We study the fully entangled fraction of a quantum state. An upper bound is obtained for arbitrary bipartite system. This upper bound only depends on the Frobenius norm of the state.
This paper proposes a dynamic model to forecast intraday volume percentages by decomposing the trade volume into two parts: The average part as the intraday volume pattern and the residual term as the abnormal changes...This paper proposes a dynamic model to forecast intraday volume percentages by decomposing the trade volume into two parts: The average part as the intraday volume pattern and the residual term as the abnormal changes. An empirical test on data spanning half-a-year gold futures and S&P 500 futures reveals that a rolling average of the previous days' volume percentages shows great predictive ability for the average part. An SVM approach with the input pattern consisting of two categories is employed to forecast the residual term. One is the previous days' volume percentages in the same time interval and the other is the most recent volume percentages. The study shows that this dynamic SVM-based forecasting approach outperforms the other commonly used statistical methods and enhances the tracking performance of a VWAP strategy greatly.展开更多
The traditional Cauchy-Stokes(C-S)decomposition states that the velocity gradient tensor can be decomposed into a symmetric tensor and an anti-symmetric tensor,namely the strain-rate tensor and the vorticity tensor.Ho...The traditional Cauchy-Stokes(C-S)decomposition states that the velocity gradient tensor can be decomposed into a symmetric tensor and an anti-symmetric tensor,namely the strain-rate tensor and the vorticity tensor.However,there are two problems with the C-S decomposition.One is that the anti-symmetric(vorticity)tensor cannot represent the fluid rotation or vortex.Another is that the symmetric(strain-rate)tensor cannot distinguish the stretching/compression and shear.Since vorticity cannot distinguish between the non-rotational shear and the rigid rotation,vorticity has been decomposed into a rigid rotation part called“Liutex”and an anti-symmetric shear in our previous work.A Liutex-based principal coordinate system has been proposed,and the corresponding velocity gradient tensor decomposition,called the principal decomposition,is presented in this principal coordinate system,which results in:(1)a Liutex tensor that represents rigid rotation,(2)a tensor that represents pure shear and(3)a tensor that represents stretching/compression.However,each point has its own principal coordinate system,which implies that the principal decomposition is performed in different principal coordinate systems,not the original(global)coordinate system.To address this issue,the principal decomposition in the original coordinate system is derived in this paper,and,therefore,provides a new kinematic approach to study the local rigid rotation,pure shear,and stretching/compression.The principal decomposition is unique,Galilean invariant and has clear physical meaning.The new velocity gradient tensor decomposition could become a foundation for new fluid kinematics.展开更多
Fluid kinematics describes the fluid motion without consideration of any force.Classical fluid kinematics adopts Helmholtz velocity decomposition,which is equivalent to Cauchy-Stokes(CS)velocity gradient tensor decomp...Fluid kinematics describes the fluid motion without consideration of any force.Classical fluid kinematics adopts Helmholtz velocity decomposition,which is equivalent to Cauchy-Stokes(CS)velocity gradient tensor decomposition.CS decomposes the velocity gradient tensor into a strain-rate(symmetric)tensor and a vorticity(anti-symmetric)tensor.However,several questions arise:(1)since vorticity cannot represent fluid rotation,the vorticity tensor is a mixture of vorticity shear and rigid rotation,(2)since the strain-rate tensor cannot represent fluid shear,the strain-rate tensor is a mixture of stretching and shear,(3)the stretching and shear in the CS decomposition are dependent on the selection of coordinate system and are therefore not Galilean invariant.On the other hand,Liutex is a new physical quantity to represent the rigid fluid rotation and a principal coordinate system can be set up based on Liutex.A principal decomposition of the velocity gradient tensor,or the rotation-stretching-shear decomposition,can be easily carried out in the principal coordinate system with a clear physical meaning,which represents the rigid rotation,stretching(compression)and shear(symmetric and anti-symmetric shear).In the principal decomposition,all elements in three sub-tensors are Galilean invariant and,therefore,the principal decomposition is unique,Galilean invariant and independent of coordinate system.The principal decomposition is then transformed back to the original xyz coordinate system.The Liutex-based principal decomposition creates the new fluid kinematics which is ready for building up new fluid dynamics.Since fluid kinematics is the foundation of the fluid dynamics,the new fluid kinematics could replace the classical fluid kinematics,Helmholtz or CS decomposition,and open a new gate to develop new fluid dynamics especially for vortex science and turbulence research.展开更多
The fluid kinematics of Liutex decomposes a velocity gradient tensor(VGT)of∇v into four components,including rotation(R),stretching/compressing(SC),anti-symmetric shear(Santi-sym)and symmetric shear(Ssym),as oppose to...The fluid kinematics of Liutex decomposes a velocity gradient tensor(VGT)of∇v into four components,including rotation(R),stretching/compressing(SC),anti-symmetric shear(Santi-sym)and symmetric shear(Ssym),as oppose to the traditional Cauchy-Stokes decomposition where a VGT was decomposed into the strain rate and vorticity tensors.The current study limpidly clarified the physical meanings of these deformations in the newly-proposed decomposition from the perspectives of both fluid kinematics and dynamics.With in-depth understanding the physical connotations of these deformations,the present study further suggests that the Ssym be the only deformation appropriately correlated to the stress tensor,leading to the establishment of a new constitutive relation for Newtonian fluids with the modified model assumptions originated from Stokes in 1845.Moreover,the present research finds that the“principal decomposition”proposed by Liu is not mathematically unique when a VGT has three real eigenvalues(TR).Within the context,a new decomposition method is introduced to avoid the non-uniqueness issue arising from using the principal decomposition to establish fluid dynamics equations.Based on the modified Stokes assumptions and the novel VGT decomposition method,a set of new fluid dynamics momentum equations are obtained for Newtonian fluid.The added stress tensor of Fadd is identified as the key difference between the newly-derived governing equations and the conventional Navier-Stokes(N-S)equations,which is caused by excluding the SC correlation to the stress tensor in the new constitutive equation.Finally,a preliminary analysis of Fadd is conducted using the existing channel turbulence direct numerical simulations(DNS)data based on the traditional N-S equations.The Fadd is found widely existing in turbulence and is of the same order of magnitude with the other force terms.Therefore,the Fadd is expected to have some nonnegligible effects on altering the current DNS data based on the traditional N-S equations,which will be further verified by performing the“DNS”simulation using the newly-derived fluid dynamics equations in near future.展开更多
基金Supported by the National Natural Science Foundation of China under Grant Nos.11401032,1150115311271138,and 11531004+2 种基金the Natural Science Foundation of Hainan Province under Grant Nos.20151010,114006 and 20161006the Scientific Research Foundation for Colleges of Hainan Province under Grant No.Hnky2015-18Simons Foundation under Grant No.198129
文摘We study the fully entangled fraction of a quantum state. An upper bound is obtained for arbitrary bipartite system. This upper bound only depends on the Frobenius norm of the state.
文摘This paper proposes a dynamic model to forecast intraday volume percentages by decomposing the trade volume into two parts: The average part as the intraday volume pattern and the residual term as the abnormal changes. An empirical test on data spanning half-a-year gold futures and S&P 500 futures reveals that a rolling average of the previous days' volume percentages shows great predictive ability for the average part. An SVM approach with the input pattern consisting of two categories is employed to forecast the residual term. One is the previous days' volume percentages in the same time interval and the other is the most recent volume percentages. The study shows that this dynamic SVM-based forecasting approach outperforms the other commonly used statistical methods and enhances the tracking performance of a VWAP strategy greatly.
基金The authors are thankful for the support by the UTA Department of Mathematics which houses the UTA Vortex and Turbulence Research Team.The authors are also grateful to Texas Advanced Computing Center(TACC)for providing computation time.The DNSUTA code was released by Chaoqun Liu in 2009 and the Liutex code was released by Chaoqun Liu in 2018 which can be downloaded from the UTA web site at https://www.uta.edu/math/cnsm/public_html/cnsm/cnsm.html.
文摘The traditional Cauchy-Stokes(C-S)decomposition states that the velocity gradient tensor can be decomposed into a symmetric tensor and an anti-symmetric tensor,namely the strain-rate tensor and the vorticity tensor.However,there are two problems with the C-S decomposition.One is that the anti-symmetric(vorticity)tensor cannot represent the fluid rotation or vortex.Another is that the symmetric(strain-rate)tensor cannot distinguish the stretching/compression and shear.Since vorticity cannot distinguish between the non-rotational shear and the rigid rotation,vorticity has been decomposed into a rigid rotation part called“Liutex”and an anti-symmetric shear in our previous work.A Liutex-based principal coordinate system has been proposed,and the corresponding velocity gradient tensor decomposition,called the principal decomposition,is presented in this principal coordinate system,which results in:(1)a Liutex tensor that represents rigid rotation,(2)a tensor that represents pure shear and(3)a tensor that represents stretching/compression.However,each point has its own principal coordinate system,which implies that the principal decomposition is performed in different principal coordinate systems,not the original(global)coordinate system.To address this issue,the principal decomposition in the original coordinate system is derived in this paper,and,therefore,provides a new kinematic approach to study the local rigid rotation,pure shear,and stretching/compression.The principal decomposition is unique,Galilean invariant and has clear physical meaning.The new velocity gradient tensor decomposition could become a foundation for new fluid kinematics.
基金This work was mainly supported by the Department of Mathematics of University of Texas at Arlington。
文摘Fluid kinematics describes the fluid motion without consideration of any force.Classical fluid kinematics adopts Helmholtz velocity decomposition,which is equivalent to Cauchy-Stokes(CS)velocity gradient tensor decomposition.CS decomposes the velocity gradient tensor into a strain-rate(symmetric)tensor and a vorticity(anti-symmetric)tensor.However,several questions arise:(1)since vorticity cannot represent fluid rotation,the vorticity tensor is a mixture of vorticity shear and rigid rotation,(2)since the strain-rate tensor cannot represent fluid shear,the strain-rate tensor is a mixture of stretching and shear,(3)the stretching and shear in the CS decomposition are dependent on the selection of coordinate system and are therefore not Galilean invariant.On the other hand,Liutex is a new physical quantity to represent the rigid fluid rotation and a principal coordinate system can be set up based on Liutex.A principal decomposition of the velocity gradient tensor,or the rotation-stretching-shear decomposition,can be easily carried out in the principal coordinate system with a clear physical meaning,which represents the rigid rotation,stretching(compression)and shear(symmetric and anti-symmetric shear).In the principal decomposition,all elements in three sub-tensors are Galilean invariant and,therefore,the principal decomposition is unique,Galilean invariant and independent of coordinate system.The principal decomposition is then transformed back to the original xyz coordinate system.The Liutex-based principal decomposition creates the new fluid kinematics which is ready for building up new fluid dynamics.Since fluid kinematics is the foundation of the fluid dynamics,the new fluid kinematics could replace the classical fluid kinematics,Helmholtz or CS decomposition,and open a new gate to develop new fluid dynamics especially for vortex science and turbulence research.
基金Project supported by the Shanghai Municipal Education Commission(Grant No.AR960)the Shanghai Municipal Science and Technology Commission(Grant No.20JC1413700).
文摘The fluid kinematics of Liutex decomposes a velocity gradient tensor(VGT)of∇v into four components,including rotation(R),stretching/compressing(SC),anti-symmetric shear(Santi-sym)and symmetric shear(Ssym),as oppose to the traditional Cauchy-Stokes decomposition where a VGT was decomposed into the strain rate and vorticity tensors.The current study limpidly clarified the physical meanings of these deformations in the newly-proposed decomposition from the perspectives of both fluid kinematics and dynamics.With in-depth understanding the physical connotations of these deformations,the present study further suggests that the Ssym be the only deformation appropriately correlated to the stress tensor,leading to the establishment of a new constitutive relation for Newtonian fluids with the modified model assumptions originated from Stokes in 1845.Moreover,the present research finds that the“principal decomposition”proposed by Liu is not mathematically unique when a VGT has three real eigenvalues(TR).Within the context,a new decomposition method is introduced to avoid the non-uniqueness issue arising from using the principal decomposition to establish fluid dynamics equations.Based on the modified Stokes assumptions and the novel VGT decomposition method,a set of new fluid dynamics momentum equations are obtained for Newtonian fluid.The added stress tensor of Fadd is identified as the key difference between the newly-derived governing equations and the conventional Navier-Stokes(N-S)equations,which is caused by excluding the SC correlation to the stress tensor in the new constitutive equation.Finally,a preliminary analysis of Fadd is conducted using the existing channel turbulence direct numerical simulations(DNS)data based on the traditional N-S equations.The Fadd is found widely existing in turbulence and is of the same order of magnitude with the other force terms.Therefore,the Fadd is expected to have some nonnegligible effects on altering the current DNS data based on the traditional N-S equations,which will be further verified by performing the“DNS”simulation using the newly-derived fluid dynamics equations in near future.
