The First Cohomology of Lie Superalgebra P~(2)\tilde{P}(2) with Coefficients in Kac Modules

Over a field of characteristic p>2, firstly, the structure of Kac modules of Lie superalgebra $\tilde{P}(2)$ and the weight space decompositions are given. Secondly, the weight-derivations of $\tilde{P}(2)$ to its Kac modules are computed. Finally, the first cohomology of $\tilde{P}(2)$ with coefficients in Kac modules is determined

Dimension Reduction for Positively Curved Steady Solitons

We consider noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature. We show that such solitons always dimension reduce at infinity. This generalizes an earlier result in [CDM22] to higher dimensions. In dimension four, we classify possible reductions at infinity, which lays foundation for possible classifications of steady solitons. Moreover, we show that any tangent flow at infinity of a general noncollapsed steady soliton must split off a line. This generalizes an earlier result in [BCDMZ21] to higher dimensions. While this article is under preparation, we realized that part of our main results are proved independently in a recent post [ZZ23] under different assumptions

Tuning the Dzyaloshinskii-Moriya Interaction in Pt/Co/MgO heterostructures through MgO thickness

The interfacial Dzyaloshinskii-Moriya interaction (DMI) in the ferromagnetic/heavy metal ultra-thin film structures , has attracted a lot of attention thanks to its capability to stabilize Neel-type domain walls (DWs) and magnetic skyrmions for the realization of non-volatile memory and logic devices. In this study, we demonstrate that magnetic properties in perpendicularly magnetized Ta/Pt/Co/MgO/Pt heterostructures, such as magnetization and DMI, can be significantly influenced through both the MgO and the Co ultrathin film thickness. By using a field-driven creep regime domain expansion technique, we find that non-monotonic tendencies of DMI field appear when changing the thickness of MgO and the MgO thickness corresponding to the largest DMI field varies as a function of the Co thicknesses. We interpret this efficient control of DMI as subtle changes of both Pt/Co and Co/MgO interfaces, which provide a method to investigate ultra-thin structures design to achieve skyrmion electronics.Comment: 18 pages, 11 figure

Local smooth convergence of F\mathbb{F}-limit flows

The metric flow is introduced and extensively studied by Bamler [Bam20b, Bam20c], especially as an $\mathbb{F}$-limit of a sequence of smooth Ricci flows with uniformly bounded Nash entropy, in which case each regular point on the limit is a point of smooth convergence. In this note, we shall consider the $\mathbb{F}$-convergence of a sequence of $\mathbb{F}$-limit flows, and, like Bamler, show that each regular point on the limit is also a point of smooth convergence. The main result will be applied in a forthcoming work of the authors [CMZ23]