In this article we shall examine several different types of figurative numbers which have been studied extensively over the period of 2500 years, and currently scattered on hundreds of websites. We shall discuss their...In this article we shall examine several different types of figurative numbers which have been studied extensively over the period of 2500 years, and currently scattered on hundreds of websites. We shall discuss their computation through simple recurrence relations, patterns and properties, and mutual relationships which have led to curious results in the field of elementary number theory. Further, for each type of figurative numbers we shall show that the addition of first finite numbers and infinite addition of their inverses often require new/strange techniques. We sincerely hope that besides experts, students and teachers of mathematics will also be benefited with this article.展开更多
The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites.To this end,this paper presents a method to rapidly generate low-thrust c...The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites.To this end,this paper presents a method to rapidly generate low-thrust collision-avoidance trajectories in the formation reconfiguration using Finite Fourier Series(FFS).The FFS method can rapidly generate the collision-avoidance threedimensional trajectory.The results obtained by the FFS method are used as an initial guess in the Gauss Pseudospectral Method(GPM)solver to verify the applicability of the results.Compared with the GPM method,the FFS method needs very little computing time to obtain the results with very little difference in performance index.To verify the effectiveness,the proposed method is tested and validated by a formation control testbed.Three satellite simulators in the testbed are used to simulate two-dimensional satellite formation reconfiguration.The simulation and experimental results show that the FFS method can rapidly generate trajectories and effectively reduce the risk of collision between satellites.This fast trajectory generation method has great significance for on-line,constantly satellite formation reconfiguration.展开更多
The article presents the proof of the validity of the generalized Riemann hypothesis on the basis of adjustment and correction of the proof of the Riemanns hypothesis in the work?[1], obtained by a finite exponential ...The article presents the proof of the validity of the generalized Riemann hypothesis on the basis of adjustment and correction of the proof of the Riemanns hypothesis in the work?[1], obtained by a finite exponential functional series and finite exponential functional progression.展开更多
文摘In this article we shall examine several different types of figurative numbers which have been studied extensively over the period of 2500 years, and currently scattered on hundreds of websites. We shall discuss their computation through simple recurrence relations, patterns and properties, and mutual relationships which have led to curious results in the field of elementary number theory. Further, for each type of figurative numbers we shall show that the addition of first finite numbers and infinite addition of their inverses often require new/strange techniques. We sincerely hope that besides experts, students and teachers of mathematics will also be benefited with this article.
基金supported in part by the National Natural Science Foundation of China(Nos.11702072 and 11672093)。
文摘The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites.To this end,this paper presents a method to rapidly generate low-thrust collision-avoidance trajectories in the formation reconfiguration using Finite Fourier Series(FFS).The FFS method can rapidly generate the collision-avoidance threedimensional trajectory.The results obtained by the FFS method are used as an initial guess in the Gauss Pseudospectral Method(GPM)solver to verify the applicability of the results.Compared with the GPM method,the FFS method needs very little computing time to obtain the results with very little difference in performance index.To verify the effectiveness,the proposed method is tested and validated by a formation control testbed.Three satellite simulators in the testbed are used to simulate two-dimensional satellite formation reconfiguration.The simulation and experimental results show that the FFS method can rapidly generate trajectories and effectively reduce the risk of collision between satellites.This fast trajectory generation method has great significance for on-line,constantly satellite formation reconfiguration.
文摘The article presents the proof of the validity of the generalized Riemann hypothesis on the basis of adjustment and correction of the proof of the Riemanns hypothesis in the work?[1], obtained by a finite exponential functional series and finite exponential functional progression.
