In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 a...In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 and the sub-Laplacian in the Heisenberg group Hn.展开更多
A p-Laplacian ( p > 2 ) reaction-diffusion system on weighted graphs is introduced to a networked SIR epidemic model. After overcoming difficulties caused by the nonlinear p-Laplacian, we show that the endemic equi...A p-Laplacian ( p > 2 ) reaction-diffusion system on weighted graphs is introduced to a networked SIR epidemic model. After overcoming difficulties caused by the nonlinear p-Laplacian, we show that the endemic equilibrium is globally asymptotically stable if the basic reproduction number r<sub>0</sub> is greater than 1, while the disease-free equilibrium is globally asymptotically stable if r<sub>0</sub> is lower than 1. We extend the stability results of SIR models with graph Laplacian ( p = 2 ) to general graph p-Laplacian.展开更多
The Gauss-linking integral for disjoint oriented smooth closed curves is derived linking integrals from the Biot-Savart description of the magnetic field. DeTurck and Gluck extend this linking from 3-space <em>R...The Gauss-linking integral for disjoint oriented smooth closed curves is derived linking integrals from the Biot-Savart description of the magnetic field. DeTurck and Gluck extend this linking from 3-space <em>R</em><sup>3</sup> to <em>SU</em> (2) space of the unit 3-sphere and hyperbolic space in Minkowski <em>R</em><sup>1,3</sup>. I herein extend Gauss-linking to self-linking and develop the concept of self-dual, which is then applied to gravitomagnetic dynamics. My purpose is to redefine Wheeler’s <em>geon</em> from unstable field structures based on the electromagnetic field to self-stabilized gravitomagnetic field structures.展开更多
Predicting protein functions is an important issue in the post-genomic era. This paper studies several network-based kernels including local linear embedding (LLE) kernel method, diffusion kernel and laplacian kerne...Predicting protein functions is an important issue in the post-genomic era. This paper studies several network-based kernels including local linear embedding (LLE) kernel method, diffusion kernel and laplacian kernel to uncover the relationship between proteins functions and protein-protein interactions (PPI). The author first construct kernels based on PPI networks, then apply support vector machine (SVM) techniques to classify proteins into different functional groups. The 5-fold cross validation is then applied to the selected 359 GO terms to compare the performance of different kernels and guilt-by-association methods including neighbor counting methods and Chi-square methods. Finally, the authors conduct predictions of functions of some unknown genes and verify the preciseness of our prediction in part by the information of other data source.展开更多
In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ? n+1. Then we survey some results on the spectral zeta function which is induced by the trace of the ...In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ? n+1. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.展开更多
In this paper,we study the asymptotic behaviour of the scattering phase s(λ)of the Dirichlet Laplacian associated with obstacle,where Ω is a bounded open subset of IR<sup>n</sup>(n≥2) with non-smoot...In this paper,we study the asymptotic behaviour of the scattering phase s(λ)of the Dirichlet Laplacian associated with obstacle,where Ω is a bounded open subset of IR<sup>n</sup>(n≥2) with non-smooth boundaryΩ and connected complement Ω<sub>e</sub>=IR<sup>n</sup>\.We can prove that if Ω satisfies a certain geometrical condition,then where φ(λ)=[(4π)<sup>n/2</sup>Γ(1+(n/2)]<sup>-1</sup>|Ω|<sub>n</sub>λ<sup>n/2</sup>,d<sub>n</sub>】0 depending only on n,and |·|<sub>j</sub>(j=n-1,n)is a j-dimensional Lebesgue measure.展开更多
The trace of the wave kernel μ(t) =∑ω=1^∞ exp(-itEω^1/2), where {Eω}ω^∞=1 are the eigenvalues of the negative Laplacian -△↓2 = -∑k^3=1 (δ/δxk)^2 in the (x^1, x^2, x^3)-space, is studied for a vari...The trace of the wave kernel μ(t) =∑ω=1^∞ exp(-itEω^1/2), where {Eω}ω^∞=1 are the eigenvalues of the negative Laplacian -△↓2 = -∑k^3=1 (δ/δxk)^2 in the (x^1, x^2, x^3)-space, is studied for a variety of bounded domains, where -∞ 〈 t 〈 ∞ and i= √-1. The dependence of μ (t) on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane Ω in Ra surrounded by simply connected bounded domains Ω j with smooth bounding surfaces S j (j = 1,……, n), where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si^* (i = 1 + kj-1,……, kj) of the bounding surfaces S j are considered, such that S j = Ui-1+kj-1^kj Si^*, where k0=0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined from the asymptotic expansion ofexpansion of μ(t) for small │t│.展开更多
This paper is devoted to asymptotic formulae for functions related with the spectrum of the negative Laplacian in two and three dimensional bounded simply connected domains with impedance boundary conditions,where the...This paper is devoted to asymptotic formulae for functions related with the spectrum of the negative Laplacian in two and three dimensional bounded simply connected domains with impedance boundary conditions,where the impedances are assumed to be discontinuous functions. Moreover,asymptotic expressions for the difference of eigenvalues related to the impedance problems with different impedances are derived.Further results may be obtained.展开更多
This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each inter...This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ4u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.展开更多
In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are ...In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are proved.展开更多
基金partially supported by a William Fulbright Research Grant and a Competitive Research Grant at Georgetown University
文摘In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 and the sub-Laplacian in the Heisenberg group Hn.
文摘A p-Laplacian ( p > 2 ) reaction-diffusion system on weighted graphs is introduced to a networked SIR epidemic model. After overcoming difficulties caused by the nonlinear p-Laplacian, we show that the endemic equilibrium is globally asymptotically stable if the basic reproduction number r<sub>0</sub> is greater than 1, while the disease-free equilibrium is globally asymptotically stable if r<sub>0</sub> is lower than 1. We extend the stability results of SIR models with graph Laplacian ( p = 2 ) to general graph p-Laplacian.
文摘The Gauss-linking integral for disjoint oriented smooth closed curves is derived linking integrals from the Biot-Savart description of the magnetic field. DeTurck and Gluck extend this linking from 3-space <em>R</em><sup>3</sup> to <em>SU</em> (2) space of the unit 3-sphere and hyperbolic space in Minkowski <em>R</em><sup>1,3</sup>. I herein extend Gauss-linking to self-linking and develop the concept of self-dual, which is then applied to gravitomagnetic dynamics. My purpose is to redefine Wheeler’s <em>geon</em> from unstable field structures based on the electromagnetic field to self-stabilized gravitomagnetic field structures.
基金This research is supported in part by HKRGC Grant 7017/07P, HKU CRCG Grants, HKU strategic theme grant on computational sciences, HKU Hung Hing Ying Physical Science Research Grant, National Natural Science Foundation of China Grant No. 10971075 and Guangdong Provincial Natural Science Grant No. 9151063101000021. The preliminary version of this paper has been presented in the OSB2009 conference and published in the corresponding conference proceedings[25]. The authors would like to thank the anonymous referees for their helpful comments and suggestions.
文摘Predicting protein functions is an important issue in the post-genomic era. This paper studies several network-based kernels including local linear embedding (LLE) kernel method, diffusion kernel and laplacian kernel to uncover the relationship between proteins functions and protein-protein interactions (PPI). The author first construct kernels based on PPI networks, then apply support vector machine (SVM) techniques to classify proteins into different functional groups. The 5-fold cross validation is then applied to the selected 359 GO terms to compare the performance of different kernels and guilt-by-association methods including neighbor counting methods and Chi-square methods. Finally, the authors conduct predictions of functions of some unknown genes and verify the preciseness of our prediction in part by the information of other data source.
基金supported by National Security Agency,United States Army Research Offfice and a Hong Kong RGC Competitive Earmarked Research (Grant No. 600607)
文摘In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ? n+1. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.
基金Research partially supported by the Natural Science Foundation of Chinathe Grant of Chinese State Education Committee
文摘In this paper,we study the asymptotic behaviour of the scattering phase s(λ)of the Dirichlet Laplacian associated with obstacle,where Ω is a bounded open subset of IR<sup>n</sup>(n≥2) with non-smooth boundaryΩ and connected complement Ω<sub>e</sub>=IR<sup>n</sup>\.We can prove that if Ω satisfies a certain geometrical condition,then where φ(λ)=[(4π)<sup>n/2</sup>Γ(1+(n/2)]<sup>-1</sup>|Ω|<sub>n</sub>λ<sup>n/2</sup>,d<sub>n</sub>】0 depending only on n,and |·|<sub>j</sub>(j=n-1,n)is a j-dimensional Lebesgue measure.
文摘The trace of the wave kernel μ(t) =∑ω=1^∞ exp(-itEω^1/2), where {Eω}ω^∞=1 are the eigenvalues of the negative Laplacian -△↓2 = -∑k^3=1 (δ/δxk)^2 in the (x^1, x^2, x^3)-space, is studied for a variety of bounded domains, where -∞ 〈 t 〈 ∞ and i= √-1. The dependence of μ (t) on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane Ω in Ra surrounded by simply connected bounded domains Ω j with smooth bounding surfaces S j (j = 1,……, n), where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si^* (i = 1 + kj-1,……, kj) of the bounding surfaces S j are considered, such that S j = Ui-1+kj-1^kj Si^*, where k0=0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined from the asymptotic expansion ofexpansion of μ(t) for small │t│.
文摘This paper is devoted to asymptotic formulae for functions related with the spectrum of the negative Laplacian in two and three dimensional bounded simply connected domains with impedance boundary conditions,where the impedances are assumed to be discontinuous functions. Moreover,asymptotic expressions for the difference of eigenvalues related to the impedance problems with different impedances are derived.Further results may be obtained.
文摘This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ4u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.
基金supported by the National Natural Science Foundation of China(10871157)Research Fund for the Doctoral Program of Higher Education of China(200806990032)Keji Chuangxin Jijin of Northwestern Polytechnical University(2007KJ01012)
文摘In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are proved.