A Geometric Definition Of Schubert Polynomials and Dual Schubert Polynomials For Classical Lie Groups

In this paper, we first discuss the topological properties of projective Stiefel manifolds, we compute their cohomology rings and classify their cohomology endomorphisms; Then by embedding the flag manifold of a classical Lie group into its corresponding infinite dimensional projective Stiefel manifold(which is homotopic to the product of infinite dimensional complex projective space $\mathbb{C}P^{\infty}$), we define the Schubert polynomials and dual Schubert polynomials. Finally we discuss the property and the computation of these polynomials.Comment: This paper has been withdrawn by the author due to a crucial This paper have a vital error in Lemma 2.1. So the definition for Schubert polynomials are not valid for Lie groups of type B,C,

StarGraph: A Coarse-to-Fine Representation Method for Large-Scale Knowledge Graph

Conventional representation learning algorithms for knowledge graphs (KG) map each entity to a unique embedding vector, ignoring the rich information contained in neighbor entities. We propose a method named StarGraph, which gives a novel way to utilize the neighborhood information for large-scale knowledge graphs to get better entity representations. The core idea is to divide the neighborhood information into different levels for sampling and processing, where the generalized coarse-grained information and unique fine-grained information are combined to generate an efficient subgraph for each node. In addition, a self-attention network is proposed to process the subgraphs and get the entity representations, which are used to replace the entity embeddings in conventional methods. The proposed method achieves the best results on the ogbl-wikikg2 dataset, which validates the effectiveness of it. The code is now available at https://github.com/hzli-ucas/StarGrap

The Taihang Mountain Region of North China is Experiencing A Significant Warming Trend

The Earth’s climate has warmed by approximately 0.6 °C over the last century, but temperature change in the Taihang Mountain region—an important transition zone in North China which functions as an ecological barrier for Beijing, Tianjin, and other big cities—is still unknown. In this study, we analyze the spatial and temporal trends in the average annual and seasonal surface air temperature in the Taihang Mountain region from 1968 to 2017. The effect of elevation, longitude, latitude, percent forestland, percent farmland, and gross domestic product (GDP) on temperature was also determined. Our results show that the Taihang Mountain has warmed by 0.3 °C/decade over the past five decades. Partitioned seasonally, average warming was 0.38, 0.14, 0.21, and 0.47 °C/decade in spring, summer, fall, and winter, respectively. Elevation and latitude were significantly negatively correlated with temperature but had no correlation with the temporal warming trend (i.e., the Z value from a Mann–Kendall test). The Z value was significantly negatively correlated with percent forestland and positively correlated with GDP, indicating that economic development has induced warming, but afforestation may reduce the rate of warming increase. Together, our results provide important insights into the rates and drivers of climate change within mountainous regions

The Taihang Mountain Region of North China is Experiencing A Significant Warming Trend

The Earth’s climate has warmed by approximately 0.6 °C over the last century, but temperature change in the Taihang Mountain region—an important transition zone in North China which functions as an ecological barrier for Beijing, Tianjin, and other big cities—is still unknown. In this study, we analyze the spatial and temporal trends in the average annual and seasonal surface air temperature in the Taihang Mountain region from 1968 to 2017. The effect of elevation, longitude, latitude, percent forestland, percent farmland, and gross domestic product (GDP) on temperature was also determined. Our results show that the Taihang Mountain has warmed by 0.3 °C/decade over the past five decades. Partitioned seasonally, average warming was 0.38, 0.14, 0.21, and 0.47 °C/decade in spring, summer, fall, and winter, respectively. Elevation and latitude were significantly negatively correlated with temperature but had no correlation with the temporal warming trend (i.e., the Z value from a Mann–Kendall test). The Z value was significantly negatively correlated with percent forestland and positively correlated with GDP, indicating that economic development has induced warming, but afforestation may reduce the rate of warming increase. Together, our results provide important insights into the rates and drivers of climate change within mountainous regions