Let R be an s-unital ring, and we prove a commutativity theorem of R satisfying the following conditions: (l ) For each x, y∈ R,there exist bounded positive integers k =k (x,y), s=s (x,y), t =t (x,y)(where, at least...Let R be an s-unital ring, and we prove a commutativity theorem of R satisfying the following conditions: (l ) For each x, y∈ R,there exist bounded positive integers k =k (x,y), s=s (x,y), t =t (x,y)(where, at least one of k, s, t is not equal to 1) such展开更多
This note is to investigate the properties of strongly semipotent rings.It is proved that every strongly semipotent ring is a idempotent unit regular ring,i.e.,there exist a non-zero idempotent e and a unit u such tha...This note is to investigate the properties of strongly semipotent rings.It is proved that every strongly semipotent ring is a idempotent unit regular ring,i.e.,there exist a non-zero idempotent e and a unit u such that er=eu for all r∉J(R),where J(R)is the Jacobson radical of ring R.展开更多
To address the current issues of low reconfigurability,low integration,and high dynamic power consumption in programmable units,this study proposes a novel programmable photonic unit cell,termed MZI-cascaded-ring unit...To address the current issues of low reconfigurability,low integration,and high dynamic power consumption in programmable units,this study proposes a novel programmable photonic unit cell,termed MZI-cascaded-ring unit(MCR).The unit functions analogously to an MZI,enabling broadband routing when operating within the free spectral range(FSR)of the embedded resonator,and it transitions into a wavelength-selective mode,leveraging the micro-ring’s resonance to achieve precise amplitude and phase control for narrowband signals while outside the FSR,featuring dual operational regimes.With the implementation of spiral waveguide structures,the design achieves higher integration density and lower dynamic power consumption.Based on the hexagonal mesh extension of such a unit,the programmable photonic processor successfully demonstrates a reconfiguration of large amounts of fundamental functions with tunable performance metrics,including broadband linear operations like optical router and wavelength-selective functionalities like wavelength division multiplexing.This work establishes a new paradigm for programmable photonic integrated circuit design.展开更多
文摘Let R be an s-unital ring, and we prove a commutativity theorem of R satisfying the following conditions: (l ) For each x, y∈ R,there exist bounded positive integers k =k (x,y), s=s (x,y), t =t (x,y)(where, at least one of k, s, t is not equal to 1) such
文摘This note is to investigate the properties of strongly semipotent rings.It is proved that every strongly semipotent ring is a idempotent unit regular ring,i.e.,there exist a non-zero idempotent e and a unit u such that er=eu for all r∉J(R),where J(R)is the Jacobson radical of ring R.
基金National Natural Science Foundation of China(62075038)Postgraduate Research&Practice Innovation Program of Jiangsu Province(KYCX24_0401).
文摘To address the current issues of low reconfigurability,low integration,and high dynamic power consumption in programmable units,this study proposes a novel programmable photonic unit cell,termed MZI-cascaded-ring unit(MCR).The unit functions analogously to an MZI,enabling broadband routing when operating within the free spectral range(FSR)of the embedded resonator,and it transitions into a wavelength-selective mode,leveraging the micro-ring’s resonance to achieve precise amplitude and phase control for narrowband signals while outside the FSR,featuring dual operational regimes.With the implementation of spiral waveguide structures,the design achieves higher integration density and lower dynamic power consumption.Based on the hexagonal mesh extension of such a unit,the programmable photonic processor successfully demonstrates a reconfiguration of large amounts of fundamental functions with tunable performance metrics,including broadband linear operations like optical router and wavelength-selective functionalities like wavelength division multiplexing.This work establishes a new paradigm for programmable photonic integrated circuit design.
