We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways,discuss the discrimination of quantum channels,and show that the distinguishability can be improved by self-l...We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways,discuss the discrimination of quantum channels,and show that the distinguishability can be improved by self-linking each quantum channel n times as n grows.We also find that the maximum value of Uhlmann's theorem can be achieved for diagonal channels.展开更多
This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis techniq...This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.展开更多
We study the quantification of geometric discord for tripartite quantum systems.Firstly,we obtain the analytic formula of geometric discord for tripartite pure states.It is already known that the geometric discord of ...We study the quantification of geometric discord for tripartite quantum systems.Firstly,we obtain the analytic formula of geometric discord for tripartite pure states.It is already known that the geometric discord of pure states reduces to the geometric entanglement in bipartite systems,the results presented here show that this property is no longer true in tripartite systems.Furthermore,we provide an operational meaning for tripartite geometric discord by linking it to quantum state discrimination,that is,we prove that the geometric discord of tripartite states is equal to the minimum error probability to discriminate a set of quantum states with von Neumann measurement.Lastly,we calculate the geometric discord of three-qubit Bell diagonal states and then investigate the dynamic behavior of tripartite geometric discord under local decoherence.It is interesting that the frozen phenomenon exists for geometric discord in this scenario.展开更多
In the realm of large-scale machine learning,it is crucial to explore methods for reducing computational complexity and memory demands while maintaining generalization performance.Additionally,since the collected data...In the realm of large-scale machine learning,it is crucial to explore methods for reducing computational complexity and memory demands while maintaining generalization performance.Additionally,since the collected data may contain some sensitive information,it is also of great significance to study privacy-preserving machine learning algorithms.This paper focuses on the performance of the differentially private stochastic gradient descent(SGD)algorithm based on random features.To begin,the algorithm maps the original data into a lowdimensional space,thereby avoiding the traditional kernel method for large-scale data storage requirement.Subsequently,the algorithm iteratively optimizes parameters using the stochastic gradient descent approach.Lastly,the output perturbation mechanism is employed to introduce random noise,ensuring algorithmic privacy.We prove that the proposed algorithm satisfies the differential privacy while achieving fast convergence rates under some mild conditions.展开更多
Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely...Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.展开更多
In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explai...In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.展开更多
In this paper,we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin(DDG)method(Liu and Yan in SIAM J Numer Anal 47(1):475-698,2009),the DDG method with ...In this paper,we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin(DDG)method(Liu and Yan in SIAM J Numer Anal 47(1):475-698,2009),the DDG method with the interface correction(DDGIC)(Liu and Yan in Commun Comput Phys 8(3):541-564,2010),the symmetric DDG method(Vidden and Yan in Comput Math 31(6):638-662,2013),and the nonsymmetric DDG method(Yan in J Sci Comput 54(2):663-683,2013).We also include the study of the interior penalty DG(IPDG)method,due to its close relation to DDG methods.Error estimates are carried out for both P2 and P3 polynomial approximations.By investigating the quantitative errors at the Lobatto points,we show that the DDGIC and symmetric DDG methods are superior,in the sense of obtaining(k+2)th superconvergence orders for both P2 and P3 approximations.Superconvergence order of(k+2)is also observed for the IPDG method with P3 polynomial approximations.The errors are sensitive to the choice of the numerical flux coefficient for even degree P2 approximations,but are not for odd degree P3 approxi-mations.Numerical experiments are carried out at the same time and the numerical errors match well with the analytically estimated errors.展开更多
Loquat(Eriobotrya japonica)fruit accumulates lignin during postharvest storage under chilling conditions(0℃),while low-temperature conditioning(LTC;5℃for 6 days followed by transfer to 0℃)or heat treatment(HT;40℃f...Loquat(Eriobotrya japonica)fruit accumulates lignin during postharvest storage under chilling conditions(0℃),while low-temperature conditioning(LTC;5℃for 6 days followed by transfer to 0℃)or heat treatment(HT;40℃for 4 h followed by transfer to 0℃)can alleviate lignification.Here we compared transcriptome profiles of loquat fruit samples under LTC or HT to those stored at 0℃at five time points from day 1 to day 8 after treatment.High-throughput transcriptome sequences were de novo assembled into 53,319 unique transcripts with an N50 length of 1306 bp.A total of 2235 differentially expressed genes were identified in LTC,and 1020 were identified in HT compared to 0℃.Key genes in the lignin biosynthetic pathway,including EjPAL2,EjCAD1,EjCAD3,4CL,COMT,and HCT,were responsive to LTC or HT treatment,but they showed different expression patterns during the treatments,indicating that different structural genes could regulate lignification at different treatment stages.Coexpression network analysis showed that these candidate biosynthetic genes were associated with a number of transcription factors,including those belonging to the AP2,MYB,and NAC families.Gene ontology(GO)enrichment analysis of differentially expressed genes indicated that biological processes such as stress responses,cell wall and lignin metabolism,hormone metabolism,and metal ion transport were significantly affected under LTC or HT treatment when compared to 0℃.Our analyses provide insights into transcriptome responses to postharvest treatments in loquat fruit.展开更多
The goal of the arterial graft design problem is to find an optimal graft built on an occluded artery, which can be mathematically modeled by a fluid based shape optimization problem. The smoothness of the graft is on...The goal of the arterial graft design problem is to find an optimal graft built on an occluded artery, which can be mathematically modeled by a fluid based shape optimization problem. The smoothness of the graft is one of the important aspects in the arterial graft design problem since it affects the flow of the blood significantly. As an attractive design tool for this problem, level set methods are quite efficient for obtaining better shape of the graft. In this paper, a cubic spline level set method and a radial basis function level set method are designed to solve the arterial graft design problem. In both approaches, the shape of the arterial graft is implicitly tracked by the zero-level contour of a level set function and a high level of smoothness of the graft is achieved. Numerical results show the efficiency of the algorithms in the arterial graft design.展开更多
In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations.Conver...In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations.Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine.Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes,numerical experiments and CPU time-methodology.Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods.Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples.Numerical test examples,dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.展开更多
Abstract. In this paper, we first characterize the finiteness of fractal interpolation functions (FIFs) on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling fa...Abstract. In this paper, we first characterize the finiteness of fractal interpolation functions (FIFs) on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on the Sierpinski gasket (SG). As an application, we prove that the solution of the following Dirichlet problem on SG is a FIF with uniform vertical scaling factor 1/5 :△=0 on SG / {q1, q2, q3}, and u(qi)=ai, i = 1, 2, 3, where qi, i=1, 2, 3, are boundary points of SG.展开更多
In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline th...In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.展开更多
In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation(δ2 gij)/δt2+μ/((1 + t)λ)(δ gij)/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ...In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation(δ2 gij)/δt2+μ/((1 + t)λ)(δ gij)/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ ≤ 1, μ 〉 λ + 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t, x) of the solution metric gij remains uniformly bounded.展开更多
Define the incremental fractional Brownian field ZH(τ, s) = BH(s+τ) -BH(s),where BH(s) is a standard fractional Brownian motion with Hurst parameter H ∈ (0, 1). Inthis paper, we first derive an exact asy...Define the incremental fractional Brownian field ZH(τ, s) = BH(s+τ) -BH(s),where BH(s) is a standard fractional Brownian motion with Hurst parameter H ∈ (0, 1). Inthis paper, we first derive an exact asymptotic of distribution of the maximum MH(Tu) =supτ∈[0,1],s∈[0,xτu] ZH(τ, s), which holds uniformly for x ∈ [A, B] with A, B two positive con-stants. We apply the findings to analyse the tail asymptotic and limit theorem of MH (τ) witha random index τ. In the end, we also prove an almost sure limit theorem for the maximum M1/2(τ) with non-random index T.展开更多
We investigate the evolution of interfacial gravity-capillary waves propagating along the interface be-tween two dielectric fluids under the action of a horizontal electric field.There is a uniform backgroundflow in e...We investigate the evolution of interfacial gravity-capillary waves propagating along the interface be-tween two dielectric fluids under the action of a horizontal electric field.There is a uniform backgroundflow in each layer,and the relative motion tends to induce Kelvin-Helmholtz(KH)instability.The com-bined effects of gravity,surface tension and electrically induced forces are all taken into account.Underthe short-wave assumption,the expansion and truncation method of Dirichlet-Neumann(DN)operatorsis applied to derive a reduced dynamical model.When KH instability is suppressed linearly by a consider-ably large electric field,our numerical results reveal that in certain regions of parameter space,nonlinearsymmetric traveling wave solutions can be found near the minimum phase speed.Additionally,the de-tailed bifurcation structures are presented together with typical wave profiles.展开更多
Although the isogeometric collocation(IGA-C)method has been successfully utilized in practical applications due to its simplicity and efficiency,only a little theoretical results have been established on the numerical...Although the isogeometric collocation(IGA-C)method has been successfully utilized in practical applications due to its simplicity and efficiency,only a little theoretical results have been established on the numerical analysis of the IGA-C method.In this paper,we deduce the convergence rate of the consistency of the IGA-C method.Moreover,based on the formula of the convergence rate,the necessary and sufficient condition for the consistency of the IGA-C method is developed.These results advance the numerical analysis of the IGA-C method.展开更多
The parity operator P and time reversal operator T are two important operators in the quantum theory, in particular, in the PT -symmetric quantum theory. By using the concrete forms of P and T , we discuss their geome...The parity operator P and time reversal operator T are two important operators in the quantum theory, in particular, in the PT -symmetric quantum theory. By using the concrete forms of P and T , we discuss their geometrical properties in two dimensional spaces. It is showed that if T is given, then all P links with the quadric surfaces;if P is given, then all T links with the quadric curves. Moreover, we give out the generalized unbroken PT -symmetric condition of an operator. The unbroken PT -symmetry of a Hermitian operator is also showed in this way.展开更多
In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with "small" initial data. We first present some estimates on...In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with "small" initial data. We first present some estimates on solutions of linear wave equations in several space variables. Then, we derive a lower bound of the life-span of the classical solutions to the equations with "small" initial data.展开更多
In this paper,we investigate theλ-nuclearity in the system of completely 1-summing mapping spaces(Π1(⋅,⋅),π1).In Section 2,we obtain that C is the unique operator space that is nuclear in the system(Π1(⋅,⋅),π1).W...In this paper,we investigate theλ-nuclearity in the system of completely 1-summing mapping spaces(Π1(⋅,⋅),π1).In Section 2,we obtain that C is the unique operator space that is nuclear in the system(Π1(⋅,⋅),π1).We generalize some results in Section 2 toλ-nuclearity in Section 3.展开更多
Isogeometric analysis(IGA)is introduced to establish the direct link between computer-aided design and analysis.It is commonly implemented by Galerkin formulations(isogeometric Galerkin,IGA-G)through the use of nonuni...Isogeometric analysis(IGA)is introduced to establish the direct link between computer-aided design and analysis.It is commonly implemented by Galerkin formulations(isogeometric Galerkin,IGA-G)through the use of nonuniform rational B-splines(NURBS)basis functions for geometric design and analysis.Another promising approach,isogeometric collocation(IGA-C),working directly with the strong form of the partial differential equation(PDE)over the physical domain defined by NURBS geometry,calculates the derivatives of the numerical solution at the chosen collocation points.In a typical IGA,the knot vector of the NURBS numerical solution is only determined by the physical domain.A new perspective on the IGAmethod is proposed in this study to improve the accuracy and convergence of the solution.Solving the PDE with IGA can be regarded as fitting the load function defined on the NURBS geometry(right-hand side)with derivatives of the NURBS numerical solution(left-hand side).Moreover,the design of the knot vector has a close relationship to theNURBS functions to be fitted in the area of data fitting in geometric design.Therefore,the detected feature points of the load function are integrated into the initial knot vector of the physical domainto construct thenewknot vector of thenumerical solution.Then,they are connected seamlessly with the IGA-C framework for its great potential combining the accuracy and smoothness merits with the computational efficiency,which we call isogeometric collocation by fitting load function(IGACL).In numerical experiments,we implement our method to solve 1D,2D,and 3D PDEs and demonstrate the improvement in accuracy by comparing it with the standard IGA-C method.We also verify the superiority in the accuracy of our knot selection scheme when employed in the IGA-G method,which we call isogeometric Galerkin by fitting load function(IGA-GL).展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61877054,12031004,and 12271474).
文摘We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways,discuss the discrimination of quantum channels,and show that the distinguishability can be improved by self-linking each quantum channel n times as n grows.We also find that the maximum value of Uhlmann's theorem can be achieved for diagonal channels.
基金supported by the National Natural Science Foundation of China(Grant Nos.11871428 and 12071214)the Natural Science Foundation for Colleges and Universities of Jiangsu Province of China(Grant No.20KJB110011)+1 种基金supported by the National Science Foundation(Grant No.DMS-1620335)and the Simons Foundation(Grant No.637716)supported by the National Natural Science Foundation of China(Grant Nos.11871428 and 12272347).
文摘This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.
基金Project supported by the National Natural Science Foundation of China(Grant No.12201555)China Postdoctoral Science Foundation(Grant No.2021M702864)。
文摘We study the quantification of geometric discord for tripartite quantum systems.Firstly,we obtain the analytic formula of geometric discord for tripartite pure states.It is already known that the geometric discord of pure states reduces to the geometric entanglement in bipartite systems,the results presented here show that this property is no longer true in tripartite systems.Furthermore,we provide an operational meaning for tripartite geometric discord by linking it to quantum state discrimination,that is,we prove that the geometric discord of tripartite states is equal to the minimum error probability to discriminate a set of quantum states with von Neumann measurement.Lastly,we calculate the geometric discord of three-qubit Bell diagonal states and then investigate the dynamic behavior of tripartite geometric discord under local decoherence.It is interesting that the frozen phenomenon exists for geometric discord in this scenario.
基金supported by Zhejiang Provincial Natural Science Foundation of China(LR20A010001)National Natural Science Foundation of China(12271473 and U21A20426)。
文摘In the realm of large-scale machine learning,it is crucial to explore methods for reducing computational complexity and memory demands while maintaining generalization performance.Additionally,since the collected data may contain some sensitive information,it is also of great significance to study privacy-preserving machine learning algorithms.This paper focuses on the performance of the differentially private stochastic gradient descent(SGD)algorithm based on random features.To begin,the algorithm maps the original data into a lowdimensional space,thereby avoiding the traditional kernel method for large-scale data storage requirement.Subsequently,the algorithm iteratively optimizes parameters using the stochastic gradient descent approach.Lastly,the output perturbation mechanism is employed to introduce random noise,ensuring algorithmic privacy.We prove that the proposed algorithm satisfies the differential privacy while achieving fast convergence rates under some mild conditions.
基金supported by the National Natural Science Foundation of China(12131015,12071422).
文摘Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.
基金Supported by the National Natural Science Foundation of China(Grant No.11471253 and No.11571311)
文摘In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.
基金the National Science Foundation grant DMS-1620335 and Simons Foundation Grant 637716Research work of Xinghui Zhong is supported by the National Natural Science Foundation of China(NSFC)(Grant no.11871428).
文摘In this paper,we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin(DDG)method(Liu and Yan in SIAM J Numer Anal 47(1):475-698,2009),the DDG method with the interface correction(DDGIC)(Liu and Yan in Commun Comput Phys 8(3):541-564,2010),the symmetric DDG method(Vidden and Yan in Comput Math 31(6):638-662,2013),and the nonsymmetric DDG method(Yan in J Sci Comput 54(2):663-683,2013).We also include the study of the interior penalty DG(IPDG)method,due to its close relation to DDG methods.Error estimates are carried out for both P2 and P3 polynomial approximations.By investigating the quantitative errors at the Lobatto points,we show that the DDGIC and symmetric DDG methods are superior,in the sense of obtaining(k+2)th superconvergence orders for both P2 and P3 approximations.Superconvergence order of(k+2)is also observed for the IPDG method with P3 polynomial approximations.The errors are sensitive to the choice of the numerical flux coefficient for even degree P2 approximations,but are not for odd degree P3 approxi-mations.Numerical experiments are carried out at the same time and the numerical errors match well with the analytically estimated errors.
基金supported by grants from the National Natural Science Foundation of China(31630067,11771393,and 11632015)the Natural Science Foundation of Zhejiang Province,China(LR16C150001 and LZ14A010002)+1 种基金the Project of the Science and Technology Department of Zhejiang Province(2016C04001)the US National Science Foundation(IOS-1539831).
文摘Loquat(Eriobotrya japonica)fruit accumulates lignin during postharvest storage under chilling conditions(0℃),while low-temperature conditioning(LTC;5℃for 6 days followed by transfer to 0℃)or heat treatment(HT;40℃for 4 h followed by transfer to 0℃)can alleviate lignification.Here we compared transcriptome profiles of loquat fruit samples under LTC or HT to those stored at 0℃at five time points from day 1 to day 8 after treatment.High-throughput transcriptome sequences were de novo assembled into 53,319 unique transcripts with an N50 length of 1306 bp.A total of 2235 differentially expressed genes were identified in LTC,and 1020 were identified in HT compared to 0℃.Key genes in the lignin biosynthetic pathway,including EjPAL2,EjCAD1,EjCAD3,4CL,COMT,and HCT,were responsive to LTC or HT treatment,but they showed different expression patterns during the treatments,indicating that different structural genes could regulate lignification at different treatment stages.Coexpression network analysis showed that these candidate biosynthetic genes were associated with a number of transcription factors,including those belonging to the AP2,MYB,and NAC families.Gene ontology(GO)enrichment analysis of differentially expressed genes indicated that biological processes such as stress responses,cell wall and lignin metabolism,hormone metabolism,and metal ion transport were significantly affected under LTC or HT treatment when compared to 0℃.Our analyses provide insights into transcriptome responses to postharvest treatments in loquat fruit.
基金Supported by National Foundation of Natural Science(11471092)Natural Science Foundation of Zhejiang Province(LZ13A010003)Foundation of Zhejiang Educational Committee(Y201121891)
文摘The goal of the arterial graft design problem is to find an optimal graft built on an occluded artery, which can be mathematically modeled by a fluid based shape optimization problem. The smoothness of the graft is one of the important aspects in the arterial graft design problem since it affects the flow of the blood significantly. As an attractive design tool for this problem, level set methods are quite efficient for obtaining better shape of the graft. In this paper, a cubic spline level set method and a radial basis function level set method are designed to solve the arterial graft design problem. In both approaches, the shape of the arterial graft is implicitly tracked by the zero-level contour of a level set function and a high level of smoothness of the graft is achieved. Numerical results show the efficiency of the algorithms in the arterial graft design.
基金the Natural Science Foundation of China(Grant Nos.61673169,11301127,11701176,11626101,and 11601485)The Natural Science Foundation of Huzhou City(Grant No.2018YZ07).
文摘In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations.Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine.Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes,numerical experiments and CPU time-methodology.Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods.Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples.Numerical test examples,dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.
基金Supported by the National Natural Science Foundation of China(11271327)Zhejiang Provincial National Science Foundation of China(LR14A010001)
文摘Abstract. In this paper, we first characterize the finiteness of fractal interpolation functions (FIFs) on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on the Sierpinski gasket (SG). As an application, we prove that the solution of the following Dirichlet problem on SG is a FIF with uniform vertical scaling factor 1/5 :△=0 on SG / {q1, q2, q3}, and u(qi)=ai, i = 1, 2, 3, where qi, i=1, 2, 3, are boundary points of SG.
基金Supported by the National Natural Science Foundation of China(61572430,61303144)the Natural Science Foundation of Zhejiang Province(LY15F020002,LY16F020020)the Ningbo Natural Science Foundation(2016A610223)
文摘In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.
基金supported in part by the NNSF of China(11271323,91330105)the Zhejiang Provincial Natural Science Foundation of China(LZ13A010002)the Science Foundation in Higher Education of Henan(18A110036)
文摘In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation(δ2 gij)/δt2+μ/((1 + t)λ)(δ gij)/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ ≤ 1, μ 〉 λ + 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t, x) of the solution metric gij remains uniformly bounded.
基金supported by National Science Foundation of China(11501250)Natural Science Foundation of Zhejiang Province of China(LQ14A010012,LY15A010019)+2 种基金Postdoctoral Research Program of Zhejiang ProvinceNatural Science Foundation of Jiangsu Higher Education Institution of China(14KJB110023)Research Foundation of SUST
文摘Define the incremental fractional Brownian field ZH(τ, s) = BH(s+τ) -BH(s),where BH(s) is a standard fractional Brownian motion with Hurst parameter H ∈ (0, 1). Inthis paper, we first derive an exact asymptotic of distribution of the maximum MH(Tu) =supτ∈[0,1],s∈[0,xτu] ZH(τ, s), which holds uniformly for x ∈ [A, B] with A, B two positive con-stants. We apply the findings to analyse the tail asymptotic and limit theorem of MH (τ) witha random index τ. In the end, we also prove an almost sure limit theorem for the maximum M1/2(τ) with non-random index T.
基金supported by the National Natural Science Foundation of China(Grant 12071429).
文摘We investigate the evolution of interfacial gravity-capillary waves propagating along the interface be-tween two dielectric fluids under the action of a horizontal electric field.There is a uniform backgroundflow in each layer,and the relative motion tends to induce Kelvin-Helmholtz(KH)instability.The com-bined effects of gravity,surface tension and electrically induced forces are all taken into account.Underthe short-wave assumption,the expansion and truncation method of Dirichlet-Neumann(DN)operatorsis applied to derive a reduced dynamical model.When KH instability is suppressed linearly by a consider-ably large electric field,our numerical results reveal that in certain regions of parameter space,nonlinearsymmetric traveling wave solutions can be found near the minimum phase speed.Additionally,the de-tailed bifurcation structures are presented together with typical wave profiles.
基金supported by the National Natural Science Foundation of China(61872316)the Natural Science Foundation of Zhejiang Province,China(LY19F020004)
文摘Although the isogeometric collocation(IGA-C)method has been successfully utilized in practical applications due to its simplicity and efficiency,only a little theoretical results have been established on the numerical analysis of the IGA-C method.In this paper,we deduce the convergence rate of the consistency of the IGA-C method.Moreover,based on the formula of the convergence rate,the necessary and sufficient condition for the consistency of the IGA-C method is developed.These results advance the numerical analysis of the IGA-C method.
基金Supported by Research Fund,Kumoh National Institute of Technology
文摘The parity operator P and time reversal operator T are two important operators in the quantum theory, in particular, in the PT -symmetric quantum theory. By using the concrete forms of P and T , we discuss their geometrical properties in two dimensional spaces. It is showed that if T is given, then all P links with the quadric surfaces;if P is given, then all T links with the quadric curves. Moreover, we give out the generalized unbroken PT -symmetric condition of an operator. The unbroken PT -symmetry of a Hermitian operator is also showed in this way.
基金supported in part by the NNSF of China(11271323,91330105)the Zhejiang Provincial Natural Science Foundation of China(LZ13A010002)the Henan Provincial Natural Science Foundation of China(152300410226)
文摘In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with "small" initial data. We first present some estimates on solutions of linear wave equations in several space variables. Then, we derive a lower bound of the life-span of the classical solutions to the equations with "small" initial data.
基金the National Natural Science Foundation of China(11871423).
文摘In this paper,we investigate theλ-nuclearity in the system of completely 1-summing mapping spaces(Π1(⋅,⋅),π1).In Section 2,we obtain that C is the unique operator space that is nuclear in the system(Π1(⋅,⋅),π1).We generalize some results in Section 2 toλ-nuclearity in Section 3.
基金supported by the National Natural Science Foundation of China under Grant Nos.61872316,62272406,61932018the National Key R&D Plan of China under Grant No.2020YFB1708900.
文摘Isogeometric analysis(IGA)is introduced to establish the direct link between computer-aided design and analysis.It is commonly implemented by Galerkin formulations(isogeometric Galerkin,IGA-G)through the use of nonuniform rational B-splines(NURBS)basis functions for geometric design and analysis.Another promising approach,isogeometric collocation(IGA-C),working directly with the strong form of the partial differential equation(PDE)over the physical domain defined by NURBS geometry,calculates the derivatives of the numerical solution at the chosen collocation points.In a typical IGA,the knot vector of the NURBS numerical solution is only determined by the physical domain.A new perspective on the IGAmethod is proposed in this study to improve the accuracy and convergence of the solution.Solving the PDE with IGA can be regarded as fitting the load function defined on the NURBS geometry(right-hand side)with derivatives of the NURBS numerical solution(left-hand side).Moreover,the design of the knot vector has a close relationship to theNURBS functions to be fitted in the area of data fitting in geometric design.Therefore,the detected feature points of the load function are integrated into the initial knot vector of the physical domainto construct thenewknot vector of thenumerical solution.Then,they are connected seamlessly with the IGA-C framework for its great potential combining the accuracy and smoothness merits with the computational efficiency,which we call isogeometric collocation by fitting load function(IGACL).In numerical experiments,we implement our method to solve 1D,2D,and 3D PDEs and demonstrate the improvement in accuracy by comparing it with the standard IGA-C method.We also verify the superiority in the accuracy of our knot selection scheme when employed in the IGA-G method,which we call isogeometric Galerkin by fitting load function(IGA-GL).