Nonunique solutions of kinetic equations

Two very hard particle models are solved and the nonuniqueness of the initial value problem for these (model) kinetic equations is explicitly demonstrated, when distribution functions decaying sufficiently slowly are permitted. The intimate connection between nonuniqueness and violation of conservation laws is made evident. The associated eigenvalue problems are solved. Finally, the general implications of these results for kinetic equations with transition rates that are increasing functions of the state variable, are stated in the form of a number of conjectures. They affect the solution of the Boltzmann equation for realistic intermolecular interactions when the collision rategI(g, ) is an increasing function of the relative velocityg.     

Study of the Anode Region of a DC Glow Discharge by a Spatially One-Dimensional Hybrid Method

The anode region of a dc glow discharge at low pressure and current is studied by a new self-consistent, spatially one-dimensional hybrid method. The method consists of the coupled solution of the steady-state fluid equations of electrons, ions, and excited atoms and the Poisson equation using the electron transport properties as well as the excitation and ionization frequencies from a strict solution of the non-uniform kinetic equation of the electrons. Results such as the electric potential, the electron velocity distribution function, and the densities of the charge carriers and excited atoms are reported for the anode region of neon glow discharges. Physical properties of the plasma in the anode region known from experiments have been confirmed by the model such as the occurrence of the anode fall, a formation of plateau-like areas of the potential profile in front of the anode, the anode dark space, and the anode glow.     

Impact of Electron–Electron Collisions on the Spatial Electron Relaxation in Non-Isothermal Plasmas

The impact of electron–electron collisions on the spatial relaxation of electrons in the column-anode plasma of a glow discharge, acted upon by a space-independent electric field and initiated by a constant influx at the cathode side of the plasma, is investigated in inert gas plasmas. The investigations are based on a new method for numerically solving the one-dimensional inhomogeneous Boltzmann equation of the electrons including electron–electron interaction in weakly ionized, collision-dominated plasmas. A detailed analysis of the spatial behaviour of the velocity distribution function and relevant macroscopic properties of the electrons is given for various degrees of ionization and electric field strengths. A significant impact of the electron–electron collisions on the relaxation structure and the resultant relaxation length already at relatively low ionization degrees has been found for low to medium electric fields.     

Ligand based virtual screening using SVM on GPU

In silico methods play an essential role in modern drug discovery methods. Virtual screening, an in silico method, is used to filter out the chemical space on which actual wet lab experiments are need to be conducted. Ligand based virtual screening is a computational strategy using which one can build a model of the target protein based on the knowledge of the ligands that bind successfully to the target. This model is then used to predict if the new molecule is likely to bind to the target. Support vector machine, a supervised learning algorithm used for classification, can be utilized for virtual screening the ligand data. When used for virtual screening purpose, SVM could produce interesting results. But since we have a huge ligand data, the time taken for training the SVM model is quite high compared to other learning algorithms. By parallelizing these algorithms on multi-core processors, one can easily expedite these discoveries. In this paper, a GPU based ligand based virtual screening tool (GpuSVMScreen) which uses SVM have been proposed and bench-marked. This data parallel virtual screening tool provides high throughput by running in short time. The proposed GpuSVMScreen can successfully screen large number of molecules (billions) also. The source code of this tool is available at http://ccc.nitc.ac.in/project/GPUSVMSCREEN.     

Global existence of solutions for a model Boltzmann equation

A model recently introduced by Ianiro and Lebowitz is shown to have a global solution for initial data having a finiteH-functional and belonging toL ¹ (L _x ). Methods previously introduced by Tartar to deal with discrete velocity models are used.     

Construction of positive exact (2+1)-dimensional shock wave solutions for two discrete Boltzmann models

It is proved that (2+1)-dimensional (spacex, y; timet) positive exact shock wave solutions of two discrete Boltzmann models exist. For each densityN _i, these solutions are linear combinations of three similarity shock waves,N _i =n _0i + _j n _ji /[1+d _j exp( _j y+y _j x+ _j t)],j=1,2,3. Two models with four independent densities are investigated: the square discrete-velocity Boltzmann model and the model with eight velocities oriented toward the eight corners of a cube.The positivity problem for the densities is nontrivial. Two classes of solutions are considered for which the two first similarity shock wave components depend on only one spatial dimension, _j=const· _j ,j=1,2. For the positivity, if ₁₂>0, it is sufficient to prove that the 16 asymptotic shock limitsn _0i ,n _0i +n _3i , _j=0 ² n _ji , _j=0 ³ n _ji are positive. The density solutions are built up with five arbitrary parameters and we prove that there exist subdomains of the arbitrary parameter space in which the 16 shock limits are positive. We study numerically two explicit shock wave solutions. We are interested in the movement of the shock front when the time is growing and in the possible appearance of bumps. In the space, at intermediate times, these bumps represent populations of particles which are larger than at initial time or at equilibrium time.     

Classical solution of the nonlinear Boltzmann equation in allR 3: Asymptotic behavior of solutions

Proof is given of the existence of a classical solution to the nonlinear Boltzmann equation in allR ³. The solution, which is global in time, exists if the initial data go to zero fast enough at infinity and the mean free path is sufficiently large. The solution is smooth in the space variable if the initial value is smooth. The asymptotic behavior of solutions is also given. It is shown that ast the solution to the Boltzmann equation can be approximated by the solution to the free motion problem.     

基于消失点约束的多相机标定方法

针对多摄像机一维标定算法精度低、抗噪性和稳定性差等问题,提出了基于消失点之间互相约束的多摄像机标定方法.为了避免径向畸变对成像造成的影响,利用欧式空间位置约束的几何特性,进行畸变参数的求取.通过靶标特征点约束结合摄像机的射影不变性排除杂点的干扰,再利用空间消失点之间夹角一致性,以及靶标特征点所构成的直线和消失点的反向射线平行性来求解摄像机的参数.当一维靶标任意运动时,存在着无法区分靶标特征点对应的成像点临界问题,采取反推理论数学分析法可事先避免该问题的出现.通过构建多摄像机系统进行标定实验,可以看出该方法具有较高的标定精度,且随着噪声的增加,标定结果具有一定的抗噪性和稳定性;由相对误差值可知,该方法可应用于多摄像机系统.     

Case-Specific Random Forests

Random forest (RF) methodology is a nonparametric methodology for prediction problems. A standard way to use RFs includes generating a global RF to predict all test cases of interest. In this article, we propose growing different RFs specific to different test cases, namely case-specific random forests (CSRFs). In contrast to the bagging procedure in the building of standard RFs, the CSRF algorithm takes weighted bootstrap resamples to create individual trees, where we assign large weights to the training cases in close proximity to the test case of interest a priori. Tuning methods are discussed to avoid overfitting issues. Both simulation and real data examples show that the weighted bootstrap resampling used in CSRF construction can improve predictions for specific cases. We also propose a new case-specific variable importance (CSVI) measure as a way to compare the relative predictor variable importance for predicting a particular case. It is possible that the idea of building a predictor case-specifically can be generalized in other areas.