In two previous papers <a href="#ref1">[1]</a> and <a href="#ref2">[2]</a>, a structure for vector products in <em>n</em> dimensions was presented, and at the sa...In two previous papers <a href="#ref1">[1]</a> and <a href="#ref2">[2]</a>, a structure for vector products in <em>n</em> dimensions was presented, and at the same time it was possible to propose the existence of a vector analogous to the curl of a vector field, for a space of four dimensions. In continuation of these works, the objective is to develop, through dimensional analogy, the idea of a hypothetical vector field, associated with the classical electromagnetic wave. This hypothetical field has a possible mathematical existence only when considering a space of four dimensions. The properties of the electromagnetic wave are preserved and equations with mathematical forms analogous to those of Maxwell’s equations are presented.展开更多
A new method for solving electromagnetic field boundary value problem is given.Byusing this method,the boundary value problem of the vector wave equation can be transformedinto the independent boundary value problem o...A new method for solving electromagnetic field boundary value problem is given.Byusing this method,the boundary value problem of the vector wave equation can be transformedinto the independent boundary value problem of scalar wave equations and the two additionalvector differential operations.All the dyadic Green’s functions got by eigenfunction expansionof the dyadic Green’s function can be got by this method easily and some of the dyadic Green’sfunctions for complex systems which are very difficult to get by the ordinary method have beengot by this new method.The dyadic Green’s function for a dielectric loaded cavity is one of thegiven examples.展开更多
We know from Noether’s theorem that there is a conserved charge for every continuous symmetry. In General Relativity, Killing vectors describe the spacetime symmetries and to each such Killing vector field, we can as...We know from Noether’s theorem that there is a conserved charge for every continuous symmetry. In General Relativity, Killing vectors describe the spacetime symmetries and to each such Killing vector field, we can associate conserved charge through stress-energy tensor of matter which is mentioned in the article. In this article, I show that under simple set of canonical transformation of most general class of Bogoliubov transformation between creation, annihilation operators, those charges associated with spacetime symmetries are broken. To do that, I look at stress-energy tensor of real scalar field theory (as an example) in curved spacetime and show how it changes under simple canonical transformation which is enough to justify our claim. Since doing Bogoliubov transformation is equivalent to coordinate transformation which according to Einstein’s equivalence principle is equivalent to turn on effect of gravity, therefore, we can say that under the effect of gravity those charges are broken.展开更多
In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebe...In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander＇s condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p（x）-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander＇s condition, but they also hold for Grushin vector fields as well with obvious modifications.展开更多
We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution ...We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on K¨ahler manifolds will be studied by Li and Liu(2014).展开更多
In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichart...In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz＇s inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal展开更多
We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem wit...We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.展开更多
Exact self-similar solutions to Einstein’s field equations for the Kantowski-Sachs space-time are determined. The self-similarity property is applied to determine the functional form of the unknown functions that def...Exact self-similar solutions to Einstein’s field equations for the Kantowski-Sachs space-time are determined. The self-similarity property is applied to determine the functional form of the unknown functions that define the gravitational model and to reduce the order of the field equations. The consequences of matter, described by the energy-momentum tensor, are investigated in the case of a perfect fluid. Some physical features and kinematical properties of the obtained model are studied.展开更多
A concrete numerical example of Z6-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcati...A concrete numerical example of Z6-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number H(2k + 1) ? (2k + I)2 - 1 for the perturbed Hamiltonian systems.展开更多
太阳高能活动爆发与活动区内的电流结构有着密切的联系,安培(Ampere)定律j_(z)=1/μ_(0)(▽×B)_(z)是测量活动区内视向电流密度的理论基础。由于实测的矢量磁场中不可避免地存在随机噪声,因此,应用安培定律的不同形式计算的电流密...太阳高能活动爆发与活动区内的电流结构有着密切的联系,安培(Ampere)定律j_(z)=1/μ_(0)(▽×B)_(z)是测量活动区内视向电流密度的理论基础。由于实测的矢量磁场中不可避免地存在随机噪声,因此,应用安培定律的不同形式计算的电流密度存在显著的差异。为了比较不同形式计算结果的差异并从中探索一种实用的电流计算方法,基于太阳动力学天文台(Solar Dynamic Observatory,SDO)/日震学与磁场成像仪(Helioseismic and Magnetic Imager,HMI)在2011年2月15日测量的活动区AR11158的矢量磁图,利用安培定律的微分算法和积分算法分别计算了活动区内视向电流密度的分布图。结果显示,微分算法获得的视向电流密度分布图受随机噪声的影响要远比积分算法获得的结果大,电流分布图中的电流结构没有积分算法获得的结果清晰。另外,在扩大积分环路半径的情况下,所计算的电流分布图中的噪声信号快速降低,视向电流分布图中的电流结构更清晰。但是当继续扩大积分环路半径时,在获得清晰电流分布图的同时,部分精细结构也随之失真。该研究结果论证了适当扩大积分环路计算视向电流分布图可以降低计算结果受随机噪声的影响,从而获得清晰真实的视向电流分布图,但是积分路径的半径过大在消除噪声影响的同时会丢失电流分布中的一些精细结构。因此在实际计算电流的过程中,应该利用高分辨率的矢量磁图,选定合适的积分路径,利用安培定律的积分算法来计算活动区的视向电流,从而帮助我们探索耀斑爆发与活动区内电流结构的关系。展开更多
文摘In two previous papers <a href="#ref1">[1]</a> and <a href="#ref2">[2]</a>, a structure for vector products in <em>n</em> dimensions was presented, and at the same time it was possible to propose the existence of a vector analogous to the curl of a vector field, for a space of four dimensions. In continuation of these works, the objective is to develop, through dimensional analogy, the idea of a hypothetical vector field, associated with the classical electromagnetic wave. This hypothetical field has a possible mathematical existence only when considering a space of four dimensions. The properties of the electromagnetic wave are preserved and equations with mathematical forms analogous to those of Maxwell’s equations are presented.
基金This project is supported by the National Science Fundation of China
文摘A new method for solving electromagnetic field boundary value problem is given.Byusing this method,the boundary value problem of the vector wave equation can be transformedinto the independent boundary value problem of scalar wave equations and the two additionalvector differential operations.All the dyadic Green’s functions got by eigenfunction expansionof the dyadic Green’s function can be got by this method easily and some of the dyadic Green’sfunctions for complex systems which are very difficult to get by the ordinary method have beengot by this new method.The dyadic Green’s function for a dielectric loaded cavity is one of thegiven examples.
文摘We know from Noether’s theorem that there is a conserved charge for every continuous symmetry. In General Relativity, Killing vectors describe the spacetime symmetries and to each such Killing vector field, we can associate conserved charge through stress-energy tensor of matter which is mentioned in the article. In this article, I show that under simple set of canonical transformation of most general class of Bogoliubov transformation between creation, annihilation operators, those charges associated with spacetime symmetries are broken. To do that, I look at stress-energy tensor of real scalar field theory (as an example) in curved spacetime and show how it changes under simple canonical transformation which is enough to justify our claim. Since doing Bogoliubov transformation is equivalent to coordinate transformation which according to Einstein’s equivalence principle is equivalent to turn on effect of gravity, therefore, we can say that under the effect of gravity those charges are broken.
基金supported by NSFC(Grant No.11371056)supported by a US NSF grant
文摘In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander＇s condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p（x）-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander＇s condition, but they also hold for Grushin vector fields as well with obvious modifications.
基金supported by National Natural Science Foundation of China(Grant No.11401374)Shanghai YangFan Project(Grant No.14YF1401400)
文摘We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on K¨ahler manifolds will be studied by Li and Liu(2014).
基金Supported by the National Natural Science Foundation of China the Doctoral Foundation of NEM of China
文摘In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz＇s inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal
文摘We consider the five-point boundary value problem for a fifth-order differential equation, where the nonlinearity is superlinear at both the origin and +infinity. Our method of proof combines the Kneser’s theorem with the well-known from combinatorial topology Sperner’s lemma. We also notice that our geometric approach is strongly based on the associated vector field.
文摘Exact self-similar solutions to Einstein’s field equations for the Kantowski-Sachs space-time are determined. The self-similarity property is applied to determine the functional form of the unknown functions that define the gravitational model and to reduce the order of the field equations. The consequences of matter, described by the energy-momentum tensor, are investigated in the case of a perfect fluid. Some physical features and kinematical properties of the obtained model are studied.
基金This work was supported by the Strategic Research (Grant No. 7000934) from the City University of Hong Kong.
文摘A concrete numerical example of Z6-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number H(2k + 1) ? (2k + I)2 - 1 for the perturbed Hamiltonian systems.
文摘太阳高能活动爆发与活动区内的电流结构有着密切的联系,安培(Ampere)定律j_(z)=1/μ_(0)(▽×B)_(z)是测量活动区内视向电流密度的理论基础。由于实测的矢量磁场中不可避免地存在随机噪声,因此,应用安培定律的不同形式计算的电流密度存在显著的差异。为了比较不同形式计算结果的差异并从中探索一种实用的电流计算方法,基于太阳动力学天文台(Solar Dynamic Observatory,SDO)/日震学与磁场成像仪(Helioseismic and Magnetic Imager,HMI)在2011年2月15日测量的活动区AR11158的矢量磁图,利用安培定律的微分算法和积分算法分别计算了活动区内视向电流密度的分布图。结果显示,微分算法获得的视向电流密度分布图受随机噪声的影响要远比积分算法获得的结果大,电流分布图中的电流结构没有积分算法获得的结果清晰。另外,在扩大积分环路半径的情况下,所计算的电流分布图中的噪声信号快速降低,视向电流分布图中的电流结构更清晰。但是当继续扩大积分环路半径时,在获得清晰电流分布图的同时,部分精细结构也随之失真。该研究结果论证了适当扩大积分环路计算视向电流分布图可以降低计算结果受随机噪声的影响,从而获得清晰真实的视向电流分布图,但是积分路径的半径过大在消除噪声影响的同时会丢失电流分布中的一些精细结构。因此在实际计算电流的过程中,应该利用高分辨率的矢量磁图,选定合适的积分路径,利用安培定律的积分算法来计算活动区的视向电流,从而帮助我们探索耀斑爆发与活动区内电流结构的关系。
