Mixing time is de fined as the time required for achieving a certain degree of homogeneity of injected tracer in a unit operation vessel. It has been used as a key parameter for assessing the performance of a mixing s...Mixing time is de fined as the time required for achieving a certain degree of homogeneity of injected tracer in a unit operation vessel. It has been used as a key parameter for assessing the performance of a mixing system. From an experimental standpoint, several techniques have been developed for measuring the mixing time. Based on the disturbances to fl ow, they can be classi fied into two groups: non-intrusive and intrusive. However, depending on the type of data generated, they can be also classi fied into direct measurements and indirect measurements(Eulerian and Lagrangian). Since the techniques available for measuring mixing times in an agitated tank do not provide the same information, its choice depends on several factors, namely: accuracy, reproducibility,suitability, cost, sampling speed, type of data, and processing time. A review of the experimental techniques reported in the literature in the last 50 years for the measurement of mixing time in stirred vessels under single and gas–liquid fl ow conditions with Newtonian and non-Newtonian fl uids in the laminar and turbulent regime is made, and a comparison between these techniques is also presented.展开更多
Abstract： The existence of periodic solutions of a class of non- autonomous differential delay equations with the form x′（t）=-∑k=1^n-1f（t,x（t-kr）） is considered, where r 〉 0 is a given constant and f∈C（R&#...Abstract： The existence of periodic solutions of a class of non- autonomous differential delay equations with the form x′（t）=-∑k=1^n-1f（t,x（t-kr）） is considered, where r 〉 0 is a given constant and f∈C（R×R,R） is odd in x, r-periodic in t and satisfies some superlinear conditions at origin and at infinity. First, the delay system is changed to an equivalent Hamiltonian system. Then the existence of periodic solutions of the Hamiltonian system is studied. Periodic solutions of the Hamiltonian system can be obtained by critical points of a functional defined on a Hilbert space, i.e. , points satisfying φ′（z）=0. By using a linking theorem in critical point theory, the existence of critical points of the functional is obtained. Therefore, the existence of periodic solutions for the Hamiltonian system and its equivalent differential delay equation is established.展开更多
We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is timedependent ...We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is timedependent Hamiltonian system. The quantum unitary operator relevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.展开更多
A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration o...A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration of these equations is usually complicated in practice for complex systems, where both the computational time and the memory storage become limit- ing factors. For this reason, low-storage Runge-Kutta methods become increasingly popular for the time integration. This work suggests a series of s-stage sth-order explicit Runge- Kutta methods specific for autonomous linear equations, which only requires two times of the memory storage for the state vector. We also introduce a 13-stage eighth-order scheme for autonomous linear equations, which has optimized stability region and is reduced to a fifth-order method for general equations. These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes. As an example, we ap- ply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.展开更多
The method of path integral is employed to calculate the time evolution of the eigenstates of a charged particle under the Fock-Darwin(FD) Hamiltonian subjected to a time-dependent electric field in the plane of the s...The method of path integral is employed to calculate the time evolution of the eigenstates of a charged particle under the Fock-Darwin(FD) Hamiltonian subjected to a time-dependent electric field in the plane of the system.An exact analytical expression is established for the evolution of the eigenstates.This result then provides a general solution to the time-dependent Schro¨dinger equation.展开更多
In the study of long time asymptotic behaviors of the solutions to a class system of the incompressible non-Newtonian fluid flows in R3, it is proved that the weak solutions decay in L2 norm at （1 ＋ t）- 3/4 and the...In the study of long time asymptotic behaviors of the solutions to a class system of the incompressible non-Newtonian fluid flows in R3, it is proved that the weak solutions decay in L2 norm at （1 ＋ t）- 3/4 and the error of difference between non-Newtonian fluid and linear equation is also investigated. The findings are mainly based on the classic Fourier splitting methods.展开更多
The exact short time propagator, in a form similar to the Crank-Nicholson method but in the spirit of spectrally transformed Hamiltonian, was proposed to solve the triatomic reactive time-dependent schrodinger equatio...The exact short time propagator, in a form similar to the Crank-Nicholson method but in the spirit of spectrally transformed Hamiltonian, was proposed to solve the triatomic reactive time-dependent schrodinger equation. This new propagator is exact and unconditionally convergent for calculating reactive scattering processes with large time step sizes. In order to improve the computational efficiency, the spectral difference method was applied. This resulted the Hamiltonian with elements confined in a narrow diagonal band. In contrast to our previous theoretical work, the discrete variable representation was applied and resulted in full Hamiltonian matrix. As examples, the collision energy-dependent probability of the triatomic H＋H2 and O＋O2 reaction are calculated. The numerical results demonstrate that this new propagator is numerically accurate and capable of propagating the wave packet with large time steps. However, the efficiency and accuracy of this new propagator strongly depend on the mathematical method for solving the involved linear equations and the choice of preconditioner.展开更多
This paper, that has been introduced at the annual meeting of the Renaissance Society of America （Washington, March 22, 2012）, is a little part of a wider research about migration and movements of people between Wes...This paper, that has been introduced at the annual meeting of the Renaissance Society of America （Washington, March 22, 2012）, is a little part of a wider research about migration and movements of people between Western and Eastern Europe （and vice versa） that, started one year ago, is still in progress. Despite a common thought that had considered, still in 15th century, Hungarians as unculturished and violent people, the town of Ferrara, ruled by the Estes, had welcomed many of them during the 15-16th centuries. They were, above all, and as the sources testify, literati and students. This paper tries to show and analyze the cultural reasons and the background that have determined Hungarians＇ presence in Ferrara during the Renaissance, with the consciousness that if many sources have been studied, many others must be展开更多
If the uncertainty principle applies to the Verlinde entropic idea, it leads to a new term in the Newton＇s second law of mechanics in the Planck＇s scale. This curious velocity dependent term inspires a frictional fe...If the uncertainty principle applies to the Verlinde entropic idea, it leads to a new term in the Newton＇s second law of mechanics in the Planck＇s scale. This curious velocity dependent term inspires a frictional feature of the gravity. In this short letter we address that this new term modifies the effective mass and the Newtonian constant as the time dependent quantities. Thus we must have a running on the value of the effective mass on the particle mass m near the holographic screen and the G. This result has a nigh relation with the Dirac hypothesis about the large numbers hypothesis （L.N.H.）. We propose that the corrected entropie terms via Verlinde idea can be brought as a holographic evidence for the authenticity of the Dirac idea.展开更多
Identifying Hamiltonian of a quantum system is of vital importance for quantum information processing.In this article, we realized and benchmarked a quantum Hamiltonian identification algorithm recently proposed(Zhang...Identifying Hamiltonian of a quantum system is of vital importance for quantum information processing.In this article, we realized and benchmarked a quantum Hamiltonian identification algorithm recently proposed(Zhang and Sarovar, 2014). we realized the algorithm on a liquid nuclear magnetic resonance quantum information processor using two types of working media with different forms of Hamiltonian. Our experiment realized the quantum identification algorithm based on free induction decay signals. We also showed how to process data obtained in a practical experiment. We studied the influence of decoherence by numerical simulations. Our experiments and simulations demonstrate that the algorithm is effective and robust.展开更多
This paper considers the problem of L2-disturbance attenuation for a class of time-delay port-controlled Hamiltonian systems. A v-dissipative inequality is established by using a proper control law and a storage funct...This paper considers the problem of L2-disturbance attenuation for a class of time-delay port-controlled Hamiltonian systems. A v-dissipative inequality is established by using a proper control law and a storage function. Then based on the Razumikhin stability theorem, a sufficient condition is proposed for the asymptotically stability of the closed-loop system. Finally, the authors investigate the case that there are time-invariant uncertainties belonging to some convex bounded polytypic domain and an L2 disturbance attenuation control law is proposed. Study of illustrative example with simulation shows that the presented method in this paper works very well in the disturbance attenuation of time-delay Hamiltonian systems.展开更多
We study the optimal quantum control of heteronuclear two-qubit systems described by a Hamiltonian containing both nonlocal internal drift and local control terms.We derive an explicit formula to compute the minimum t...We study the optimal quantum control of heteronuclear two-qubit systems described by a Hamiltonian containing both nonlocal internal drift and local control terms.We derive an explicit formula to compute the minimum time required to steer the system from an initial state to a specified final state.As applications the minimal time to implement Controlled-NOT gate,SWAP gate and Controlled-U gate is calculated in detail.The experimental realizations of these quantum gates are explicitly presented.展开更多
The performance in finite time of a quantum-mechanical Brayton engine cycle is discussed, without intro- duction of temperature. The engine model consists of two quantum isoenergetic and two quantum isobaric processes...The performance in finite time of a quantum-mechanical Brayton engine cycle is discussed, without intro- duction of temperature. The engine model consists of two quantum isoenergetic and two quantum isobaric processes, and works with a single particle in a harmonic trap. Directly employing the finite-time thermodynamics, the efficiency at maximum power output is determined. Extending the harmonic trap to a power-law trap, we find that the efficiency at max/mum power is independent of any parameter involved in the model, but depends on the confinement of the trapping potential.展开更多
We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general conv...We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t →∞.展开更多
We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system.Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system...We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system.Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory.As an application of the results,we study a class of non-autonomous differential delay equation which can be changed to non-canonical Hamiltonian system and obtain the existence of multiple periodic solutions for the equation by employing variational method.展开更多
基金Supported by DGAPA-UNAM through the grant IN-108312
文摘Mixing time is de fined as the time required for achieving a certain degree of homogeneity of injected tracer in a unit operation vessel. It has been used as a key parameter for assessing the performance of a mixing system. From an experimental standpoint, several techniques have been developed for measuring the mixing time. Based on the disturbances to fl ow, they can be classi fied into two groups: non-intrusive and intrusive. However, depending on the type of data generated, they can be also classi fied into direct measurements and indirect measurements(Eulerian and Lagrangian). Since the techniques available for measuring mixing times in an agitated tank do not provide the same information, its choice depends on several factors, namely: accuracy, reproducibility,suitability, cost, sampling speed, type of data, and processing time. A review of the experimental techniques reported in the literature in the last 50 years for the measurement of mixing time in stirred vessels under single and gas–liquid fl ow conditions with Newtonian and non-Newtonian fl uids in the laminar and turbulent regime is made, and a comparison between these techniques is also presented.
文摘Abstract： The existence of periodic solutions of a class of non- autonomous differential delay equations with the form x′（t）=-∑k=1^n-1f（t,x（t-kr）） is considered, where r 〉 0 is a given constant and f∈C（R×R,R） is odd in x, r-periodic in t and satisfies some superlinear conditions at origin and at infinity. First, the delay system is changed to an equivalent Hamiltonian system. Then the existence of periodic solutions of the Hamiltonian system is studied. Periodic solutions of the Hamiltonian system can be obtained by critical points of a functional defined on a Hilbert space, i.e. , points satisfying φ′（z）=0. By using a linking theorem in critical point theory, the existence of critical points of the functional is obtained. Therefore, the existence of periodic solutions for the Hamiltonian system and its equivalent differential delay equation is established.
基金Supported by the Korea Science and Engineering Foundation (KOSEF) Grant Funded by the Korea Government (MOST) under Grant No.F01-2007-000-10075-0
文摘We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is timedependent Hamiltonian system. The quantum unitary operator relevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.
基金This work is supported by the National Natural Science Foundation of China （No.21373064）, the Program for Innovative Research Team of Guizhou Province （No.QKTD[2014]4021）, and the Natural Sci- entific Foundation from Guizhou Provincial Department of Education （No.ZDXK[2014]IS）. All the calculations were performed at Guizhou Provincial High- Performance Computing Center of Condensed Mate- rials and Molecular Simulation in Guizhou Education University.
文摘A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration of these equations is usually complicated in practice for complex systems, where both the computational time and the memory storage become limit- ing factors. For this reason, low-storage Runge-Kutta methods become increasingly popular for the time integration. This work suggests a series of s-stage sth-order explicit Runge- Kutta methods specific for autonomous linear equations, which only requires two times of the memory storage for the state vector. We also introduce a 13-stage eighth-order scheme for autonomous linear equations, which has optimized stability region and is reduced to a fifth-order method for general equations. These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes. As an example, we ap- ply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.
基金Supported by the National Natural Science Foundation of China under Grant No. 10805029Zhejiang Natural Science Foundation underGrant No. R6090717the K.C. Wong Magna Foundation of Ningbo University
文摘The method of path integral is employed to calculate the time evolution of the eigenstates of a charged particle under the Fock-Darwin(FD) Hamiltonian subjected to a time-dependent electric field in the plane of the system.An exact analytical expression is established for the evolution of the eigenstates.This result then provides a general solution to the time-dependent Schro¨dinger equation.
文摘In the study of long time asymptotic behaviors of the solutions to a class system of the incompressible non-Newtonian fluid flows in R3, it is proved that the weak solutions decay in L2 norm at （1 ＋ t）- 3/4 and the error of difference between non-Newtonian fluid and linear equation is also investigated. The findings are mainly based on the classic Fourier splitting methods.
文摘The exact short time propagator, in a form similar to the Crank-Nicholson method but in the spirit of spectrally transformed Hamiltonian, was proposed to solve the triatomic reactive time-dependent schrodinger equation. This new propagator is exact and unconditionally convergent for calculating reactive scattering processes with large time step sizes. In order to improve the computational efficiency, the spectral difference method was applied. This resulted the Hamiltonian with elements confined in a narrow diagonal band. In contrast to our previous theoretical work, the discrete variable representation was applied and resulted in full Hamiltonian matrix. As examples, the collision energy-dependent probability of the triatomic H＋H2 and O＋O2 reaction are calculated. The numerical results demonstrate that this new propagator is numerically accurate and capable of propagating the wave packet with large time steps. However, the efficiency and accuracy of this new propagator strongly depend on the mathematical method for solving the involved linear equations and the choice of preconditioner.
文摘This paper, that has been introduced at the annual meeting of the Renaissance Society of America （Washington, March 22, 2012）, is a little part of a wider research about migration and movements of people between Western and Eastern Europe （and vice versa） that, started one year ago, is still in progress. Despite a common thought that had considered, still in 15th century, Hungarians as unculturished and violent people, the town of Ferrara, ruled by the Estes, had welcomed many of them during the 15-16th centuries. They were, above all, and as the sources testify, literati and students. This paper tries to show and analyze the cultural reasons and the background that have determined Hungarians＇ presence in Ferrara during the Renaissance, with the consciousness that if many sources have been studied, many others must be
文摘If the uncertainty principle applies to the Verlinde entropic idea, it leads to a new term in the Newton＇s second law of mechanics in the Planck＇s scale. This curious velocity dependent term inspires a frictional feature of the gravity. In this short letter we address that this new term modifies the effective mass and the Newtonian constant as the time dependent quantities. Thus we must have a running on the value of the effective mass on the particle mass m near the holographic screen and the G. This result has a nigh relation with the Dirac hypothesis about the large numbers hypothesis （L.N.H.）. We propose that the corrected entropie terms via Verlinde idea can be brought as a holographic evidence for the authenticity of the Dirac idea.
基金supported by the National Natural Science Foundation of China(11175094 and 91221205)the National Basic Research Program of China(2011CB9216002)
文摘Identifying Hamiltonian of a quantum system is of vital importance for quantum information processing.In this article, we realized and benchmarked a quantum Hamiltonian identification algorithm recently proposed(Zhang and Sarovar, 2014). we realized the algorithm on a liquid nuclear magnetic resonance quantum information processor using two types of working media with different forms of Hamiltonian. Our experiment realized the quantum identification algorithm based on free induction decay signals. We also showed how to process data obtained in a practical experiment. We studied the influence of decoherence by numerical simulations. Our experiments and simulations demonstrate that the algorithm is effective and robust.
基金supported by the National Natural Science Foundation of China under Grant Nos.61074068, 61004013 and 61034007the Research Fund the Doctoral Program of Chinese Higher Education under Grant No.200804220028+2 种基金China Postdoctoral Science Foundation under Grant No.20100481300the Postdoctoral Innovation Program of Shandong Province under Grant No.200902014the Natural Science Foundation of Shandong Province under Grant No.ZB2010FM013
文摘This paper considers the problem of L2-disturbance attenuation for a class of time-delay port-controlled Hamiltonian systems. A v-dissipative inequality is established by using a proper control law and a storage function. Then based on the Razumikhin stability theorem, a sufficient condition is proposed for the asymptotically stability of the closed-loop system. Finally, the authors investigate the case that there are time-invariant uncertainties belonging to some convex bounded polytypic domain and an L2 disturbance attenuation control law is proposed. Study of illustrative example with simulation shows that the presented method in this paper works very well in the disturbance attenuation of time-delay Hamiltonian systems.
基金supported by the National Natural Science Foundation of China(Grant No.11275131)the National Research Foundation for the Doctoral Program of Higher Education of China
文摘We study the optimal quantum control of heteronuclear two-qubit systems described by a Hamiltonian containing both nonlocal internal drift and local control terms.We derive an explicit formula to compute the minimum time required to steer the system from an initial state to a specified final state.As applications the minimal time to implement Controlled-NOT gate,SWAP gate and Controlled-U gate is calculated in detail.The experimental realizations of these quantum gates are explicitly presented.
基金Supported by the National Natural Science Foundation of China under Grant No. 11265010, the Jiangxi Provincial Natural Science Foundation under Grant No. 20132BAB212009, University Young Teacher Training Program of the SMEC under Grant No. egdll005, and by Innovation Program of the SMEC under Grant No. 12YZ177
文摘The performance in finite time of a quantum-mechanical Brayton engine cycle is discussed, without intro- duction of temperature. The engine model consists of two quantum isoenergetic and two quantum isobaric processes, and works with a single particle in a harmonic trap. Directly employing the finite-time thermodynamics, the efficiency at maximum power output is determined. Extending the harmonic trap to a power-law trap, we find that the efficiency at max/mum power is independent of any parameter involved in the model, but depends on the confinement of the trapping potential.
基金supported by National Natural Science Foundation of China(Grant Nos.1132510311301106 and 11201288)+1 种基金China Postdoctoral Science Foundation(Grant No.2014M550210)Guangxi Experiment Center of Information Science(Grant No.YB1410)
文摘We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM(Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t →∞.
基金supported by Natural Science Foundation of the Jiangsu Higher Education Institutions(Grant No.12KJB110015)
文摘We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system.Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory.As an application of the results,we study a class of non-autonomous differential delay equation which can be changed to non-canonical Hamiltonian system and obtain the existence of multiple periodic solutions for the equation by employing variational method.
