User:Ulf Rehmann/Table of automatically generated TeX code

This page gives an analysis of the code here, generated automatically from some png files underlying our old wiki pages. As this page does contain a lot of $\TeX$ code, it loads slowly.

Under the name of some of our EoM-pages the table below lists some png files, displaying their image and their $TeX$ rendering (automatically retrieved and corrected by hand). The first column gives the running number in this table, followed (in parentheses) by the number used here. The last column gives the confidence and the name of the png file, followed (in parentheses) by the number it has in the sequence of all png files called by its calling EoM-page.

Here is a short survey of the more systematic errors which seem to occur:

1. Trailing punctuation is dismissed.

[concerns almost all images] ; technically: pixels in sparse last pixel columns of bit images are suppressed/ignored?

2. "Displayed" images are not recognized as such.

[concerns almost all images]

Therefore these are displayed too small, and like "inline" $\TeX$ format.

Remark: This cannot be discovered from the png file, it has to be retrieved from the html markup in the calling file: Displayed images are embedded in some html <table> markup.

3. Sparse initial column pixels of the bit image are dismissed

(in parts this affects essential symbols), [see nr. 15,16,36,43,58,59,60,61,62,63,97,109]

4. Some fonts are not recognized

\cal: [7.12.25.26,30,31,32,33,95,111] \mathbf: [30,83,111,127] \bf:[ 133,148,149]

5. Semi-colon is interpreted as double pipe = "||"

[33,49,86,101]

6. Some code is not displayed at all.

(This seems to be a bug of our MathJax TeX interpreter.) [67,74,78,81,83,94,101,106]

This seems to happen when a string "\text {" is involved, can apparently be fixed by using "\text{", but still unclear.

7. Questions

The different interpretation of the matrix delimiters in [56-63] is a bit surprising. Should be checked!

Also, the vanishing of some '-' signs in the first column of some matrices, maybe that is related to 3.?

Algebraic curve

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
1.(23.)		$g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$	$$g\leq \left\{ \begin {array}{ll} {\frac {(n-2)^2}4} &{\text{ for even }n,}\\ {\frac {(n-1)(n-3)}4} &{\text{ for odd }n,} \end {array} \right.$$	conf 0.698 a01145065.png (65)

Algebraic geometry

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
2.(116.)		$\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$	$$\theta =\int\limits _ 0^{\lambda }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$	conf 0.997 a01150014.png (14)
3.(133.)		$\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$	$$\omega =2\int\limits _ 0^{1/c}\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$	conf 0.973 a01150021.png (21)
4.(67.)		$\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$	$$\widetilde w=2\int\limits _ 0^{1/\varepsilon }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$	conf 0.107 a01150022.png (22)
5.(105.)		$\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$	$$\theta (v+\pi i r )=\theta (r),\quad \theta (v+\alpha _ j)=e^{L_j(v)}\theta (v),$$	conf 0.775 a01150044.png (44)
6.(17.)		$\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$	$$\left( \begin {array}{ll} {\alpha } &b\\ c &d \end {array} \right)\equiv \left( \begin {array}{ll} 1&0\\ 0&1 \end {array} \right)(\operatorname {mod}7).$$	conf 0.440 a01150078.png (78)

Algebraic surface

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
7.(144.)		$0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$	$$0\rightarrow {\cal O}_V\rightarrow E _ {\alpha }\rightarrow T _ V\rightarrow 0$$	conf 0.981 a011640132.png (132)
8.(73.)		$M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$	$$M=\operatorname {dim}\operatorname {Im}(H^1(V,E_{\alpha })\rightarrow H ^1(V,T_V)).$$	conf 0.997 a011640137.png (137)
9.(88.)		$\operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } )$	$$\operatorname {dim}_kH^2(V,E_{\alpha })+\operatorname {dim}_kH^2(V,T_V).$$	conf 0.996 a011640139.png (139)
10.(117.)		$N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$	$$N_m=\left(\begin {array}c{m+3}\\ 3 \end {array} \right)-dm+2t+\tau +p-1.$$	conf 0.369 a01164027.png (27)
11.(72.)		$p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$	$$p_{\alpha }(V)=\left(\begin {array}c{n-1}\\ 3 \end {array} \right)-d(n-1)+2t+\tau +p-1$$	conf 0.396 a01164029.png (29)
12.(68.)*		$p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$	$$p_{\alpha }(V)=-\operatorname {dim}_kH_1(V,{\cal O}_V)+\operatorname {dim}_kH^2(V,{\cal O}_V)=$$	conf 0.756 F a01164047.png (47)
13.(93.)*		$1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$	$$1+p_{\alpha }(V)=\frac {\operatorname {deg}(c_1^2)+\operatorname {deg}(c_2)}{12},$$	conf 0.752 F a01164053.png (53)

Cartan subalgebra

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
14.(33.)*		$\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$	$$\mathfrak g_0=\big\{X\in \mathfrak g:\forall H \in \mathfrak t\exists n_{X,H}\in {\mathbb Z}((\text{ ad }H)^{n_{X,H}}(X)=0)\big\},$$	conf 0.110 F c0205509.png (9)

Cartan theorem

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
15.(49.)*		$f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$	$$[e_i,f_j]=\delta _ {ij}h_i,\quad [h_i,e_j]=\alpha _ {ij}e_j,\quad [h_i,f_j]=-\alpha _ {ij}f_j,$$	conf 0.149 F c0205704.png (4)
16.(55.)*		$\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$	$$\dots \rightarrow H ^p(X,S)\rightarrow H ^p(X,F)\stackrel {\phi_p }{\rightarrow }H^p(X,G)\rightarrow $$	conf 0.853 F c02057064.png (64)

Comitant

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
17.(7.)		$H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$	$$H=\frac 1{36}\left| \begin {array}{cc} {\frac {\partial ^2f}{\partial x ^2}} &{\frac {\partial ^2f}{\partial x \partial y }}\\ {\frac {\partial ^2f}{\partial x \partial y }} &{\frac {\partial ^2f}{\partial y ^2}} \end {array} \right|=$$	conf 0.956 c02333033.png (33)
18.(76.)		$= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$	$$=(a_0a_2-a_1^2)x^2+(a_0a_3-a_1a_2)xy+(a_1a_3-a_2^2)y^2$$	conf 0.549 c02333034.png (34)
19.(11.)*		$( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$	$$(\alpha _ 0,\alpha _ 1,\alpha _ 2,\alpha _ 3)\mapsto (\alpha _ 0\alpha _ 2-\alpha _ 1^2,\frac 12(\alpha _ 0\alpha _ 3-\alpha _ 1\alpha _ 2),\alpha _ 1\alpha _ 3-\alpha _ 2^2)$$	conf 0.521 F c02333035.png (35)

Deformation

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
20.(26.)		$\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$	$$\operatorname {Aut}_{R^{\prime }}(X^{\prime }|X_0)\rightarrow \operatorname {Aut}_R(X_{R^{\prime }}^{\prime }\otimes R |X_0)$$	conf 0.683 \ d030700175.png (175)
21.(27.)		$\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$	$$\operatorname {dim}_kH^1(X_0,T_{X_0})-\operatorname {dim}M_{X_0}\leq \operatorname {dim}_kH^2(X_0,T_{X_0}).$$	conf 0.944 d030700190.png (190)
22.(78.)*		$\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$	$$\alpha \circ b =\alpha b +\sum _ {i=1}^{\infty }\phi _ i(\alpha ,b)t^i,\quad \alpha ,b\in V,$$	conf 0.097 F d030700263.png (263)
23.(96.)*		$\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$	$$\Phi (\alpha )=\alpha +\sum _ {i=1}^{\infty }t^i\phi _ i(\alpha ),\quad \alpha \in V,$$	conf 0.873 F d030700270.png (270)

Differential algebra

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
24.(106.)		$S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$	$$S^tF=\sum _ {j=1}^rc_jA^{p_j}A_1^{i_{1j}}\dots A _ {m-l}^{i_{{m-l},j}},$$	conf 0.149 d031830107.png (107)
25.(146.)*		$( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$	$(\eta _ 1,\ldots ,\eta _ k)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ k)$	conf 0.562 F d031830141.png (141)
26.(145.)$^F$*		$( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$	$(\eta _ 1,\ldots ,\eta _ n)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ n)$	conf 0.376 F d031830150.png (150)
27.(57.)		$\omega _ { V } = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$	$$\omega _ V=\sum _ {0\leq i \leq m }\alpha _ i\left( \begin {array}c{x+i}\\ i \end {array} \right),$$	conf 0.780 d03183016.png (16)
28.(111.)		$e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$	$$e_{ij}=\operatorname {ord}_{Y_j}F_i,\quad 1 \leq i \leq n ,\quad i \leq j \leq n,$$	conf 0.187 d03183043.png (43)

Dimension polynomial

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
29.(48.)		$\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$	$$\omega _ {\eta /F}(x)=\sum _ {0\leq i \leq m }\alpha _ i\left(\begin {array}c{x+i}\\ i \end {array} \right),$$	conf 0.968 d03249029.png (29)

Duality

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
30.(118.)*		$H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$	$$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow {\mathbf C},$$	conf 0.824 F d034120173.png (173)
31.(59.)*		$H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow H _ { c } ^ { n } ( X , \Omega )$	$$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow H _ c^n(X,\Omega )$$	conf 0.921 F d034120175.png (175)
32.(124.)*		$( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$	$$(H^p(X,{\cal F}))^{\prime }\cong H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega )).$$	conf 0.829 F d034120184.png (184)
33.(29.)*		$\beta : \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X F , \Omega ) \rightarrow \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X \backslash Y || F , \Omega )$	$$\beta :\operatorname {Ext}_c^{n-p-1}(X;{\cal F},\Omega )\rightarrow \operatorname {Ext}_c^{n-p-1}(X\backslash Y ;{\cal F},\Omega ).$$	conf 0.634	F d034120236.png (236)
34.(77.)*		$\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$	$$\underset {n\rightarrow \infty }{\overline {\lim }}|\alpha _ n|^{1/n}=\sigma <+\infty.$$	conf 0.521 F d034120247.png (247)
35.(58.)*		$h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$	$$h(\phi )=\underset {n\rightarrow \infty }{\overline {\lim }}\frac {\operatorname {ln}|A(re^{i\phi })|}r$$	conf 0.861 F d034120253.png (253)
36.(69.)*		$\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$	$$\operatorname*{sup}_{l\in E^\perp \atop \|l\|\le 1 }|l(\omega )|=\operatorname*{inf}_{x\in E }\|\omega -x\|,$$	conf 0.293 F d034120360.png (360)
37.(15.)		$\operatorname { sup } _ { f \in B ^ { 1 } } | \int _ { \partial G } f ( \zeta ) \omega ( \zeta ) d \zeta | = \operatorname { inf } _ { \phi \in E ^ { 1 } } \int _ { \partial G } | \omega ( \zeta ) - \phi ( \zeta ) \| d \zeta |$	$$\operatorname*{sup}_{f\in B ^1}\big|\int\limits _ {\partial G }f(\zeta )\omega (\zeta )d\zeta \big|=\operatorname*{inf}_{\phi \in E ^1}\int\limits _ {\partial G }|\omega (\zeta )-\phi (\zeta ) ||d\zeta |.$$	conf 0.508 d034120376.png (376)
38.(52.)		$f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$	$$f=\{f_{\alpha }\}\in \prod _ {\alpha }F_{\alpha },\quad g =\{g_{\alpha }\}\in \operatorname*\oplus _ {\alpha }G_{\alpha }.$$	conf 0.491 d034120509.png (509)
39.(140.)		$f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$	$$f^{*}(x^{*})=\operatorname*{sup}_{x\in X }(\langle x ^{*},x\rangle -f(x))$$	conf 0.900 d034120535.png (535)
40.(94.)		$f _ { 0 } ( x ) \rightarrow \text { inf, } \quad f _ { i } ( x ) \leq 0 , \quad i = 1 , \ldots , m , \quad x \in B$	$$f_0(x)\rightarrow \text{ inf, }\quad f _ i(x)\leq 0 ,\quad i =1,\ldots ,m,\quad x \in B,$$	conf 0.810 d034120555.png (555)
41.(74.)*		$( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$	$$(c_{\gamma },c^r)=\sum _ {t^r\in K }c_r(t^{\prime })c^r(t^r)\operatorname {mod}1$$	conf 0.117 F d03412079.png (79)

Extension of a differential field

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
42.(63.)		$F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$	$$F_1F_2=F_1\langle F _ 2\rangle =F_1(F_2)=F_2(F_1)=F_2\langle F _ 1\rangle,$$	conf 0.628 e03696024.png (24)

Formal group

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
43.(120.)*		$\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$	$$\operatorname {log}F_{\rm MU }(X)=\sum _ {i=1}^{\infty }i^{-1}[{\rm CP}^{i-1}]X^i,$$	conf 0.098 F f040820118.png (118)
44.(147.)*		$( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$	$$(x_1,\ldots ,x_n)\circ (y_1,\ldots ,y_n)=(z_1,\ldots ,z_n),$$	conf 0.553 F f04082059.png (59)

Gel'fond-Schneider method

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
45.(148.)		$\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$	$\alpha ^{\beta }=\operatorname {exp}\{\beta \operatorname {log}\alpha \}$	conf 0.979 g1300205.png (5)

Group

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
46.(22.)*		$\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$	source incomplete	conf 0.226 F g04521075.png (75)

Homogeneous space

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
47.(89.)		$\mathfrak { g } = \mathfrak { f } + \mathfrak { m } , \quad \mathfrak { f } \cap \mathfrak { m } = \{ 0 \}$	$$\mathfrak g=\mathfrak f+\mathfrak m,\quad \mathfrak f\cap \mathfrak m=\{0\},$$	conf 0.793 h04769069.png (69)

Hopf algebra

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
48.(103.)		$m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$	$m\circ (\iota \otimes 1 )\circ \mu =m\circ (1\otimes \iota )\circ \mu =e\circ \epsilon$	conf 0.618 h047970129.png (129)
49.(107.)*		$F _ { 1 } ( X || Y ) , \ldots , F _ { n } ( X || Y ) \in K [ X _ { 1 } , \ldots , X _ { n } || Y _ { 1 } , \ldots , Y _ { n } ] \}$	$F_1(X;Y),\ldots ,F_n(X;Y)\in K [X_1,\ldots ,X_n;Y_1,\ldots ,Y_n]\}$	conf 0.353 F h047970139.png (139)
50.(97.)		$\epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g }$	$$\epsilon (x)=0,\quad \delta (x)=x\otimes 1 +1\otimes x ,\quad x \in \mathfrak g.$$	conf 0.213 h04797042.png (42)

Invariants, theory of

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
51.(149.)*		$\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$	$$\alpha _ {i_1,\dots,i_n}\rightarrow \alpha _ {i_1,\dots,i_n}^{\prime }.$$	conf 0.142 F i05235015.png (15)

Jordan algebra

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
52.(150.)		$H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$	$$(C_3,\Gamma )=\big\{X\in C _ 3:X=\Gamma ^{-1}X\square ^{\prime }\Gamma \big\},$$	conf 0.651 j05427030.png (30)
53.(42.)		$\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$	$$\Gamma =\operatorname {diag}\{\gamma _ 1,\gamma _ 2,\gamma _ 3\},\quad \gamma _ i\neq 0 ,\quad \gamma _ i\in F,$$	conf 0.987 j05427031.png (31)
54.(125.)*		$\mathfrak { g } = \mathfrak { g } - 1 + \mathfrak { g } \mathfrak { d } + \mathfrak { g } _ { 1 }$	$\mathfrak g=\mathfrak g_{-1}+\mathfrak g_0+\mathfrak g_1$	conf 0.598 F j05427077.png (77)

Jordan matrix

Nr.	Image of png File	$\TeX$, automatically generated version	$\TeX$, manually corrected version	Confidence, F? png file
55.(6.)*		$J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$	$$J=\left\| \begin {array}{cccc} J_{n_1}(\lambda_1) &0 &0 &0\\ 0 &\ddots &\ddots &0\\ 0 &\ddots &\ddots &0\\ 0 &0 &0 &J_{n_s}(\lambda_s) \end {array} \right\|,$$	conf 0.072 F j0543403.png (3)
56.(64.)		$C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$	$$C_m(\lambda )=\operatorname {rk}(A-\lambda E )^{m-1}-2\operatorname {rk}(A-\lambda E )^m+$$	conf 0.955 j05434030.png (30)
57.(1.)*		$J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$	$$J_m(\lambda)=\left\| \begin {array}{cccccc} \lambda &1 &\square &\square &\square &\square \\ \square &\lambda &1 &\square &0 &\square \\ \square &\square &\ddots &\ddots &\square &\square\\ \square &\square &\square &\ddots &\ddots &\square \\ \square &0 &\square &\square &\lambda &1\\ \square &\square &\square &\square &\square &\lambda \end {array} \right\|,$$	conf 0.098 F j0543406.png (6)

Lie algebra, semi-simple

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58.(5.)		$\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$	$$B_n:\quad \left\| \begin {array}{rrrrrr} 2 &{-1} &0 &{\dots } &0 &0\\ {-1} &2 &{-1} &{\dots } &0 &0\\ 0 &{-1} &2 &{\dots } &0 &0\\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot \\ 0 &0 &0 &{\dots } &{-1} &0\\ 0 &0 &0 &{\dots } &2 &{-2}\\ 0 &0 &0 &{\dots } &{-1} &2 \end {array} \right\|,$$	conf 0.232 l058510127.png (127)
59.(3.)*		$\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$	$$D_n:\quad \left\| \begin {array}{rrrrrrr} 2 &{-1} &0 &{\dots } &0 &0 &0 &0\\ {-1} &2 &{-1} &{\dots } &0 &0 &0 &0\\ 0 &{-1} &2 &{\dots } &0 &0 &0 &0\\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot &\cdot &\cdot \\ 0 &0 &0 &{\dots } &2 &{-1} &0 &0\\ 0 &0 &0 &{\dots } &{-1} &2 &{-1} &{-1}\\ 0 &0 &0 &{\dots } &0 &{-1} &2 &0\\ 0 &0 &0 &{\dots } &0 &{-1} &0 &2 \end {array} \right\|,$$	conf 0.055 F l058510129.png (129)
60.(8.)*		$\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$	$$E_6: \quad \left\| \begin {array}{rrrrrr} 2 &0 &{-1} &0 &0 &0\\ 0 &2 &0 &{-1} &0 &0\\ {-1} &0 &2 &{-1} &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0\\ 0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$	conf 0.628 F l058510130.png (130)
61.(4.)		$\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$	$$E_7:\quad \left\| \begin {array}{rrrrrrr} 2 &0 &{-1} &0 &0 &0 &0\\ 0 &2 &0 &{-1} &0 &0 &0\\ {-1} &0 &2 &{-1} &0 &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0 &0\\ 0 &0 &0 &{-1} &2 &{-1} &0\\ 0 &0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$	conf 0.278 l058510131.png (131)
62.(2.)*		$\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$	$$E_8:\quad \left\| \begin {array}{rrrrrrrr} 2 &0 &{-1} &0 &0 &0 &0 & 0\\ 0 &2 &0 &{-1} &0 &0 &0 &0\\ {-1} &0 &2 &{-1} &0 &0 &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0 &0 &0\\ 0 &0 &0 &{-1} &2 &{-1} &0 &0\\ 0 &0 &0 &0 &{-1} &2 &{-1} &0\\ 0 &0 &0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$	conf 0.354 F l058510132.png (132)
63.(10.)*		$\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$	$$F_4:\quad \left\| \begin {array}{rrrr} 2 &{-1} &0 &0\\ {-1} &2 &{-2} &0\\ 0 &{-1} &2 &{-1}\\ 0 &0 &{-1} &2 \end {array} \right\|,\quad G _ 2:\quad \left\| \begin {array}{rr} 2&{-1}\\ {-3}&2 \end {array} \right\|.$$	conf 0.374 F l058510133.png (133)
64.(98.)		$\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$	$$\mathfrak g_{\alpha }=\{X\in \mathfrak g:[H,X]=\alpha (H)X,H\in \mathfrak h\}.$$	conf 0.976 l05851030.png (30)
65.(126.)		$\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$	$$\mathfrak g=\mathfrak h+\sum _ {\alpha \in \Sigma }\mathfrak g_{\alpha }.$$	conf 0.945 l05851037.png (37)
66.(61.)*		$\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$	$$\mathfrak g_{\alpha }=\operatorname {dim}[\mathfrak g_{\alpha },\mathfrak g_{-\alpha }]=1.$$	conf 0.520 F l05851044.png (44)
67.(65.)*		$[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$	$$[H_{\alpha },X_{\alpha }]=2X_{\alpha }\quad {\rm and }\quad [H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }.$$	conf 0.539 F l05851050.png (50)
68.(70.)		$\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$	$$\beta (H_{\alpha })=\frac {2(\alpha ,\beta )}{(\alpha ,\alpha )},\quad \alpha ,\beta \in \Sigma,$$	conf 0.997 l05851051.png (51)
69.(112.)		$[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$	$$[\mathfrak g_{\alpha },\mathfrak g_{\beta }]=\mathfrak g_{\alpha +\beta }$$	conf 0.917 l05851057.png (57)
70.(127.)		$H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$	$$H_{\alpha _ 1},\ldots ,H_{\alpha _ k},X_{\alpha }\quad (\alpha \in \Sigma )$$	conf 0.432 l05851064.png (64)
71.(113.)*		$[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$	$$[[X_{\alpha _ i},X_{-}\alpha _ i],X_{-\alpha _ j}]=-n(i,j)X_{\alpha _ j},$$	conf 0.628 F l05851069.png (69)
72.(79.)		$n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$	$$n(i,j)=\alpha _ j(H_i)=\frac {2(\alpha _ i,\alpha _ j)}{(\alpha _ j,\alpha _ j)}.$$	conf 0.992 l05851073.png (73)
73.(13.)		$[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$	$$[X_{\alpha },X_{\beta }]=\left\{ \begin {array}{ll} {N_{\alpha ,\beta }X_{\alpha +\beta }} &{\text{ if }\alpha +\beta \in \Sigma,}\\ 0 &{\text{ if }\alpha +\beta \notin \Sigma,} \end {array} \right.$$	conf 0.988 l05851074.png (74)
74.(80.)		$N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$	$$N_{\alpha ,\beta }=-N_{-\alpha ,-\beta }\quad {\rm and }\quad N _ {\alpha ,\beta }=\pm (p+1),$$	conf 0.961 l05851078.png (78)
75.(85.)*		$X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$	$$iH_\alpha,X_{\alpha }-X_{-\alpha },\quad i (X_{\alpha }+X_{-\alpha })\quad (\alpha \in \Sigma _ {+})$$	conf 0.691 F l05851085.png (85)

Lie algebra, solvable

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76.(119.)*		$[ \mathfrak { g } _ { i } , \mathfrak { g } _ { i } ] \subset \mathfrak { g } _ { \mathfrak { i } } + 1$	$[\mathfrak g_i,\mathfrak g_i]\subset \mathfrak g_{i+1}$	conf 0.276 F l05852011.png (11)
77.(141.)		$\operatorname { dim } \mathfrak { g } _ { i } = \operatorname { dim } \mathfrak { g } - i$	$\operatorname {dim}\mathfrak g_i=\operatorname {dim}\mathfrak g-i$	conf 0.901 l05852046.png (46)

Lie group

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78.(62.)*		$( G ) \cong \operatorname { Aut } ( L ( G ) ) \quad \text { and } \quad L ( \operatorname { Aut } ( G ) ) \cong D ( L ( G ) )$	$$\operatorname {Aut}(G)\cong \operatorname {Aut}(L(G))\quad {\rm and }\quad L (\operatorname {Aut}(G))\cong D (L(G)),$$	conf 0.693 F l058590115.png (115)
79.(50.)		$( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$	$$(X,Y)\rightarrow \operatorname {exp}^{-1}(\operatorname {exp}X\operatorname {exp}Y),\quad X ,Y\in L (G),$$	conf 0.856 l05859086.png (86)

Lie group, compact

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80.(121.)*		$J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$	$$J=\left\| \begin {array}{cc} 0 &{E_x}\\ {-E_x} &0 \end {array} \right\|,$$	conf 0.364 F l05861012.png (12)

Lie group, nilpotent

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81.(83.)		$N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$	$$N(F)=\{g\in GL (V):gv\equiv v \operatorname {mod}V_i\;\text{for all }v\in V _ i,\;i\geq 1 \}$$	conf 0.466 l0586604.png (4)

Lie group, semi-simple

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82.(35.)*		$L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$	$$L(\mathfrak g)\cong \Gamma _ 0(\mathfrak u)\cap \mathfrak h^{\prime }/\Gamma _ 0([\mathfrak k,\mathfrak k])$$	conf 0.659 F l058680102.png (102)
83.(81.)*		$\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$	$$\Gamma _ 1=\Gamma _ 1(g)=\{X\in h :\alpha (X)\in 2 \pi i {\mathbf Z}\;\text{for all }\alpha \in \Sigma \}.$$	conf 0.183 F l05868032.png (32)

Lie p-algebra

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84.(36.)		$( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$	$$(\operatorname {ad}x)^ny=\sum _ {j=1}^n(-1)^j\left(\begin {array}cn\\ j \end {array} \right)x^{n-j}yx^j$$	conf 0.356 l05872026.png (26)
85.(99.)		$\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$	$$\pi (x+y)=\pi (x)+\pi (y),\quad \pi (\lambda x )=\lambda ^p\pi (x),\quad \lambda \in k .$$	conf 0.964 l05872078.png (78)

Lie theorem

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86.(134.)		$y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$	$$y_i=f_i(g_1,\ldots ,g_i;x_1,\ldots ,x_n),\quad i =1,\ldots ,n$$	conf 0.276 l05876010.png (10)
87.(86.)		$X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$	$$X_i=\sum _ {j=1}^n\xi _ {ij}(x)\frac {\partial }{\partial x _ j},\quad i =1,\ldots ,r,$$	conf 0.656 l05876016.png (16)
88.(66.)*		$\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$	$$\frac {\partial f _ j}{\partial g _ i}(g,x)=\sum _ {k=1}^r\xi _ {kj}(f(g_sx))\psi _ {ki}(g),$$	conf 0.336 F l05876030.png (30)
89.(19.)*		$\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$	$$\sum _ {k=1}^N(\xi _ {ik}\frac {\partial \xi _ {jl}}{\partial x _ k}-\xi _ {jk}\frac {\partial \xi _ {il}}{\partial x _ k})=\sum _ {k=1}^rc_{ij}^k\xi _ {kl},$$	conf 0.157 F l05876037.png (37)
90.(14.)		$\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$	$$\left.\begin {array}c{c_{ij}^k=-c_{ji}^k},\\ {\displaystyle\sum _ {l=1}^r(c_{il}^mc_{jk}^l+c_{kl}^mc_{ij}^l+c_{jl}^mc_{ki}^l)=0,\quad 1 \leq i ,j,k,l,m\leq r,} \end {array} \right\}$$	conf 0.085 l05876052.png (52)

Maximal torus

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91.(95.)		$F ( x _ { 1 } f _ { 1 } + \ldots + x _ { x } f _ { n } ) = x _ { 1 } x _ { n } + x _ { 2 } x _ { n } - 1 + \ldots + x _ { p } x _ { n } - p + 1$	$$F(x_1f_1+\ldots +x_xf_n)=x_1x_n+x_2x_{n-1}+\ldots +x_px_{n-p+1},$$	conf 0.198 m06301072.png (72)

Non-Abelian cohomology

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92.(114.)*		$\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$	$$\phi (g_1)\phi (g_2)\phi (g_1g_2)^{-1}=\operatorname {Int}m(g_1,g_2),$$	conf 0.443 F n066900110.png (110)
93.(90.)*		$( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$	$$m'(g_1,g_2)=h(g_1)(\phi (g_1)(h(g_2)))m(g_1,g_2)h(g_1,g_2)^{-1}.$$	conf 0.764 F n066900118.png (118)
94.(44.)		$\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$	$$\delta (e)=e\quad \;\text{and }\quad \delta (\rho (a)b)=\sigma (a)\delta (b),\quad \alpha \in C ^0,\quad b \in C ^1,$$	conf 0.400 n06690016.png (16)
95.(60.)*		$C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$	$$C^{*}(\mathfrak U,{\cal F})=(C^0(\mathfrak U,{\cal F}),C^1(\mathfrak U,{\cal F}),C^2(\mathfrak U,{\cal F})),$$	conf 0.205 F n06690028.png (28)

Picard scheme

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96.(39.)*		$\operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Fic } ( X \times k S ^ { \prime } ) / \operatorname { Fic } ( S ^ { \prime } )$	$$\operatorname {Pic}_{X/k}(S^{\prime })=\operatorname {Pic}(X\times_k S ^{\prime })/\operatorname {Pic}(S^{\prime })$$	conf 0.345 F + p07267025.png (25)

Principal analytic fibration

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97.(100.)*		$g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$	$$g_j:U_i\cap U _ j\rightarrow G ,\quad i ,j\in I ,\quad U _ i\cap U _ j\neq \emptyset,$$	conf 0.184 F p07464025.png (25)

Quantum groups

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98.(101.)		$\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$	$$\phi ^{*}:\mathfrak g^{*}\otimes \mathfrak g^{*}\rightarrow \mathfrak g^{*}$$	conf 0.837 q07631062.png (62)
99.(108.)		$\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$	$$\delta :U_{\mathfrak g}\rightarrow U _ {\mathfrak g}\otimes U _ {\mathfrak g}$$	conf 0.648 q07631071.png (71)
100.(56.)*		$\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$	$$\delta (\alpha )=\operatorname {lim}_{h\rightarrow 0 }h^{-1}(\Delta (a)-\Delta ^{\prime }(\alpha ))$$	conf 0.304 F q07631072.png (72)
101.(129.)*		$[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$	$$[\alpha ,X_i^{\pm }]=\pm \alpha _ i(\alpha )X_i^{\pm }\quad \text{for }a\in \mathfrak h;$$	conf 0.544 F q07631088.png (88)
102.(128.)		$[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$	$$[X_i^{+},X_j^{-}]=2\delta _ {ij}h^{-1}\operatorname {sinh}(hH_i/2).$$	conf 0.893 q07631089.png (89)
103.(20.)		$\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$	$$\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\ k \end {array} \right)q^{-k(n-k)/2}(X_i^{\pm })^kX_j^{\pm }\cdot (X_i^{\pm })^{n-k}=0.$$	conf 0.055 q07631092.png (92)
104.(30.)		$\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$	$$\left( \begin {array}ln\\ k \end {array} \right)_q=\frac {(q^n-1)\ldots (q^{n-k+1}-1)}{(q^k-1)\ldots (q-1)} .$$	conf 0.443 q07631095.png (95)
105.(21.)*		$\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$	$$\Delta (X_i^{\pm })=X_i^{\pm }\otimes \operatorname {exp}(\frac {hH_i}4)+\operatorname {exp}(\frac {-hH_i}4)\otimes X _ i^{\pm }.$$	conf 0.212 F q07631099.png (99)

Rational representation

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106.(91.)		$0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$	$$0\leq \frac {2(\chi ,\alpha )}{(\alpha ,\alpha )}<p\quad \text{for all }\alpha \in \Delta.$$	conf 0.879 r077630100.png (100)
107.(135.)		$\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$	$$\phi _ 0\otimes \phi _ 1^{Fr}\otimes \ldots \otimes \phi _ d^{{Fr}^d},$$	conf 0.136 r077630104.png (104)
108.(45.)*		$\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$	$$\chi =\delta _ {\phi }-\sum _ {\alpha \in \Delta }m_{\alpha }\alpha ,\quad m _ {\alpha }\in Z ,\quad m _ {\alpha }\geq 0.$$	conf 0.862 F r07763055.png (55)

Singular point

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109.(31.)		$\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$	$$\sum _ {k_1,\ldots ,k_n=0}^{\infty }c_{k_1\cdots k _ n}(z_1-\zeta _ 1)^{k_1}\ldots (z_n-\zeta _ n)^{k_n}$$	conf 0.324 s085590225.png (225)
110.(46.)		$\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$	$$\frac {m_1}{n_1}<\frac {m_2}{n_1n_2}<\ldots <\frac {m_g}{n_1\ldots n _ g}=\frac {m_g}n$$	conf 0.459 s085590404.png (404)
111.(115.)*		$p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$	$$p(Z)=1-\operatorname {dim}H^0({\mathbf Z},{\cal O}_{\mathbf Z })+\operatorname {dim}H^1({\mathbf Z},{\cal O}_{\mathbf Z })$$	conf 0.997 F s085590429.png (429)
112.(136.)*		$X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$	$$X_{\epsilon }=\{(x_0,\ldots ,x_x):f(x_0,\ldots ,x_x)=\epsilon \}$$	conf 0.433 F s085590440.png (440)
113.(12.)		$= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$	$$=\left\{ \begin {array}{ll} {(x+\lambda )^2\ldots (x+k\lambda )^2} &{\text{ if }\mu =2k,}\\ {(x+\lambda )^2\ldots (x+k\lambda )^2(x+(k+1)\lambda )} &{\text{ if }\mu =2k+1,} \end {array} \right.$$	conf 0.870 s085590458.png (458)
114.(75.)		$( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$	$$\big(\frac {\partial F (x,y,\lambda )}{\partial x },\frac {\partial F (x,y,\lambda )}{\partial y }\big)$$	conf 0.986 s085590482.png (482)
115.(137.)		$\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$	$$\frac {dx_i}{dx_{i_0}}=f_i(x),\quad f _ i\in C (U),\quad i \neq i _ 0.$$	conf 0.594 s085590515.png (515)
116.(142.)*		$A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$	$$A=\left\| \begin {array}{ll} {\alpha } &b\\ c &e \end {array} \right\|$$	conf 0.506 F s085590527.png (527)
117.(53.)		$\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$	$$\Delta =(F_{xx}^{\prime \prime })_0(F_{yy}^{\prime \prime })_0-(F_{xy}^{\prime \prime })_0^2$$	conf 0.920 s085590634.png (634)
118.(16.)*		$\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$	$$\left\| \begin {array}{lll} {F_x^{\prime }} &{F_y^{\prime }} &{F_z^{\prime }}\\ {G_x^{\prime }} &{G_y^{\prime }} &{G_Z^{\prime }} \end {array} \right\|$$	conf 0.230 F s085590645.png (645)
119.(92.)		$( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$	$$(F_x^{\prime })_0=0,\quad (F_y^{\prime })_0=0,\quad (F_z^{\prime })_0=0.$$	conf 0.300 s085590653.png (653)

Solv manifold

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120.(138.)		$\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$	$$\{e\}\rightarrow \Delta \rightarrow \pi \rightarrow {\mathbf Z}^s\rightarrow \{e\}$$	conf 0.972 s08610054.png (54)

Stability theorems in algebraic K-theory

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121.(71.)		$\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$	$$\psi _ {t_1,\ldots ,t_n}^{\prime }:SK_1(R)\rightarrow S K _ 1(R(t_1,\ldots ,t_n)).$$	conf 0.379 s08706033.png (33)

Steinberg module

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122.(130.)		$e = \frac { | U | } { | G | } ( \sum _ { b \in B } b ) ( \sum _ { w \in W } \operatorname { sign } ( w ) w )$	$$e=\frac {|U|}{|G|}\big(\sum _ {b\in B }b\big)\big(\sum _ {w\in W }\operatorname {sign}(w)w\big)$$	conf 0.138 s13053016.png (16)

Steinberg symbol

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123.(24.)*		$( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$	$$(x_{ij}(a),x_{kl}(b))=\left\{ \begin {array}{ll} 1 &{\text{ if }i\neq l ,j\neq k },\\ {x_{il}(ab)} &{\text{ if }i\neq l ,j=k}. \end {array} \right.$$	conf 0.381 F s13054017.png (17)

Tilting theory

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124.(84.)		$0 \rightarrow \Lambda \rightarrow T _ { 1 } \rightarrow \ldots \rightarrow T _ { n } \rightarrow 0$	$$0\rightarrow \Lambda \rightarrow T _ 1\rightarrow \ldots \rightarrow T _ n\rightarrow 0 $$	conf 0.946 t130130105.png (105)

Tits quadratic form

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125.(18.)		$q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$	$$q_R(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}x_ix_j+\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}r_{i,j}x_ix_j,$$	conf 0.112 t130140104.png (104)
126.(40.)		$[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$	$$[X]\mapsto \chi _ R([X])=\sum _ {m=0}^{\infty }(-1)^m\operatorname {dim}_K\operatorname {Ext}_R^m(X,X)$$	conf 0.116 t130140118.png (118)
127.(132.)*		$\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$	$$\underline {\dim }:K_0(\operatorname {mod}R)\rightarrow {\mathbf Z}^{Q_0}$$	conf 0.287 F t130140119.png (119)
128.(37.)*		$q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$	$$q_I(x)=\sum _ {i\in I }x_i^2+\sum _ {i\prec j \atop j\in I\setminus {\rm max}I}x_ix_j-\sum _ {p\in \operatorname {max}I}\big(\sum _ {i\prec p }x_i\big)x_p$$	conf 0.197 F t130140140.png (140)
129.(131.)*		$X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$	$$X\mapsto \underline {\dim }X=(\operatorname {dim}_KX_j)_{j\in Q _ 0}$$	conf 0.819 F t13014044.png (44)
130.(25.		$[ X ] \mapsto \chi _ { Q } ( [ X ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( X ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( X , X )$	$$[X]\mapsto \chi _ Q([X])=\operatorname {dim}_K\operatorname {End}_Q(X)-\operatorname {dim}_K\operatorname {Ext}_Q^1(X,X)$$	conf 0.661 t13014048.png (48)
131.(38.)*		$A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$	$$A_Q(v)=\prod _ {i,j\in Q _ 0}\prod _ {\langle \beta :j\rightarrow i \rangle \in Q _ 1}M_{v_i\times v _ j}(K)_{\beta }$$	conf 0.481 F t13014056.png (56)
132.(139.)*		$\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$	$$q_Q(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {i,j\in Q _ 0}d_{ij}x_ix_j,$$	conf 0.648 F t1301406.png (6)

Torus

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133.(41.)*		$r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$	$$r=\alpha \operatorname {sin}u{\bf k}+l(1+\epsilon \operatorname {cos}u)({\bf i}\operatorname {cos}v+{\bf j}\operatorname {sin}v)$$	conf 0.585 F t0933502.png (2)
134.(122.)*		$d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$	$$ds^2=\alpha ^2du^2+l^2(1+\epsilon \operatorname {cos}u)^2dv^2,$$	conf 0.696 F t0933507.png (7)

Uniform distribution

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135.(9.)		$u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$	$$u_3(x)=\left\{ \begin {array}{ll} {\frac {x^2}2,} &{0\leq x <1,}\\ {\frac {[x^2-3(x-1)^2]}2,} &{1\leq x <2,}\\ {\frac {[x^2-3(x-1)^2+3(x-2)^2]}2,} &{2\leq x <3,}\\ {0,} &{x\notin [0,3].} \end {array} \right.$$	conf 0.733 u09524027.png (27)
136.(32.)*		$p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$	$$p(x)=\left\{ \begin {array}{ll} {\frac 1{b-\alpha },} &{x\in [\alpha ,b],}\\ {0,} &{x\notin [\alpha ,b].} \end {array} \right.$$	conf 0.681 F u0952403.png (3)
137.(34.)		$u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$	$$u_n(x)=\frac 1{(n-1)!}\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\ k \end {array} \right)(x-k)_{+}^{n-1}$$	conf 0.569 u09524030.png (30)
138.(109.)		$z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$	$$z_{+}=\left\{ \begin {array}{ll} {z,} &{z>0}.\\ {0,} &{z\leq 0 }. \end {array} \right.$$	conf 0.676 u09524034.png (34)
139.(43.)		$F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$	$$F(x)=\left\{ \begin {array}{ll} {0,} &{x\leq a },\\ {\frac {x-a}{b-a},} &{a<x\leq b },\\ {1,} &{x>b}, \end {array} \right.$$	conf 0.468 u0952407.png (7)
140.(47.)		$p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$	$$p(x_1,\ldots ,x_n)=\left\{ \begin {array}{ll} {C\neq 0 ,} &{x\in D },\\ {0,} &{x\notin D }, \end {array} \right.$$	conf 0.705 u09524072.png (72)

Unipotent group

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141.(143.)		$\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$	$$\{g\in \operatorname {GL}(V):(1-g)^n=0\},\quad n =\operatorname {dim}V,$$	conf 0.287 u0954106.png (6)

Weyl module

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142.(51.)		$\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$	$$\operatorname {diag}(t_1,\ldots ,t_n)\mapsto t _ 1^{\lambda _ 1}\ldots t _ n^{\lambda _ n}\in K,$$	conf 0.507 w120090122.png (122)
143.(54.)*		$\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$	$$\chi _ {\lambda }=\sum _ {\mu \in \Lambda (n)}\operatorname {dim}_K(\Delta (\lambda )^{\mu })_{e_{\mu }},$$	conf 0.461 F w120090135.png (135)
144.(110.)		$\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$	$$\mathfrak B=\{e_{\pm }\alpha ,h_{\beta }:\alpha \in \Phi ^{+},\beta \in \Sigma \}.$$	conf 0.381 w120090259.png (259)
145.(82.)		$\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$	$$\left( \begin {array}ch\\ i \end {array} \right)=\frac {h(h-1)\ldots (h-i+1)}{i!} $$	conf 0.487 w120090342.png (342)
146.(28.)*		$\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$	$$\mathfrak S_{\{1,\ldots ,\lambda _ 1\}}\times \mathfrak S_{\{\lambda _ 1+1,\ldots ,\lambda _ 1+\lambda _ 2\}}\times \dots $$	conf 0.312 F w12009095.png (95)
147.(104.)		$\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$	$$\ldots \times \mathfrak S_{\{\lambda _ 1+\ldots +\lambda _ {n-1}+1,\ldots ,r\}},$$	conf 0.259 w12009096.png (96)

Witt vector

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148.(87.)*		$\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$	$$\langle \alpha ><b\rangle =\langle \alpha b \rangle ,\quad \langle {\bf 1}\rangle ={\bf f}_1={\bf V}_1=\text{ unit element}1,$$	conf 0.351 F w098100172.png (172)
149.(123.)*		$\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$	$$\langle \alpha +b\rangle =\sum _ {n=1}^{\infty }{\bf V}_n\langle r _ n(\alpha ,b){\bf f}_n.$$	conf 0.143 F w098100177.png (177)
150.(102.)		$\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$	$$\sigma (\alpha _ 1,\alpha _ 2,\ldots )=(\alpha _ 1^p,\alpha _ 2^p,\ldots )$$	conf 0.771 w098100190.png (190)