We introduce a total order and an absolute value function for dual numbers.The absolute value function of dual numbers takes dual number values,and has properties similar to those of the absolute value function of rea...We introduce a total order and an absolute value function for dual numbers.The absolute value function of dual numbers takes dual number values,and has properties similar to those of the absolute value function of real numbers.We define the magnitude of a dual quaternion,as a dual number.Based upon these,we extend 1-norm,co-norm,and 2-norm to dual quaternion vectors.展开更多
In this paper,we investigate some properties of dual complex numbers,dual complex vectors,and dual complex matrices.First,based on the magnitude of the dual complex number,we study the Young inequality,the Hölder...In this paper,we investigate some properties of dual complex numbers,dual complex vectors,and dual complex matrices.First,based on the magnitude of the dual complex number,we study the Young inequality,the Hölder inequality,and the Minkowski inequality in the setting of dual complex numbers.Second,we define the p-norm of a dual complex vector,which is a nonnegative dual number,and show some related properties.Third,we study the properties of eigenvalues of unitary matrices and unitary triangulation of arbitrary dual complex matrices.In particular,we introduce the operator norm of dual complex matrices induced by the p-norm of dual complex vectors,and give expressions of three important operator norms of dual complex matrices.展开更多
The dual-layer granular bed filter packed with randomly arranged granules was simulated to study the effects of bed depth of the lower layer of fine granules and the inlet gas velocity on the collection mechanism.The ...The dual-layer granular bed filter packed with randomly arranged granules was simulated to study the effects of bed depth of the lower layer of fine granules and the inlet gas velocity on the collection mechanism.The computational results show that the collection efficiency is much better from this granular bed than a single-layer granular bed,especially for particle diameters of 1-10μm.The inlet gas velocity has less effect on the grade collection efficiency of the dual-layer granular bed than of the single-layer granular bed.The dual-layer granular bed provides a high collection efficiency and low pressure drop.The relationship between the grade collection efficiency and the Stokes number(St)based on the inlet gas velocity is obtained.If St is below a threshold,the grade collection efficiency remains stable;if St is in value above threshold,the grade collection efficiency increases linearly with lg(St).As the bed depth of the lower layer of fine granules increases,the threshold for St shifts forward.展开更多
The tangent linear（TL） models and adjoint（AD） models have brought great difficulties for the development of variational data assimilation system. It might be impossible to develop them perfectly without great effo...The tangent linear（TL） models and adjoint（AD） models have brought great difficulties for the development of variational data assimilation system. It might be impossible to develop them perfectly without great efforts, either by hand, or by automatic differentiation tools. In order to break these limitations, a new data assimilation system, dual-number data assimilation system（DNDAS）, is designed based on the dual-number automatic differentiation principles. We investigate the performance of DNDAS with two different optimization schemes and subsequently give a discussion on whether DNDAS is appropriate for high-dimensional forecast models. The new data assimilation system can avoid the complicated reverse integration of the adjoint model, and it only needs the forward integration in the dual-number space to obtain the cost function and its gradient vector concurrently. To verify the correctness and effectiveness of DNDAS, we implemented DNDAS on a simple ordinary differential model and the Lorenz-63 model with different optimization methods. We then concentrate on the adaptability of DNDAS to the Lorenz-96 model with high-dimensional state variables. The results indicate that whether the system is simple or nonlinear, DNDAS can accurately reconstruct the initial condition for the forecast model and has a strong anti-noise characteristic. Given adequate computing resource, the quasi-Newton optimization method performs better than the conjugate gradient method in DNDAS.展开更多
This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of p...This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear La- grange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. It is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Moreover, the paper develops the dual algorithm associ- ated with the proposed nonlinear Lagrangians. Numerical results reported suggest that the dual algorithm based on proposed nonlinear Lagrangians is effective for solving some nonlinear optimization problems.展开更多
The dual algorithm for minimax problems is further studied in this paper.The resulting theoretical analysis shows that the condition number of the corresponding Hessian of the smooth modified Lagrange function with ch...The dual algorithm for minimax problems is further studied in this paper.The resulting theoretical analysis shows that the condition number of the corresponding Hessian of the smooth modified Lagrange function with changing parameter in the dual algorithm is proportional to the reciprocal of the parameter,which is very important for the efficiency of the dual algorithm.At last,the numerical experiments are reported to validate the analysis results.展开更多
A self-invented atomization process, in which molten metal is atomized into powder by a high-velocity gas stream carrying solid particles as the atomization medium, was introduced. The characteristics of powders prepa...A self-invented atomization process, in which molten metal is atomized into powder by a high-velocity gas stream carrying solid particles as the atomization medium, was introduced. The characteristics of powders prepared by common gas atomization and dual-phase flow atomization under similar conditions were compared. The experimental results show that the dual-phase flow-atomized powders have average particle sizes that are one-half that of the common gas-atomized particles;additionally, they possess a finer microstructure and higher cooling rate under the same atomization gas pressure and the same gas flow. The Weber number in the crash criteria of liquid atomization is adopted to measure the crash ability of the atomization media. The Weber number of the dual-phase flow atomization medium is the sum of that of the gas and the solid particles. Furthermore, the critical equation of the crash model in dual-phase flow atomization is established, and the main regularities associated with this process were analyzed.展开更多
基金supported by Hong Kong Innovation and Technology Commission(InnoHK Project CIMDA)supported by the National Natural Science Foundation of China(No.11971138)+3 种基金the Natural Science Foundation of Zhejiang Province of China(Nos.LY19A010019,LD19A010002)supported by Hong Kong Research Grants Council(Project 11204821)Hong Kong Innovation and Technology Commission(InnoHK Project CIMDA)City University of Hong Kong(Project 9610034).
文摘We introduce a total order and an absolute value function for dual numbers.The absolute value function of dual numbers takes dual number values,and has properties similar to those of the absolute value function of real numbers.We define the magnitude of a dual quaternion,as a dual number.Based upon these,we extend 1-norm,co-norm,and 2-norm to dual quaternion vectors.
基金the National Natural Science Foundation of China(Grant No.11871051).
文摘In this paper,we investigate some properties of dual complex numbers,dual complex vectors,and dual complex matrices.First,based on the magnitude of the dual complex number,we study the Young inequality,the Hölder inequality,and the Minkowski inequality in the setting of dual complex numbers.Second,we define the p-norm of a dual complex vector,which is a nonnegative dual number,and show some related properties.Third,we study the properties of eigenvalues of unitary matrices and unitary triangulation of arbitrary dual complex matrices.In particular,we introduce the operator norm of dual complex matrices induced by the p-norm of dual complex vectors,and give expressions of three important operator norms of dual complex matrices.
基金This work was supported by the National Key R&D Program of China(Grant No.2016YFB0601101)the National Natural Sci-ence Foundation of China(Grant No.51576194).
文摘The dual-layer granular bed filter packed with randomly arranged granules was simulated to study the effects of bed depth of the lower layer of fine granules and the inlet gas velocity on the collection mechanism.The computational results show that the collection efficiency is much better from this granular bed than a single-layer granular bed,especially for particle diameters of 1-10μm.The inlet gas velocity has less effect on the grade collection efficiency of the dual-layer granular bed than of the single-layer granular bed.The dual-layer granular bed provides a high collection efficiency and low pressure drop.The relationship between the grade collection efficiency and the Stokes number(St)based on the inlet gas velocity is obtained.If St is below a threshold,the grade collection efficiency remains stable;if St is in value above threshold,the grade collection efficiency increases linearly with lg(St).As the bed depth of the lower layer of fine granules increases,the threshold for St shifts forward.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.41475094 and 41375113)
文摘The tangent linear（TL） models and adjoint（AD） models have brought great difficulties for the development of variational data assimilation system. It might be impossible to develop them perfectly without great efforts, either by hand, or by automatic differentiation tools. In order to break these limitations, a new data assimilation system, dual-number data assimilation system（DNDAS）, is designed based on the dual-number automatic differentiation principles. We investigate the performance of DNDAS with two different optimization schemes and subsequently give a discussion on whether DNDAS is appropriate for high-dimensional forecast models. The new data assimilation system can avoid the complicated reverse integration of the adjoint model, and it only needs the forward integration in the dual-number space to obtain the cost function and its gradient vector concurrently. To verify the correctness and effectiveness of DNDAS, we implemented DNDAS on a simple ordinary differential model and the Lorenz-63 model with different optimization methods. We then concentrate on the adaptability of DNDAS to the Lorenz-96 model with high-dimensional state variables. The results indicate that whether the system is simple or nonlinear, DNDAS can accurately reconstruct the initial condition for the forecast model and has a strong anti-noise characteristic. Given adequate computing resource, the quasi-Newton optimization method performs better than the conjugate gradient method in DNDAS.
文摘This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear La- grange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. It is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Moreover, the paper develops the dual algorithm associ- ated with the proposed nonlinear Lagrangians. Numerical results reported suggest that the dual algorithm based on proposed nonlinear Lagrangians is effective for solving some nonlinear optimization problems.
文摘The dual algorithm for minimax problems is further studied in this paper.The resulting theoretical analysis shows that the condition number of the corresponding Hessian of the smooth modified Lagrange function with changing parameter in the dual algorithm is proportional to the reciprocal of the parameter,which is very important for the efficiency of the dual algorithm.At last,the numerical experiments are reported to validate the analysis results.
文摘A self-invented atomization process, in which molten metal is atomized into powder by a high-velocity gas stream carrying solid particles as the atomization medium, was introduced. The characteristics of powders prepared by common gas atomization and dual-phase flow atomization under similar conditions were compared. The experimental results show that the dual-phase flow-atomized powders have average particle sizes that are one-half that of the common gas-atomized particles;additionally, they possess a finer microstructure and higher cooling rate under the same atomization gas pressure and the same gas flow. The Weber number in the crash criteria of liquid atomization is adopted to measure the crash ability of the atomization media. The Weber number of the dual-phase flow atomization medium is the sum of that of the gas and the solid particles. Furthermore, the critical equation of the crash model in dual-phase flow atomization is established, and the main regularities associated with this process were analyzed.
