In this paper, decentralized methods of optimally rigid graphs generation for formation control are researched. The notion of optimally rigid graph is first defined in this paper to describe a special kind of rigid gr...In this paper, decentralized methods of optimally rigid graphs generation for formation control are researched. The notion of optimally rigid graph is first defined in this paper to describe a special kind of rigid graphs. The optimally rigid graphs can be used to decrease the topology complexity of graphs while maintaining their shapes. To minimize the communication complexity of formations, we study the theory of optimally rigid formation generation. First, four important propositions are presented to demonstrate the feasibility of using a decentralized method to generate optimally rigid graphs. Then, a formation algorithm for multi-agent systems based on these propositions is proposed. At last, some simulation examples are given to show the efficiency of the proposed algorithm.展开更多
In this paper,the (?)-equivariant (s, t)-equivalence relation and (?)-equivariant infinitesimally (r, s)-stability of (?)-equivariant bifurcation problem are defined. The criterion for (?)-equivariant infinitesimally ...In this paper,the (?)-equivariant (s, t)-equivalence relation and (?)-equivariant infinitesimally (r, s)-stability of (?)-equivariant bifurcation problem are defined. The criterion for (?)-equivariant infinitesimally (r, s)-stability is proven when (?) is a compact finite Lie group .Transversality condition is used to characterize the stability.展开更多
We prove that any linear multi-step method G1^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（Zk） with odd order u （u≥ 3） cannot be conjugate to a symplectic method G2^T of order w （w 〉 u） via any generalize...We prove that any linear multi-step method G1^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（Zk） with odd order u （u≥ 3） cannot be conjugate to a symplectic method G2^T of order w （w 〉 u） via any generalized linear multi-step method G3^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（∑l=0^mγklZl）. We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when G3^T is a more general operator.展开更多
In [1], tile universal unfolding of Coo map germs under a subgroup of the group A,which is well known group defined by J.Mather [2], was discussed. In this papersSome conditions are given to characterize the infinites...In [1], tile universal unfolding of Coo map germs under a subgroup of the group A,which is well known group defined by J.Mather [2], was discussed. In this papersSome conditions are given to characterize the infinitesimally stability of unfoldings.展开更多
In one of his astronomical works the prominent arabic medieval scientists Thabit ibn Qurra （836-901） studied the visible motion of the Sun and found the points, where its velocity is maximum or minimum. He also lbun...In one of his astronomical works the prominent arabic medieval scientists Thabit ibn Qurra （836-901） studied the visible motion of the Sun and found the points, where its velocity is maximum or minimum. He also lbund the points on the ecliptic, where this velocity is equal to the average velocity of the Sun over all the ecliptic. For this purpose he used the idea of infinitely small arcs and their ratios in different points of the circle. The great scientist Leonard Euler （1707-1783） introduced in his works on spherical trigonometry the line-element ds of the surface of the sphere, i.e. the differential of the arc length. He constructed the spherical trigonometry as an inner geometry on the surface of the sphere. He replaced the trigonometry lines, which were in use befbre him, by trigonometric functions.展开更多
基金supported by National Natural Science Foundation of China (No. 60934003, No. 61074065)Key Project for Natural Science Research of Hebei Education Department (No. ZD200908)
文摘In this paper, decentralized methods of optimally rigid graphs generation for formation control are researched. The notion of optimally rigid graph is first defined in this paper to describe a special kind of rigid graphs. The optimally rigid graphs can be used to decrease the topology complexity of graphs while maintaining their shapes. To minimize the communication complexity of formations, we study the theory of optimally rigid formation generation. First, four important propositions are presented to demonstrate the feasibility of using a decentralized method to generate optimally rigid graphs. Then, a formation algorithm for multi-agent systems based on these propositions is proposed. At last, some simulation examples are given to show the efficiency of the proposed algorithm.
基金Supported by the National Nature Science Foundation of China (10261002)
文摘In this paper,the (?)-equivariant (s, t)-equivalence relation and (?)-equivariant infinitesimally (r, s)-stability of (?)-equivariant bifurcation problem are defined. The criterion for (?)-equivariant infinitesimally (r, s)-stability is proven when (?) is a compact finite Lie group .Transversality condition is used to characterize the stability.
基金Acknowledgements. We would like to thank the editors for their valuable suggestions and corrections. This research is supported by the National Natural Science Foundation of China （Grant Nos. 10471145 and 10672143）, and by Morningside Center of Mathematics, Chinese Academy of Sciences.
文摘We prove that any linear multi-step method G1^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（Zk） with odd order u （u≥ 3） cannot be conjugate to a symplectic method G2^T of order w （w 〉 u） via any generalized linear multi-step method G3^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（∑l=0^mγklZl）. We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when G3^T is a more general operator.
文摘In [1], tile universal unfolding of Coo map germs under a subgroup of the group A,which is well known group defined by J.Mather [2], was discussed. In this papersSome conditions are given to characterize the infinitesimally stability of unfoldings.
文摘In one of his astronomical works the prominent arabic medieval scientists Thabit ibn Qurra （836-901） studied the visible motion of the Sun and found the points, where its velocity is maximum or minimum. He also lbund the points on the ecliptic, where this velocity is equal to the average velocity of the Sun over all the ecliptic. For this purpose he used the idea of infinitely small arcs and their ratios in different points of the circle. The great scientist Leonard Euler （1707-1783） introduced in his works on spherical trigonometry the line-element ds of the surface of the sphere, i.e. the differential of the arc length. He constructed the spherical trigonometry as an inner geometry on the surface of the sphere. He replaced the trigonometry lines, which were in use befbre him, by trigonometric functions.
