As data becomes increasingly complex,measuring dependence among variables is of great interest.However,most existing measures of dependence are limited to the Euclidean setting and cannot effectively characterize the ...As data becomes increasingly complex,measuring dependence among variables is of great interest.However,most existing measures of dependence are limited to the Euclidean setting and cannot effectively characterize the complex relationships.In this paper,we propose a novel method for constructing independence tests for random elements in Hilbert spaces,which includes functional data as a special case.Our approach is using distance covariance of random projections to build a test statistic that is computationally efficient and exhibits strong power performance.We prove the equivalence between testing for independence expressed on the original and the projected covariates,bridging the gap between measures of testing independence in Euclidean spaces and Hilbert spaces.Implementation of the test involves calibration by permutation and combining several p-values from different projections using the false discovery rate method.Simulation studies and real data examples illustrate the finite sample properties of the proposed method under a variety of scenarios.展开更多
The main purpose of this paper is to introduce and deal with a self adaptive inertial subgradient extragradient iterative algorithm with a new and interesting stepsize rule in real Hilbert spaces.Under some proper con...The main purpose of this paper is to introduce and deal with a self adaptive inertial subgradient extragradient iterative algorithm with a new and interesting stepsize rule in real Hilbert spaces.Under some proper control conditions imposed on the coefficients and operators,we prove a new strong convergence result for solving variational inequalities with regard to pseudomonotone and Lipschitzian operators.Moreover,some numerical simulation results are given to show the rationality and validity of our algorithm.展开更多
The purpose of this article is to introduce a general split feasibility problems for two families of nonexpansive mappings in Hilbert spaces. We prove that the sequence generated by the proposed new algorithm converge...The purpose of this article is to introduce a general split feasibility problems for two families of nonexpansive mappings in Hilbert spaces. We prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.展开更多
In this paper, we prove that the generalized Roper-Suffridge extension operator can be embeded in Loewner chains on the unit ball in Hibert spaces, and obtain the fact that the operator keeps the properties of almost ...In this paper, we prove that the generalized Roper-Suffridge extension operator can be embeded in Loewner chains on the unit ball in Hibert spaces, and obtain the fact that the operator keeps the properties of almost spirallike mapping of typeβ and order α, almost starlikeness of order α, spirallikeness of type ofβ and starlikeness.展开更多
Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi＇s res...Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi＇s results on H∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.展开更多
A system of generalized mixed equilibrium-like problems is introduced and the existence of its solutions is shown by using the auxiliary principle technique in Hilbert spaces.
In this paper,we consider the solvability of generalized variational inequalities involving multi-valued relaxed monotone operators in the framework of Hilbert spaces.Our results mainly improve the corresponding resul...In this paper,we consider the solvability of generalized variational inequalities involving multi-valued relaxed monotone operators in the framework of Hilbert spaces.Our results mainly improve the corresponding results announced by Verma[R U Verma,Generalized variational inequalities involving multivalued relaxed monotone operators,Appl Math Lett,1997,10：107-109]and many others.展开更多
The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the s...The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.展开更多
The separability and the entanglement(that is,inseparability)of the composite quantum states play important roles in quantum information theory.Mathematically,a quantum state is a trace-class positive operator with tr...The separability and the entanglement(that is,inseparability)of the composite quantum states play important roles in quantum information theory.Mathematically,a quantum state is a trace-class positive operator with trace one acting on a complex separable Hilbert space.In this paper,in more general frame,the notion of separability for quantum states is generalized to bounded positive operators acting on tensor product of Hilbert spaces.However,not like the quantum state case,there are different kinds of separability for positive operators with different operator topologies.Four types of such separability are discussed;several criteria such as the finite rank entanglement witness criterion,the positive elementary operator criterion and PPT criterion to detect the separability of the positive operators are established;some methods to construct separable positive operators by operator matrices are provided.These may also make us to understand the separability and entanglement of quantum states better,and may be applied to find new separable quantum states.展开更多
In this paper, we study the relation between the ordered reproducing Hilbert space and its reproducing kernel. A complete description of a similar and unitary equivalence of two quasi-invariant subspaces generated by ...In this paper, we study the relation between the ordered reproducing Hilbert space and its reproducing kernel. A complete description of a similar and unitary equivalence of two quasi-invariant subspaces generated by polynomials with leading terms is given.展开更多
The Mann iterations have no strong convergence even for nonexpansive mappings in Hilbert spaces. The aim of this paper is to propose a modification of the Mann iterations for strictly asymptotically pseudocontractive ...The Mann iterations have no strong convergence even for nonexpansive mappings in Hilbert spaces. The aim of this paper is to propose a modification of the Mann iterations for strictly asymptotically pseudocontractive maps in Hilbert spaces to have strong convergence. Our results extend those of Kim, Xu, Nakajo, Takahashi and many others.展开更多
We study the norm retrieval by projections on an infinite-dimensional Hilbert space H. Let {ei}i∈I be an orthonormal basis in H and Wi = {ei}^⊥ for all i ∈ I. We show that {Wi}i∈I does norm retrieval if and only i...We study the norm retrieval by projections on an infinite-dimensional Hilbert space H. Let {ei}i∈I be an orthonormal basis in H and Wi = {ei}^⊥ for all i ∈ I. We show that {Wi}i∈I does norm retrieval if and only if I is an infinite subset of N. We also give some properties of norm retrieval by projections.展开更多
In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 〈 s 〈 1/2. Th...In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 〈 s 〈 1/2. Then for every bounded linear operator T ： H → H and x0 ∈ H with ｜｜T｜｜ = 1 = ｜｜xo｜｜ such that ｜｜Txo｜｜ 〉 1-6, there exist xε ∈ H and a bounded linear operator S ： H → H with ｜｜S｜｜ = 1 = ｜｜xε｜｜ such that ｜｜Sxε｜｜=1, ｜｜x-ε0｜｜≤√2ε＋4√2ε, ｜｜S-T｜｜≤√2ε.展开更多
We consider a gradient iteration algorithm for prediction of functional linear regression under the framework of reproducing kernel Hilbert spaces.In the algorithm,we use an early stopping technique,instead of the cla...We consider a gradient iteration algorithm for prediction of functional linear regression under the framework of reproducing kernel Hilbert spaces.In the algorithm,we use an early stopping technique,instead of the classical Tikhonov regularization,to prevent the iteration from an overfitting function.Under mild conditions,we obtain upper bounds,essentially matching the known minimax lower bounds,for excess prediction risk.An almost sure convergence is also established for the proposed algorithm.展开更多
This article derives the relation between universal interpolating sequences and some spectral properties of the multiplication operator by the independent variable z in case the underlying space is a Hilbert space of ...This article derives the relation between universal interpolating sequences and some spectral properties of the multiplication operator by the independent variable z in case the underlying space is a Hilbert space of functions analytic on the open unit disk.展开更多
In the paper, we study the stability of Hilbert space generalized frames under perturbations. We get some results that are in spirit close to classical results for discrete frames, due to OLE Christensen.
In open quantum systems,the Liouvillian gap characterizes the relaxation time toward the steady state.However,accurately computing this quantity is notoriously difficult due to the exponential growth of the Hilbert sp...In open quantum systems,the Liouvillian gap characterizes the relaxation time toward the steady state.However,accurately computing this quantity is notoriously difficult due to the exponential growth of the Hilbert space and the non-Hermitian nature of the Liouvillian superoperator.In this work,we propose a variational quantum algorithm for efficiently estimating the Liouvillian gap.By utilizing the Choi-Jamio lkowski isomorphism,we reformulate the problem as finding the first excitation energy of an effective non-Hermitian Hamiltonian.Our method employs variance minimization with an orthogonality constraint to locate the first excited state and adopts a two-stage optimization scheme to enhance convergence.Moreover,to address scenarios with degenerate steady states,we introduce an iterative energy-offset scanning technique.Numerical simulations on the dissipative XXZ model confirm the accuracy and robustness of our algorithm across a range of system sizes and dissipation strengths.These results demonstrate the promise of variational quantum algorithms for simulating open quantum many-body systems on near-term quantum hardware.展开更多
Deep learning algorithms based on neural networks make remarkable achievements in machine fault diagnosis,while the noise mixed in measured signals harms the prediction accuracy of networks.Existing denoising methods ...Deep learning algorithms based on neural networks make remarkable achievements in machine fault diagnosis,while the noise mixed in measured signals harms the prediction accuracy of networks.Existing denoising methods in neural networks,such as using complex network architectures and introducing sparse techniques,always suffer from the difficulty of estimating hyperparameters and the lack of physical interpretability.To address this issue,this paper proposes a novel interpretable denoising layer based on reproducing kernel Hilbert space(RKHS)as the first layer for standard neural networks,with the aim to combine the advantages of both traditional signal processing technology with physical interpretation and network modeling strategy with parameter adaption.By investigating the influencing mechanism of parameters on the regularization procedure in RKHS,the key parameter that dynamically controls the signal smoothness with low computational cost is selected as the only trainable parameter of the proposed layer.Besides,the forward and backward propagation algorithms of the designed layer are formulated to ensure that the selected parameter can be automatically updated together with other parameters in the neural network.Moreover,exponential and piecewise functions are introduced in the weight updating process to keep the trainable weight within a reasonable range and avoid the ill-conditioned problem.Experiment studies verify the effectiveness and compatibility of the proposed layer design method in intelligent fault diagnosis of machinery in noisy environments.展开更多
In this paper the authors give a definite meaning to any formal trigonometrical series and generalize it to all abstract Hilbert space. Then in the case L-2(-infinity + infinity) they discussed extensively the general...In this paper the authors give a definite meaning to any formal trigonometrical series and generalize it to all abstract Hilbert space. Then in the case L-2(-infinity + infinity) they discussed extensively the generalized expansion problem by Hermite functions, and applied to a non-strictly nonlinear hyperbolic system.展开更多
基金Supported by the Grant of National Science Foundation of China(11971433)Zhejiang Gongshang University“Digital+”Disciplinary Construction Management Project(SZJ2022B004)+1 种基金Institute for International People-to-People Exchange in Artificial Intelligence and Advanced Manufacturing(CCIPERGZN202439)the Development Fund for Zhejiang College of Shanghai University of Finance and Economics(2023FZJJ15).
文摘As data becomes increasingly complex,measuring dependence among variables is of great interest.However,most existing measures of dependence are limited to the Euclidean setting and cannot effectively characterize the complex relationships.In this paper,we propose a novel method for constructing independence tests for random elements in Hilbert spaces,which includes functional data as a special case.Our approach is using distance covariance of random projections to build a test statistic that is computationally efficient and exhibits strong power performance.We prove the equivalence between testing for independence expressed on the original and the projected covariates,bridging the gap between measures of testing independence in Euclidean spaces and Hilbert spaces.Implementation of the test involves calibration by permutation and combining several p-values from different projections using the false discovery rate method.Simulation studies and real data examples illustrate the finite sample properties of the proposed method under a variety of scenarios.
文摘The main purpose of this paper is to introduce and deal with a self adaptive inertial subgradient extragradient iterative algorithm with a new and interesting stepsize rule in real Hilbert spaces.Under some proper control conditions imposed on the coefficients and operators,we prove a new strong convergence result for solving variational inequalities with regard to pseudomonotone and Lipschitzian operators.Moreover,some numerical simulation results are given to show the rationality and validity of our algorithm.
基金Supported by the Scientific Research Fund of Sichuan Provincial Department of Science and Technology(2015JY0165,2011JYZ011)the Scientific Research Fund of Sichuan Provincial Education Department(14ZA0271)+2 种基金the Scientific Research Project of Yibin University(2013YY06)the Natural Science Foundation of China Medical University,Taiwanthe National Natural Science Foundation of China(11361070)
文摘The purpose of this article is to introduce a general split feasibility problems for two families of nonexpansive mappings in Hilbert spaces. We prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.
基金Foundation item： Supported by the National Natural Science Foundation of China（10626015 10571044） Supported by the Fundamental Research of National Natural Science Foundation of Henan University（04ZDZR004）
文摘In this paper, we prove that the generalized Roper-Suffridge extension operator can be embeded in Loewner chains on the unit ball in Hibert spaces, and obtain the fact that the operator keeps the properties of almost spirallike mapping of typeβ and order α, almost starlikeness of order α, spirallikeness of type ofβ and starlikeness.
文摘Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi＇s results on H∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.
文摘A system of generalized mixed equilibrium-like problems is introduced and the existence of its solutions is shown by using the auxiliary principle technique in Hilbert spaces.
基金Supported by the Natural Science Foundation of Hebei Province（A2010001943） Supported by the Science Grant of Beijing Jiaotong University（2011YJS075）
文摘In this paper,we consider the solvability of generalized variational inequalities involving multi-valued relaxed monotone operators in the framework of Hilbert spaces.Our results mainly improve the corresponding results announced by Verma[R U Verma,Generalized variational inequalities involving multivalued relaxed monotone operators,Appl Math Lett,1997,10：107-109]and many others.
基金the NSFC(60473034)the Science Foundation of Zhejiang Province(Y604003).
文摘The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.
基金Supported by National Natural Science Foundation of China(Grant No.11171249)。
文摘The separability and the entanglement(that is,inseparability)of the composite quantum states play important roles in quantum information theory.Mathematically,a quantum state is a trace-class positive operator with trace one acting on a complex separable Hilbert space.In this paper,in more general frame,the notion of separability for quantum states is generalized to bounded positive operators acting on tensor product of Hilbert spaces.However,not like the quantum state case,there are different kinds of separability for positive operators with different operator topologies.Four types of such separability are discussed;several criteria such as the finite rank entanglement witness criterion,the positive elementary operator criterion and PPT criterion to detect the separability of the positive operators are established;some methods to construct separable positive operators by operator matrices are provided.These may also make us to understand the separability and entanglement of quantum states better,and may be applied to find new separable quantum states.
基金NNSFC in China,No.10301019a Jiangsu Natural Science Foundation No.BK2007049
文摘In this paper, we study the relation between the ordered reproducing Hilbert space and its reproducing kernel. A complete description of a similar and unitary equivalence of two quasi-invariant subspaces generated by polynomials with leading terms is given.
基金Research Foundation of Henan University (No.06YBZR034)
文摘The Mann iterations have no strong convergence even for nonexpansive mappings in Hilbert spaces. The aim of this paper is to propose a modification of the Mann iterations for strictly asymptotically pseudocontractive maps in Hilbert spaces to have strong convergence. Our results extend those of Kim, Xu, Nakajo, Takahashi and many others.
基金supported in part by the Foundation of Fuzhou University(Grant No.JA15059)
文摘We study the norm retrieval by projections on an infinite-dimensional Hilbert space H. Let {ei}i∈I be an orthonormal basis in H and Wi = {ei}^⊥ for all i ∈ I. We show that {Wi}i∈I does norm retrieval if and only if I is an infinite subset of N. We also give some properties of norm retrieval by projections.
基金supported by Natural Science Foundation of China (Grant No. 11071201)supported by Natural Science Foundation of China (Grant No. 11001231)
文摘In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 〈 s 〈 1/2. Then for every bounded linear operator T ： H → H and x0 ∈ H with ｜｜T｜｜ = 1 = ｜｜xo｜｜ such that ｜｜Txo｜｜ 〉 1-6, there exist xε ∈ H and a bounded linear operator S ： H → H with ｜｜S｜｜ = 1 = ｜｜xε｜｜ such that ｜｜Sxε｜｜=1, ｜｜x-ε0｜｜≤√2ε＋4√2ε, ｜｜S-T｜｜≤√2ε.
基金supported in part by National Natural Science Foundation of China(Grant No.11871438)supported in part by the HKRGC GRF Nos.12300218,12300519,17201020,17300021,C1013-21GF,C7004-21GFJoint NSFC-RGC N-HKU76921。
文摘We consider a gradient iteration algorithm for prediction of functional linear regression under the framework of reproducing kernel Hilbert spaces.In the algorithm,we use an early stopping technique,instead of the classical Tikhonov regularization,to prevent the iteration from an overfitting function.Under mild conditions,we obtain upper bounds,essentially matching the known minimax lower bounds,for excess prediction risk.An almost sure convergence is also established for the proposed algorithm.
文摘This article derives the relation between universal interpolating sequences and some spectral properties of the multiplication operator by the independent variable z in case the underlying space is a Hilbert space of functions analytic on the open unit disk.
文摘In the paper, we study the stability of Hilbert space generalized frames under perturbations. We get some results that are in spirit close to classical results for discrete frames, due to OLE Christensen.
基金supported by the National Natural Science Foundation of China(Grant Nos.12375013 and 12275090)the Guangdong Basic and Applied Basic Research Fund(Grant No.2023A1515011460)Guangdong Provincial Quantum Science Strategic Initiative(Grant No.GDZX2200001)。
文摘In open quantum systems,the Liouvillian gap characterizes the relaxation time toward the steady state.However,accurately computing this quantity is notoriously difficult due to the exponential growth of the Hilbert space and the non-Hermitian nature of the Liouvillian superoperator.In this work,we propose a variational quantum algorithm for efficiently estimating the Liouvillian gap.By utilizing the Choi-Jamio lkowski isomorphism,we reformulate the problem as finding the first excitation energy of an effective non-Hermitian Hamiltonian.Our method employs variance minimization with an orthogonality constraint to locate the first excited state and adopts a two-stage optimization scheme to enhance convergence.Moreover,to address scenarios with degenerate steady states,we introduce an iterative energy-offset scanning technique.Numerical simulations on the dissipative XXZ model confirm the accuracy and robustness of our algorithm across a range of system sizes and dissipation strengths.These results demonstrate the promise of variational quantum algorithms for simulating open quantum many-body systems on near-term quantum hardware.
基金Supported by National Natural Science Foundation of China(Grant Nos.12072188,11632011,11702171,11572189,51121063)Shanghai Municipal Natural Science Foundation of China(Grant No.20ZR1425200).
文摘Deep learning algorithms based on neural networks make remarkable achievements in machine fault diagnosis,while the noise mixed in measured signals harms the prediction accuracy of networks.Existing denoising methods in neural networks,such as using complex network architectures and introducing sparse techniques,always suffer from the difficulty of estimating hyperparameters and the lack of physical interpretability.To address this issue,this paper proposes a novel interpretable denoising layer based on reproducing kernel Hilbert space(RKHS)as the first layer for standard neural networks,with the aim to combine the advantages of both traditional signal processing technology with physical interpretation and network modeling strategy with parameter adaption.By investigating the influencing mechanism of parameters on the regularization procedure in RKHS,the key parameter that dynamically controls the signal smoothness with low computational cost is selected as the only trainable parameter of the proposed layer.Besides,the forward and backward propagation algorithms of the designed layer are formulated to ensure that the selected parameter can be automatically updated together with other parameters in the neural network.Moreover,exponential and piecewise functions are introduced in the weight updating process to keep the trainable weight within a reasonable range and avoid the ill-conditioned problem.Experiment studies verify the effectiveness and compatibility of the proposed layer design method in intelligent fault diagnosis of machinery in noisy environments.
文摘In this paper the authors give a definite meaning to any formal trigonometrical series and generalize it to all abstract Hilbert space. Then in the case L-2(-infinity + infinity) they discussed extensively the generalized expansion problem by Hermite functions, and applied to a non-strictly nonlinear hyperbolic system.
