Escape times for rigid Brownian rotators in a bistable potential from the time evolution of the Green function and the characteristic time of the probability evolution

The greatest relaxation time for an assembly of three- dimensional rigid rotators in an axially symmetric bistable potential is obtained exactly in terms of continued fractions as a sum of the zero frequency decay functions (averages of the Legendre polynomials) of the system. This is accomplished by studying the entire time evolution of the Green function (transition probability) by expanding the time dependent distribution as a Fourier series and proceeding to the zero frequency limit of the Laplace transform of that distribution. The procedure is entirely analogous to the calculation of the characteristic time of the probability evolution (the integral of the configuration space probability density function with respect to the position co-ordinate) for a particle undergoing translational diffusion in a potential; a concept originally used by Malakhov and Pankratov (Physica A 229 (1996) 109). This procedure allowed them to obtain exact solutions of the Kramers one-dimensional translational escape rate problem for piecewise parabolic potentials. The solution was accomplished by posing the problem in terms of the appropriate Sturm-Liouville equation which could be solved in terms of the parabolic cylinder functions. The method (as applied to rotational problems and posed in terms of recurrence relations for the decay functions, i.e., the Brinkman approach c.f. Blomberg, Physica A 86 (1977) 49, as opposed to the Sturm-Liouville one) demonstrates clearly that the greatest relaxation time unlike the integral relaxation time which is governed by a single decay function (albeit coupled to all the others in non-linear fashion via the underlying recurrence relation) is governed by a sum of decay functions. The method is easily generalized to multidimensional state spaces by matrix continued fraction methods allowing one to treat non-axially symmetric potentials, where the distribution function is governed by two state variables. (C) 2001 Elsevier Science B.V. All rights reserved.

Error sensitive historical GIS: Identifying areal interpolation errors in time series data

Historical GIS has the potential to re-invigorate our use of statistics from historical censuses and related sources. In particular, areal interpolation can be used to create long-run time-series of spatially detailed data that will enable us to enhance significantly our understanding of geographical change over periods of a century or more. The difficulty with areal interpolation, however, is that the data that it generates are estimates which will inevitably contain some error. This paper describes a technique that allows the automated identification of possible errors at the level of the individual data values.

Graphs of relations and Hilbert series

We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by $n$ generators and $\frac {n(n-1)}{2}$ relations for $n \leq 7$. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.

Acoustic, volumetric, compressibility and refractivity properties and Flory’s reduction parameters of some homologous series of alkyl alkanoates from 298.15 to 333.15 K

The speeds of sound u in, densities ? and refractive indices nD of some homologous series, such as n-alkyl ethanoates, n-alkyl propionates, methyl alkanoates, ethyl alkanoates, dialkyl malonates, and alkyl haloalkanoates, were measured in the temperature range from 298.15 to 333.15 K. Molar volume V, isentropic and isothermal compressibilities ?S and ?T, molar refraction Rm, Eykman’s constant Cm, molecular radius r, Rao’s molar function R, thermal expansion coefficient a, thermal pressure coefficient ?, and Flory’s characteristic parameters image, P*, V*, and T* have been calculated from the measured experimental data. Applicability of Rao theory and Flory–Patterson–Pandey (FPP) theory have been examined and discussed for these alkanoates.

Acoustic, volumetric, compressibility and refractivity properties and reduction parameters for the ERAS and Flory models of some homologous series of amines from 298.15 to 328.15 K

The speeds of sound u, densities ? and refractive indices nD of homologous series of mono-, di-, and tri-alkylamines were measured in the temperature range from 298.15 to 328.15 K. Isentropic and isothermal compressibilities ?S and ?T, molar refraction Rm, Eykman’s constant Cm, Rao’s molar sound function R, thermal expansion coefficient a, thermal pressure coefficient ?, and reduction parameters P*, V*, and T* in frameworks of the ERAS model for associated amines and Flory model for tertiary amines have been calculated from the measured experimental data. Applicability of the Rao theory and the ERAS and Flory models have been examined and discussed for the alkyl amines.