The approachability ideal without a maximal set

We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the ${\omega }_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[{\omega }_2]$ does not have a maximal set modulo clubs.     

A simple, accurate, geometrical approximation to the Keplerian motion

A new method which shows, simply and accurately, the planetary position as a geometrical point in the orbit as a function of time is presented. By comparison with the historical hypotheses of planetary motion, it is thus visually recognizable how such hypotheses, especially those propagated in the 17th century, approached the Keplerian motion geometrically by means of observations. The reason why such a mathematically accurate hypothesis as that presented here was not developed previously, say in the 17th century, was due mainly to the inaccurate values for the solar parallax involved in the observations of that time.     

Mean Spherical Approximation-Based Partitioned Density Functional Theory

Previous literature claims that the density functional theory for non-uniform non-hard sphere interaction potential fluid can be improved on by treating the tail part by the third order functional perturbation expansion approximation (FPEA) with the symmetrical and intuitive consideration-based simple function C0(3)(r1, r2, r3) =ζ∫ dr4a(r4 - r1)a(r4 - r2)a(r4 - r3) as the uniform third order direct correlation function (DCF) for the tail part,here kernel function a(r) = (6/πσ3)Heaviside(σ/2 - r). The present contribution concludes that for the mean spherical approximation-based second order DCF, the terms higher than second order in the FPEA of the tail part of the non-uniform first order DCF are exactly zero. The reason for the partial success of the previous a kernel function-based third order FPEA for the tail part is due to the adjustable parameter ζ and the short range of the a kernel function.Improvement over the previous theories is proposed and tested.     

The Furstenberg lemma characterizes amenability

We characterize amenability in terms of the existence of equivariant assignments of measures for cocycles into the homeomorphism group of a single compact metric space.

     

Estimating the Mean of Heavy-Tailed Distributions

An asymptotically normally distributed estimate for the expected value of a positive random variable with infinite variance is introduced. Its behavior relative to estimation using the sample mean is investigated by simulations. An example of how to apply the estimate to file-size measurements on Internet traffic is also shown.     

The Fundamentals of Particle Size Analysis by Transmission Fluctuation Spectrometry.

A transmission signal measured on a flowing suspension of particles with a high spatial and temporal resolution shows significant fluctuations, which contain the complete information on particle size distribution and particle concentration. In Parts 1 and 2, the basic properties of signal fluctuations were studied for temporal averaging by a gliding time window and for spatial averaging by a circular beam of uniform intensity. However, the experimental implementation of such conditions is difficult. Now, the theory is extended for Gaussian beams of variable diameter and averaging by signal filtering in the frequency domain. This provides the basis for an experimental implementation by transmission of a laser beam and analog signal processing by an array of low pass filters.     

Statistical mechanics of the nonlinear Schrödinger equation. II. Mean field approximation

We investigate a mean field approximation to the statistical mechanics of complex fields with dynamics governed by the nonlinear Schrödinger equation. Such fields, whose Hamiltonian is unbounded below, may model plasmas, lasers, and other physical systems. Restricting ourselves to one-dimensional systems with periodic boundary conditions, we find in the mean field approximation a phase transition from a uniform regime to a regime in which the system is dominated by solitons. We compute explicitly, as a function of temperature and density (L ² norm), the transition point at which the uniform configuration becomes unstable to local perturbations; static and dynamic mean field approximations yield the same result.     

The mean association number: Ultraviolet spectroscopic study of PbCl2−H2O−C2H5OH

The concept of mean asociation number (MAN) has been proposed to describe the association state of an electrolyte. Both the mean spherical approximation (MSA) and the generalized mean spherical approximation (GMSA) are used to calculate the MAN. The theoretical results obtained are compared to the experiment results from an ultraviolet spectroscopic study of PbCl₂–H₂O–C₂H₅OH solutions and also to the predictions of the Bjerrum theory as given by Friedman.