Determining the minimal distance between the target state and the convex combination of given states is a fundamental problem in quantum resource theory,offering critical guidance for experimental implementations.In t...Determining the minimal distance between the target state and the convex combination of given states is a fundamental problem in quantum resource theory,offering critical guidance for experimental implementations.In this paper,we embark on an in-depth exploration of the use of a quantum state prepared by the convex combination of given qubit states to optimally approximate the l_(1)-norm of coherence of the target quantum state,striving to make the prepared state and the target state as similar as possible.Here,we present the analytical solution for the optimal distance for any N given quantum states.We find that the optimal approximation problem for any N>4 quantum states can be transformed into an optimal approximation problem for no more than four quantum states,which not only significantly streamlines the problem but also proves advantageous for laboratories in terms of material conservation.Ultimately,a one-to-one comparison between the analytical and numerical solutions verifies the effectiveness of our approach.展开更多
The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises.The n...The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises.The noise term is approximated through the spectral projection of the covariance operator,which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises,the well-posedness of the SPDE is established under certain covariance operator-dependent conditions.These SPDEs with projected noises are then numerically approximated with the finite element method.A general error estimate framework is established for the finite element approximations.Based on this framework,optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained.It is shown that with the proposed approach,convergence order of white noise driven SPDEs is improved by half for one-dimensional problems,and by an infinitesimal factor for higher-dimensional problems.展开更多
Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, co...Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, complement operators, so the rough set theory has been enriched from the operator-oriented and set-oriented views. Approximate power set spaces are defined, and it is proved that the approximation operators are epimorphisms from power set space to approximate power set spaces. Some basic properties of approximate power set space are got by epimorphisms in contrast to power set space.展开更多
This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation techn...This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...展开更多
The problem of strong uniqueness of best approximation from an RS set in a Banach space is considered. For a fixed RS set G and an element x∈X , we proved that the best approximation g * to x from ...The problem of strong uniqueness of best approximation from an RS set in a Banach space is considered. For a fixed RS set G and an element x∈X , we proved that the best approximation g * to x from G is strongly unique.展开更多
Based on the moving least square （MLS） approximations and the boundary integral equations （BIEs）, a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection ope...Based on the moving least square （MLS） approximations and the boundary integral equations （BIEs）, a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection operator is used to tackle the nonlinear boundary inequality conditions. The Signorini problem is then reformulated as BIEs and the unknown boundary variables are approximated by the MLS approximations. Accordingly, only a nodal data structure on the boundary of a domain is required. The convergence of the algorithm is proven. Numerical examples are given to show the high convergence rate and high computational efficiency of the presented algorithm.展开更多
We review and compare two definitions of rough set approximations.One is defined by a pair of sets in the universe and the other by a pair of sets in the quotient universe.The latter definition,although less studied,i...We review and compare two definitions of rough set approximations.One is defined by a pair of sets in the universe and the other by a pair of sets in the quotient universe.The latter definition,although less studied,is semantically superior for interpreting rule induction and is closely related to granularity switching in granular computing.Numerical measures about the accuracy and quality of approximations are examined.Several semantics difficulties are commented.展开更多
A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in th...A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in the L2 norm and nodal supercnnvergence. These results generalize those obtained earlier by Hulme for continuous piecevjise polynomials and by Delfour-Dubeau for discontinuous pieceuiise polynomials. A duality relationship for the two types of approximations is also given.展开更多
In this work it is shown that the kinetic energy and the exchange-correlation energy are mutual dependent on each other.This aspect is first derived in an orbital-free context.It is shown that the total Fermi potentia...In this work it is shown that the kinetic energy and the exchange-correlation energy are mutual dependent on each other.This aspect is first derived in an orbital-free context.It is shown that the total Fermi potential depends on the density only,the individual parts,the Pauli kinetic energy and the exchange-correlation energy,however,are orbital dependent and as such mutually influence each other.The numerical investigation is performed for the orbital-based non-interacting Kohn-Sham system in order to avoid additional effects due to further approximations of the kinetic energy.The numerical influence of the exchange-correlation functional on the non-interacting kinetic energy is shown to be of the orderof a few Hartrees.For chemical purposes,however,the energetic performance as a function of the nuclear coordinates is much more important than total energies.Therefore,the effect on the bond dissociation curve was studied exemplarily for the carbon monoxide.The data reveals that,the mutual influence between the exchange-correlation functional and the kinetic energy has a significant influence on bond dissociation energies and bond distances.Therefore,the effect of the exchange-correlation treatment must be considered in the design of orbital-free density functional approximations for the kinetic energy.展开更多
Using an improved approximate formula to the centrifugal term, we present arbitrary l-state scattering solutions of the hyperbolic potential. The approximate analytical formula of scattering phase shifts and normalize...Using an improved approximate formula to the centrifugal term, we present arbitrary l-state scattering solutions of the hyperbolic potential. The approximate analytical formula of scattering phase shifts and normalized wavefunctions are presented. All data calculated by the above approximate analytical formula are compared with those obtained by using the numerical integration method in the scattering state cases. We find that this improved approximate formula is better than previous one since the calculated results are in good agreement with those exact ones.展开更多
Studying the dynamics of solitons in nonlinear time-fractional partial differential equations has received substantial attention,in the last decades.The main aim of the current investigation is to consider the time-fr...Studying the dynamics of solitons in nonlinear time-fractional partial differential equations has received substantial attention,in the last decades.The main aim of the current investigation is to consider the time-fractional Sharma–Tasso–Olver–Burgers(STOB)equation in the Caputo–Fabrizio(CF)context and obtain its valid approximations through adopting a mixed approach composed of the homotopy analysis method(HAM)and the Laplace transform.The existence and uniqueness of the solution of the time-fractional STOB equation in the CF context are investigated by demonstrating the Lipschitz condition forφ(x,t;u)as the kernel and giving some theorems.To illustrate the CF operator effect on the dynamics of the obtained solitons,several two-and threedimensional plots are formally considered.It is shown that the mixed approach is capable of producing valid approximations to the time-fractional STOB equation in the CF context.展开更多
A generalized multi-layered granulation structure used by neighborhood systems is proposed. With granulated views, the concepts of approximations under incomplete information systems are studied, which are represented...A generalized multi-layered granulation structure used by neighborhood systems is proposed. With granulated views, the concepts of approximations under incomplete information systems are studied, which are represented by covering of the universe. With respect to different levels of granulations, a pair of lower and upper approximations is defined and an approximation structure is investigated, which lead to a more general approximation structure. The generalized multi-layered granulation structure provides a basis of the proposed framework of granular computing. Using this framework, the interesting and useful results about information granulation and approximation reasoning can be obtained. This paper presents some useful explorations about the incomplete information systems from information views.展开更多
In this paper we investigate weighted polynomial approximations with several variables. Our study relates to the approximation for by weighted polynomial. Then we will give some results relating to the Lagrange interp...In this paper we investigate weighted polynomial approximations with several variables. Our study relates to the approximation for by weighted polynomial. Then we will give some results relating to the Lagrange interpolation, the best approximation, the Markov-Bernstein inequality and the Nikolskii- type inequality.展开更多
In this paper, we present a method for solving coupled problem. This method is mainly based on the successive approximations method. The external force acting on the structure is replaced by λ = p (x1, H + u (x1, λ)...In this paper, we present a method for solving coupled problem. This method is mainly based on the successive approximations method. The external force acting on the structure is replaced by λ = p (x1, H + u (x1, λ)). Then we have a nonlinear equation of unknown?λ to solve by successive approximations method. By this method, we obtain easily the analytic expression of the displacement. In addition, good results are obtained with only a few iterations.展开更多
Using a modified saddle-point method we obtained asymptotic approximation of the r-Stirling numbers of the first and second kind. Here we used a nontrivial trans formation of the same type as that discussed by N.M. Te...Using a modified saddle-point method we obtained asymptotic approximation of the r-Stirling numbers of the first and second kind. Here we used a nontrivial trans formation of the same type as that discussed by N.M. Temme[13], to put the integral representations in volved into a form on which we can apply the saddle point展开更多
We address the problem of convergence of approximations obtained from two versions of the piecewise power-law representations arisen in Systems Biology. The most important cases of mean-square and uniform convergence ...We address the problem of convergence of approximations obtained from two versions of the piecewise power-law representations arisen in Systems Biology. The most important cases of mean-square and uniform convergence are studied in detail. Advantages and drawbacks of the representations as well as properties of both kinds of convergence are discussed. Numerical approximation algorithms related to piecewise power-law representations are described in Appendix.展开更多
In this paper, we introduce the concepts of g and b approximations via general ordered topological approximation spaces. Also, increasing (decreasing) g, b boundary, positive and negative regions are given in general ...In this paper, we introduce the concepts of g and b approximations via general ordered topological approximation spaces. Also, increasing (decreasing) g, b boundary, positive and negative regions are given in general ordered topological approximation spaces (GOTAS, for short). Some important properties of them were investigated. From this study, we can say that studying any properties of rough set concepts via GOTAS is a generalization of Pawlak approximation spaces and general approximation spaces.展开更多
Given a continuous function f defined on the unit cube of R^n and a convex function _t,_t(0)-0,_t(x)>0,for x>0,we prove that the set of best L^(t)-approximations by monotone functions has exactly one element ft,...Given a continuous function f defined on the unit cube of R^n and a convex function _t,_t(0)-0,_t(x)>0,for x>0,we prove that the set of best L^(t)-approximations by monotone functions has exactly one element ft,which is also a continuous function.Moreover if the family of convex functions {_t}t>0 converges uniformly on compact sets to a function _0, then the best approximation f_t→f_0 uniformly,as t→0,where fo is the best approximation of f within the Orlicz space L^(0) The best approxima- tions{f_t}are obtained as well as minimizing integrals or the Luxemburg norm展开更多
In this paper a new approach to construction of iterative methods of bilateral approximations of eigenvalue is proposed and investigated. The conditions on initial approximation, which ensure the convergence of iterat...In this paper a new approach to construction of iterative methods of bilateral approximations of eigenvalue is proposed and investigated. The conditions on initial approximation, which ensure the convergence of iterative processes, are obtained.展开更多
基金supported by the Fundamental Research Projects of Shanxi Province(Grant No.202203021222225)the National Natural Science Foundation of China(Grant Nos.12175029,12011530014,and 11775040)the Key Research and Development Project of Liaoning Province(Grant No.2020JH2/10500003).
文摘Determining the minimal distance between the target state and the convex combination of given states is a fundamental problem in quantum resource theory,offering critical guidance for experimental implementations.In this paper,we embark on an in-depth exploration of the use of a quantum state prepared by the convex combination of given qubit states to optimally approximate the l_(1)-norm of coherence of the target quantum state,striving to make the prepared state and the target state as similar as possible.Here,we present the analytical solution for the optimal distance for any N given quantum states.We find that the optimal approximation problem for any N>4 quantum states can be transformed into an optimal approximation problem for no more than four quantum states,which not only significantly streamlines the problem but also proves advantageous for laboratories in terms of material conservation.Ultimately,a one-to-one comparison between the analytical and numerical solutions verifies the effectiveness of our approach.
基金partially supported by U.S.National Science Foundation,No.DMS1620150U.S.Army ARDEC,No.W911SR-14-2-0001+2 种基金partially supported by National Natural Science Foundation of China,No.91130003,No.11021101,and No.11290142partially supported by Hong Kong RGC General Research Fund,No.16307319the UGC–Research Infrastructure Grant,No.IRS20SC39。
文摘The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises.The noise term is approximated through the spectral projection of the covariance operator,which is not required to be commutative with the Laplacian operator.Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises,the well-posedness of the SPDE is established under certain covariance operator-dependent conditions.These SPDEs with projected noises are then numerically approximated with the finite element method.A general error estimate framework is established for the finite element approximations.Based on this framework,optimal error estimates of finite element approximations for elliptic SPDEs driven by power-law noises are obtained.It is shown that with the proposed approach,convergence order of white noise driven SPDEs is improved by half for one-dimensional problems,and by an infinitesimal factor for higher-dimensional problems.
基金Supported by the National Natural Science Foundation of China (No.69803007)
文摘Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, complement operators, so the rough set theory has been enriched from the operator-oriented and set-oriented views. Approximate power set spaces are defined, and it is proved that the approximation operators are epimorphisms from power set space to approximate power set spaces. Some basic properties of approximate power set space are got by epimorphisms in contrast to power set space.
基金supported by the NSF China#10571075NSF-Guangdong China#04010473+1 种基金The research of the second author was supported by Jinan University Foundation#51204033the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State education Ministry#2005-383
文摘This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...
文摘The problem of strong uniqueness of best approximation from an RS set in a Banach space is considered. For a fixed RS set G and an element x∈X , we proved that the best approximation g * to x from G is strongly unique.
基金supported by the National Natural Science Foundation of China(Grant No.11101454)the Natural Science Foundation of Chongqing CSTC,China(Grant No.cstc2014jcyjA00005)the Program of Innovation Team Project in University of Chongqing City,China(Grant No.KJTD201308)
文摘Based on the moving least square （MLS） approximations and the boundary integral equations （BIEs）, a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection operator is used to tackle the nonlinear boundary inequality conditions. The Signorini problem is then reformulated as BIEs and the unknown boundary variables are approximated by the MLS approximations. Accordingly, only a nodal data structure on the boundary of a domain is required. The convergence of the algorithm is proven. Numerical examples are given to show the high convergence rate and high computational efficiency of the presented algorithm.
文摘We review and compare two definitions of rough set approximations.One is defined by a pair of sets in the universe and the other by a pair of sets in the quotient universe.The latter definition,although less studied,is semantically superior for interpreting rule induction and is closely related to granularity switching in granular computing.Numerical measures about the accuracy and quality of approximations are examined.Several semantics difficulties are commented.
基金This research has been supported in part by the Natural Sciences and Engineering Research Council of Canada(Grant OGPIN-336)and by the"Ministere de l'Education du Quebec"(FCAR Grant-ER-0725)
文摘A formulation of a differential equation as projection and fixed point pi-Mem alloivs approximations using general piecnvise functions. We prone existence and uniqueness of the up proximate solution* convergence in the L2 norm and nodal supercnnvergence. These results generalize those obtained earlier by Hulme for continuous piecevjise polynomials and by Delfour-Dubeau for discontinuous pieceuiise polynomials. A duality relationship for the two types of approximations is also given.
基金The project was supported by the Fund for Scientific Research in Flanders (FWO-Vlaanderen) for Research Grant G021115N.
文摘In this work it is shown that the kinetic energy and the exchange-correlation energy are mutual dependent on each other.This aspect is first derived in an orbital-free context.It is shown that the total Fermi potential depends on the density only,the individual parts,the Pauli kinetic energy and the exchange-correlation energy,however,are orbital dependent and as such mutually influence each other.The numerical investigation is performed for the orbital-based non-interacting Kohn-Sham system in order to avoid additional effects due to further approximations of the kinetic energy.The numerical influence of the exchange-correlation functional on the non-interacting kinetic energy is shown to be of the orderof a few Hartrees.For chemical purposes,however,the energetic performance as a function of the nuclear coordinates is much more important than total energies.Therefore,the effect on the bond dissociation curve was studied exemplarily for the carbon monoxide.The data reveals that,the mutual influence between the exchange-correlation functional and the kinetic energy has a significant influence on bond dissociation energies and bond distances.Therefore,the effect of the exchange-correlation treatment must be considered in the design of orbital-free density functional approximations for the kinetic energy.
基金Supported by the National Natural Science Foundation of China under Grant No.11275165the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province,China under Grant No.11KJD430007Jiangsu Overseas Research and Training Program for University Prominent Young and Middle-aged Teachers and Presidents
文摘Using an improved approximate formula to the centrifugal term, we present arbitrary l-state scattering solutions of the hyperbolic potential. The approximate analytical formula of scattering phase shifts and normalized wavefunctions are presented. All data calculated by the above approximate analytical formula are compared with those obtained by using the numerical integration method in the scattering state cases. We find that this improved approximate formula is better than previous one since the calculated results are in good agreement with those exact ones.
文摘Studying the dynamics of solitons in nonlinear time-fractional partial differential equations has received substantial attention,in the last decades.The main aim of the current investigation is to consider the time-fractional Sharma–Tasso–Olver–Burgers(STOB)equation in the Caputo–Fabrizio(CF)context and obtain its valid approximations through adopting a mixed approach composed of the homotopy analysis method(HAM)and the Laplace transform.The existence and uniqueness of the solution of the time-fractional STOB equation in the CF context are investigated by demonstrating the Lipschitz condition forφ(x,t;u)as the kernel and giving some theorems.To illustrate the CF operator effect on the dynamics of the obtained solitons,several two-and threedimensional plots are formally considered.It is shown that the mixed approach is capable of producing valid approximations to the time-fractional STOB equation in the CF context.
文摘A generalized multi-layered granulation structure used by neighborhood systems is proposed. With granulated views, the concepts of approximations under incomplete information systems are studied, which are represented by covering of the universe. With respect to different levels of granulations, a pair of lower and upper approximations is defined and an approximation structure is investigated, which lead to a more general approximation structure. The generalized multi-layered granulation structure provides a basis of the proposed framework of granular computing. Using this framework, the interesting and useful results about information granulation and approximation reasoning can be obtained. This paper presents some useful explorations about the incomplete information systems from information views.
文摘In this paper we investigate weighted polynomial approximations with several variables. Our study relates to the approximation for by weighted polynomial. Then we will give some results relating to the Lagrange interpolation, the best approximation, the Markov-Bernstein inequality and the Nikolskii- type inequality.
文摘In this paper, we present a method for solving coupled problem. This method is mainly based on the successive approximations method. The external force acting on the structure is replaced by λ = p (x1, H + u (x1, λ)). Then we have a nonlinear equation of unknown?λ to solve by successive approximations method. By this method, we obtain easily the analytic expression of the displacement. In addition, good results are obtained with only a few iterations.
文摘Using a modified saddle-point method we obtained asymptotic approximation of the r-Stirling numbers of the first and second kind. Here we used a nontrivial trans formation of the same type as that discussed by N.M. Temme[13], to put the integral representations in volved into a form on which we can apply the saddle point
文摘We address the problem of convergence of approximations obtained from two versions of the piecewise power-law representations arisen in Systems Biology. The most important cases of mean-square and uniform convergence are studied in detail. Advantages and drawbacks of the representations as well as properties of both kinds of convergence are discussed. Numerical approximation algorithms related to piecewise power-law representations are described in Appendix.
文摘In this paper, we introduce the concepts of g and b approximations via general ordered topological approximation spaces. Also, increasing (decreasing) g, b boundary, positive and negative regions are given in general ordered topological approximation spaces (GOTAS, for short). Some important properties of them were investigated. From this study, we can say that studying any properties of rough set concepts via GOTAS is a generalization of Pawlak approximation spaces and general approximation spaces.
文摘Given a continuous function f defined on the unit cube of R^n and a convex function _t,_t(0)-0,_t(x)>0,for x>0,we prove that the set of best L^(t)-approximations by monotone functions has exactly one element ft,which is also a continuous function.Moreover if the family of convex functions {_t}t>0 converges uniformly on compact sets to a function _0, then the best approximation f_t→f_0 uniformly,as t→0,where fo is the best approximation of f within the Orlicz space L^(0) The best approxima- tions{f_t}are obtained as well as minimizing integrals or the Luxemburg norm
文摘In this paper a new approach to construction of iterative methods of bilateral approximations of eigenvalue is proposed and investigated. The conditions on initial approximation, which ensure the convergence of iterative processes, are obtained.
