This is a series of studies on Wu＇s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy＇s function f（z） （z = x ＋ iy） and its integral ...This is a series of studies on Wu＇s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy＇s function f（z） （z = x ＋ iy） and its integral J[f（z）]≡（2πi）-∮cf（t）（t-z）-1dt taken along the unit circle as contour C,inside which（the open domain D＋） f（z） is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f（z） can be determined in analytical form in terms of the values f（t） of f（z） numerically prescribed on C（｜t｜ = 1） ,as so enunciated by Wu＇s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle,for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction,and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics,engineering and technology in analogy with resolving the inverse problem presented here.展开更多
Unmanned aerial vehicle(UAV)path planning plays an important role in power systems.In order to address the challenge in UAV path planning,an improved crested porcupine optimizer(ICPO)combining the Cauchy inverse cumul...Unmanned aerial vehicle(UAV)path planning plays an important role in power systems.In order to address the challenge in UAV path planning,an improved crested porcupine optimizer(ICPO)combining the Cauchy inverse cumulative distribution function and JAYA algorithm is proposed in this paper.First,the traditional random initialization is replaced by sine chaotic mapping,making the initial population more evenly distributed in the search space and improving the quality of the initial solution.Since the global search ability of the crested porcupine optimizer(CPO)is limited,the Cauchy inverse cumulative distribution strategy is introduced.In addition,as CPO is prone to fall into local optima in later stages,a weighted JAYA-CPO attack strategy is proposed to balance the global exploration and local exploitation,thereby improving the algorithm’s ability to escape from local optima.Finally,ICPO is compared with another 10 algorithms on the cec2017 and cec2020 test sets.The experimental results show that ICPO has excellent competitiveness and optimization performance.The ICPO algorithm is applied to the path planning problem of power inspection UAV and is compared with four algorithms.The results show that the algorithm can generate more feasible path trajectories across two terrains with varying complexity,demonstrating the effectiveness and significance of the ICPO algorithm for UAV power inspection path planning.展开更多
文摘This is a series of studies on Wu＇s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy＇s function f（z） （z = x ＋ iy） and its integral J[f（z）]≡（2πi）-∮cf（t）（t-z）-1dt taken along the unit circle as contour C,inside which（the open domain D＋） f（z） is regular but has singularities distributed in open domain Doutside C. Resolution is given to the inverse problem that the singularities of f（z） can be determined in analytical form in terms of the values f（t） of f（z） numerically prescribed on C（｜t｜ = 1） ,as so enunciated by Wu＇s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle,for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction,and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics,engineering and technology in analogy with resolving the inverse problem presented here.
基金supported by the National Natural Sci-ence Foundation of China(No.62102373,No.62273243,and No.62473341)Henan Province Key R&D Project(No.241111210400)Joint Fund Key Project of science and Technology R&D Plan of Henan Province(No.235200810022).
文摘Unmanned aerial vehicle(UAV)path planning plays an important role in power systems.In order to address the challenge in UAV path planning,an improved crested porcupine optimizer(ICPO)combining the Cauchy inverse cumulative distribution function and JAYA algorithm is proposed in this paper.First,the traditional random initialization is replaced by sine chaotic mapping,making the initial population more evenly distributed in the search space and improving the quality of the initial solution.Since the global search ability of the crested porcupine optimizer(CPO)is limited,the Cauchy inverse cumulative distribution strategy is introduced.In addition,as CPO is prone to fall into local optima in later stages,a weighted JAYA-CPO attack strategy is proposed to balance the global exploration and local exploitation,thereby improving the algorithm’s ability to escape from local optima.Finally,ICPO is compared with another 10 algorithms on the cec2017 and cec2020 test sets.The experimental results show that ICPO has excellent competitiveness and optimization performance.The ICPO algorithm is applied to the path planning problem of power inspection UAV and is compared with four algorithms.The results show that the algorithm can generate more feasible path trajectories across two terrains with varying complexity,demonstrating the effectiveness and significance of the ICPO algorithm for UAV power inspection path planning.
