In this paper, the heat, resolvent and wave kernels associated to the Schr?dinger operator with multi-inverse square potential on the Euclidian space Rn are given in explicit forms.
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
面对高度复杂和多变的现代战场,如何快速直观地发现战场上的态势热点,降低认知负载是指挥员所面临的巨大挑战。针对战场时变条件下的态势热点发现问题,在分析热点形成机理的基础上,提出关注势理论,构建了符合人类认知特性的改进的Logis...面对高度复杂和多变的现代战场,如何快速直观地发现战场上的态势热点,降低认知负载是指挥员所面临的巨大挑战。针对战场时变条件下的态势热点发现问题,在分析热点形成机理的基础上,提出关注势理论,构建了符合人类认知特性的改进的Logistic时间衰减函数(improved Logistic time decay function,ILTDF),进而提出随时间衰减的加权核密度估计(weighted kernel density estimation decay over time,W-KDE-DOT)法以及基于关注势的战场态势热力图构建方法。实验结果表明,基于关注势理论的热力图构建方法,能够更准确地刻画出态势热点的时变特性,更符合战场态势热点发现客观规律,能够更好地为指挥决策人员准确把握战场态势热点提供支撑。展开更多
The asymptotic expansion of the heat kernel Θ(t)=sum from ∞ to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞ are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R^n(n=2 or 3) is studied for sho...The asymptotic expansion of the heat kernel Θ(t)=sum from ∞ to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞ are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R^n(n=2 or 3) is studied for short-time t for a generalbounded domain Ω with a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J+1,...,k) andthe Robin conditions ((?)+γ_i)φ=0 on Γ_i (i=k+1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ω consists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.展开更多
We consider the Hyers-Ulam stability problem of the generalized quadratic functional equationuoA＋voB-2woP1 - 2ko P2 =0, which is a distributional version of the classical generalized quadratic functional equation f（...We consider the Hyers-Ulam stability problem of the generalized quadratic functional equationuoA＋voB-2woP1 - 2ko P2 =0, which is a distributional version of the classical generalized quadratic functional equation f（x＋y）＋g（x - y） - 2h（x） - 2k（y）=0展开更多
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
假定Lf(x)=−1/ω(x)∑i,j∂i(aij(⋅)∂jf)(x)+V(x)f(x)为退化Schrödinger算子,其中ω是来自Muckenhoupt class A_(2)的权.又设V是非负位势,属于与ω(x)dx有关的反Hölder不等式.基于分数阶热半群{e^(−tLα)}_(t>0)的正则性估...假定Lf(x)=−1/ω(x)∑i,j∂i(aij(⋅)∂jf)(x)+V(x)f(x)为退化Schrödinger算子,其中ω是来自Muckenhoupt class A_(2)的权.又设V是非负位势,属于与ω(x)dx有关的反Hölder不等式.基于分数阶热半群{e^(−tLα)}_(t>0)的正则性估计,我们通过两个面积函数S_(α)^(L)和g_(α)^(L)来刻画与L有关的Hardy空间.展开更多
文摘In this paper, the heat, resolvent and wave kernels associated to the Schr?dinger operator with multi-inverse square potential on the Euclidian space Rn are given in explicit forms.
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.
文摘面对高度复杂和多变的现代战场,如何快速直观地发现战场上的态势热点,降低认知负载是指挥员所面临的巨大挑战。针对战场时变条件下的态势热点发现问题,在分析热点形成机理的基础上,提出关注势理论,构建了符合人类认知特性的改进的Logistic时间衰减函数(improved Logistic time decay function,ILTDF),进而提出随时间衰减的加权核密度估计(weighted kernel density estimation decay over time,W-KDE-DOT)法以及基于关注势的战场态势热力图构建方法。实验结果表明,基于关注势理论的热力图构建方法,能够更准确地刻画出态势热点的时变特性,更符合战场态势热点发现客观规律,能够更好地为指挥决策人员准确把握战场态势热点提供支撑。
文摘The asymptotic expansion of the heat kernel Θ(t)=sum from ∞ to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞ are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R^n(n=2 or 3) is studied for short-time t for a generalbounded domain Ω with a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J+1,...,k) andthe Robin conditions ((?)+γ_i)φ=0 on Γ_i (i=k+1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ω consists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.
基金Supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (Grant No. KRF-2007-521-C00016)
文摘We consider the Hyers-Ulam stability problem of the generalized quadratic functional equationuoA＋voB-2woP1 - 2ko P2 =0, which is a distributional version of the classical generalized quadratic functional equation f（x＋y）＋g（x - y） - 2h（x） - 2k（y）=0
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.
基金Supported by Natural Science Foundation of Shandong Province(No.ZR2020MA004)NSFC(No.11871293)。
文摘假定Lf(x)=−1/ω(x)∑i,j∂i(aij(⋅)∂jf)(x)+V(x)f(x)为退化Schrödinger算子,其中ω是来自Muckenhoupt class A_(2)的权.又设V是非负位势,属于与ω(x)dx有关的反Hölder不等式.基于分数阶热半群{e^(−tLα)}_(t>0)的正则性估计,我们通过两个面积函数S_(α)^(L)和g_(α)^(L)来刻画与L有关的Hardy空间.
