Uniquely circular colourable and uniquely fractional colourable graphs of large girth

Given any rational numbers $r \geq r' >2$ and an integer $g$, we prove that there is a graph $G$ of girth at least $g$, which is uniquely $r$-colourable and uniquely $r'$-fractional colourable

Subsampling and Jackknifing: A Practically Convenient Solution for Large Data Analysis with Limited Computational Resources

Modern statistical analysis often encounters datasets with large sizes. For these datasets, conventional estimation methods can hardly be used immediately because practitioners often suffer from limited computational resources. In most cases, they do not have powerful computational resources (e.g., Hadoop or Spark). How to practically analyze large datasets with limited computational resources then becomes a problem of great importance. To solve this problem, we propose here a novel subsampling-based method with jackknifing. The key idea is to treat the whole sample data as if they were the population. Then, multiple subsamples with greatly reduced sizes are obtained by the method of simple random sampling with replacement. It is remarkable that we do not recommend sampling methods without replacement because this would incur a significant cost for data processing on the hard drive. Such cost does not exist if the data are processed in memory. Because subsampled data have relatively small sizes, they can be comfortably read into computer memory as a whole and then processed easily. Based on subsampled datasets, jackknife-debiased estimators can be obtained for the target parameter. The resulting estimators are statistically consistent, with an extremely small bias. Finally, the jackknife-debiased estimators from different subsamples are averaged together to form the final estimator. We theoretically show that the final estimator is consistent and asymptotically normal. Its asymptotic statistical efficiency can be as good as that of the whole sample estimator under very mild conditions. The proposed method is simple enough to be easily implemented on most practical computer systems and thus should have very wide applicability

Deterministic Spin-Orbit Torque Switching of Mn3Sn with the Interplay between Spin Polarization and Kagome Plane

Previous studies have demonstrated spin-orbit torque (SOT) switching of Mn3Sn where the spin polarization lies in the kagome plane (configuration I). However, the critical current density ($J_{crit}$) is unrealistically large ($J_{crit}$=$10^{14}$ A/$m^2$) and independent on the external field ($H_{ext}$). The stabilized magnetic state also depends on the initial state. These features conflict with the ferromagnet (FM) switching scheme as claimed in those studies, and thus call for other explanations. Alternatively, the system with the spin polarization perpendicular to the kagome plane (configuration II) is more like the FM based system since the spin polarization is orthogonal to all magnetic moments. In this work, we show SOT switching of Mn3Sn in configuration II. Similar to the FM, Jcrit and Hext are in the order of $10^{10}$ A/$m^2$ and hundreds of Oersted, respectively. The switching result is also independent of the initial state. Interestingly, the unique spin structure of Mn3Sn also leads to distinct features from FM systems. We demonstrate that Jcrit increases linearly with Hext, and extrapolation gives ultralow $J_{crit}$ for the field-free switching system. In addition, the switching polarity is opposite to the FM. We also provide the switching phase diagram as a guideline for experimental demonstration. Our work provides comprehensive understanding for the switching mechanism in both configurations. The switching protocol proposed in this work is more advantageous in realistic spintronic applications. We also clearly reveal the fundamental difference between FM and noncollinear antiferromagnetic switching

Select2Col: Leveraging Spatial-Temporal Importance of Semantic Information for Efficient Collaborative Perception

Collaboration by leveraging the shared semantic information plays a crucial role in overcoming the perception capability limitations of isolated agents. However, existing collaborative perception methods tend to focus solely on the spatial features of semantic information, while neglecting the importance of the temporal dimension. Consequently, the potential benefits of collaboration remain underutilized. In this article, we propose Select2Col, a novel collaborative perception framework that takes into account the {s}patial-t{e}mpora{l} importanc{e} of semanti{c} informa{t}ion. Within the Select2Col, we develop a collaborator selection method that utilizes a lightweight graph neural network (GNN) to estimate the importance of semantic information (IoSI) in enhancing perception performance, thereby identifying contributive collaborators while excluding those that bring negative impact. Moreover, we present a semantic information fusion algorithm called HPHA (historical prior hybrid attention), which integrates multi-scale attention and short-term attention modules to capture the IoSI in feature representation from the spatial and temporal dimensions respectively, and assigns IoSI-consistent weights for efficient fusion of information from selected collaborators. Extensive experiments on two open datasets demonstrate that our proposed Select2Col significantly improves the perception performance compared to state-of-the-art approaches. The code associated with this research is publicly available at https://github.com/huangqzj/Select2Col/