Koszul property of a class of graded algebras with nonpure resolutions

Given any integers a,b,c, and d with a 〉 1, c ≥ 0, b ≥ a＋c, and d ≥ b ＋ c, the notion of （a, b, c, d）-Koszul algebra is introduced, which is another class of standard graded algebras with ＂nonpure＂ resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of （a, b, c, d）-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be （a, b, c, d）- Koszul are provided.

On BiHom-L-R Smash Products

Let(H,αH,βH,ψH,ωH,SH)be a BiHom-Hopf algebra and(A,αA,βA)be an(H,αH,βH,ψH,ωH)-BiHom-bimodule algebra,where the mapsαH,βH,ψH,ωH,αA,βA are bijective.We first prove the Maschke-type theorem for the BiHom-L-R smash product over a finite-dimensional semisimple BiHom-Hopf algebra.Next we give a Morita context between the BiHom-subalgebra A^(biH)and the BiHom-L-R smash product A#H.

Tailoring chiral optical properties by femtosecond laser direct writing in silica

An object that possesses chirality,that is,having its mirror image not overlayed on itself by rotation and translation,can provide a different optical response to a left-or right-handed circular polarized light.Chiral nanostructures may exhibit polarization-selective optical properties that can be controlled for micro-to-nano optical element engineering.An attractive way to induce such complex nanostructures in three-dimension in glass is femtosecond laser direct writing.However,the mechanism of femtosecond laser induced chirality remains to be unveiled due to complex physical and chemical processes occurring during the ultrashort light-matter interaction.Here,a phenomenological model is proposed and is built on two-layers phase shifters to account for this laser-induced optical chirality in an initially achiral material(silica glass).This model is based on the observation that femtosecond laser induced nanogratings own two principal contributions to its aggregate birefringent response:a form and a stress-related one.By refining this formalism,a multilayer approach is developed to imprint on demand optical rotation.Values up to+/-60°at 550 nm within an optimal 80μm thickness in silica glass are possible,corresponding to the highest value in a glass to date.These results provide new insights of circular-optical control in micro-nano optical manufacturing and open new opportunities for photonics applications.

Real-time observation of vortex mode switching in a narrow-linewidth mode-locked fiber laser

Temporal and spatial resonant modes are always possessed in physical systems with energy oscillation.In ultrafast fiber lasers,enormous progress has been made toward controlling the interactions of many longitudinal modes,which results in temporally mode-locked pulses.Recently,optical vortex beams have been extensively investigated due to their quantized orbital angular momentum,spatially donut-like intensity,and spiral phase front.In this paper,we have demonstrated the first to our knowledge observation of optical vortex mode switching and their corresponding pulse evolution dynamics in a narrow-linewidth mode-locked fiber laser.The spatial mode switching is achieved by incorporating a dual-resonant acousto-optic mode converter in the vortex mode-locked fiber laser.The vortex mode-switching dynamics have four stages,including quiet-down,relaxation oscillation,quasi mode-locking,and energy recovery prior to the stable mode-locking of another vortex mode.The evolution dynamics of the wavelength shifting during the switching process are observed via the time-stretch dispersion Fourier transform method.The spatial mode competition through optical nonlinearity induces energy fluctuation on the time scale of ultrashort pulses,which plays an essential role in the mode-switching dynamic process.The results have great implications in the study of spatial mode-locking mechanisms and ultrashort laser applications.

Quasi-d-Koszul Modules and Applications

Let R be a Noetherian semiperfect algebra. A necessary and sufficient condition for a finitely generated R-module to be quasi-d-Koszul is given, which improves one of the main results in [1]. The authors also give a necessary and sufficient condition for the Minimal Horseshoe Lemma holding in mod(R). As an application, it is proved that the "Minimal Horseshoe Lemma" is true in the category of quasi-d-Koszul modules under certain conditions.