AN IMPROVED ERROR ANALYSIS FOR FINITE ELEMENT APPROXIMATION OF BIOLUMINESCENCE TOMOGRAPHY

This paper is concerned with an ill-posed problem which results from the area of molecular imaging and is known as BLT problem. Using Tikhonov regularization technique, a quadratic optimization problem can be formulated. We provide an improved error estimate for the finite element approximation of the regularized optimization problem. Some numerical examples are presented to demonstrate our theoretical results.     

On Fluorophore Imaging by Diffusion Equation Model:Decompositions and Optimizations

Fluorescence imaging is a target-specific molecular imaging technology using absorption coefficient of fluorophore.For this imaging model governed by an inverse problem for the coupled diffusion system,which describes the interaction of the excitation field from several boundary sources and the corresponding emission field,we reformulate it as an optimization problem.For solving this non-quadratic optimizing problem,we propose a decomposition scheme,which extracts the horizontal information of the target from the boundary measurement data directly.The realizability of this hybrid imaging scheme is rigorously proved mathematically for cubic and ellipsoid targets,by constructing an indicator function for the horizontal location of the target explicitly.Then based on this horizonal location as a good initial guess for the iteration process,the cost functional is optimized efficiently using the trust domain scheme.Numerical implementations are provided to show the validity of the proposed scheme.     

FAST ACOUSTIC IMAGING FOR A 3D PENETRABLE OBJECT IMMERSED IN A SHALLOW WATER WAVEGUIDE＊

The paper is concerned with the inverse problem for reconstructing a 3D penetrable ob- ject in a shallow water waveguide from the far-field data of the scattered fields with many acoustic point source incidences. An indicator sampling method is analyzed and presented for fast imaging the size, shape and location of such a penetrable object. The method has the advantages that a priori knowledge is avoided for the geometrical and material proper- ties of the penetrable obstacle and the much complicated iterative techniques are avoided during the inversion. Numerical examples are given of successful shape reconstructions for several 3D penetrable obstacles having a variety of shapes. In particular, numerical results show that the proposed method is able to produce a good reconstruction of the size, shape and location of the penetrable target even for the case where the incident and observation points are restricted to some limited apertures.     

THE SURVIVAL ANALYSIS FOR SINGLE-SPECIES SYSTEM IN A POLLUTED ENVIRONMENT

This article concentrates on the study of a mathematical model for the effect of toxicant levels on a single-species ecosystem in the case where there is a constant emission of a toxicant. Some sufficient conditions for weak persistence and extinction are found. The threshold between persistence and extinction can be established in some cases.     

Inequalities on the Triangle

The article deals with generalizations of the inequalities for convex functions on the triangle. The Jensen and the Hermite-Hadamard inequality are included in the study. Considering a convex function on the triangle, we obtain a generalization of the Jensen-Mercer inequality, and a refinement of the Hermite-Hadamard inequality.     

Borderline Case of Traces and Extensions for Weighted Sobolev Spaces

In this paper,we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases.We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces,and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces(on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.     

A Note on Property(ω)

In this note we study the property(ω),a variant of Weyl's theorem introduced by Rakoevic,by means of the new spectrum.We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property(ω) and approximate Weyl's theorem hold.As a consequence of the main result,we study the property(ω) and approximate Weyl's theorem for a class of operators which we call the λ-weak-H(p) operators.     

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds Dedicated to Professor Zhiming Ma on His Seventieth Birthday

In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a(K, m)-super Ricci flow, where K ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds.Our result improves an important result due to Lott and Villani(2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.     

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.     

性质$(\omega)$的注解

In this note we study the property （ω）, a variant of Weyl＇s theorem introduced by Rakocevic, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property （ω） and approximate Weyl＇s theorem hold. As a consequence of the main result, we study the property （ω） and approximate Weyl＇s theorem for a class of operators which we call the λ-weak-H（p） operators.     

Mean Index for Non-periodic Orbits in Hamiltonian Systems

In this paper,we define mean index for non-periodic orbits in Hamiltonian systems and study its properties.In general,the mean index is an interval in R which is uniformly continuous on the systems.We show that the index interval is a point for a quasi-periodic orbit.The mean index can be considered as a generalization of rotation number defined by Johnson and Moser in the study of almost periodic Schr¨odinger operators.Motivated by their works,we study the relation of Fredholm property of the linear operator and the mean index at the end of the paper.     

一类卷积型算子在端点Triebel-Lizorkin空间上的有界性

This paper focuses on the study of the boundedness of convolution-type Calderón-Zygmund operators on some endpoint Triebel-Lizorkin spaces. Applying wavelets, molecular decomposition and interpolation theory, the author establishes the boundedness on certain endpoint Triebel-Lizorkin spaces F˙10 ,q（2 q ≤ ∞） under a very weak pointwise regularity condition.     

OPTIMAL LOGISTICS FOR MULTIPLE JEEPS

We consider variations of the classical jeep problems： the optimal logistics for a caravan of jeeps which travel together in the desert. The main purpose is to arrange the travels for the one-way trip and the round trip of a caravan of jeeps so that the chief jeep visits the farthest destination. Based on the dynamic program principle, the maximum distances for the caravan when only part of the jeeps should return and when all drivers should return are obtained. Some related results such as the efficiency of the abandoned jeeps, and the advantages of more jeeps in the caravan are also presented.     

THE DYNAMIC PROPERTIES FOR THE PROTEIN MOLECULAR SYSTEM

In this paper, a new theory for the dynamic properties of protein molecular system have been proposed by means of the analysis of the characteristics of the collective excitations generated under the localized fluctuation and the deformation Of structure of the protein. Some new results obtained from this study show that the Davydov theory is an approximate and disadvantegeous theory.     

ON THE SPECTRUM OF SINGULAR HAMILTONIAN DIFFERENTIAL SYSTEMS

In this paper, we study a bounded-below singular Hamiltonian system. Sufficient and necessary conditions are obtained for the existence and the number of eigenvalues on the left-axis. The main results of this paper are the extension and improvement of Weyl spectral theorem for singular second order differential equations.     

THE EXISTENCE, UNIQUENESS AND STABILITY OF ALMOST PERIODIC SOLUTION FOR A CLASS OF NONLINEAR SYSTEM

In this paper, we study the problems on the existence, uniqueness and stability of almost periodic solution for a class of nonlinear system. Using fixed point theorem and Lyapunov functional, the sufficient conditions are given which guarantee the existence, uniqueness and stability of almost periodic solution for the system.     

SPECTRAL MAPPING THEOREM FOR ASCENT,ESSENTIAL ASCENT,DESCENT AND ESSENTIAL DESCENT SPECTRUM OF LINEAR RELATIONS

In [7], Cross showed that the spectrum of a linear relation T on a normed space satisfies the spectral mapping theorem. In this paper, we extend the notion of essential ascent and descent for an operator acting on a vector space to linear relations acting on Banach spaces. We focus to define and study the descent, essential descent, ascent and essential ascent spectrum of a linear relation everywhere defined on a Banach space X. In particular, we show that the corresponding spectrum satisfy the polynomial version of the spectral mapping theorem.     

ASYMPTOTIC BEHAVIOR OF THE TE AND TM APPROXIMATIONS TO SECOND HARMONIC GENERATION

We consider a Second Harmonic Generation （SHG） problem of an optical signal wave with an optical pump in a medium represented by a smooth bounded domain Ω→∩ IR^d which is assumed to contain a heterogeneous material： a compactly imbedded subdomain B^r →∩∩ in the shape of a small ball contains a nonlinear material, while Ω／ B^r is filled with a linear material. We begin by proving existence and uniqueness of the solution to the TE approximation of SHG for arbitrary bounded susceptibilities, thus improving the result obtained by Bao and Dobson （ Eur. J. Appl. Math. 6 （1995）, 573-590） under small enough susceptibilities assuption. We then establish an existence and uniqueness result of a solution to the TM approximation problem. In both parts we study the asymptotic behavior of the system as the size of the nonlinear material vanishes： error estimates and asymptotic expansion of the solution are derived for both TE and TM approximations.     

A semiparametric additive rates model for clustered recurrent event data

Recurrent event data often arises in biomedical studies, and individuals within a cluster might not be independent. We propose a semiparametric additive rates model for clustered recurrent event data, wherein the covariates are assumed to add to the unspecified baseline rate. For the inference on the model parameters, estimating equation approaches are developed, and both large and finite sample properties of the proposed estimators are established.