UNIQUENESS FOR SOLUTIONS OF NONHOMOGENEOUS A -HARMONIC EQUATIONS WITH VERY WEAK BOUNDARY VALUES

A uniqueness result for nonhomogeneous quasilinear elliptic partial differential equations with very weak boundary values was proved.

EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY

This paper is concerned with the harmonic equation(P;) : ?u = 0, u > 0 in B;and ?u/?ν+(（n-2）/2)u =(（n-2）/2) Ku;on S;where B;is the unit ball in R;, n ≥ 4 with Euclidean metric g;, ?B;= S;is its boundary, K is a function on S;and ε is a small positive parameter. We construct solutions of the subcritical equation(P;) which blow up at one critical point of K. We give also a sufficient condition on the function K to ensure the nonexistence of solutions for(P;) which blow up at one point. Finally, we prove a nonexistence result of single peaked solutions for the supercritical equation(P;).

A multiscale Galerkin method for the hypersingular integral equation reduced by the harmonic equation

The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot（θ/2）. Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j＋1 × 2j＋1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.

Adaptive hp finite element method for fluorescence molecular tomography with simplified spherical harmonics approximation

Recently,the simplified spherical harmonics equations(SP)model has at tracted much att entionin modeling the light propagation in small tissue ggeometriesat visible and near-infrared wave-leng ths.In this paper,we report an eficient numerical method for fluorescence moleeular tom-ography(FMT)that combines the advantage of SP model and adaptive hp finite elementmethod(hp-FEM).For purposes of comparison,hp-FEM and h-FEM are,respectively applied tothe reconstruction pro cess with diffusion approximation and SPs model.Simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom are designed to evaluate thereconstruction methods in terms of the location and the reconstructed fluorescent yield.Theexperimental results demonstrate that hp-FEM with SPy model,yield more accurate results thanh-FEM with difusion approximation model does.The phantom experiments show the potentialand feasibility of the proposed approach in FMT applications.

Research on the Structural Rigidity Characteristics of a Reconfigurable TBM Thrust Mechanism

To improve the adaptability of TBMs in diverse geological environments,this paper proposes a reconfigurable Type-V thrust mechanism(V-TM)with rearrangeable working states,in which structural stiffness can be automatically altered during operation.Therefore,millions of configurations can be obtained,and thousands of instances of working status per configuration can be set respectively.Nonetheless,the complexity of configurations and diversity of working states contributes to further complications for the structural stiffness algorithm.This results in challenges such as difficulty calculating the payload compliance index and the environment adaptability index.To solve this problem,we use the configuration matrix to describe the relationship between propelling jacks under reconfiguration and adopt pattern vectors to describe the working state of each hydraulic cylinder.Then,both the dynamic compatible equation between propeller forces of the hydraulic cylinders and driving forces,and the kinematic harmonizing equation between the hydraulic cylinder displacements and their deformations are established.Next,we derive the stiffness analytical equation using Hooke’s law and the Jacobian Matrix.The proposed approach provides an effective algorithm to support structural rigidity analysis,and lays a solid theoretical foundation for calculating the performance indexes of the V-TM.We then analyze the rigidity characteristics of typical configurations under different working states,and obtain the main factors affecting structural stiffness of the V-TM.The results show the deviation degree of structural parameters in hydraulic cylinders within the same group,and the working status of propelling jacks.Finally,our constructive conclusions contribute valuable information for matching and optimization by drawing on the factors that affect the structural rigidity of the V-TM.

The α-orthogonal complements of regular Dirichlet subspaces for one-dimensional Brownian motion

Roughly speaking a regular Dirichlet subspace of a Dirichlet form is a subspace which is also a regular Dirichlet form on the same state space. In particular, the domain of regular Dirichlet subspace is a closed subspace of the Hilbert space induced by the domain and a-inner product of original Dirichlet form. We investigate the orthogonal complement of regular Dirichlet subspace for one-dimensional Brownian motion in this paper. Our main results indicate that this orthogonal complement has a very close connection with the a-harmonic equation under Neumann type condition.

Symmetry of the Point Spectrum of Infinite Dimensional Hamiltonian Operators and Its Applications

This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp（H） = σp （A） U σp1 （-A＊）. Using the characteristic of the set σp1（-A＊）, we divide the point spectrum σp （d） of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σp1（-A＊） and one part of σp（A） are symmetric with respect to the real axis each other. Based on this result, the symmetry of σp（H） is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.

Operator Factorization forMultiple-Scattering Problems and an Application to Periodic Media

This work concerns multiple-scattering problems for time-harmonic equations in a reference genericmedia.We consider scatterers that can be sources,obstacles or compact perturbations of the reference media.Our aim is to restrict the computational domain to small compact domains containing the scatterers.We use Robin-to-Robin(RtR)operators(in the most general case)to express boundary conditions for the interior problem.We show that one can always factorize the RtR map using only operators defined using single-scatterer problems.This factorization is based on a decomposition of the diffracted field,on thewhole domainwhere it is defined.Assuming that there exists a good method for solving single-scatterer problems,it then gives a convenient way to compute RtR maps for a random number of scatterers.