Real double flag varieties for the symplectic group

In this paper we study a key example of a Hermitian symmetric space and a natural associated double flag variety, namely for the real symplectic group G and the symmetric subgroup L, the Levi part of the Siegel parabolic $P_S$. We give a detailed treatment of the case of the maximal parabolic subgroups Q of L corresponding to Grassmannians and the product variety of $G/P_S$ and $L/Q$; in particular we classify the L-orbits here, and find natural explicit integral transforms between degenerate principal series of L and G.     

Robust nonlinear optimization with conic representable uncertainty set

The robust optimization methodology is known as a popular method dealing with optimization problems with uncertain data and hard constraints. This methodology has been applied so far to various convex conic optimization problems where only their inequality constraints are subject to uncertainty. In this paper, the robust optimization methodology is applied to the general nonlinear programming (NLP) problem involving both uncertain inequality and equality constraints. The uncertainty set is defined by conic representable sets, the proposed uncertainty set is general enough to include many uncertainty sets, which have been used in literature, as special cases. The robust counterpart (RC) of the general NLP problem is approximated under this uncertainty set. It is shown that the resulting approximate RC of the general NLP problem is valid in a small neighborhood of the nominal value. Furthermore a rather general class of programming problems is posed that the robust counterparts of its problems can be derived exactly under the proposed uncertainty set. Our results show the applicability of robust optimization to a wider area of real applications and theoretical problems with more general uncertainty sets than those considered so far. The resulting robust counterparts which are traditional optimization problems make it possible to use existing algorithms of mathematical optimization to solve more complicated and general robust optimization problems.     

The role of problem representation and feature knowledge in algebraic equation-solving

Domain experts have two major advantages over novices with regard to problem solving: experts more accurately encode deep problem features (feature encoding) and demonstrate better conceptual understanding of critical problem features (feature knowledge). In the current study, we explore the relative contributions of encoding and knowledge of problem features (e.g., negative signs, the equals sign, variables) when beginning algebra students solve simple algebraic equations. Thirty-two students completed problems designed to measure feature encoding, feature knowledge and equation solving. Results indicate that though both feature encoding and feature knowledge were correlated with equation-solving success, only feature knowledge independently predicted success. These results have implications for the design of instruction in algebra, and suggest that helping students to develop feature knowledge within a meaningful conceptual context may improve both encoding and problem-solving performance.     

Symmetric Powers of Nat 𝔰𝔩2(𝕂)

We identify the spaces of homogeneous polynomials in two variables 𝕂[Y^k, XY^k?1, ?, X^k] among representations of the Lie ring 𝔰𝔩₂(𝕂). This amounts to constructing a compatible 𝕂-linear structure on some abstract 𝔰𝔩₂(𝕂)-modules, where 𝔰𝔩₂(𝕂) is viewed as a Lie ring.     

Classroom note: The area under x α by geometry

By stretching the area under the curve x ^α it is shown to be of the form x ^{α + 1} p(α). Geometry is then used to prove p(α) = 1/(α + 1).     

等束宽超短脉冲光束的远场特性

利用角谱法和复解析信号法推导出了等束宽超短高斯脉冲光束在远场的解析传输公式及其傅里叶谱.研究了若干个甚至一个光周期数的等束宽超短高斯脉冲光束的远场特性，并与等衍射长度超短高斯脉冲光束作了比较.研究表明，等束宽超短高斯脉冲光束在远场存在脉冲展宽、光谱变窄、离轴光谱红移、轴上光谱蓝移等特性. 关键词： 角谱法 复解析信号法 等束宽超短高斯脉冲光束     

一类位置不变的Pickands型估计量

当极值指标大于0时, 本文提出了一种位置不变的Pickands型估计量,证明了该估计量的强弱相合性, 给出了其渐近展式和强收敛速度,并对$k_2$的最优选择进行了讨论,最后利用自适应性方法对该估计量和其它Pickands型估计量进行随机模拟分析,比较该估计量的优越性.     

A construction of weakly inverse semigroups

Let S° be an inverse semigroup with semilattice biordered set E° of idempotents and E a weakly inverse biordered set with a subsemilattice Ep = { e ∈ E ｜ arbieary f ∈ E, S（f , e） loheain in w（e）} isomorphic to E° by θ：Ep→E°. In this paper, it is proved that if arbieary f, g ∈E, f ←→ g→→ f°θD^s° g°θand there exists a mapping φ from Ep into the symmetric weakly inverse semigroup P J（E∪ S°） satisfying six appropriate conditions, then a weakly inverse semigroup ∑ can be constructed in P J（S°）, called the weakly inverse hull of a weakly inverse system （S°, E, θ, φ） with I（∑） ≌ S°, E（∑） ∽- E. Conversely, every weakly inverse semigroup can be constructed in this way. Furthermore, a sufficient and necessary condition for two weakly inverse hulls to be isomorphic is also given.     

基于稀疏表示和特征加权的离格双耳声源定位*

基于头相关传递函数数据库的传统双耳声源定位方法的定位角度往往被限定在头相关传递函数数据库的离散测量点上。当头相关传递函数数据库的测量方位角间隔较大时,这类算法的性能会显著下降,这就是典型的离格问题。该文提出了基于加权宽带稀疏贝叶斯学习的离格双耳声源定位算法。首先该算法建立离格双耳信号的稀疏表示模型,然后利用双耳相干与扩散能量比特征对各个频点进行加权以降低噪声和混响的影响,最后通过加权宽带稀疏贝叶斯学习方法估计离格声源的方位角。实验结果表明,该算法在各种复杂的声学环境下都有着较高的定位精度和鲁棒性,特别是提高了离格条件下的声源定位性能。     

Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle H = （Hu）u∈G^0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle （G^0, （Hu）,μ）, then dimHu = 1 （u ∈ G^0）. Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.