Volume of a compact Lie group - 편집 역사

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imported>Pythagoras0: /* expositions */

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expositions

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2015-11-05T10:51:14Z

articles

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expositions

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imported>Pythagoras0: 새 문서: ==expositions== * Zhang, Lin. “Volumes of Orthogonal Groups and Unitary Groups.” arXiv:1509.00537 [math-Ph, Physics:quant-Ph], September 1, 2015. http://arxiv.org/abs/1509.00537. ...

2015-09-03T01:19:46Z

새 문서: ==expositions== * Zhang, Lin. “Volumes of Orthogonal Groups and Unitary Groups.” arXiv:1509.00537 [math-Ph, Physics:quant-Ph], September 1, 2015. http://arxiv.org/abs/1509.00537. ...

새 문서

==expositions==
* Zhang, Lin. “Volumes of Orthogonal Groups and Unitary Groups.” arXiv:1509.00537 [math-Ph, Physics:quant-Ph], September 1, 2015. http://arxiv.org/abs/1509.00537.

==articles==
* Macdonald, I. G. “The Volume of a Compact Lie Group.” Inventiones Mathematicae 56, no. 2 (February 1980): 93–95. doi:10.1007/BF01392542.

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 ==introduction==
-* Define $J_\lambda(x)=  h(\lambda)^{-1}\det(e^{\lambda_i x_j})$, where $ h(\lambda)=\prod_{i<j}(\lambda_i-\lambda_j)$.
+* Define <math>J_\lambda(x)=  h(\lambda)^{-1}\det(e^{\lambda_i x_j})</math>, where <math> h(\lambda)=\prod_{i<j}(\lambda_i-\lambda_j)</math>.
-* For each $x$, $J_\lambda(x)$ is an analytic function of $\lambda$; in particular, $J_0(x)=\left(\prod_{j=1}^{n-1} j!\right)  h(x)$.
+* For each <math>x</math>, <math>J_\lambda(x)</math> is an analytic function of <math>\lambda</math>; in particular, <math>J_0(x)=\left(\prod_{j=1}^{n-1} j!\right)  h(x)</math>.
-* The functions $J_\lambda(x)$ play a central role in random matrix theory.
+* The functions <math>J_\lambda(x)</math> play a central role in random matrix theory.
-* For example, if $\Lambda$ and $X$ are Hermitian matrices with eigenvalues given by $\lambda$ and $x$, respectively, then
+* For example, if <math>\Lambda</math> and <math>X</math> are Hermitian matrices with eigenvalues given by <math>\lambda</math> and <math>x</math>, respectively, then
 \begin{equation}\label{iz}
 \int_{U(n)}e^{\rm tr \Lambda U X U^*} dU=\frac{J_\lambda(x)}{J_0(x)},

← 이전 판		2020년 11월 13일 (금) 10:31 판
29번째 줄:		29번째 줄:

	[[분류:Lie theory]]		[[분류:Lie theory]]
		+	[[분류:migrate]]

← 이전 판		2016년 1월 1일 (금) 12:43 판
1번째 줄:		1번째 줄:
		+	==introduction==
		+	* Define $J_\lambda(x)= h(\lambda)^{-1}\det(e^{\lambda_i x_j})$, where $ h(\lambda)=\prod_{i<j}(\lambda_i-\lambda_j)$.
		+	* For each $x$, $J_\lambda(x)$ is an analytic function of $\lambda$; in particular, $J_0(x)=\left(\prod_{j=1}^{n-1} j!\right) h(x)$.
		+	* The functions $J_\lambda(x)$ play a central role in random matrix theory.
		+	* For example, if $\Lambda$ and $X$ are Hermitian matrices with eigenvalues given by $\lambda$ and $x$, respectively, then
		+	\begin{equation}\label{iz}
		+	\int_{U(n)}e^{\rm tr \Lambda U X U^*} dU=\frac{J_\lambda(x)}{J_0(x)},
		+	\end{equation}
		+	where the integral is with respect to normalised Haar measure on the unitary group.
		+	* This is known as the Harish-Chandra, or Itzykson-Zuber, formula.
		+
		+
	==related items==		==related items==
	* [[Random matrix]]		* [[Random matrix]]

@@ 1번째 줄: / 1번째 줄: @@
 ==expositions==
 * Diaconis, Persi, and Peter J. Forrester. “A. Hurwitz and the Origins of Random Matrix Theory in Mathematics.” arXiv:1512.09229 [math-Ph], December 31, 2015. http://arxiv.org/abs/1512.09229.
@@ 11번째 줄: / 15번째 줄: @@
 * Macdonald, I. G. “The Volume of a Compact Lie Group.” Inventiones Mathematicae 56, no. 2 (February 1980): 93–95. doi:10.1007/BF01392542.
 * Itzykson, C., and J. B. Zuber. “The Planar Approximation. II.” Journal of Mathematical Physics 21, no. 3 (1980): 411–21. doi:10.1063/1.524438.

@@ 6번째 줄: / 6번째 줄: @@
 ==articles==
 * Hashimoto, Y. “On Macdonald’s Formula for the Volume of a Compact Lie Group.” Commentarii Mathematici Helvetici 72, no. 4 (April 3, 2014): 660–62. doi:10.1007/s000140050040.
 * Macdonald, I. G. “The Volume of a Compact Lie Group.” Inventiones Mathematicae 56, no. 2 (February 1980): 93–95. doi:10.1007/BF01392542.
 * Itzykson, C., and J. B. Zuber. “The Planar Approximation. II.” Journal of Mathematical Physics 21, no. 3 (1980): 411–21. doi:10.1063/1.524438.

@@ 2번째 줄: / 2번째 줄: @@
 * https://terrytao.wordpress.com/2013/02/08/the-harish-chandra-itzykson-zuber-integral-formula/
 * Zhang, Lin. “Volumes of Orthogonal Groups and Unitary Groups.” arXiv:1509.00537 [math-Ph, Physics:quant-Ph], September 1, 2015. http://arxiv.org/abs/1509.00537.
 * Boya, Luis J., E. C. G. Sudarshan, and Todd Tilma. “Volumes of Compact Manifolds.” Reports on Mathematical Physics 52, no. 3 (December 2003): 401–22. doi:10.1016/S0034-4877(03)80038-1. http://arxiv.org/abs/math-ph/0210033v3
 ==articles==