BCn interpolation polynomials - 편집 역사

2020년 11월 14일 (토) 05:37에 Pythagoras0님의 편집

2020-11-14T05:37:27Z

2020년 11월 13일 (금) 13:27에 imported>Pythagoras0님의 편집

2020-11-13T13:27:25Z

imported>Pythagoras0: /* introduction */

2017-08-16T02:22:08Z

introduction

2017년 8월 16일 (수) 02:21에 imported>Pythagoras0님의 편집

2017-08-16T02:21:17Z

imported>Pythagoras0: /* related items */

2017-08-02T04:40:58Z

2017년 6월 27일 (화) 02:56에 imported>Pythagoras0님의 편집

2017-06-27T02:56:11Z

imported>Pythagoras0: /* related items */

2017-05-18T08:29:15Z

imported>Pythagoras0: /* binomial coefficient */

2017-05-12T04:13:57Z

binomial coefficient

imported>Pythagoras0: /* properties */

2017-05-08T04:15:04Z

properties

imported>Pythagoras0: /* related items */

2017-04-26T07:39:57Z

@@ 4번째 줄: / 4번째 줄: @@
 ===notation===
-* We define relations $\prec$ and $\succ$ such that $\kappa\prec\lambda$ (equivalently $\lambda\succ\kappa$) for two partitions iff $\lambda/\kappa$ is a vertical strip; that is, $\kappa_i\le \lambda_i\le \kappa_i+1$ for all $i$.
+* We define relations <math>\prec</math> and <math>\succ</math> such that <math>\kappa\prec\lambda</math> (equivalently <math>\lambda\succ\kappa</math>) for two partitions iff <math>\lambda/\kappa</math> is a vertical strip; that is, <math>\kappa_i\le \lambda_i\le \kappa_i+1</math> for all <math>i</math>.
 * we frequently use the product of the form
 \[
 \prod_{(i,j)\in \lambda} f(i,j),
 \]
-where $(i,j)\in \lambda$ means that $1\le i$ and $1\le j\le
+where <math>(i,j)\in \lambda</math> means that <math>1\le i</math> and <math>1\le j\le
-\lambda'_i$
+\lambda'_i</math>
 * let us define [[Generalized q-shifted factorials]]
 \begin{align}
@@ 35번째 줄: / 35번째 줄: @@
 \prod_{1\le i\le l} (t^{1-i} x;q)_{\lambda_i}.
 \end{align}
-* $\bar{P}^{*(n)}_\lambda(\mu;q,t,s):=\bar{P}^{*(n)}_\lambda(q^{\mu_i} t^{n-i} s;q,t,s)=\bar{P}^{*(n)}_\lambda(q^{\mu_1}t^{n-1},q^{\mu_2}t^{n-2},\cdots, q^{\mu_n}t^{0};q,t,s)$
+* <math>\bar{P}^{*(n)}_\lambda(\mu;q,t,s):=\bar{P}^{*(n)}_\lambda(q^{\mu_i} t^{n-i} s;q,t,s)=\bar{P}^{*(n)}_\lambda(q^{\mu_1}t^{n-1},q^{\mu_2}t^{n-2},\cdots, q^{\mu_n}t^{0};q,t,s)</math>
 ==definition==
-* Let $\lambda$ be a partition with at most $n$ parts
+* Let <math>\lambda</math> be a partition with at most <math>n</math> parts
-* The BCn interpolation $\bar{P}^{*(n)}_\lambda(x_1,\dots,x_n;q,t,s)$ is the unique polynomial in $\Lambda_{t,s}$ satisfying the following conditions :
+* The BCn interpolation <math>\bar{P}^{*(n)}_\lambda(x_1,\dots,x_n;q,t,s)</math> is the unique polynomial in <math>\Lambda_{t,s}</math> satisfying the following conditions :
-# $\deg \bar{P}^{*(n)}_\lambda(x;q,t,s)\leq |\lambda|$
+# <math>\deg \bar{P}^{*(n)}_\lambda(x;q,t,s)\leq |\lambda|</math>
-# $\bar{P}^{*(n)}_\lambda(\mu;q,t,s)=0$ if $\quad\lambda\not\subset\mu$
+# <math>\bar{P}^{*(n)}_\lambda(\mu;q,t,s)=0</math> if <math>\quad\lambda\not\subset\mu</math>
-# $\bar{P}^{*(n)}_\lambda(\lambda;q,t,s)=\dots$ (normalization)
+# <math>\bar{P}^{*(n)}_\lambda(\lambda;q,t,s)=\dots</math> (normalization)
@@ 88번째 줄: / 88번째 줄: @@
 ==BCn q-binomial coefficient==
 * see [[BCn q-binomial coefficient]]
-$$
+:<math>
 {\lambda \brack \mu}_{q,t,s}
 :=
 \frac{\bar{P}^{*(n)}_\mu(\lambda;q,t,s t^{1-n})}
 {\bar{P}^{*(n)}_\mu(\mu;q,t,s t^{1-n})}
-$$
+</math>

← 이전 판		2020년 11월 13일 (금) 13:27 판
110번째 줄:		110번째 줄:

	[[분류:symmetric polynomials]]		[[분류:symmetric polynomials]]
		+	[[분류:migrate]]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
 ===notation===
 * We define relations $\prec$ and $\succ$ such that $\kappa\prec\lambda$ (equivalently $\lambda\succ\kappa$) for two partitions iff $\lambda/\kappa$ is a vertical strip; that is, $\kappa_i\le \lambda_i\le \kappa_i+1$ for all $i$.
@@ 33번째 줄: / 36번째 줄: @@
 \end{align}
 * $\bar{P}^{*(n)}_\lambda(\mu;q,t,s):=\bar{P}^{*(n)}_\lambda(q^{\mu_i} t^{n-i} s;q,t,s)=\bar{P}^{*(n)}_\lambda(q^{\mu_1}t^{n-1},q^{\mu_2}t^{n-2},\cdots, q^{\mu_n}t^{0};q,t,s)$
 ==definition==

@@ 32번째 줄: / 32번째 줄: @@
 \prod_{1\le i\le l} (t^{1-i} x;q)_{\lambda_i}.
 \end{align}
-* $\bar{P}^{*(n)}_\lambda(\mu;q,t,s):=\bar{P}^{*(n)}_\lambda(q^{\mu_i} t^{n-i} s;q,t,s)$
+* $\bar{P}^{*(n)}_\lambda(\mu;q,t,s):=\bar{P}^{*(n)}_\lambda(q^{\mu_i} t^{n-i} s;q,t,s)=\bar{P}^{*(n)}_\lambda(q^{\mu_1}t^{n-1},q^{\mu_2}t^{n-2},\cdots, q^{\mu_n}t^{0};q,t,s)$
-===properties===
+==definition==
-* let $\bar{P}^{*(n)}_\lambda(\mu;q,t,s):=\bar{P}^{*(n)}_\lambda(q^{\mu_1}t^{n-1},q^{\mu_2}t^{n-2},\cdots, q^{\mu_n}t^{0};q,t,s)$
+* Let $\lambda$ be a partition with at most $n$ parts
-* extra vanishing
+* The BCn interpolation $\bar{P}^{*(n)}_\lambda(x_1,\dots,x_n;q,t,s)$ is the unique polynomial in $\Lambda_{t,s}$ satisfying the following conditions :
-$$
+# $\deg \bar{P}^{*(n)}_\lambda(x;q,t,s)\leq |\lambda|$
-\bar{P}^{*(n)}_\lambda(\mu;q,t,s)=0
+# $\bar{P}^{*(n)}_\lambda(\mu;q,t,s)=0$ if $\quad\lambda\not\subset\mu$
-\quad\lambda\not\subset\mu,
+# $\bar{P}^{*(n)}_\lambda(\lambda;q,t,s)=\dots$ (normalization)
 $$
 ==branching rule==

@@ 99번째 줄: / 99번째 줄: @@
 * [[Lifted BCn interpolation polynomials]]
 * [[Koornwinder polynomials]]
 ==articles==

@@ 41번째 줄: / 41번째 줄: @@
 \bar{P}^{*(n)}_\lambda(\mu;q,t,s)=0
 \quad\lambda\not\subset\mu,
 $$
@@ 96번째 줄: / 83번째 줄: @@
 (v+1/v-q^{j-1} t^{1-i}s-q^{1-j}t^{i-1}/s)
 \]

@@ 44번째 줄: / 44번째 줄: @@
 ==binomial coefficient==
-* $BC_n$ $q$-binomial coefficient
+* [[BCn q-binomial coefficient]]
 $$
 {\lambda \brack \mu}_{q,t,s}

@@ 102번째 줄: / 102번째 줄: @@
 * [[Branching rules for Macdonald polynomials]]
 * [[Interpolation Macdonald polynomials]]
 ==articles==