GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

Let （M,g, e^-fdv） be a smooth metric measure space. In this paper, we con- sider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the following nonlinear weighted p-heat equationand f is a smooth function on M under the assumptionthat the m-dimensional nonnegative Bakry-Emery Ricci curvature. Secondly, we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-Emery Ricci curva- ture which is a generalization to the results of Kotschwar and Ni [9], Li [7].

Hamilton-Souplet-Zhang's gradient estimates for two weighted nonlinear parabolic equations

In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-Émery Ricci tensor bounded below: One is $${u_t} = {\Delta _f}u + au\log u + bu$$ with a, b two real constants, and another is $${u_t} = {\Delta _f}u + \lambda {u^\alpha }$$ with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.

A Visual Attention Model for Robot Object Tracking

Inspired by human behaviors, a robot object tracking model is proposed on the basis of visual attention mechanism, which is fit for the theory of topological perception. The model integrates the image-driven, bottom-up attention and the object-driven, top-down attention, whereas the previous attention model has mostly focused on either the bottom-up or top-down attention. By the bottom-up component, the whole scene is segmented into the ground region and the salient regions. Guided by top-down strategy which is achieved by a topological graph, the object regions are separated from the salient regions. The salient regions except the object regions are the barrier regions. In order to estimate the model, a mobile robot platform is developed, on which some experiments are implemented. The experimental results indicate that processing an image with a resolution of 752 × 480 pixels takes less than 200 ms and the object regions are unabridged. The analysis obtained by comparing the proposed model with the existing model demonstrates that the proposed model has some advantages in robot object tracking in terms of speed and efficiency.

Further study of a weighted elliptic equation

A Liouville type result is established for non-negative entire solutions of a weighted elliptic equation.This provides a positive answer to a problem left open by Du and Guo(2015) and Phan and Souplet(2012)(see(CJ) by Du and Guo(2015) and Conjecture B by Phan and Souplet(2012)). Meanwhile, some regularity results are also obtained. The main results in this paper imply that the number ps is the critical value of the Dirichlet problems of the related equation, even though there are still some open problems left. Our results also apply for the equation with a Hardy potential.

Research on the optimal dynamical systems of three-dimensional Navier-Stokes equations based on weighted residual

In this paper, the theory of constructing optimal dynamical systems based on weighted residual presented by Wu & Sha is applied to three-dimensional Navier-Stokes equations, and the optimal dynamical system modeling equations are derived. Then the multiscale global optimization method based on coarse graining analysis is presented, by which a set of approximate global optimal bases is directly obtained from Navier-Stokes equations and the construction of optimal dynamical systems is realized. The optimal bases show good properties, such as showing the physical properties of complex flows and the turbulent vortex structures, being intrinsic to real physical problem and dynamical systems, and having scaling symmetry in mathematics, etc.. In conclusion, using fewer terms of optimal bases will approach the exact solutions of Navier-Stokes equations, and the dynamical systems based on them show the most optimal behavior.

Weighted Pseudo Almost Periodic Solutions of N-th Order Neutral Differential Equations with Piecewise Constant Arguments

In this work, we present some existence theorems of weighted pseudo almost periodic solutions for N-th order neutral differential equations with piecewise constant argument by means of weighted pseudo almost periodic solutions of relevant difference equations.

A class of weighted estimating equations for additive hazards models with covariates missing at random

Missing covariate data arise frequently in biomedical studies.In this article,we propose a class of weighted estimating equations for the additive hazards regression model when some of the covariates are missing at random.Time-specific and subject-specific weights are incorporated into the formulation of weighted estimating equations.Unified results are established for estimating selection probabilities that cover both parametric and non-parametric modelling schemes.The resulting estimators have closed forms and are shown to be consistent and asymptotically normal.Simulation studies indicate that the proposed estimators perform well for practical settings.An application to a mouse leukemia study is illustrated.

Embeddings of weighted Sobolev spaces and degenerate elliptic problems

New embeddings of some weighted Sobolev spaces with weights a(x)and b(x)are established.The weights a(x)and b(x)can be singular.Some applications of these embeddings to a class of degenerate elliptic problems of the form-div(a(x)?u)=b(x)f(x,u)in?,u=0 on??,where?is a bounded or unbounded domain in RN,N 2,are presented.The main results of this paper also give some generalizations of the well-known Caffarelli-Kohn-Nirenberg inequality.

On a Quasilinear Elliptic Equation with Superlinear Nonlinearities

This work is devoted to studying a quasilinear elliptic boundary value problem with superlinear nonlinearities in a weighted Sobolev space in a domain of R^N. Based on the Galerkin method, Brouwer's theorem and the weighted compact Sobolev-type embedding theorem, a new result about the existence of solutions is revealed to the problem.

Weighted quantile regression for longitudinal data using empirical likelihood

This paper proposes a new weighted quantile regression model for longitudinal data with weights chosen by empirical likelihood(EL). This approach efficiently incorporates the information from the conditional quantile restrictions to account for within-subject correlations. The resulted estimate is computationally simple and has good performance under modest or high within-subject correlation. The efficiency gain is quantified theoretically and illustrated via simulation and a real data application.

Qualitative properties and classification of solutions to elliptic equations with Stein-Weiss type convolution part

In this paper,we study the qualitative properties and classification of the solutions to the elliptic equations with Stein-Weiss type convolution part.Firstly,we study the qualitative properties,such as the symmetry,regularity and asymptotic behavior of the positive solutions.Secondly,we classify the non-positive solutions by proving some Liouville type theorems for the finite Morse index solutions and stable solutions to the nonlocal elliptic equations with double weights.

Additive Hazards Regression under Generalized Case-Cohort Sampling

Case-cohort design usually requires the disease rate to be low in large cohort study,although it has been extensively used in practice.However,the disease with high rate is frequently observed in many clinical studies.Under such circumstances,it is desirable to consider a generalized case-cohort design,where only a fraction of cases are sampled.In this article,we propose the inference procedure for the additive hazards regression under the generalized case-cohort sampling.Asymptotic properties of the proposed estimators for the regression coefcients are established.To demonstrate the efectiveness of the generalized case-cohort sampling,we compare it with simple random sampling in terms of asymptotic relative efciency.Furthermore,we derive the optimal allocation of the subsamples for the proposed design.The fnite sample performance of the proposed method is evaluated through simulation studies.

Sobolev type embedding and weak solutions with a prescribed singular set

New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed singular set for a weighted elliptic equation. Our main results in this paper also provide positive weak solutions with a prescribed singular set to an equation with Hardy potential.

An additive-multiplicative rates model for multivariate recurrent events with event categories missing at random

Multivariate recurrent event data arises when study subjects may experience more than one type of recurrent events. In some situations, however, although event times are always observed, event categories may be partially missing. In this paper, an additive-multiplicative rates model is proposed for the analysis of multivariate recurrent event data when event categories are missing at random. A weighted estimating equations approach is developed for parameter estimation, and the resulting estimators are shown to be consistent and asymptotically normal. In addition, a model-checking technique is presented to assess the adequacy of the model. Simulation studies are conducted to evaluate the finite sample behavior of the proposed estimators, and an application to a platelet transfusion reaction study is provided.

Orbit determination using incremental phase and TDOA of X-ray pulsar

X-ray pulsars offer stable, periodic X-ray pulse sequences that can be used in spacecraft positioning systems. A method using X-ray pulsars to determine the initial orbit of a satellite is presented in this paper. This method suggests only one detector to be equipped on the satellite and assumes that the detector observes three pulsars in turn. To improve the performance, the use of incremental phase in one observation duration is proposed, and the incremental phase is combined with the time difference of arrival(TDOA). Then, a weighted least squares(WLS) algorithm is formulated to calculate the initial orbit. Numerical simulations are performed to assess the proposed orbit determination method.