On the stability of the additive Cauchy functional equation in random normed spaces

Some stability results for the functional equations of Cauchy and Jensen in probabilistic setting are proved by using the fixed point method.     

Random coefficient differential equation models for bacterial growth

In the mathematical modeling of population growth, and in particular of bacterial growth, parameters are either measured directly or determined by curve fitting. These parameters have large variability that depends on the experimental method and its inherent error, on differences in the actual population sample size used, as well as other factors that are difficult to account for. In this work the parameters that appear in the Monod kinetics growth model are considered random variables with specified distributions. A stochastic spectral representation of the parameters is used, together with the polynomial chaos method, to obtain a system of differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time.     

The distribution of the domination number of class cover catch digraphs for non-uniform one-dimensional data

For two or more classes of points in R^d with d≥1, the class cover catch digraphs (CCCDs) can be constructed using the relative positions of the points from one class with respect to the points from one or all of the other classes. The CCCDs were introduced by Priebe et al. [C.E. Priebe, J.G. DeVinney, D.J. Marchette, On the distribution of the domination number of random class catch cover digraphs. Statistics and Probability Letters 55 (2001) 239-246] who investigated the case of two classes, X and Y. They calculated the exact (i.e., finite sample) distribution of the domination number of the CCCDs based on X points relative to Y points both of which were uniformly distributed on a bounded interval. We investigate the distribution of the domination number of the CCCDs based on data from non-uniform X points on an interval with end points from Y. Then we extend these calculations for multiple Y points on bounded intervals.     

Geometry of random sequential adsorption

By sequentially adding line segments to a line or disks to a surface at random positions without overlaps, we obtain configurations of the one- and two-dimensional random sequential adsorption (RSA) problem. We have simulated the one- and two-dimensional problem with periodic boundary condition. The one-dimensional simulations are compared with the exact analytical solutions to give an estimate of the accuracy of the simulation. In two dimensions the geometrical properties of the RSA configuration are discussed and in addition known results of the RSA process are reproduced. Various statistical distributions of the Voronoi-Dirichlet (VD) network corresponding to the RSA disk configuration are analyzed. In order to characterize pores in the RSA configuration, we introduce circular holes. There is a direct correspondence between vertices of the VD network and these holes, and also between direct/indirect geometrical neighbors and these holes. The hole size distribution is found to be a parabola. We also find general relations that connect the asymptotic behavior of the surface coverage, the correlation function, and the hole size distribution.     

Estimation of surface desorption times in hydrophobically coated nanochannels and their effect on shear-driven and pressure-driven chromatography

The present paper provides a detailed analysis of the analyte-wall adsorption effects in nanochannels, including a random walk study of the analyte-wall collision frequency, and uses these insights to estimate wall desorption times from chromatographic experiments in nanochannels. Using coumarin dye analytes and using a methanol/water mixture buffered at pH 3 in 120-nm deep channels, the surface desorption times on naked fused-silica glass were found to be maximally of the order of 60 to 150 μs, while they were found to be on the order of 100 to 500 μs on a hydrophobically coated wall. These nonzero adsorption and desorption times lead to an additional band broadening when conducting chromatographic separations. Shear-driven flows, requiring a noncoated moving wall and a stationary coated wall, intrinsically turn out to be more prone to this effect than pressure-driven or electro-driven flows for example. The present study also shows that, interestingly, the number of analyte-wall collisions increases with the inverse of the channel depth and not with its second power, as would be expected from the Einstein–Smoluchowski relationship for molecular diffusion.     

Self-avoiding walks in slits and slabs with interactive walls

Self-avoiding walk models of a polymer confined between two parallel attractive walls in two and three dimensions (slits and slabs, respectively) have recently had a revival of interest. They were first studied as simple models of steric stabilisation and sensitised flocculation in colloids. The revival has been catalysed by new exact solution techniques, that have allowed the solution of directed walk models in two dimensions in full generality, and by new Monte Carlo techniques that have allowed the simulation of the full parameter space in the three-dimensional slab model. Additionally, rigorous techniques applied to the slab problem have also yielded new results. The contributions to the study of this problem that have been recently added include a novel phase diagram for the “infinite-slab” (when the walls are a macroscopic distance apart but both walls may still “see” the polymer) the delineation of the repulsive and attractive regimes of the parameter space, and a conjectured scaling theory for the problem in general dimensions.     

Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications

In this paper, we first present some sufficient conditions for the existence of a global random attractor for general stochastic lattice dynamical systems. These sufficient conditions provide a convenient approach to obtain an upper bound of Kolmogorov ε-entropy for the global random attractor. Then we apply the abstract result to the stochastic lattice sine-Gordon equation.     

The consistency index in reciprocal matrices: Comparison of deterministic and statistical approaches

When checking the inconsistency level of a positive reciprocal matrix Saaty uses a deterministic criterion based on two parameters, a benchmark (the average), and a consistency level, usually 10%. Using results from a simulation experiment with 100,000 positive random reciprocal matrices of size varying from 3 to 15, we developed a probabilistic criterion and compare it to Saaty’s index. We found that if a positive reciprocal matrix is consistent according to the deterministic criterion is also consistent according to the probabilistic criterion only if we accept a higher than usual probability of Type I error. Reducing this error implies that the benchmark must be a small percentile of the probability distribution of the consistency index.     

Stationary State Entanglement of Two Atoms Inside an Optical Cavity under Noise

In this paper, the entanglement properties of a system of two atoms inside an optical cavity in a stochastic interaction with field are studied by the Jaynes-Cummings Model. The phase telegraph noise is considered as a noise term and an exact solution to the model is obtained. The solution reveals the resulting decoherence effects of the noise on the entanglement properties of the system. It shows that under the noise the individual atoms do not entangle with the cavity field. However, a strong atom-atom entanglement is observed in a stationary state. It is seen that a relatively strong noise is cooperative in the construction of the steady state atom-atom entanglement.     

EDGEWORTH EXPANSION FOR NEAREST NEIGHBOR- KERNEL ESTIMATE AND RANDOM WEIGHTING APPROXIMATION OF CONDITIONAL DENSITY

§ 1　 IntroductionThere are many discussions on the estimate of density,for example,the kernel andnearest neighbor estimates.Reference [1 ] gave the nearest neighbor-kernel estimate ofconditional density.[2 ] discussed the properties on the nearest neighbor-kernel estimateand bootstrap approximation of conditional density.[3 ] studied some asymptotical proper-ties on kernel estimate and random weighting approximation of density.In this paper,wewill discuss the properties of the nearest neighb…