The singularity categories of the Cluster-tilted algebras of Dynkin type

In this paper, we use the stable categories of some selfinjective algebras to describe the singularity categories of the cluster-tilted algebras of Dynkin type. Furthermore, in this way, we settle the problem of singularity equivalence classification of the cluster-tilted algebra of type $A$, $D$ and $E$ respectively.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1012.4661 by other author

Exchange graphs of cluster algebras have the non-leaving-face property

The claim in the title is proved

Ultrafast Process Characterization of Laser-Induced Damage in Fused Silica Using Pump-Probe Shadow Imaging Techniques

This study delves into the intricate dynamics of laser-induced damage in fused silica using a time-resolved pump-probe (TRPP) shadowgraph. Three typical ultra-fast processes, laser-induced plasma evolution, shockwave propagation and material fracture splashing, were quantitatively investigated. The results indicate that the diameter of plasma is proportional to the pulse laser energy and increases linearly during the pulse laser duration with an expansion rate of approximately 6 km/s. The maximum shockwave velocity on the air side is 9 km/s, occurring at the end of the pulse duration, and then rapidly decreases due to air resistance, reaching approximately 1 km/s around a 300 ns delay. After hundreds of nanoseconds, there is a distinct particle splashing phenomenon, with the splashing particle speed distribution ranging from 0.15 km/s to 2.0 km/s. The particle sizes of the splashing particles range from 4 μm to 15 μm. Additionally, the smaller the delay, the faster the speed of the splashing particles. Overall, TRPP technology provides crucial insights into the temporal evolution of laser-induced damage in fused silica, contributing to a comprehensive understanding essential for optimizing the performance and safety of laser systems