Since the quadruped robot possesses predominant environmental adaptability,it is expected to be employed in nature environments. In some situations,such as ice surface and tight space,the quadruped robot is required t...Since the quadruped robot possesses predominant environmental adaptability,it is expected to be employed in nature environments. In some situations,such as ice surface and tight space,the quadruped robot is required to lower the height of center of gravity( COG) to enhance the stability and maneuverability. To properly handle these situations,a quadruped controller based on the central pattern generator( CPG) model,the discrete tracking differentiator( TD) and proportional-derivative( PD) sub-controllers is presented. The CPG is used to generate basic rhythmic motion for the quadruped robot. The discrete TD is not only creatively employed to implement the transition between two different rhythmic medium values of the CPG which results in the adjustment of the height of COG of the quadruped robot,but also modified to control the transition duration which enables the quadruped robot to achieve the stable transition. Additionally,two specific PD sub-controllers are constructed to adjust the oscillation amplitude of the CPG,so as to avoid the severe deviation in the transverse direction during transition locomotion. Finally,the controller is validated on a quadruped model. A tunnel with variable height is built for the quadruped model to travel through. The simulation demonstrates the severe deviation without the PD sub-controllers,and the reduced deviation with the PD sub-controllers.展开更多
This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles)...This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.展开更多
A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forc...A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.展开更多
This paper proposes a novel application of cohomology to protein structure analysis. Since proteins interact each other by forming transient protein complexes, their shape (e.g., shape complementarity) plays an import...This paper proposes a novel application of cohomology to protein structure analysis. Since proteins interact each other by forming transient protein complexes, their shape (e.g., shape complementarity) plays an important role in their functions. In our mathematical toy models, proteins are represented as a loop of triangles (2D model) or tetrahedra (3D model), where their interactions are defined as fusion of loops. The purpose of this paper is to describe the conditions for loop fusion using the language of cohomology. In particular, this paper uses cohomology to describe the conditions for “allosteric regulation”, which has been attracted attention in safer drug discovery. I hope that this paper will provide a new perspective on the mechanism of allosteric regulation. Advantages of the model include its topological nature. That is, we can deform the shape of loops by deforming the shape of triangles (or tetrahedra) as long as their folded structures are preserved. Another advantage is the simplicity of the “allosteric regulation” mechanism of the model. Furthermore, the effect of the “post-translational modification” can be understood as a resolution of singularities of a flow of triangles (or tetrahedra). No prior knowledge of either protein science, exterior calculus, or cohomology theory is required. The author hopes that this paper will facilitate the interaction between mathematics and protein science.展开更多
Recently, a novel kind of quantum key distribution called the round-robin differential phase-shift （RRDPS） protocol was proposed, which bounds the amount of leakage without monitoring signal disturbance. The protoco...Recently, a novel kind of quantum key distribution called the round-robin differential phase-shift （RRDPS） protocol was proposed, which bounds the amount of leakage without monitoring signal disturbance. The protocol can be implemented by a weak coherent source. The security of this protocol with a simply characterized source has been proved. The application of a common phase shift can improve the secret key rate of the protocol. In practice, the randomized phase is discrete and the secret key rate is deviated from the continuous case. In this study, we analyze security of the RRDPS protocol with discrete-phase-randomized coherent state source and bound the secret key rate. We fix the length of each packet at 32 and 64, then simulate the secret key rates of the RRDPS protocol with discrete-phase randomization and continuous-phase randomization. Our simulation results show that the performance of the discrete-phase randomization case is close to the continuous counterpart with only a small number of discrete phases. The research is practically valuable for experimental implementation.展开更多
The nonlinear responses of planar motions of a fluid-conveying pipe embedded in nonlinear elastic foundations are investigated via the differential quadrature method discretization （DQMD） of the governing partial di...The nonlinear responses of planar motions of a fluid-conveying pipe embedded in nonlinear elastic foundations are investigated via the differential quadrature method discretization （DQMD） of the governing partial differential equation. For the analytical model, the effect of the nonlinear elastic foundation is modeled by a nonlinear restraining force. By using an iterative algorithm, a set of ordinary differential dynamical equations derived from the equation of motion of the system are solved numerically and then the bifurcations are analyzed. The numerical results, in which the existence of chaos is demonstrated, are presented in the form of phase portraits of the oscillations. The intermittency transition to chaos has been found to arise.展开更多
Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a gi...Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a given shape. In the mathematical model, molecules are represented as loops of n-simplices (2-simplices are triangles and 3-simplices are tetrahedra). We design a new molecule of a given shape by patching together a set of smaller molecules that cover the shape. The covering set of small molecules is defined using a binary relation between sets of molecules. A new molecule is then obtained as a sum of the smaller molecules, where addition of molecules is defined using transformations acting on a set of (n + 1)-dimensional cones. Due to page limitations, only the two-dimensional case (i.e., loops of triangles) is considered. No prior knowledge of Sheaf Theory, Category Theory, or Protein Science is required. The author hopes that this paper will encourage further collaboration between Mathematics and Protein Science.展开更多
For the five-point discrete formulae of directional derivatives in the finite point method,overcoming the challenge resulted from scattered point sets and making full use of the explicit expressions and accuracy of th...For the five-point discrete formulae of directional derivatives in the finite point method,overcoming the challenge resulted from scattered point sets and making full use of the explicit expressions and accuracy of the formulae,this paper obtains a number of theoretical results:(1)a concise expression with definite meaning of the complicated directional difference coefficient matrix is presented,which characterizes the correlation between coefficients and the connection between coefficients and scattered geometric characteristics;(2)various expressions of the discriminant function for the solvability of numerical differentials along with the estimation of its lower bound are given,which are the bases for selecting neighboring points and making analysis;(3)the estimations of combinatorial elements and of each element in the directional difference coefficient matrix are put out,which exclude the existence of singularity.Finally,the theoretical analysis results are verified by numerical calculations.The results of this paper have strong regularity,which lay the foundation for further research on the finite point method for solving partial differential equations.展开更多
We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is rem...We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is removed from the numerical formulation. The SFWG scheme is very simple and easy to implement on polygonal meshes. We prove the well-posedness of the scheme and derive optimal order error estimates in energy and L2 norm. The error results are independent of the permeability tensor, hence the SFWG method is stable and accurate for both the Stokes and Darcy dominated problems. Finally, we present some numerical experiments to verify the efficiency and stability of the SFWG method.展开更多
The stochastic resource allocation(SRA) problem is an extensive class of combinatorial optimization problems widely existing in complex systems such as communication networks and unmanned systems. In SRA, the ability ...The stochastic resource allocation(SRA) problem is an extensive class of combinatorial optimization problems widely existing in complex systems such as communication networks and unmanned systems. In SRA, the ability of a resource to complete a task is described by certain probability,and the objective is to maximize the reward by appropriately assigning available resources to different tasks. This paper is aimed at an important branch of SRA, that is, stochastic SRA(SSRA) for which the probability for resources to complete tasks is also uncertain. Firstly, a general SSRA model with multiple independent uncertain parameters(GSSRA-MIUP) is built to formulate the problem. Then,a scenario-based reformulation which can address multi-source uncertainties is proposed to facilitate the problem-solving process. Secondly, in view of the superiority of the differential evolution algorithm in real-valued optimization, a discrete version of this algorithm was originally proposed and further combined with a specialized local search to create an efficient hybrid optimizer. The hybrid algorithm is compared with the discrete differential evolution algorithm, a pure random sampling method, as well as a restart local search method. Experimental results show that the proposed hybrid optimizer has obvious advantages in solving GSSRA-MIUP problems.展开更多
This survey reviews the recent development of gradient domain mesh deformation method. Different to other deformation methods, the gradient domain deformation method is a surface-based, variational optimization method...This survey reviews the recent development of gradient domain mesh deformation method. Different to other deformation methods, the gradient domain deformation method is a surface-based, variational optimization method. It directly encodes the geometric details in differential coordinates, which are also called Laplacian coordinates in literature. By preserving the Laplacian coordinates, the mesh details can be well preserved during deformation. Due to the locality of the Laplacian coordinates, the variational optimization problem can be casted into a sparse linear system. Fast sparse linear solver can be adopted to generate deformation result interactively, or even in real-time. The nonlinear nature of gradient domain mesh deformation leads to the development of two categories of deformation methods： linearization methods and nonlinear optimization methods. Basically, the linearization methods only need to solve the linear least-squares system once. They are fast, easy to understand and control, while the deformation result might be suboptimal. Nonlinear optimization methods can reach optimal solution of deformation energy function by iterative updating. Since the computation of nonlinear methods is expensive, reduced deformable models should be adopted to achieve interactive performance. The nonlinear optimization methods avoid the user burden to input transformation at deformation handles, and they can be extended to incorporate various nonlinear constraints, like volume constraint, skeleton constraint, and so on. We review representative methods and related approaches of each category comparatively and hope to help the user understand the motivation behind the algorithms. Finally, we discuss the relation between physical simulation and gradient domain mesh deformation to reveal why it can achieve physically plausible deformation result. Kun Zhou is currently a Cheung Kong professor in the Department of Computer Science, Zhejiang Uni- versity, and a member of the State Key Laboratory of CAD＆CG. He received his B.S. degree and Ph.D. degree from Zhejiang University in 1997 and 2002, respectively. Af- ter graduation, he joined Microsoft Research Asia as an associate re-展开更多
A stabilizer-free weak Galerkin finite element method is proposed for the Stokes equations in this paper.Here we omit the stabilizer term in the new method by increasing the degree of polynomial approximating spaces f...A stabilizer-free weak Galerkin finite element method is proposed for the Stokes equations in this paper.Here we omit the stabilizer term in the new method by increasing the degree of polynomial approximating spaces for the weak gradient operators.The new algorithm is simple in formulation and the computational complexity is also reduced.The corresponding approximating spaces consist of piecewise polynomials of degree k≥1 for the velocity and k-1 for the pressure,respectively.Optimal order error estimates have been derived for the velocity in both H^(1) and L^(2) norms and for the pressure in L^(2) norm.Numerical examples are presented to illustrate the accuracy and convergency of the method.展开更多
基金Sponsored by the National Natural Science Foundation of China(Grant No.61375101)
文摘Since the quadruped robot possesses predominant environmental adaptability,it is expected to be employed in nature environments. In some situations,such as ice surface and tight space,the quadruped robot is required to lower the height of center of gravity( COG) to enhance the stability and maneuverability. To properly handle these situations,a quadruped controller based on the central pattern generator( CPG) model,the discrete tracking differentiator( TD) and proportional-derivative( PD) sub-controllers is presented. The CPG is used to generate basic rhythmic motion for the quadruped robot. The discrete TD is not only creatively employed to implement the transition between two different rhythmic medium values of the CPG which results in the adjustment of the height of COG of the quadruped robot,but also modified to control the transition duration which enables the quadruped robot to achieve the stable transition. Additionally,two specific PD sub-controllers are constructed to adjust the oscillation amplitude of the CPG,so as to avoid the severe deviation in the transverse direction during transition locomotion. Finally,the controller is validated on a quadruped model. A tunnel with variable height is built for the quadruped model to travel through. The simulation demonstrates the severe deviation without the PD sub-controllers,and the reduced deviation with the PD sub-controllers.
文摘This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.
基金supported by the Natural Science Basic Research Plan in Shaanxi Province of China(2013JM1022)the Fundamental Research Funds for the Central Universities(K50511700004)
文摘A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.
文摘This paper proposes a novel application of cohomology to protein structure analysis. Since proteins interact each other by forming transient protein complexes, their shape (e.g., shape complementarity) plays an important role in their functions. In our mathematical toy models, proteins are represented as a loop of triangles (2D model) or tetrahedra (3D model), where their interactions are defined as fusion of loops. The purpose of this paper is to describe the conditions for loop fusion using the language of cohomology. In particular, this paper uses cohomology to describe the conditions for “allosteric regulation”, which has been attracted attention in safer drug discovery. I hope that this paper will provide a new perspective on the mechanism of allosteric regulation. Advantages of the model include its topological nature. That is, we can deform the shape of loops by deforming the shape of triangles (or tetrahedra) as long as their folded structures are preserved. Another advantage is the simplicity of the “allosteric regulation” mechanism of the model. Furthermore, the effect of the “post-translational modification” can be understood as a resolution of singularities of a flow of triangles (or tetrahedra). No prior knowledge of either protein science, exterior calculus, or cohomology theory is required. The author hopes that this paper will facilitate the interaction between mathematics and protein science.
基金Supported by the National Basic Research Program of China under Grant No 2013CB338002the National Natural Science Foundation of China under Grant Nos 11304397 and 61505261
文摘Recently, a novel kind of quantum key distribution called the round-robin differential phase-shift （RRDPS） protocol was proposed, which bounds the amount of leakage without monitoring signal disturbance. The protocol can be implemented by a weak coherent source. The security of this protocol with a simply characterized source has been proved. The application of a common phase shift can improve the secret key rate of the protocol. In practice, the randomized phase is discrete and the secret key rate is deviated from the continuous case. In this study, we analyze security of the RRDPS protocol with discrete-phase-randomized coherent state source and bound the secret key rate. We fix the length of each packet at 32 and 64, then simulate the secret key rates of the RRDPS protocol with discrete-phase randomization and continuous-phase randomization. Our simulation results show that the performance of the discrete-phase randomization case is close to the continuous counterpart with only a small number of discrete phases. The research is practically valuable for experimental implementation.
基金the National Natural Science Foundation of China(No.10772071)the Scientific Research Foundation of HUST(No.2006Q003B).
文摘The nonlinear responses of planar motions of a fluid-conveying pipe embedded in nonlinear elastic foundations are investigated via the differential quadrature method discretization （DQMD） of the governing partial differential equation. For the analytical model, the effect of the nonlinear elastic foundation is modeled by a nonlinear restraining force. By using an iterative algorithm, a set of ordinary differential dynamical equations derived from the equation of motion of the system are solved numerically and then the bifurcations are analyzed. The numerical results, in which the existence of chaos is demonstrated, are presented in the form of phase portraits of the oscillations. The intermittency transition to chaos has been found to arise.
文摘Proteins perform a variety of functions in living organisms and their functions are largely determined by their shape. In this paper, we propose a novel mathematical method for designing protein-like molecules of a given shape. In the mathematical model, molecules are represented as loops of n-simplices (2-simplices are triangles and 3-simplices are tetrahedra). We design a new molecule of a given shape by patching together a set of smaller molecules that cover the shape. The covering set of small molecules is defined using a binary relation between sets of molecules. A new molecule is then obtained as a sum of the smaller molecules, where addition of molecules is defined using transformations acting on a set of (n + 1)-dimensional cones. Due to page limitations, only the two-dimensional case (i.e., loops of triangles) is considered. No prior knowledge of Sheaf Theory, Category Theory, or Protein Science is required. The author hopes that this paper will encourage further collaboration between Mathematics and Protein Science.
基金supported by the National Natural Science Foundation of China(11671049)the Foundation of LCP,and the CAEP Foundation(CX2019026).
文摘For the five-point discrete formulae of directional derivatives in the finite point method,overcoming the challenge resulted from scattered point sets and making full use of the explicit expressions and accuracy of the formulae,this paper obtains a number of theoretical results:(1)a concise expression with definite meaning of the complicated directional difference coefficient matrix is presented,which characterizes the correlation between coefficients and the connection between coefficients and scattered geometric characteristics;(2)various expressions of the discriminant function for the solvability of numerical differentials along with the estimation of its lower bound are given,which are the bases for selecting neighboring points and making analysis;(3)the estimations of combinatorial elements and of each element in the directional difference coefficient matrix are put out,which exclude the existence of singularity.Finally,the theoretical analysis results are verified by numerical calculations.The results of this paper have strong regularity,which lay the foundation for further research on the finite point method for solving partial differential equations.
基金supported by the National Natural Science Foundation of China(Grant Nos.1901015,12271208,11971198,91630201,11871245,11771179,11826101)by the Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education,Jilin University.
文摘We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is removed from the numerical formulation. The SFWG scheme is very simple and easy to implement on polygonal meshes. We prove the well-posedness of the scheme and derive optimal order error estimates in energy and L2 norm. The error results are independent of the permeability tensor, hence the SFWG method is stable and accurate for both the Stokes and Darcy dominated problems. Finally, we present some numerical experiments to verify the efficiency and stability of the SFWG method.
基金supported by the National Natural Science Foundation of China under Grant No.71361130011
文摘The stochastic resource allocation(SRA) problem is an extensive class of combinatorial optimization problems widely existing in complex systems such as communication networks and unmanned systems. In SRA, the ability of a resource to complete a task is described by certain probability,and the objective is to maximize the reward by appropriately assigning available resources to different tasks. This paper is aimed at an important branch of SRA, that is, stochastic SRA(SSRA) for which the probability for resources to complete tasks is also uncertain. Firstly, a general SSRA model with multiple independent uncertain parameters(GSSRA-MIUP) is built to formulate the problem. Then,a scenario-based reformulation which can address multi-source uncertainties is proposed to facilitate the problem-solving process. Secondly, in view of the superiority of the differential evolution algorithm in real-valued optimization, a discrete version of this algorithm was originally proposed and further combined with a specialized local search to create an efficient hybrid optimizer. The hybrid algorithm is compared with the discrete differential evolution algorithm, a pure random sampling method, as well as a restart local search method. Experimental results show that the proposed hybrid optimizer has obvious advantages in solving GSSRA-MIUP problems.
文摘This survey reviews the recent development of gradient domain mesh deformation method. Different to other deformation methods, the gradient domain deformation method is a surface-based, variational optimization method. It directly encodes the geometric details in differential coordinates, which are also called Laplacian coordinates in literature. By preserving the Laplacian coordinates, the mesh details can be well preserved during deformation. Due to the locality of the Laplacian coordinates, the variational optimization problem can be casted into a sparse linear system. Fast sparse linear solver can be adopted to generate deformation result interactively, or even in real-time. The nonlinear nature of gradient domain mesh deformation leads to the development of two categories of deformation methods： linearization methods and nonlinear optimization methods. Basically, the linearization methods only need to solve the linear least-squares system once. They are fast, easy to understand and control, while the deformation result might be suboptimal. Nonlinear optimization methods can reach optimal solution of deformation energy function by iterative updating. Since the computation of nonlinear methods is expensive, reduced deformable models should be adopted to achieve interactive performance. The nonlinear optimization methods avoid the user burden to input transformation at deformation handles, and they can be extended to incorporate various nonlinear constraints, like volume constraint, skeleton constraint, and so on. We review representative methods and related approaches of each category comparatively and hope to help the user understand the motivation behind the algorithms. Finally, we discuss the relation between physical simulation and gradient domain mesh deformation to reveal why it can achieve physically plausible deformation result. Kun Zhou is currently a Cheung Kong professor in the Department of Computer Science, Zhejiang Uni- versity, and a member of the State Key Laboratory of CAD＆CG. He received his B.S. degree and Ph.D. degree from Zhejiang University in 1997 and 2002, respectively. Af- ter graduation, he joined Microsoft Research Asia as an associate re-
基金supported in part by China Natural National Science Foundation(Nos.91630201,U1530116,11726102,11771179,93K172018Z01,11701210,JJKH20180113KJ,20190103029JH)by the Program for Cheung Kong Scholars of Ministry of Education of China,Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education.The research of Liu was partially supported by China Natural National Science Foundation(No.12001306)Guangdong Provincial Natural Science Foundation(No.2017A030310285).
文摘A stabilizer-free weak Galerkin finite element method is proposed for the Stokes equations in this paper.Here we omit the stabilizer term in the new method by increasing the degree of polynomial approximating spaces for the weak gradient operators.The new algorithm is simple in formulation and the computational complexity is also reduced.The corresponding approximating spaces consist of piecewise polynomials of degree k≥1 for the velocity and k-1 for the pressure,respectively.Optimal order error estimates have been derived for the velocity in both H^(1) and L^(2) norms and for the pressure in L^(2) norm.Numerical examples are presented to illustrate the accuracy and convergency of the method.
