No abstract uploaded!

We find some necessary conditions for a real Banach space to be an almost CL-space. We also discuss the stability of CL-spaces and almost CL-spaces by$c$_{0}- and$l$_{1}-sums. Finally, we address the question if a space of vector-valued continuous functions can be a CL-space or an almost CL-space.

Local polynomial estimators are popular techniques for nonparametric regression estimation and have received great attention in the literature. Their simplest version, the local constant estimator, can be easily extended to the errors-in-variables context by exploiting its similarity with the deconvolution kernel density estimator. The generalization of the higher order versions of the estimator, however, is not straightforward and has remained an open problem for the last 15 years. We propose an innovative local polynomial estimator of any order in the errors-in-variables context, derive its design-adaptive asymptotic properties and study its finite sample performance on simulated examples. We provide not only a solution to a long-standing open problem, but also provide methodological contributions to error-in-variable regression, including local polynomial estimation of derivative functions.

In this paper, we are concerned with the one-species VlasovPoissonBoltzmann system with a nonconstant background density in full space. There exists a stationary solution when the background density is a small perturbation of a positive constant state. We prove the nonlinear stability of solutions to the Cauchy problem near the stationary state in some Sobolev space without any time derivatives. This result is nontrivial even when the background density is a constant state. In the proof, the macroscopic balance laws are essentially used to deal with the a priori estimates on both the microscopic and macroscopic parts of the solution. Moreover, some interactive energy functionals are introduced to overcome difficulty stemming from the absence of time derivatives in the energy functional.

We present a stochastic Justh–Krishnaprasad flocking model describing interactions among individuals in a planar domain with their positions and heading angles. The deterministic counterpart of the proposed model describes the formation of nematic alignment in an ensemble of planar particles moving with a unit speed. When the noise is turned off, we show that the nematic alignment state, in which all heading angles are either same or the opposite, is nonlinearly stable using a Lyapunov functional approach. We employed a diameter-like functional via the rearrangement of heading angles in the 2π-interval. In contrast, under the additive noise, a continuous angle configuration will be deviated asymptotically from the nematic state. Nevertheless, in any finite-time interval, we will see that some part of angle configuration will stay close to the nematic state with a positive probability, where we call this phenomenon as stochastic persistency. We provide a quantitative estimate on the probability for stochastic persistency and compare several numerical examples with analytical results.