This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold M^{2mn} (resp. a smooth manifold M^{2mn} ), if there exists a partition M^{2mn} of weight M^{2mn} such that the Chern number M^{2mn} (resp. Pontrjagin number M^{2mn} ) is nonzero, then\emph {any} circle action on M^{2mn} (resp. M^{2mn} ) has at least M^{2mn} fixed points. When an even-dimensional smooth manifold M^{2mn} admits a semi-free action with isolated fixed points, we show that M^{2mn} bounds, which generalizes a well-known fact in the free case. We also provide a topological obstruction, in terms of the first Chern class, to the existence of semi-free circle action with\emph {nonempty} isolated fixed points on almost-complex manifolds. The main ingredients of our proofs are Bott's residue formula and rigidity theorem.

Errors-in-variables regression is the study of the association between covariates and responses where covariates are observed with errors. In this paper, we consider the estimation of multivariate regression functions for dependent data with errors in covariates. Nonparametric deconvolution technique is used to account for errors-in-variables. The asymptotic behavior of regression estimators depends on the smoothness of the error distributions, which are characterized as either ordinarily smooth or super smooth. Asymptotic normality is established for both strongly mixing and -mixing processes, when the error distribution function is either ordinarily smooth or super smooth.

In this paper, we study the Li stability of perturbation of constant states for 2 x 2 systems of conservation laws. We introduce a nonlinear functional which is equivalent to the Li norm of the difference between the constant state and the approximate solutions consisting of elementary waves and is non-increasing in time for the limiting weak solution. This yields the Li stability of the constant states. The approximate solutions are constructed with the aid of wave tracing for the random choice method. Our functional reveals the aspects of nonlinear wave behavior, particularly the coupling of waves pertaining to different characteristic families, which affects the Li norm of the solutions.

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.

There have been many attempts to define quasilocal energy/mass for a spacelike 2-surface in a spacetime by the Hamilton–Jacobi method. The main difficulty in this approach is the subtle choice of the background configuration to be subtracted from the physical Hamiltonian. Quasilocal mass should be positive for general surfaces, but on the other hand should be zero for surfaces in the flat spacetime. In this paper, we survey the work in a series of papers [6, 25–27] in which a new notion of quasilocal mass/energy–momentum is proposed and investigated. In particular, the notion of energy observed by a 'quasilocal observer' will be discussed.