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剑桥雅思英语考试真题1-11PDF下载[转载+更新]

新增剑桥雅思英语考试真题11PDF及听力音频下载—见文尾

雅思英语考试(IELTS, International English Language Testing System)是很多人已经参加或者将要参加的一门英语考试,尤其是留学申请欧美大学时,几乎是一份必须的英语成绩证明。英国连移民、工作签证都需要雅思成绩。

而准备雅思考试的技巧其实不外乎研究雅思考试的真题。雅思考试的出题方:剑桥大学,每隔一段时间就会放出一本剑桥雅思全真试题参考书(Cambridge IELTS Student’s Book with Answers Official Examination Papers from University of Cambridge ESOL Examinations),其试题基本上是过去雅思考试中出现过的真题,不过真题中出现过的题目在以后的雅思考试是不会出现的。但是剑桥雅思真题系列参考书的重要性是绝对不能忽视的。

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Gelfand-Tsetlin basis for the representations of $\gl_n$

1. Finite dimensional representations of $\gl_n$

Let $e_{ij}$, $i,j=1,\dots,n$ denote the standard basis of the general linear Lie algebra $\gl_n$ over the field of complex numbers. The subalgebra $\gl_{n-1}$ is spanned by the basis elements $e_{ij}$ with $i,j=1,\dots,n-1$. Denote by $\h=\h_n$ the diagonal Cartan subalgebra in $\gl_n$. The elements $e_{11},\dots,e_{nn}$ form a basis of $\h$.

Finite-dimensional irreducible representations of $\gl_n$ are in a one-to-one correspondence with $n$-tuples of complex numbers $\la=(\la_1,\dots,\la_n)$ such that
\[\la_i-\la_{i+1}\in\Z^+\quad\text{for }i=1,\dots,n-1. \]Such an $n$-tuple $\la$ is called the highest weight of the corresponding representation which we shall denote by $L(\la)$. It contains unique, up to a multiple, nonzero vector $v^+$ (the highest vector) such that $e_{ii}v^+=\la_iv^+$ for $i=1,\dots,n$ and $e_{ij}v^+=0$ for $1\lle i < j \lle n$.

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李代数及其表示论书籍推荐

本文推荐主要针对李代数, 偶尔李群部分. 主要是李代数的入门比较容易, 有扎实的线性代数基础和少量的抽象代数知识就可以入门. 而至于和李群相结合的部分需要微分流形相关知识, 因此对一般的本科学生来说有一定难度. 推荐指数带个人感情色彩, 切勿太过认真.

基础理论篇

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Properties of Wronskians

This is a note for the article Critical points of master functions and flag varieties. Most of the properties below are from the appendix of this article. Some are not written in the paper but used there. I did not write the well-known relations between differential equations and Wronskians, such as Abel’s identity.

Here we assume all functions in this page are functions of $x$ with sufficiently many derivatives.

Define the Wronskian of functions $g_1,\dots,g_n$ by $W(g_1,\dots,g_n)=\det(g_i^{(j-1)})_{i,j=1}^n$. We follow the convention that for $n=0$ the corresponding Wronskian is equal to $1$. We also write $W_n(g_1,\dots,g_n)$ for $W(g_1,\dots,g_n)$ in order to stress the order of the Wronskian.

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OS X 安装 SAS UNIVERSITY EDITION 图文教程

SASStatistical Analysis System,统计分析系统,是美国 SAS 软件研究所研制的一套大型集成应用软件系统,具有完备的数据存取、数据管理、数据分析和数据展现功能。尤其是统计分析系统部分,具由于有强大的数据分析能力,一直为业界著名软件,在数据处理和统计分析领域,被誉为国际上的标准软件和最权威的优秀统计软件包,广泛应用于政府行政管 理、科研、教育、生产和金融等不同领域,发挥着重要的作用。SAS 系统中提供的主要分析功能包括统计分析、经济计量分析、时间序列分析、决策分析、财务分 析和全面质量管理工具等。

SAS 是收费软件,版权费很贵,但是对于学习来说,可以免费下载、安装和使用 SAS University Edition。其过程如下:

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