An additional lower hybrid wave (LHW) with a higher refractive index (N//) was investigated in the HT-7 tokamak to bridge the spectral gap. It was found that the spectral gap between the wave and the electrons in ...An additional lower hybrid wave (LHW) with a higher refractive index (N//) was investigated in the HT-7 tokamak to bridge the spectral gap. It was found that the spectral gap between the wave and the electrons in the outer region was bridged by the additional wave with a higher N// spectrum. The results showed that the sawteeth oscillation was suppressed by launching the additional wave, and that the power deposition profile was moved outwards and the current profile was broadened due to the application of the additional wave. Our study indicates that the spectral gap may be bridged by an additional wave with a higher N// spectrum in the outer region.展开更多
The study of the convergent rate (spectra gap) in the L^2-sense is motivated from several different fields: probability statistics, mathematical physics, computer science and so on and it is now an artive research top...The study of the convergent rate (spectra gap) in the L^2-sense is motivated from several different fields: probability statistics, mathematical physics, computer science and so on and it is now an artive research topic. Based on a new approach (the coupling technique) introduced in [7] for the estimate of the convergent rate and as a continuation of [4],[5],[7-9],[23] and [24], this paper studies the estimate of the rate for time-continuous Markov chains. Two variational formulas for the rate are presented here for the first time for birth-death processes. For diffusions, similar results are presented in an accompany paper [10]. The new formulas enable us to recover or improve the main known results. The connection between the sharp estimate and the corresponding eigenfunction is explored and illustrated by various examples. A previous result on optimal Markovian couplings is also extended in the paper.展开更多
The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap'of finite dimensional systems is given ...The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap'of finite dimensional systems is given by Theorem 1.1, in terms of the spectral gap for one-dimensional marginals. The study of this topic provides us a chance, and it is indeed another aim of the paper, to justify the power of the results obtained previously. The exact order in dimension one (Proposition 1.4), and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic Sobolev constant for two typical infinite-dimensional models are presented (Theorems 6.2 and 6.3). Since we are interested in explicit estimates, the computations become quite involved. A long section (Section 4) is devoted to the study of the spectral gap in dimension one.展开更多
Abstract Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L2(π)-space, denoted by gap(P), is defined as the distance between 1 and the rest of the spectrum of...Abstract Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L2(π)-space, denoted by gap(P), is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap(P) and the convergence rate of P^n. When P is transient, the convergence rate of pn is equal to 1 - gap(P). When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of pn in terms of gap(P). These results are extended to L^∞ (π)-space.展开更多
This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable ...This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable restriction on the initial distributions, the rate is bounded from below by the spectral gap. Furthemore, in the compact case or for birth-death processes or half-line diffosions, the rate is shown to be equal to the spectral gap.展开更多
We generalize the decomposition method of the finite Markov chains for Poincare inequality in Jerrum et al.(Ann.Appl.Probab.,14,1741-1765(2004)) to the reversible continuous-time Markov chains.And inductively,we g...We generalize the decomposition method of the finite Markov chains for Poincare inequality in Jerrum et al.(Ann.Appl.Probab.,14,1741-1765(2004)) to the reversible continuous-time Markov chains.And inductively,we give the lower bound of spectral gap for the ergodic open Jackson network by the decomposition method and the symmetrization procedure.The upper bound of the spectral gap is also presented.展开更多
We give an easy proof of Andrews and Clutterbuck's main results [J. Amer. Math. Soc., 2011, 24(3): 899-916], which gives both a sharp lower bound for the spectral gap of a Schr5dinger operator and a sharp modulus ...We give an easy proof of Andrews and Clutterbuck's main results [J. Amer. Math. Soc., 2011, 24(3): 899-916], which gives both a sharp lower bound for the spectral gap of a Schr5dinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We arrive directly at the same estimates by the 'double coordinate' approach and asymptotic behavior of parabolic flows. Although using the techniques appeared in the above paper, we partly simplify the method and argument. This maybe help to provide an easy way for estimating spectral gap. Besides, we also get a new lower bound of spectral gap for a class of SchSdinger operator.展开更多
For a reversible quasi-birth and death process, we generalize and refine the decomposition method, by constructing a birth-death process and a sequence of restriction processes. The spectral gap for the quasi-birth an...For a reversible quasi-birth and death process, we generalize and refine the decomposition method, by constructing a birth-death process and a sequence of restriction processes. The spectral gap for the quasi-birth and death process is estimated in terms of the spectral gaps for these processes, and in some special cases, the estimation is sharp. With the aid of the symmetrization procedure, the result is also applied to two queueing models: M/M/1 in random environment and MIMIc with synchronous vacation.展开更多
We give a two sided estimate on the spectral gap for the Boltzmann measures μh on the circle. We prove that the spectral gap is greater than 1 for any h∈R and the spectral gap tends to the positive infinity as h→∞...We give a two sided estimate on the spectral gap for the Boltzmann measures μh on the circle. We prove that the spectral gap is greater than 1 for any h∈R and the spectral gap tends to the positive infinity as h→∞ with speed |h|.展开更多
THIS is the second one of a series of three reviews. The ideas introduced in the last review are used to study the estimate of spectral gap and four classes of typical eigenvalue problems on manifolds. The comparison ...THIS is the second one of a series of three reviews. The ideas introduced in the last review are used to study the estimate of spectral gap and four classes of typical eigenvalue problems on manifolds. The comparison with the known optimal estimates is given, some new progress is reported and some open problems are proposed.展开更多
THIS is the last one of a series of three papers. Here, we discuss six topics related to the spectral gap: the gradient estimate, the heat kernel and Harnack inequality, the logarithmic Sobolev inequality, the converg...THIS is the last one of a series of three papers. Here, we discuss six topics related to the spectral gap: the gradient estimate, the heat kernel and Harnack inequality, the logarithmic Sobolev inequality, the convergence in total variation, the algebraic convergence and the infinite-dimensional case. The perturbation of spectral gap and the logarithmic Sobolev constant under a linear transform is given (Theorem 5). A new proof for computing the logarithmic Sobolev constant in a basic case is also presented (Theorem 7).展开更多
THIS is the first of a series of three reviews. They are partially surveys on three aspects: (ⅰ) explaining the main ideas of our recent application of the coupling method to the estimation of spectral gap, (ⅱ) intr...THIS is the first of a series of three reviews. They are partially surveys on three aspects: (ⅰ) explaining the main ideas of our recent application of the coupling method to the estimation of spectral gap, (ⅱ) introducing some more recent progress on the study on some related topics,(ⅲ) collecting some open problems for the further study.展开更多
Let (M, g) be a complete Riemannian manifold of dimension d and Ricci curvaturebounded below by-K for some K∈. Let dx and p(x,y) be respectively the Riemannianvolum element and Riemannian distance. Consider L=△+...Let (M, g) be a complete Riemannian manifold of dimension d and Ricci curvaturebounded below by-K for some K∈. Let dx and p(x,y) be respectively the Riemannianvolum element and Riemannian distance. Consider L=△+V with V∈C<sup>2</sup>(M) satisfyingZ=∫<sub>m</sub>exp[V]dx 【 ∞. Then the L-diffusion process is reversible with respect to μ(dx)=Z<sup>-l</sup>exp[V]dx. It is well known that the exponentially L<sup>2</sup>-convergence of the L-diffusion pro-cess equivalent to a spectral gap展开更多
The Cheeger’s inequalities and some existence criteria for spectral gap and for general symmetric forms are established. The criteria are also extended to general reversible Markov processes but not reported here. Ev...The Cheeger’s inequalities and some existence criteria for spectral gap and for general symmetric forms are established. The criteria are also extended to general reversible Markov processes but not reported here. Even though in the past several decades, the topics have been widely studied, as far as we know the first problem in the unbounded case and the second one in the general case remain open.展开更多
雷达信号分选是现代电子战中的重要环节.为了解决传统算法鲁棒性较差的问题,提出一种基于层次密度聚类和谱间隙的雷达信号分选算法.使用载频和脉宽参数进行层次密度聚类,根据重新定义的簇间距得到赋权邻接矩阵,计算赋权邻接矩阵的拉普...雷达信号分选是现代电子战中的重要环节.为了解决传统算法鲁棒性较差的问题,提出一种基于层次密度聚类和谱间隙的雷达信号分选算法.使用载频和脉宽参数进行层次密度聚类,根据重新定义的簇间距得到赋权邻接矩阵,计算赋权邻接矩阵的拉普拉斯谱间隙,通过k-means聚类的超参数k对信号进行分选.仿真实验结果表明:该文算法的平均分选准确率达0.9960、平均召回率达0.9560;相对于密度聚类(density-based spatial clustering of applications with noise,简称DBSCAN)和Meanshift算法,该文算法对杂乱脉冲、漏脉冲及超参数的干扰均有最强的鲁棒性.展开更多
基金the Knowledge Innovation Program of the Chinese Academy of Sciences (No.075FCQ0127)National Natural Science Foundation of China (Nos.10575104,10805057)
文摘An additional lower hybrid wave (LHW) with a higher refractive index (N//) was investigated in the HT-7 tokamak to bridge the spectral gap. It was found that the spectral gap between the wave and the electrons in the outer region was bridged by the additional wave with a higher N// spectrum. The results showed that the sawteeth oscillation was suppressed by launching the additional wave, and that the power deposition profile was moved outwards and the current profile was broadened due to the application of the additional wave. Our study indicates that the spectral gap may be bridged by an additional wave with a higher N// spectrum in the outer region.
基金Research supported in part by NSFC, Qin Shi Sci & Tech. Foundationthe State Education Commission of China.
文摘The study of the convergent rate (spectra gap) in the L^2-sense is motivated from several different fields: probability statistics, mathematical physics, computer science and so on and it is now an artive research topic. Based on a new approach (the coupling technique) introduced in [7] for the estimate of the convergent rate and as a continuation of [4],[5],[7-9],[23] and [24], this paper studies the estimate of the rate for time-continuous Markov chains. Two variational formulas for the rate are presented here for the first time for birth-death processes. For diffusions, similar results are presented in an accompany paper [10]. The new formulas enable us to recover or improve the main known results. The connection between the sharp estimate and the corresponding eigenfunction is explored and illustrated by various examples. A previous result on optimal Markovian couplings is also extended in the paper.
基金the Creative Research Group Fund of the National Natural Science Foundation of China (No.10121101)the"985"Project from the Ministry of Education of China
文摘The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap'of finite dimensional systems is given by Theorem 1.1, in terms of the spectral gap for one-dimensional marginals. The study of this topic provides us a chance, and it is indeed another aim of the paper, to justify the power of the results obtained previously. The exact order in dimension one (Proposition 1.4), and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic Sobolev constant for two typical infinite-dimensional models are presented (Theorems 6.2 and 6.3). Since we are interested in explicit estimates, the computations become quite involved. A long section (Section 4) is devoted to the study of the spectral gap in dimension one.
基金Supported in part by 985 Project,973 Project(Grant No.2011CB808000)National Natural Science Foundation of China(Grant No.11131003)+1 种基金Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20100003110005)the Fundamental Research Funds for the Central Universities
文摘Abstract Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L2(π)-space, denoted by gap(P), is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap(P) and the convergence rate of P^n. When P is transient, the convergence rate of pn is equal to 1 - gap(P). When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of pn in terms of gap(P). These results are extended to L^∞ (π)-space.
基金Research supported in part by NSFC, Qiu Shi Sci. & Tech. Found. DPFIHE, MCSEC and Univ. of Rome I, Italy
文摘This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable restriction on the initial distributions, the rate is bounded from below by the spectral gap. Furthemore, in the compact case or for birth-death processes or half-line diffosions, the rate is shown to be equal to the spectral gap.
基金Supported in part by 985 Project973 Project(Grant No.2011CB808000)+2 种基金NSFC(Grant No.11131003)SRFDP(Grant No.20100003110005)the Fundamental Research Funds for the Central Universities
文摘We generalize the decomposition method of the finite Markov chains for Poincare inequality in Jerrum et al.(Ann.Appl.Probab.,14,1741-1765(2004)) to the reversible continuous-time Markov chains.And inductively,we give the lower bound of spectral gap for the ergodic open Jackson network by the decomposition method and the symmetrization procedure.The upper bound of the spectral gap is also presented.
文摘We give an easy proof of Andrews and Clutterbuck's main results [J. Amer. Math. Soc., 2011, 24(3): 899-916], which gives both a sharp lower bound for the spectral gap of a Schr5dinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We arrive directly at the same estimates by the 'double coordinate' approach and asymptotic behavior of parabolic flows. Although using the techniques appeared in the above paper, we partly simplify the method and argument. This maybe help to provide an easy way for estimating spectral gap. Besides, we also get a new lower bound of spectral gap for a class of SchSdinger operator.
基金Supported in part by Program for New Century Excellent Talents in University (NCET)973 Project (Grant No. 2011CB808000)NSFC (Grant No. 10721091)
文摘For a reversible quasi-birth and death process, we generalize and refine the decomposition method, by constructing a birth-death process and a sequence of restriction processes. The spectral gap for the quasi-birth and death process is estimated in terms of the spectral gaps for these processes, and in some special cases, the estimation is sharp. With the aid of the symmetrization procedure, the result is also applied to two queueing models: M/M/1 in random environment and MIMIc with synchronous vacation.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.10871008,11571043,11671076,11871382)and 985 projects.
文摘We give a two sided estimate on the spectral gap for the Boltzmann measures μh on the circle. We prove that the spectral gap is greater than 1 for any h∈R and the spectral gap tends to the positive infinity as h→∞ with speed |h|.
文摘THIS is the second one of a series of three reviews. The ideas introduced in the last review are used to study the estimate of spectral gap and four classes of typical eigenvalue problems on manifolds. The comparison with the known optimal estimates is given, some new progress is reported and some open problems are proposed.
文摘THIS is the last one of a series of three papers. Here, we discuss six topics related to the spectral gap: the gradient estimate, the heat kernel and Harnack inequality, the logarithmic Sobolev inequality, the convergence in total variation, the algebraic convergence and the infinite-dimensional case. The perturbation of spectral gap and the logarithmic Sobolev constant under a linear transform is given (Theorem 5). A new proof for computing the logarithmic Sobolev constant in a basic case is also presented (Theorem 7).
文摘THIS is the first of a series of three reviews. They are partially surveys on three aspects: (ⅰ) explaining the main ideas of our recent application of the coupling method to the estimation of spectral gap, (ⅱ) introducing some more recent progress on the study on some related topics,(ⅲ) collecting some open problems for the further study.
基金Project partly supported by the National Natural Science Foundation of China and the State Education Commission of China.
文摘Let (M, g) be a complete Riemannian manifold of dimension d and Ricci curvaturebounded below by-K for some K∈. Let dx and p(x,y) be respectively the Riemannianvolum element and Riemannian distance. Consider L=△+V with V∈C<sup>2</sup>(M) satisfyingZ=∫<sub>m</sub>exp[V]dx 【 ∞. Then the L-diffusion process is reversible with respect to μ(dx)=Z<sup>-l</sup>exp[V]dx. It is well known that the exponentially L<sup>2</sup>-convergence of the L-diffusion pro-cess equivalent to a spectral gap
基金Project supported in part by the National Natural Science Foundation of China (Grant No. 19631060)Beijing Normal University and the State Education Commission of China.
文摘Under certain curvature condition, the existence of spectral gap is proved on path spaces with infinite time-interval.
文摘The Cheeger’s inequalities and some existence criteria for spectral gap and for general symmetric forms are established. The criteria are also extended to general reversible Markov processes but not reported here. Even though in the past several decades, the topics have been widely studied, as far as we know the first problem in the unbounded case and the second one in the general case remain open.
文摘雷达信号分选是现代电子战中的重要环节.为了解决传统算法鲁棒性较差的问题,提出一种基于层次密度聚类和谱间隙的雷达信号分选算法.使用载频和脉宽参数进行层次密度聚类,根据重新定义的簇间距得到赋权邻接矩阵,计算赋权邻接矩阵的拉普拉斯谱间隙,通过k-means聚类的超参数k对信号进行分选.仿真实验结果表明:该文算法的平均分选准确率达0.9960、平均召回率达0.9560;相对于密度聚类(density-based spatial clustering of applications with noise,简称DBSCAN)和Meanshift算法,该文算法对杂乱脉冲、漏脉冲及超参数的干扰均有最强的鲁棒性.