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Retinoates are obtained in high yield by the reaction of N-retinylimidazolium p-toluenesulfonate with alcohols. 相似文献
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We establish strict entropy production bounds for the Boltzmann equation with the hard-sphere collision kernel. Using these entropy production bounds, we prove results asserting that the rate at which strongL
1 convergence to equilibrium occurs is uniform in wide classes of initial data. This extends our previous results in this direction, which applied only to a very special collision kernel. Moreover, the present results provide computable lower bounds; compactness arguments are entirely avoided. The uniformity is an important ingredient in our study of scaling limits of solutions of the non-spatially homogeneous Boltzmann equation, and is the main focus of this paper. However, the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous Boltzmann equation with initial data far from equilibrium to reach any given small strongL
1 neighborhood of equilibrium. 相似文献
25.
We give a simple and entirely elementary proof of Gasper's Theorem on the Markov sequence problem for Jacobi polynomials. It is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by Marc Kac, and the methods developed here yield new estimates relevant to the collision model. 相似文献
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E. A. Carlen M. C. Carvalho R. Esposito J. L. Lebowitz R. Marra 《Journal of Geometric Analysis》2006,16(2):233-264
We prove theorems characterizing the minimizers for the Cahn-Hilliard free energy functional, which is used to describe the
liquid vapor phase transition (or the 2 state magnetization transition). In particular, we exactly determine the critical
density for droplet formation, and the geometry of the droplets. 相似文献
28.
The paper concerns L
1-convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials (−4≤γ<0), with and without angular cutoff. We prove the time-averaged L
1-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater
than 2+|γ|. For the usual L
1-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial
data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cutoff
on the collision kernel with −1≤γ<0, there are algebraic upper and lower bounds on the rate of L
1-convergence to equilibrium. Our methods of proof are based on entropy inequalities and moment estimates.
E.A. Carlen work partially supported by US National Science Foundation grant DMS 06-00037. M.C. Carvalho work partially supported
by POCI/MAT/61931/2004.
X. Lu work partially supported by NSF of China grant 10571101. 相似文献
29.
Reconstruction by irregular sampling and sampling at rates below the Nyquist rate are important issues for the experimental
characterization of dynamical systems. For the space of functions that can be approximated by chirps, we prove a reconstruction
theorem by random sampling at arbitrary rates. 相似文献
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