In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Be...In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.展开更多
Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet ...Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.展开更多
Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =（X1,X2,... ,Xm） be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmande...Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =（X1,X2,... ,Xm） be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander＇s condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.展开更多
As part of a study of star formation history along the Hubble sequence, we present here the results for 11 elliptical galaxies with strong nebular emission lines. After removing the dilution from the underlying old st...As part of a study of star formation history along the Hubble sequence, we present here the results for 11 elliptical galaxies with strong nebular emission lines. After removing the dilution from the underlying old stellar populations by use of stellar population synthesis model, we derive the accurate fluxes of all the emission lines in these objects, which are then classified, using emission line ratios, into one Seyfert 2, six LINERs and four HII galaxies. We also identify one HII galaxy （A1216＋04） as a hitherto unknown Wolf-Rayet galaxy from the presence of the WolfRayet broad bump at 4650A. We propose that the star-forming activities in elliptical galaxies are triggered by either galaxy-galaxy interaction or the merging of a small satellite/a massive star cluster, as has been suggested by recent numerical simulations.展开更多
Let Q be a bounded open domain in R^n with smooth boundaryаΩ.Let X=(X1,X2…,Xm)be a system of general Grushin type vector fields defined onΩand the boundaryаΩis non-characteristic for X.For△x=∑j=1^mXj^2,we den...Let Q be a bounded open domain in R^n with smooth boundaryаΩ.Let X=(X1,X2…,Xm)be a system of general Grushin type vector fields defined onΩand the boundaryаΩis non-characteristic for X.For△x=∑j=1^mXj^2,we denoteλk as the k-th eigenvalue for the bi-subelliptic operator△X2^2 onΩ.In this paper,by using the sharp sub-elliptic estimates and maximally hypoeliptic estimates,we give the optimal lower bound estimates ofλk for the operatork△X^2.展开更多
Two-dimensional bulge/disk light decomposition with GIM2D in both the r- and g-bands has been applied to a sample of 129 early-type galaxies brighter than 13.5 magnitude in the r-band, selected from the Sloan Digital ...Two-dimensional bulge/disk light decomposition with GIM2D in both the r- and g-bands has been applied to a sample of 129 early-type galaxies brighter than 13.5 magnitude in the r-band, selected from the Sloan Digital Sky Survey Data Release 2. Intensity-weighted Fourier coefficient （α4/α） was also derived for each sample galaxy. Our analysis shows that there are correlations between bulge-to-total light ratio （B/T） with bulge Sersic index nB and between bulge and disk scale sizes. Isophotal shape parameter （α4/α） is not correlated with BIT and riB. Both bulge and disk components satisfy a color-magnitude relation. The space Fundamental Plane analysis shows that galaxies with larger B/T tend to lie tighter and closer to the line of k1 ＋ k2 = 8 （the so-called ＂zone of avoidance＂） than the galaxies with smaller B/T. It indicates that existence of the disk component may lead to scatter of the distribution on the Fundamental Plane. Our analysis also shows that k1 ＋ k2 correlates with （g-r） color and B/T, but does not correlate with （α4/α） for early-type galaxies. The fitted parameters and other retrieved parameters used in this paper for all sample galaxies are available online.展开更多
Early-type galaxies （ETGs） are very important for understanding the formation and evolution of galaxies. Recent observations suggest that ETGs are not simply old stellar spheroids as we previously thought. Widesprea...Early-type galaxies （ETGs） are very important for understanding the formation and evolution of galaxies. Recent observations suggest that ETGs are not simply old stellar spheroids as we previously thought. Widespread recent star formation, cool gas and dust have been detected in a substantial fraction of ETGs. We make use of the radial profiles of 9 - r color and the concentration index from the Sloan Digital Sky Survey database to pick out 31 peculiar ETGs with central blue cores. By analyzing the photometric and spectroscopic data, we suggest that the blue cores are caused by star formation activities rather than the central weak active galactic nucleus. From the results of stellar population synthesis, we find that the stellar population of the blue cores is relatively young, spreading from several Myr to less than one Gyr. In 14 galaxies with H I observations, we find that the average gas fraction of these galaxies is about 0.55. The bluer galaxies show a higher gas fraction, and the total star forma- tion rate （SFR） correlates very well with the H I gas mass. The star formation history of these ETGs is affected by the environment, e.g. in the denser environment the H I gas is less and the total SFR is lower. We also discuss the origin of the central star formation of these early-type galaxies.展开更多
In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the nat...In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $ \bar \partial _b $ in these spaces using integral kernels and approximate inverses. In the 1990’s, Rumin introduced a differential complex for compact contact manifolds, showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator, and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper, we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.展开更多
Background: It has been postulated that elliptical cutaneous excisions must possess a length-to-width ratio of 3 to 4 and a vertex angle of 30o or less in order to be closed primarily without creating a “dog ear”. T...Background: It has been postulated that elliptical cutaneous excisions must possess a length-to-width ratio of 3 to 4 and a vertex angle of 30o or less in order to be closed primarily without creating a “dog ear”. These dimensions became axiomatic in cutaneous surgery and have been taught in the apprenticeship model for years. The present article examines the validity of that paradigm. Methods: We collected data from two sources: ellipses described in the literature (57 cases);and elliptical excisions performed at the authors’ outpatient clinic (83 cases). The surgical ellipse lengths, widths, and vertex angles were analyzed, and the data were compared to a mathematical formula used to generate a fusiform ellipse. Results: The length-to-width ratio of 3 - 4 was found to be inconsistent with the recommended vertex angle of 30o. In fact, a length-to-width ratio of 3 - 4 determines a vertex angle of 48o - 63o. A 30o vertex angle is only feasible with long length-to-width ration of about 7.5. Conclusions: The paradigm that surgical ellipses should have a vertex angle of 30o with length-to-width ratio of 3 - 4 is incorrect. Evidence from actual surgical practice and from mathematical formulation shows that either the length-to-width ratio must be larger than 3 - 4 or the vertex angle must be larger than 30 degrees.展开更多
In this paper, we study the Dirichlet problems for the following quasilinear secondorder sub-elliptic equation, sum from i,j=1 to m(X~*(A(x,u)Xu)+sum from j=1 to m(B(x,u)Xu+C(x,u)=0 in Ω, u=φ on Ω,where X={X, …, ...In this paper, we study the Dirichlet problems for the following quasilinear secondorder sub-elliptic equation, sum from i,j=1 to m(X~*(A(x,u)Xu)+sum from j=1 to m(B(x,u)Xu+C(x,u)=0 in Ω, u=φ on Ω,where X={X, …, X} is a system of real smooth vector fields which satisfies the Hrmander’scondition, A(i,j), B, C∈C~∞(■×R) and (A(x, z)) is a positive definite matris. We have provedthe existence and the maximal regularity of solutions in the "non-isotropic" Hlder space associatedwith the system of vector fields X.展开更多
文摘In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.
基金the National Natural Science Foundation of China (Nos.11571238,11601332,91130014,11471312 and 91430216).
文摘Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.
基金partially supported by the NSFC(11631011,11626251)
文摘Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =（X1,X2,... ,Xm） be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander＇s condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.
基金Supported by the National Natural Science Foundation of China
文摘As part of a study of star formation history along the Hubble sequence, we present here the results for 11 elliptical galaxies with strong nebular emission lines. After removing the dilution from the underlying old stellar populations by use of stellar population synthesis model, we derive the accurate fluxes of all the emission lines in these objects, which are then classified, using emission line ratios, into one Seyfert 2, six LINERs and four HII galaxies. We also identify one HII galaxy （A1216＋04） as a hitherto unknown Wolf-Rayet galaxy from the presence of the WolfRayet broad bump at 4650A. We propose that the star-forming activities in elliptical galaxies are triggered by either galaxy-galaxy interaction or the merging of a small satellite/a massive star cluster, as has been suggested by recent numerical simulations.
基金supported by National Natural Science Foundation of China (Grants Nos. 11631011 and 11626251)
文摘Let Q be a bounded open domain in R^n with smooth boundaryаΩ.Let X=(X1,X2…,Xm)be a system of general Grushin type vector fields defined onΩand the boundaryаΩis non-characteristic for X.For△x=∑j=1^mXj^2,we denoteλk as the k-th eigenvalue for the bi-subelliptic operator△X2^2 onΩ.In this paper,by using the sharp sub-elliptic estimates and maximally hypoeliptic estimates,we give the optimal lower bound estimates ofλk for the operatork△X^2.
基金the National Natural Science Foundation of China
文摘Two-dimensional bulge/disk light decomposition with GIM2D in both the r- and g-bands has been applied to a sample of 129 early-type galaxies brighter than 13.5 magnitude in the r-band, selected from the Sloan Digital Sky Survey Data Release 2. Intensity-weighted Fourier coefficient （α4/α） was also derived for each sample galaxy. Our analysis shows that there are correlations between bulge-to-total light ratio （B/T） with bulge Sersic index nB and between bulge and disk scale sizes. Isophotal shape parameter （α4/α） is not correlated with BIT and riB. Both bulge and disk components satisfy a color-magnitude relation. The space Fundamental Plane analysis shows that galaxies with larger B/T tend to lie tighter and closer to the line of k1 ＋ k2 = 8 （the so-called ＂zone of avoidance＂） than the galaxies with smaller B/T. It indicates that existence of the disk component may lead to scatter of the distribution on the Fundamental Plane. Our analysis also shows that k1 ＋ k2 correlates with （g-r） color and B/T, but does not correlate with （α4/α） for early-type galaxies. The fitted parameters and other retrieved parameters used in this paper for all sample galaxies are available online.
基金supported by the Doctoral Fund of the Ministry of Education of China (20100091110009)the National Natural Science Foundation of China (Grant Nos. 10878010, 10221001 and 10633040)the National Basic Research Program (973 Program, No. 2007CB815405)
文摘Early-type galaxies （ETGs） are very important for understanding the formation and evolution of galaxies. Recent observations suggest that ETGs are not simply old stellar spheroids as we previously thought. Widespread recent star formation, cool gas and dust have been detected in a substantial fraction of ETGs. We make use of the radial profiles of 9 - r color and the concentration index from the Sloan Digital Sky Survey database to pick out 31 peculiar ETGs with central blue cores. By analyzing the photometric and spectroscopic data, we suggest that the blue cores are caused by star formation activities rather than the central weak active galactic nucleus. From the results of stellar population synthesis, we find that the stellar population of the blue cores is relatively young, spreading from several Myr to less than one Gyr. In 14 galaxies with H I observations, we find that the average gas fraction of these galaxies is about 0.55. The bluer galaxies show a higher gas fraction, and the total star forma- tion rate （SFR） correlates very well with the H I gas mass. The star formation history of these ETGs is affected by the environment, e.g. in the denser environment the H I gas is less and the total SFR is lower. We also discuss the origin of the central star formation of these early-type galaxies.
基金This work was supported by NSERC(Grant No.RGPIN/9319-2005)
文摘In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $ \bar \partial _b $ in these spaces using integral kernels and approximate inverses. In the 1990’s, Rumin introduced a differential complex for compact contact manifolds, showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator, and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper, we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.
文摘Background: It has been postulated that elliptical cutaneous excisions must possess a length-to-width ratio of 3 to 4 and a vertex angle of 30o or less in order to be closed primarily without creating a “dog ear”. These dimensions became axiomatic in cutaneous surgery and have been taught in the apprenticeship model for years. The present article examines the validity of that paradigm. Methods: We collected data from two sources: ellipses described in the literature (57 cases);and elliptical excisions performed at the authors’ outpatient clinic (83 cases). The surgical ellipse lengths, widths, and vertex angles were analyzed, and the data were compared to a mathematical formula used to generate a fusiform ellipse. Results: The length-to-width ratio of 3 - 4 was found to be inconsistent with the recommended vertex angle of 30o. In fact, a length-to-width ratio of 3 - 4 determines a vertex angle of 48o - 63o. A 30o vertex angle is only feasible with long length-to-width ration of about 7.5. Conclusions: The paradigm that surgical ellipses should have a vertex angle of 30o with length-to-width ratio of 3 - 4 is incorrect. Evidence from actual surgical practice and from mathematical formulation shows that either the length-to-width ratio must be larger than 3 - 4 or the vertex angle must be larger than 30 degrees.
文摘In this paper, we study the Dirichlet problems for the following quasilinear secondorder sub-elliptic equation, sum from i,j=1 to m(X~*(A(x,u)Xu)+sum from j=1 to m(B(x,u)Xu+C(x,u)=0 in Ω, u=φ on Ω,where X={X, …, X} is a system of real smooth vector fields which satisfies the Hrmander’scondition, A(i,j), B, C∈C~∞(■×R) and (A(x, z)) is a positive definite matris. We have provedthe existence and the maximal regularity of solutions in the "non-isotropic" Hlder space associatedwith the system of vector fields X.
