In this paper,we explicitly establish Poincaréinequality for 1≤p<∞ over simple geometric domains,such as segment,rectangle,triangle or tetrahedron.We obtain sharper bounds of the constant in Poincaré in...In this paper,we explicitly establish Poincaréinequality for 1≤p<∞ over simple geometric domains,such as segment,rectangle,triangle or tetrahedron.We obtain sharper bounds of the constant in Poincaré inequality and,in particular,the explicit relation between the constant and the geometric characters of the domain.展开更多
The unique phase profile and polarization distribution of the vector vortex beam(VVB)have been a subject of increasing interest in classical and quantum optics.The development of higher-order Poincarésphere(HOPS)...The unique phase profile and polarization distribution of the vector vortex beam(VVB)have been a subject of increasing interest in classical and quantum optics.The development of higher-order Poincarésphere(HOPS)and hybrid-order Poincarésphere(HyOPS)has provided a systematic description of VVB.However,the generation of arbitrary VVBs on a HOPS and a HyOPS via a metasurface lacks a unified design framework,despite numerous reported approaches.We present a unified design framework incorporating all design parameters(e.g.,focal lengths and orders)of arbitrary HOPS and HyOPS beams into a single equation.In proof-of-concept experiments,we experimentally demonstrated four metasurfaces to generate arbitrary beams on the fifth-order HOPS(nonfocused and tightly focused,NA 0.89),0-2 order,and 0-1 order HyOPS.We showed HOPS beams’propagation and focusing properties,the superresolution focusing characteristics of the first-order cylindrical VVBs,and the different focusing properties of integerorder and fractional-order cylindrical VVBs.The simplicity and feasibility of the proposed design framework make it a potential catalyst for arbitrary VVBs using metasurfaces in applications of optical imaging,communication,and optical trapping.展开更多
In this paper,we focus on studying weighted Poincare inequalities on stratified Lie groups.We derive various Poincaréinequalities in the case 1<p=q<∞ in the high order Sobolev space Wm,p.We derive several ...In this paper,we focus on studying weighted Poincare inequalities on stratified Lie groups.We derive various Poincaréinequalities in the case 1<p=q<∞ in the high order Sobolev space Wm,p.We derive several Poincare inequalities that complement existing results,which have only been proved for the case 1<p<q<∞.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.12001170 and 11601124)Innovative Funds Plan of Henan University of Technology(Grant No.2021ZKCJ11).
文摘In this paper,we explicitly establish Poincaréinequality for 1≤p<∞ over simple geometric domains,such as segment,rectangle,triangle or tetrahedron.We obtain sharper bounds of the constant in Poincaré inequality and,in particular,the explicit relation between the constant and the geometric characters of the domain.
基金the Chinese Scholarship Council is acknowledged.This work was supported by the UK’s Engineering and Physical Sciences Research Council(Grant Nos.EP/V000624/1,EP/X03495X/1,EP/X041166/1,and EP/T02643X/1)the Royal Society(Grant No.RG\R2\232531).
文摘The unique phase profile and polarization distribution of the vector vortex beam(VVB)have been a subject of increasing interest in classical and quantum optics.The development of higher-order Poincarésphere(HOPS)and hybrid-order Poincarésphere(HyOPS)has provided a systematic description of VVB.However,the generation of arbitrary VVBs on a HOPS and a HyOPS via a metasurface lacks a unified design framework,despite numerous reported approaches.We present a unified design framework incorporating all design parameters(e.g.,focal lengths and orders)of arbitrary HOPS and HyOPS beams into a single equation.In proof-of-concept experiments,we experimentally demonstrated four metasurfaces to generate arbitrary beams on the fifth-order HOPS(nonfocused and tightly focused,NA 0.89),0-2 order,and 0-1 order HyOPS.We showed HOPS beams’propagation and focusing properties,the superresolution focusing characteristics of the first-order cylindrical VVBs,and the different focusing properties of integerorder and fractional-order cylindrical VVBs.The simplicity and feasibility of the proposed design framework make it a potential catalyst for arbitrary VVBs using metasurfaces in applications of optical imaging,communication,and optical trapping.
文摘In this paper,we focus on studying weighted Poincare inequalities on stratified Lie groups.We derive various Poincaréinequalities in the case 1<p=q<∞ in the high order Sobolev space Wm,p.We derive several Poincare inequalities that complement existing results,which have only been proved for the case 1<p<q<∞.
