Rado's theorem for rings and modules

We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary domain or a noetherian ring. The crucial idea is to study partition regularity for general modules rather than only for rings. Contrary to previous techniques, our approach is independent of the characteristic of the coefficient ring.Comment: 19 page

Two hard spheres in a spherical pore: Exact analytic results in two and three dimensions

The partition function and the one- and two-body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore. The equivalent problem for hard disks is addressed too. We establish a relation valid for any dimension between these partition functions, second virial coefficient for inhomogeneous systems in a spherical pore, and third virial coefficients for polydisperse hard spheres mixtures. Using the established relation we were able to evaluate the cluster integral $b_{2}(V)$ related with the second virial coefficient for the Hard Disc system into a circular pore. Finally, we analyse the behaviour of the obtained expressions near the maximum density.Comment: def printed versio

Descriptors for terpene esters from chromatographic and partition measurements: Estimation of human odor detection thresholds

We have used gas chromatographic retention data together with other data to obtain Abraham descriptors for 30 terpene esters. These include the air-water partition coefficient, as log Kw, for which no experimental values are available for any terpene ester. The other descriptors are the ester dipolarity, S, the hydrogen bond basicity, B, (the ester hydrogen bond acidity is zero for the esters studied), and L the logarithm of the air-hexadecane partition coefficient. Both S and B are larger than those for simple aliphatic esters, as expected from the terpene ester structures that include ring systems and ethylenic double bonds. These descriptors can then be used to obtain a large number of physicochemical and environmental properties of terpene esters. We have analyzed experimental results on human odor detection thresholds and have constructed another equation for the calculation of these thresholds, to go with a previous equation that we have reported. Then the descriptors for terpene esters can be used to estimate the important odor detection thresholds

Multiscale Partition of Unity

We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite element mesh. The method modifies a given partition of unity such that optimal convergence is achieved independent of oscillation or discontinuities of the diffusion coefficient. The modification is based on an orthogonal decomposition of the solution space while preserving the partition of unity property. This precomputation involves the solution of independent problems on local subdomains of selectable size. We deduce quantitative error estimates for the method that account for the chosen amount of localization. Numerical experiments illustrate the high approximation properties even for 'cheap' parameter choices.Comment: Proceedings for Seventh International Workshop on Meshfree Methods for Partial Differential Equations, 18 pages, 3 figure

Levinson's theorem and scattering phase shift contributions to the partition function of interacting gases in two dimensions

We consider scattering state contributions to the partition function of a two-dimensional (2D) plasma in addition to the bound-state sum. A partition function continuity requirement is used to provide a statistical mechanical heuristic proof of Levinson's theorem in two dimensions. We show that a proper account of scattering eliminates singularities in thermodynamic properties of the nonideal 2D gas caused by the emergence of additional bound states as the strength of an attractive potential is increased. The bound-state contribution to the partition function of the 2D gas, with a weak short-range attraction between its particles, is found to vanish logarithmically as the binding energy decreases. A consistent treatment of bound and scattering states in a screened Coulomb potential allowed us to calculate the quantum-mechanical second virial coefficient of the dilute 2D electron-hole plasma and to establish the difference between the nearly ideal electron-hole gas in GaAs and the strongly correlated exciton/free-carrier plasma in wide-gap semiconductors such as ZnSe or GaN.Comment: 10 pages, 3 figures; new version corrects some minor typo

Nonparametric Bayesian estimation of a H\"older continuous diffusion coefficient

We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a H\"older-continuous diffusion coefficient with smoothness parameter $0<\lambda\leq 1.$ Our approach is straightforward to implement, as the posterior distributions turn out to be inverse Gamma again, and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets

Thermodynamics of an one-dimensional ideal gas with fractional exclusion statistics

We show that the particles in the Calogero-Sutherland Model obey fractional exclusion statistics as defined by Haldane. We construct anyon number densities and derive the energy distribution function. We show that the partition function factorizes in the form characteristic of an ideal gas. The virial expansion is exactly computable and interestingly it is only the second virial coefficient that encodes the statistics information.Comment: 10pp, REVTE