catalyzed reactions are fairly in agreement, and correspond as well to the T_NMR values obtained by variable‐temperature ¹³C‐NMR experiments on the substrates in the same solvent of the resolution. This result demonstrates that clustering effects in the substrate solvation manage the chemical and the enzymatic enantioselectivity, and, moreover, that the solute? solvent cluster is always the real reacting species in solution for chemical as well as for enzymatic reactions.     

Generalized Hopf Bifurcation for Planar Vector Fields via the Inverse Integrating Factor

In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point p ₀ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p ₀ being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of p ₀ the differential system can always be brought, by means of a change to (generalized) polar coordinates (r, θ), to an equation over a cylinder in which the singular point p ₀ corresponds to a limit cycle γ ₀. This equation over the cylinder always has an inverse integrating factor which is smooth and non-flat in r in a neighborhood of γ ₀. We define the notion of vanishing multiplicity of the inverse integrating factor over γ ₀. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point p ₀ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.     

Liouvillian first integrals for the planar Lotka-Volterra system

We complete the classication of all Lotka-Volterra systemsx=x(ax+by+c),y=y(Ax+By+C), having a Liouvillian first integral. In our classification we take into account the first integrals coming from the existence of exponential factors.