A Quantum-mechanical Approach for Constrained Macromolecular Chains

Many approaches to three-dimensional constrained macromolecular chains at thermal equilibrium, at about room temperatures, are based upon constrained Classical Hamiltonian Dynamics (cCHDa). Quantum-mechanical approaches (QMa) have also been treated by different researchers for decades. QMa address a fundamental issue (constraints versus the uncertainty principle) and are versatile: they also yield classical descriptions (which may not coincide with those from cCHDa, although they may agree for certain relevant quantities). Open issues include whether QMa have enough practical consequences which differ from and/or improve those from cCHDa. We shall treat cCHDa briefly and deal with QMa, by outlining old approaches and focusing on recent ones.Comment: Expands review published in The European Physical Journal (Special Topics) Vol. 200, pp. 225-258 (2011

Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs

We investigate existence and regularity of a class of semilinear, parametric elliptic PDEs with affine dependence of the principal part of the differential operator on countably many parameters. We establish a-priori estimates and analyticity of the parametric solutions. We establish summability results of coefficient sequences of polynomial chaos type expansions of the parametric solutions in terms of tensorized Taylor-, Legendre- and Chebyshev polynomials on the infinite-dimensional parameter domain. We deduce rates of convergence for N term truncated approximations of expansions of the parametric solution. We also deduce spatial regularity of the solution, and establish convergence rates of N -term discretizations of the parametric solutions with respect to these polynomials in parameter space and with respect to a multilevel hierarchy of Finite Element spaces in the spatial domain of the PDE

Gelfand-type problem for two-phase porous media

We consider a generalization of the Gelfand problem arising in Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to the classical Gelfand problem which utilizes single temperature approach, the state of the system is described by two different temperatures. We show that similar to the classical Gelfand problem the thermal explosion occurs exclusively due to the absence of stationary temperature distribution. We also show that the presence of inter-phase heat exchange delays a thermal explosion. Moreover, we prove that in the limit of infinite heat exchange between phases the problem of thermal explosion in two phase porous media reduces to the classical Gelfand problem with renormalized constants.Comment: 20 pages, 3 figure