Besov Spaces, Schatten Classes and Weighted Versions of the Quantised Derivative

In this paper, we establish the Schatten class and endpoint weak Schatten class estimates for the commutator of Riesz transforms on weighted $L^2$ spaces. It provides a weighted version for the estimate of the quantised derivative introduced by Alain Connes and studied recently by Lord--McDonald--Sukochev--Zanin.Comment: Dedicated to Professor Besov on the occasion of his 90th birthday, for special issue in Analysis Mathematic

Speed limitation of a mobile robot and methodology of tracing odor plume in airflow environments

AbstractThe methodology of tracing odor plume via a mobile robot is considered. In this research, two typical plume-tracing methods, i.e., a zigzagging method and an upwind method, are tested in four airflow fields with different long-time average wind speeds when the robot is set to possessing four different maximum speeds. According to the simulation results, it can be deduced that the zigzagging algorithms would be efficient when the robot moves faster than the odor plume or airflow, and the upwind algorithms are preferred especially when the robot is slow

Topological Dirac states beyond π\pi orbitals for silicene on SiC(0001) surface

The discovery of intriguing properties related to the Dirac states in graphene has spurred huge interest in exploring its two-dimensional group-IV counterparts, such as silicene, germanene, and stanene. However, these materials have to be obtained via synthesizing on substrates with strong interfacial interactions, which usually destroy their intrinsic $\pi$($p_z$)-orbital Dirac states. Here we report a theoretical study on the existence of Dirac states arising from the $p_{x,y}$ orbitals instead of $p_z$ orbitals in silicene on 4H-SiC(0001), which survive in spite of the strong interfacial interactions. We also show that the exchange field together with the spin-orbital coupling give rise to a detectable band gap of 1.3 meV. Berry curvature calculations demonstrate the nontrivial topological nature of such Dirac states with a Chern number $C = 2$, presenting the potential of realizing quantum anomalous Hall effect for silicene on SiC(0001). Finally, we construct a minimal effective model to capture the low-energy physics of this system. This finding is expected to be also applicable to germanene and stanene, and imply great application potentials in nanoelectronics.Comment: 6 Figures , Accepted by Nano Letter

Calder\'on's commutator on Stratified Lie groups

Motivated by the recent work of Gimperlein and Goffeng on Calder\'on's commutator on compact Heisenberg type manifolds and the related weak Schatten class estimates, we establish the characterisation of $L^p$ boundedness for Calderon's commutator on stratified Lie groups. We further study the related weak Schatten class estimates on two step stratified Lie groups, which include the Heisenberg groups