Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential

We prove the uniqueness of bounded solutions for the spatially homogeneous Fokker-Planck-Landau equation with a Coulomb potential. Since the local (in time) existence of such solutions has been proved by Arsen'ev-Peskov (1977), we deduce a local well-posedness result. The stability with respect to the initial condition is also checked

On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity

We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedeness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules. Our proof relies on the ideas of Tanaka: we give a probabilistic interpretation of the Boltzmann equation in terms of a stochastic process. Then we show how to couple two such processes started with two different initial conditions, in such a way that they almost surely remain close to each other

Results and recommendations from an intercomparison of six Hygroscopicity-TDMA systems

The performance of six custom-built Hygrocopicity-Tandem Differential Mobility Analyser (H-TDMA) systems was investigated in the frame of an international calibration and intercomparison workshop held in Leipzig, February 2006. The goal of the workshop was to harmonise H-TDMA measurements and develop recommendations for atmospheric measurements and their data evaluation. The H-TDMA systems were compared in terms of the sizing of dry particles, relative humidity (RH) uncertainty, and consistency in determination of number fractions of different hygroscopic particle groups. The experiments were performed in an air-conditioned laboratory using ammonium sulphate particles or an external mixture of ammonium sulphate and soot particles. The sizing of dry particles of the six H-TDMA systems was within 0.2 to 4.2% of the selected particle diameter depending on investigated size and individual system. Measurements of ammonium sulphate aerosol found deviations equivalent to 4.5% RH from the set point of 90% RH compared to results from previous experiments in the literature. Evaluation of the number fraction of particles within the clearly separated growth factor modes of a laboratory generated externally mixed aerosol was done. The data from the H-TDMAs was analysed with a single fitting routine to investigate differences caused by the different data evaluation procedures used for each H-TDMA. The differences between the H-TDMAs were reduced from +12/-13% to +8/-6% when the same analysis routine was applied. We conclude that a common data evaluation procedure to determine number fractions of externally mixed aerosols will improve the comparability of H-TDMA measurements. It is recommended to ensure proper calibration of all flow, temperature and RH sensors in the systems. It is most important to thermally insulate the aerosol humidification unit and the second DMA and to monitor these temperatures to an accuracy of 0.2 degrees C. For the correct determination of external mixtures, it is necessary to take into account size-dependent losses due to diffusion in the plumbing between the DMAs and in the aerosol humidification unit.Peer reviewe

Block-Wise Pseudo-Marginal Metropolis-Hastings

The pseudo-marginal Metropolis-Hastings approach is increasingly used for Bayesian inference in statistical models where the likelihood is analytically intractable but can be estimated unbiasedly, such as random effects models and state-space models, or for data subsampling in big data settings. In a seminal paper, Deligiannidis et al. (2015) show how the pseudo-marginal Metropolis-Hastings (PMMH) approach can be made much more e cient by correlating the underlying random numbers used to form the estimate of the likelihood at the current and proposed values of the unknown parameters. Their proposed approach greatly speeds up the standard PMMH algorithm, as it requires a much smaller number of particles to form the optimal likelihood estimate. We present a closely related alternative PMMH approach that divides the underlying random numbers mentioned above into blocks so that the likelihood estimates for the proposed and current values of the likelihood only di er by the random numbers in one block. Our approach is less general than that of Deligiannidis et al. (2015), but has the following advantages. First, it provides a more direct way to control the correlation between the logarithms of the estimates of the likelihood at the current and proposed values of the parameters. Second, the mathematical properties of the method are simplified and made more transparent compared to the treatment in Deligiannidis et al. (2015). Third, blocking is shown to be a natural way to carry out PMMH in, for example, panel data models and subsampling problems. We obtain theory and guidelines for selecting the optimal number of particles, and document large speed-ups in a panel data example and a subsampling problem

Derivation of the particle dynamics from kinetic equations

We consider the microscopic solutions of the Boltzmann-Enskog equation discovered by Bogolyubov. The fact that the time-irreversible kinetic equation has time-reversible microscopic solutions is rather surprising. We analyze this paradox and show that the reversibility or irreversibility property of the Boltzmann-Enskog equation depends on the considered class of solutions. If the considered solutions have the form of sums of delta-functions, then the equation is reversible. If the considered solutions belong to the class of continuously differentiable functions, then the equation is irreversible. Also, we construct the so called approximate microscopic solutions. These solutions are continuously differentiable and they are reversible on bounded time intervals. This analysis suggests a way to reconcile the time-irreversible kinetic equations with the time-reversible particle dynamics. Usually one tries to derive the kinetic equations from the particle dynamics. On the contrary, we postulate the Boltzmann-Enskog equation or another kinetic equation and treat their microscopic solutions as the particle dynamics. So, instead of the derivation of the kinetic equations from the microdynamics we suggest a kind of derivation of the microdynamics from the kinetic equations.Comment: 18 pages; some misprints have been corrected, some references have been adde

Measurement and classification of human characteristics and capabilities during interaction tasks

In this paperwe address the need to design adaptive interacting systems for advanced industrial production machines. Modern production systems have become highly complex and include many subsidiary functionalities, thus making it difficult for least skilled human operators interact with them. In this regard, adapting the behavior of the machine and of the operator interface to the characteristics of the user allows a more effective interaction process, with a positive impact on manufacturing efficiency and user's satisfaction. To this end, it is crucial to understandwhich are the user's capabilities that influence the interaction and, hence, should be measured to provide the correct amount of adaptation.Moving along these lines, in this paper we identify groups of users that, despite having different individual capabilities and features, have common needs and response to the interaction with complex production systems. As a consequence,we define clusters of users that have the same need for adaptation. Then, adaptation rules can be defined by considering such users' clusters, rather than addressing specific individual user's needs

Corneal involvement in rheumatoid arthritis: an vivo confocal study

PURPOSE. To analyze the in vivo morphology of corneal cells and nerves in patients with rheumatoid arthritis (RA), with or without secondary Sj\uf6gren\u2019s syndrome (SSII), and to investigate the correlations between corneal alterations and RA activity. METHODS. Fifty patients with RA and 30 age- and gender-matched control subjects were studied. SSII was diagnosed according to the American-European Consensus Group criteria, and RA activity was evaluated by the Lansbury index (LI). Confocal microscopy was used to investigate corneal thickness, the number of epithelial and stromal cells, and keratocyte hyperreflectivity. In addition, the sub-basal plexus was assessed for the number, tortuosity, and reflectivity of the nerve fibers and the presence of beadlike formations. RESULTS. Sixteen percent of patients with RA also had SSII. Between the SSII and non-SSII groups, no significant differences were found in the LI or in the clinical and confocal variables. Significant differences were present between patients with RA and control subjects for all the variables studied except nerve reflectivity. In patients with RA with and without SSII, LI correlated significantly with the number of beadlike formations and the number of hyperreflective, activated keratocytes. CONCLUSIONS. Confocal microscopy of patients with RA showed several changes in corneal cells and nerves. The number of beadlike formations and the number of activated keratocytes could be interpreted as confocal signs of ocular surface disease activity. These correlations with the index of systemic disease activity, LI, may provide insight regarding the pathogenic mechanisms of dry eye in patients with RA