On the realization of Riemannian symmetric spaces in Lie groups II

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.Comment: 6 page

On the realization of Riemannian symmetric spaces in Lie groups

In this paper we give a realization of some symmetric space G/K as a closed submanifold P of G. We also give several equivalent representations of the submanifold P. Some properties of the set gK\cap P are also discussed, where gK is a coset space in G.Comment: 8 page

Holomorphic harmonic analysis on complex reductive groups

We define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties like the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of $K$-admissible measures. We prove that $K$-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of $K$-admissible measures.Comment: 15 pages, revision of a preprint by the first author in 200

A Generalization of Random Matrix Ensemble I: General Theory

We give a generalization of the random matrix ensembles, including all lassical ensembles. Then we derive the joint density function of the generalized ensemble by one simple formula, which give a direct and unified way to compute the density functions for all classical ensembles and various kinds of new ensembles. An integration formula associated with the generalized ensemble is also given. We also give a classification scheme of the generalized ensembles, which will include all classical ensembles and some new ensembles which were not considered before.Comment: 15 page

From Micro to Marco: Jade Snuff Bottles in Qianlong Era

Emperor Qianlong, whose original name is Aisin Gioro Hongli, was the sixth emperor of the Qing Dynasty. Qianlong was one of the most famous emperors in Chinese history. As the word “Qianlong” means in Manchu, he was truly blessed by the god: the Qing Empire was already very strong when Qianlong came into power; meanwhile, Qianlong had a relatively stable outside environment. It allows Qianlong to be able to focus on pushing his own governing policy. Qianlong’s policy of agricultural development and tax reduction was very effective, under sixty years of reign, the empire has expanded its territory, and the economy was growing in fast paces as well. Apart from his military and economic achievement, Qianlong has also earned a reputation as one of the greatest connoisseurs in history. Inherited a court collection that was already impressive, Qianlong revaluated, cataloged, and graded the whole collection. For pieces he liked, Qianlong will comment on, or even inscribed poems on those pieces. One of the most famous pieces he inscribed was a Southern Song Ru ware bowl (Figure 1). Among all different categories of court arts, Qianlong was particularly in favor of jade wares. Instead of being just a collector, Qianlong was an accomplished connoisseur of jades. He has written many books on the connoisseurship of jades. Of course, jade ware has become the most popular place where Qianlong inscribed his poems. There are over 800 pieces of jade wares that have Qianlong\u27s poem inscribed1. No matter the wealth, no matter the social status, Jade always stands on an unshakeable position in the Chinese culture. For thousands of years, the Chinese have been particularly obsessed with the material, and emperors are no exception. Being a Manchu emperor, Qianlong knew the importance of understanding the Chinese Han culture. Thanks to the effort of Qianlong, Jade has welcomed its third golden age since the Warrior States and Western Han. Snuff bottles, on another hand, didn’t appear before the Qing Dynasty; however, it took over Qing emperors’ favor like a whirlwind. Since the Kangxi period, Qing emperors have begun to give snuff bottles as gifts to the ministers. As an emperor known for the love of jade, hordes of jade snuff bottles were produced directly from the imperial order. From different carving schools, materials, shapes, and motifs, this thesis will discuss the archeological and aesthetic value of Qianlong jade snuff bottles. Meanwhile, this thesis will use the auction results from major auction houses to analyze the market preference and make predictions for interested readers. Through these handly jade snuff bottles, we can explore the inner world of this great emperor and the rationale behind his obsession with this delicate work of art

Quantifying the Influence of Component Failure Probability on Cascading Blackout Risk

The risk of cascading blackouts greatly relies on failure probabilities of individual components in power grids. To quantify how component failure probabilities (CFP) influences blackout risk (BR), this paper proposes a sample-induced semi-analytic approach to characterize the relationship between CFP and BR. To this end, we first give a generic component failure probability function (CoFPF) to describe CFP with varying parameters or forms. Then the exact relationship between BR and CoFPFs is built on the abstract Markov-sequence model of cascading outages. Leveraging a set of samples generated by blackout simulations, we further establish a sample-induced semi-analytic mapping between the unbiased estimation of BR and CoFPFs. Finally, we derive an efficient algorithm that can directly calculate the unbiased estimation of BR when the CoFPFs change. Since no additional simulations are required, the algorithm is computationally scalable and efficient. Numerical experiments well confirm the theory and the algorithm