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 columns of its bit image 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. Question

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

Algebraic curve

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

Algebraic geometry

Nr.	Image of png File	$\TeX$, 1st version	$\TeX$, corrected version	Confidence, F? png file
2.(116.)		$\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$	$$\theta = \int\limits _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }, $$	conf 0.997 a01150014.png (14)
3.(133.)		$\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$	$$\omega = 2 \int\limits _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }, $$	conf 0.973 a01150021.png (21)
4.(67.)		$\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$	$$\widetilde{ w } = 2 \int\limits _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } },$$	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.)		$\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$	$$\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 ). $$	conf 0.440 a01150078.png (78)

Algebraic surface

Nr.	Image of png File	$\TeX$, 1st version	$\TeX$, 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.)		$\operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } )$	$$ \operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } ). $$	conf 0.996 a011640139.png (139)
10.(117.)		$N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$	$$ N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1. $$	conf 0.369 a01164027.png (27)
11.(72.)		$p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$	$$ p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1 $$	conf 0.396 a01164029.png (29)
12.(68.)*		$p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$	$$p _ { \alpha } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , {\cal O} _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , {\cal O} _ { V } ) =$$	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	$\TeX$, 1st version	$\TeX$, 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	$\TeX$, 1st version	$\TeX$, corrected version	Confidence, F? png file
15.(49.)*		$f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$	$$ [e_i, f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { ij } f _ { j }, $$	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	$\TeX$, 1st version	$\TeX$, corrected version	Confidence, F? png file
17.(7.)		$H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$	$$H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$$	conf 0.956 c02333033.png (33)
18.(76.)		$= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$	$$ = ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 } $$	conf 0.549 c02333034.png (34)
19.(11.)*		$( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$	$$ ( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } ) $$	conf 0.521 F c02333035.png (35)

Deformation

Nr.	Image of png File	$\TeX$, 1st version	$\TeX$, 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.)		$\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$	$$ \operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } ). $$	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	$\TeX$, 1st version	$\TeX$, corrected version	Confidence, F? png file
24.(106.)		$S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$	$$ S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { { m - l } , j } },$$	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.)		$e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$	$$ e_ { i j } = \operatorname { ord } _ { { Y } _ { j } } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n, $$	conf 0.187 d03183043.png (43)

Dimension polynomial

Nr.	Image of png File	$\TeX$, 1st version	$\TeX$, 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	$\TeX$, 1st version	$\TeX$, 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.)*		$h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$	$$ h ( \phi ) = \underset { n\rightarrow \infty }{\overline{ \lim } } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r } $$	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	$\TeX$, 1st version	$\TeX$, corrected version	Confidence, F? png file
42.(63.)		$F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$	$$ F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle, $$	conf 0.628 e03696024.png (24)

Formal group

Nr.	Image of png File	$\TeX$, 1st version	$\TeX$, 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	$\TeX$, 1st version	$\TeX$, 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	$\TeX$, 1st version	$\TeX$, 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	$\TeX$, 1st version	$\TeX$, 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	$\TeX$, 1st version	$\TeX$, 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

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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

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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

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55.(6.)*		$J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$	$$J = \left\| \begin{array} { c c c c } J_{n_1}(\lambda_1) & 0 & 0 & 0 \\ 0 & \ddots & \ddots & 0 \\ 0 & \ddots & \ddots & 0 \\ 0 & 0 & 0 & J_{n_s}(\lambda_s) \end{array} \right\|,$$	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.)*		$J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$	$$J_m(\lambda) = \left\| \begin{array} { c c c c c c } \lambda & 1 & \square & \square & \square & \square \\ \square & \lambda & 1 & \square & 0 & \square \\ \square & \square & \ddots & \ddots & \square & \square\\ \square & \square & \square & \ddots & \ddots & \square \\ \square & 0 & \square & \square & \lambda & 1 \\ \square & \square & \square & \square & \square & \lambda \end{array} \right\|,$$	conf 0.098 F j0543406.png (6)

Lie algebra, semi-simple

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58.(5.)		$\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$	$$B_n: \quad \left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ \cdot & \cdot & \cdot & \dots & \cdot & \cdot \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 }\\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 }\\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|,$$	conf 0.232 l058510127.png (127)
59.(3.)*		$\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$	$$D_n: \quad \left\| \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 }\\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ \cdot & \cdot & \cdot & \dots & \cdot & \cdot &\cdot & \cdot \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 }\\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right\|,$$	conf 0.055 F l058510129.png (129)
60.(8.)*		$\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$	$$E_6: \quad \left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|,$$	conf 0.628 F l058510130.png (130)
61.(4.)		$\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$	$$E_7: \quad \left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ {-1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|,$$	conf 0.278 l058510131.png (131)
62.(2.)*		$\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$	$$E_8: \quad \left\| \begin{array} { r r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ {-1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|,$$	conf 0.354 F l058510132.png (132)
63.(10.)*		$\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$	$$F_4: \quad \left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|.$$	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.)*		$[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$	$$ [ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad {\rm and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { \alpha }. $$	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.)		$[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$	$$ [ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma, } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma, } \end{array} \right. $$	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.)*		$J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$	$$ J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|, $$	conf 0.364 F l05861012.png (12)

Lie group, nilpotent

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81.(83.)		$N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$	$$ N( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \; \text{for all } v \in V _ { i } ,\; i \geq 1 \} $$	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.)		$( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$	$$ ( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j } $$	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.)		$X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$	$$ X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r ,$$	conf 0.656 l05876016.png (16)
88.(66.)*		$\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$	$$ \frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g ), $$	conf 0.336 F l05876030.png (30)
89.(19.)*		$\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$	$$ \sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l },$$	conf 0.157 F l05876037.png (37)
90.(14.)		$\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$	$$ \left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } }, \\ { \displaystyle\sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r, } \end{array} \right\} $$	conf 0.085 l05876052.png (52)

Maximal torus

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

Non-Abelian cohomology

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92.(114.)*		$\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$	$$ \phi ( g _ { 1 } ) \phi ( g_2 ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } ), $$	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.)		$[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$	$$ [ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 ). $$	conf 0.893 q07631089.png (89)
103.(20.)		$\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$	$$ \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0. $$	conf 0.055 q07631092.png (92)
104.(30.)		$\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$	$$ \left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }. $$	conf 0.443 q07631095.png (95)
105.(21.)*		$\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$	$$ \Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes X _ { i } ^ { \pm }. $$	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.)		$\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$	$$ \phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ {{ Fr }^ { d } }, $$	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.)		$\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$	$$ \frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n } $$	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.)		$= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$	$$ = \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k, } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1, } \end{array} \right. $$	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.)		$\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$	$$ \frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }. $$	conf 0.594 s085590515.png (515)
116.(142.)*		$A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$	$$ A = \left\| \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right\| $$	conf 0.506 F s085590527.png (527)
117.(53.)		$\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$	$$ \Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 } $$	conf 0.920 s085590634.png (634)
118.(16.)*		$\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$	$$ \left\| \begin{array} { l l l } { F _ { x } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { x } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right\| $$	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.)		$\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$	$$ \psi _ { t _ { 1 } , \ldots , t _ { n } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) ). $$	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.)*		$( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$	$$ ( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq l , j \neq k }, \\ { x _ {il} ( a b ) } & { \text { if } i \neq l , j = k }. \end{array} \right. $$	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.)		$q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$	$$ q_R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } r _ {i , j } x _ { i } x _ { j }, $$	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.)*		$q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$	$$ q_I ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j \atop j\in I\setminus {\rm max}I} x _ { i } x _ { j } - \sum _ { p \in \operatorname { max }I } \big( \sum _ { i \prec p } x _ { i } \big) x _ { p } $$	conf 0.197 F t130140140.png (140)
129.(131.)*		$X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$	$$ X \mapsto \underline { \dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } } $$	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.)*		$\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$	$$ q_Q ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }, $$	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.)*		$d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$	$$ d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }, $$	conf 0.696 F t0933507.png (7)

Uniform distribution

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

Unipotent group

Nr.	Image of png File	$\TeX$, 1st version	$\TeX$, corrected version	Confidence, F? png file
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

Nr.	Image of png File	$\TeX$, 1st version	$\TeX$, corrected version	Confidence, F? png file
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.)		$\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$	$$ \left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! } $$	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

Nr.	Image of png File	$\TeX$, 1st version	$\TeX$, corrected version	Confidence, F? png file
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)