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User:Ulf Rehmann/Table of automatically generated TeX code

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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.) a01145065.png $g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$ $$g\leq \left\{ \begin {array}{ll} {\frac {(n-2)^2}4} &{\text{ for even }n,}\\ {\frac {(n-1)(n-3)}4} &{\text{ for odd }n,} \end {array} \right.$$ 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.) a01150014.png $\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ $$\theta =\int\limits _ 0^{\lambda }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ conf 0.997

a01150014.png (14)

3.(133.) a01150021.png $\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ $$\omega =2\int\limits _ 0^{1/c}\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ conf 0.973

a01150021.png (21)

4.(67.) a01150022.png $\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ $$\widetilde w=2\int\limits _ 0^{1/\varepsilon }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ conf 0.107

a01150022.png (22)

5.(105.) a01150044.png $\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$ $$\theta (v+\pi i r )=\theta (r),\quad \theta (v+\alpha _ j)=e^{L_j(v)}\theta (v),$$ conf 0.775

a01150044.png (44)

6.(17.) a01150078.png $\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ $$\left( \begin {array}{ll} {\alpha } &b\\ c &d \end {array} \right)\equiv \left( \begin {array}{ll} 1&0\\ 0&1 \end {array} \right)(\operatorname {mod}7).$$ 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.) a011640132.png $0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$ $$0\rightarrow {\cal O}_V\rightarrow E _ {\alpha }\rightarrow T _ V\rightarrow 0$$ conf 0.981

a011640132.png (132)

8.(73.) a011640137.png $M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$ $$M=\operatorname {dim}\operatorname {Im}(H^1(V,E_{\alpha })\rightarrow H ^1(V,T_V)).$$ conf 0.997

a011640137.png (137)

9.(88.) a011640139.png $\operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } )$ $$\operatorname {dim}_kH^2(V,E_{\alpha })+\operatorname {dim}_kH^2(V,T_V).$$ conf 0.996

a011640139.png (139)

10.(117.) a01164027.png $N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$ $$N_m=\left(\begin {array}c{m+3}\\ 3 \end {array} \right)-dm+2t+\tau +p-1.$$ conf 0.369

a01164027.png (27)

11.(72.) a01164029.png $p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$ $$p_{\alpha }(V)=\left(\begin {array}c{n-1}\\ 3 \end {array} \right)-d(n-1)+2t+\tau +p-1$$ conf 0.396

a01164029.png (29)

12.(68.)* a01164047.png $p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$ $$p_{\alpha }(V)=-\operatorname {dim}_kH_1(V,{\cal O}_V)+\operatorname {dim}_kH^2(V,{\cal O}_V)=$$ conf 0.756 F

a01164047.png (47)

13.(93.)* a01164053.png $1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$ $$1+p_{\alpha }(V)=\frac {\operatorname {deg}(c_1^2)+\operatorname {deg}(c_2)}{12},$$ 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.)* c0205509.png $\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$ $$\mathfrak g_0=\big\{X\in \mathfrak g:\forall H \in \mathfrak t\exists n_{X,H}\in {\mathbb Z}((\text{ ad }H)^{n_{X,H}}(X)=0)\big\},$$ 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.)* c0205704.png $f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$ $$[e_i,f_j]=\delta _ {ij}h_i,\quad [h_i,e_j]=\alpha _ {ij}e_j,\quad [h_i,f_j]=-\alpha _ {ij}f_j,$$ conf 0.149 F

c0205704.png (4)

16.(55.)* c02057064.png $\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$ $$\dots \rightarrow H ^p(X,S)\rightarrow H ^p(X,F)\stackrel {\phi_p }{\rightarrow }H^p(X,G)\rightarrow $$ 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.) c02333033.png $H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$ $$H=\frac 1{36}\left| \begin {array}{cc} {\frac {\partial ^2f}{\partial x ^2}} &{\frac {\partial ^2f}{\partial x \partial y }}\\ {\frac {\partial ^2f}{\partial x \partial y }} &{\frac {\partial ^2f}{\partial y ^2}} \end {array} \right|=$$ conf 0.956

c02333033.png (33)

18.(76.) c02333034.png $= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$ $$=(a_0a_2-a_1^2)x^2+(a_0a_3-a_1a_2)xy+(a_1a_3-a_2^2)y^2$$ conf 0.549

c02333034.png (34)

19.(11.)* c02333035.png $( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$ $$(\alpha _ 0,\alpha _ 1,\alpha _ 2,\alpha _ 3)\mapsto (\alpha _ 0\alpha _ 2-\alpha _ 1^2,\frac 12(\alpha _ 0\alpha _ 3-\alpha _ 1\alpha _ 2),\alpha _ 1\alpha _ 3-\alpha _ 2^2)$$ 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.) d030700175.png $\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$ $$\operatorname {Aut}_{R^{\prime }}(X^{\prime }|X_0)\rightarrow \operatorname {Aut}_R(X_{R^{\prime }}^{\prime }\otimes R |X_0)$$ conf 0.683
\

d030700175.png (175)

21.(27.) d030700190.png $\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$ $$\operatorname {dim}_kH^1(X_0,T_{X_0})-\operatorname {dim}M_{X_0}\leq \operatorname {dim}_kH^2(X_0,T_{X_0}).$$ conf 0.944

d030700190.png (190)

22.(78.)* d030700263.png $\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$ $$\alpha \circ b =\alpha b +\sum _ {i=1}^{\infty }\phi _ i(\alpha ,b)t^i,\quad \alpha ,b\in V,$$ conf 0.097 F

d030700263.png (263)

23.(96.)* d030700270.png $\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$ $$\Phi (\alpha )=\alpha +\sum _ {i=1}^{\infty }t^i\phi _ i(\alpha ),\quad \alpha \in V,$$ 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.) d031830107.png $S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$ $$S^tF=\sum _ {j=1}^rc_jA^{p_j}A_1^{i_{1j}}\dots A _ {m-l}^{i_{{m-l},j}},$$ conf 0.149

d031830107.png (107)

25.(146.)* d031830141.png $( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$ $(\eta _ 1,\ldots ,\eta _ k)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ k)$ conf 0.562 F

d031830141.png (141)

26.(145.)$^F$* d031830150.png $( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$ $(\eta _ 1,\ldots ,\eta _ n)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ n)$ conf 0.376 F

d031830150.png (150)

27.(57.) d03183016.png $\omega _ { V } = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ $$\omega _ V=\sum _ {0\leq i \leq m }\alpha _ i\left( \begin {array}c{x+i}\\ i \end {array} \right),$$ conf 0.780

d03183016.png (16)

28.(111.) d03183043.png $e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$ $$e_{ij}=\operatorname {ord}_{Y_j}F_i,\quad 1 \leq i \leq n ,\quad i \leq j \leq n,$$ 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.) d03249029.png $\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ $$\omega _ {\eta /F}(x)=\sum _ {0\leq i \leq m }\alpha _ i\left(\begin {array}c{x+i}\\ i \end {array} \right),$$ 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.)* d034120173.png $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$ $$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow {\mathbf C},$$ conf 0.824 F

d034120173.png (173)

31.(59.)* d034120175.png $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow H _ { c } ^ { n } ( X , \Omega )$ $$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow H _ c^n(X,\Omega )$$ conf 0.921 F

d034120175.png (175)

32.(124.)* d034120184.png $( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$ $$(H^p(X,{\cal F}))^{\prime }\cong H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega )).$$ conf 0.829 F

d034120184.png (184)

33.(29.)* d034120236.png $\beta : \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X F , \Omega ) \rightarrow \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X \backslash Y || F , \Omega )$ $$\beta :\operatorname {Ext}_c^{n-p-1}(X;{\cal F},\Omega )\rightarrow \operatorname {Ext}_c^{n-p-1}(X\backslash Y ;{\cal F},\Omega ).$$ conf 0.634 F

d034120236.png (236)

34.(77.)* d034120247.png $\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$ $$\underset {n\rightarrow \infty }{\overline {\lim }}|\alpha _ n|^{1/n}=\sigma <+\infty.$$ conf 0.521 F

d034120247.png (247)

35.(58.)* d034120253.png $h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$ $$h(\phi )=\underset {n\rightarrow \infty }{\overline {\lim }}\frac {\operatorname {ln}|A(re^{i\phi })|}r$$ conf 0.861 F

d034120253.png (253)

36.(69.)* d034120360.png $\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$ $$\operatorname*{sup}_{l\in E^\perp \atop \|l\|\le 1 }|l(\omega )|=\operatorname*{inf}_{x\in E }\|\omega -x\|,$$ conf 0.293 F

d034120360.png (360)

37.(15.) d034120376.png $\operatorname { sup } _ { f \in B ^ { 1 } } | \int _ { \partial G } f ( \zeta ) \omega ( \zeta ) d \zeta | = \operatorname { inf } _ { \phi \in E ^ { 1 } } \int _ { \partial G } | \omega ( \zeta ) - \phi ( \zeta ) \| d \zeta |$ $$\operatorname*{sup}_{f\in B ^1}\big|\int\limits _ {\partial G }f(\zeta )\omega (\zeta )d\zeta \big|=\operatorname*{inf}_{\phi \in E ^1}\int\limits _ {\partial G }|\omega (\zeta )-\phi (\zeta ) ||d\zeta |.$$ conf 0.508

d034120376.png (376)

38.(52.) d034120509.png $f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$ $$f=\{f_{\alpha }\}\in \prod _ {\alpha }F_{\alpha },\quad g =\{g_{\alpha }\}\in \operatorname*\oplus _ {\alpha }G_{\alpha }.$$ conf 0.491

d034120509.png (509)

39.(140.) d034120535.png $f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$ $$f^{*}(x^{*})=\operatorname*{sup}_{x\in X }(\langle x ^{*},x\rangle -f(x))$$ conf 0.900

d034120535.png (535)

40.(94.) d034120555.png $f _ { 0 } ( x ) \rightarrow \text { inf, } \quad f _ { i } ( x ) \leq 0 , \quad i = 1 , \ldots , m , \quad x \in B$ $$f_0(x)\rightarrow \text{ inf, }\quad f _ i(x)\leq 0 ,\quad i =1,\ldots ,m,\quad x \in B,$$ conf 0.810

d034120555.png (555)

41.(74.)* d03412079.png $( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$ $$(c_{\gamma },c^r)=\sum _ {t^r\in K }c_r(t^{\prime })c^r(t^r)\operatorname {mod}1$$ 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.) e03696024.png $F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$ $$F_1F_2=F_1\langle F _ 2\rangle =F_1(F_2)=F_2(F_1)=F_2\langle F _ 1\rangle,$$ 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.)* f040820118.png $\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$ $$\operatorname {log}F_{\rm MU }(X)=\sum _ {i=1}^{\infty }i^{-1}[{\rm CP}^{i-1}]X^i,$$ conf 0.098 F

f040820118.png (118)

44.(147.)* f04082059.png $( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$ $$(x_1,\ldots ,x_n)\circ (y_1,\ldots ,y_n)=(z_1,\ldots ,z_n),$$ 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.) g1300205.png $\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$ $\alpha ^{\beta }=\operatorname {exp}\{\beta \operatorname {log}\alpha \}$ 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.)* g04521075.png $\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$ 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.) h04769069.png $\mathfrak { g } = \mathfrak { f } + \mathfrak { m } , \quad \mathfrak { f } \cap \mathfrak { m } = \{ 0 \}$ $$\mathfrak g=\mathfrak f+\mathfrak m,\quad \mathfrak f\cap \mathfrak m=\{0\},$$ 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.) h047970129.png $m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$ $m\circ (\iota \otimes 1 )\circ \mu =m\circ (1\otimes \iota )\circ \mu =e\circ \epsilon$ conf 0.618

h047970129.png (129)

49.(107.)* h047970139.png $F _ { 1 } ( X || Y ) , \ldots , F _ { n } ( X || Y ) \in K [ X _ { 1 } , \ldots , X _ { n } || Y _ { 1 } , \ldots , Y _ { n } ] \}$ $F_1(X;Y),\ldots ,F_n(X;Y)\in K [X_1,\ldots ,X_n;Y_1,\ldots ,Y_n]\}$ conf 0.353 F

h047970139.png (139)

50.(97.) h04797042.png $\epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g }$ $$\epsilon (x)=0,\quad \delta (x)=x\otimes 1 +1\otimes x ,\quad x \in \mathfrak g.$$ 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.)* i05235015.png $\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$ $$\alpha _ {i_1,\dots,i_n}\rightarrow \alpha _ {i_1,\dots,i_n}^{\prime }.$$ 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.) j05427030.png $H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$ $$(C_3,\Gamma )=\big\{X\in C _ 3:X=\Gamma ^{-1}X\square ^{\prime }\Gamma \big\},$$ conf 0.651

j05427030.png (30)

53.(42.) j05427031.png $\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$ $$\Gamma =\operatorname {diag}\{\gamma _ 1,\gamma _ 2,\gamma _ 3\},\quad \gamma _ i\neq 0 ,\quad \gamma _ i\in F,$$ conf 0.987

j05427031.png (31)

54.(125.)* j05427077.png $\mathfrak { g } = \mathfrak { g } - 1 + \mathfrak { g } \mathfrak { d } + \mathfrak { g } _ { 1 }$ $\mathfrak g=\mathfrak g_{-1}+\mathfrak g_0+\mathfrak g_1$ 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.)* j0543403.png $J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$ $$J=\left\| \begin {array}{cccc} J_{n_1}(\lambda_1) &0 &0 &0\\ 0 &\ddots &\ddots &0\\ 0 &\ddots &\ddots &0\\ 0 &0 &0 &J_{n_s}(\lambda_s) \end {array} \right\|,$$ conf 0.072 F

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56.(64.) j05434030.png $C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$ $$C_m(\lambda )=\operatorname {rk}(A-\lambda E )^{m-1}-2\operatorname {rk}(A-\lambda E )^m+$$ conf 0.955

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57.(1.)* j0543406.png $J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$ $$J_m(\lambda)=\left\| \begin {array}{cccccc} \lambda &1 &\square &\square &\square &\square \\ \square &\lambda &1 &\square &0 &\square \\ \square &\square &\ddots &\ddots &\square &\square\\ \square &\square &\square &\ddots &\ddots &\square \\ \square &0 &\square &\square &\lambda &1\\ \square &\square &\square &\square &\square &\lambda \end {array} \right\|,$$ conf 0.098 F

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Lie algebra, semi-simple

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58.(5.) l058510127.png $\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$ $$B_n:\quad \left\| \begin {array}{rrrrrr} 2 &{-1} &0 &{\dots } &0 &0\\ {-1} &2 &{-1} &{\dots } &0 &0\\ 0 &{-1} &2 &{\dots } &0 &0\\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot \\ 0 &0 &0 &{\dots } &{-1} &0\\ 0 &0 &0 &{\dots } &2 &{-2}\\ 0 &0 &0 &{\dots } &{-1} &2 \end {array} \right\|,$$ conf 0.232

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59.(3.)* l058510129.png $\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$ $$D_n:\quad \left\| \begin {array}{rrrrrrr} 2 &{-1} &0 &{\dots } &0 &0 &0 &0\\ {-1} &2 &{-1} &{\dots } &0 &0 &0 &0\\ 0 &{-1} &2 &{\dots } &0 &0 &0 &0\\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot &\cdot &\cdot \\ 0 &0 &0 &{\dots } &2 &{-1} &0 &0\\ 0 &0 &0 &{\dots } &{-1} &2 &{-1} &{-1}\\ 0 &0 &0 &{\dots } &0 &{-1} &2 &0\\ 0 &0 &0 &{\dots } &0 &{-1} &0 &2 \end {array} \right\|,$$ conf 0.055 F

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60.(8.)* l058510130.png $\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ $$E_6: \quad \left\| \begin {array}{rrrrrr} 2 &0 &{-1} &0 &0 &0\\ 0 &2 &0 &{-1} &0 &0\\ {-1} &0 &2 &{-1} &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0\\ 0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ conf 0.628 F

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61.(4.) l058510131.png $\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ $$E_7:\quad \left\| \begin {array}{rrrrrrr} 2 &0 &{-1} &0 &0 &0 &0\\ 0 &2 &0 &{-1} &0 &0 &0\\ {-1} &0 &2 &{-1} &0 &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0 &0\\ 0 &0 &0 &{-1} &2 &{-1} &0\\ 0 &0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ conf 0.278

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62.(2.)* l058510132.png $\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$ $$E_8:\quad \left\| \begin {array}{rrrrrrrr} 2 &0 &{-1} &0 &0 &0 &0 & 0\\ 0 &2 &0 &{-1} &0 &0 &0 &0\\ {-1} &0 &2 &{-1} &0 &0 &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0 &0 &0\\ 0 &0 &0 &{-1} &2 &{-1} &0 &0\\ 0 &0 &0 &0 &{-1} &2 &{-1} &0\\ 0 &0 &0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ conf 0.354 F

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63.(10.)* l058510133.png $\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$ $$F_4:\quad \left\| \begin {array}{rrrr} 2 &{-1} &0 &0\\ {-1} &2 &{-2} &0\\ 0 &{-1} &2 &{-1}\\ 0 &0 &{-1} &2 \end {array} \right\|,\quad G _ 2:\quad \left\| \begin {array}{rr} 2&{-1}\\ {-3}&2 \end {array} \right\|.$$ conf 0.374 F

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64.(98.) l05851030.png $\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$ $$\mathfrak g_{\alpha }=\{X\in \mathfrak g:[H,X]=\alpha (H)X,H\in \mathfrak h\}.$$ conf 0.976

l05851030.png (30)

65.(126.) l05851037.png $\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$ $$\mathfrak g=\mathfrak h+\sum _ {\alpha \in \Sigma }\mathfrak g_{\alpha }.$$ conf 0.945

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66.(61.)* l05851044.png $\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$ $$\mathfrak g_{\alpha }=\operatorname {dim}[\mathfrak g_{\alpha },\mathfrak g_{-\alpha }]=1.$$ conf 0.520 F

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67.(65.)* l05851050.png $[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$ $$[H_{\alpha },X_{\alpha }]=2X_{\alpha }\quad {\rm and }\quad [H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }.$$ conf 0.539 F

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68.(70.) l05851051.png $\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$ $$\beta (H_{\alpha })=\frac {2(\alpha ,\beta )}{(\alpha ,\alpha )},\quad \alpha ,\beta \in \Sigma,$$ conf 0.997

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69.(112.) l05851057.png $[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$ $$[\mathfrak g_{\alpha },\mathfrak g_{\beta }]=\mathfrak g_{\alpha +\beta }$$ conf 0.917

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70.(127.) l05851064.png $H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$ $$H_{\alpha _ 1},\ldots ,H_{\alpha _ k},X_{\alpha }\quad (\alpha \in \Sigma )$$ conf 0.432

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71.(113.)* l05851069.png $[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$ $$[[X_{\alpha _ i},X_{-}\alpha _ i],X_{-\alpha _ j}]=-n(i,j)X_{\alpha _ j},$$ conf 0.628 F

l05851069.png (69)

72.(79.) l05851073.png $n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$ $$n(i,j)=\alpha _ j(H_i)=\frac {2(\alpha _ i,\alpha _ j)}{(\alpha _ j,\alpha _ j)}.$$ conf 0.992

l05851073.png (73)

73.(13.) l05851074.png $[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$ $$[X_{\alpha },X_{\beta }]=\left\{ \begin {array}{ll} {N_{\alpha ,\beta }X_{\alpha +\beta }} &{\text{ if }\alpha +\beta \in \Sigma,}\\ 0 &{\text{ if }\alpha +\beta \notin \Sigma,} \end {array} \right.$$ conf 0.988

l05851074.png (74)

74.(80.) l05851078.png $N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$ $$N_{\alpha ,\beta }=-N_{-\alpha ,-\beta }\quad {\rm and }\quad N _ {\alpha ,\beta }=\pm (p+1),$$ conf 0.961

l05851078.png (78)

75.(85.)* l05851085.png $X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$ $$iH_\alpha,X_{\alpha }-X_{-\alpha },\quad i (X_{\alpha }+X_{-\alpha })\quad (\alpha \in \Sigma _ {+})$$ conf 0.691 F

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Lie algebra, solvable

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76.(119.)* l05852011.png $[ \mathfrak { g } _ { i } , \mathfrak { g } _ { i } ] \subset \mathfrak { g } _ { \mathfrak { i } } + 1$ $[\mathfrak g_i,\mathfrak g_i]\subset \mathfrak g_{i+1}$ conf 0.276 F

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77.(141.) l05852046.png $\operatorname { dim } \mathfrak { g } _ { i } = \operatorname { dim } \mathfrak { g } - i$ $\operatorname {dim}\mathfrak g_i=\operatorname {dim}\mathfrak g-i$ conf 0.901

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Lie group

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78.(62.)* l058590115.png $( G ) \cong \operatorname { Aut } ( L ( G ) ) \quad \text { and } \quad L ( \operatorname { Aut } ( G ) ) \cong D ( L ( G ) )$ $$\operatorname {Aut}(G)\cong \operatorname {Aut}(L(G))\quad {\rm and }\quad L (\operatorname {Aut}(G))\cong D (L(G)),$$ conf 0.693 F

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79.(50.) l05859086.png $( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$ $$(X,Y)\rightarrow \operatorname {exp}^{-1}(\operatorname {exp}X\operatorname {exp}Y),\quad X ,Y\in L (G),$$ conf 0.856

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Lie group, compact

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

l05861012.png (12)

Lie group, nilpotent

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

l0586604.png (4)

Lie group, semi-simple

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82.(35.)* l058680102.png $L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$ $$L(\mathfrak g)\cong \Gamma _ 0(\mathfrak u)\cap \mathfrak h^{\prime }/\Gamma _ 0([\mathfrak k,\mathfrak k])$$ conf 0.659 F

l058680102.png (102)

83.(81.)* l05868032.png $\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$ $$\Gamma _ 1=\Gamma _ 1(g)=\{X\in h :\alpha (X)\in 2 \pi i {\mathbf Z}\;\text{for all }\alpha \in \Sigma \}.$$ conf 0.183 F

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Lie p-algebra

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84.(36.) l05872026.png $( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$ $$(\operatorname {ad}x)^ny=\sum _ {j=1}^n(-1)^j\left(\begin {array}cn\\ j \end {array} \right)x^{n-j}yx^j$$ conf 0.356

l05872026.png (26)

85.(99.) l05872078.png $\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$ $$\pi (x+y)=\pi (x)+\pi (y),\quad \pi (\lambda x )=\lambda ^p\pi (x),\quad \lambda \in k .$$ conf 0.964

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Lie theorem

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86.(134.) l05876010.png $y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$ $$y_i=f_i(g_1,\ldots ,g_i;x_1,\ldots ,x_n),\quad i =1,\ldots ,n$$ conf 0.276

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87.(86.) l05876016.png $X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$ $$X_i=\sum _ {j=1}^n\xi _ {ij}(x)\frac {\partial }{\partial x _ j},\quad i =1,\ldots ,r,$$ conf 0.656

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88.(66.)* l05876030.png $\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$ $$\frac {\partial f _ j}{\partial g _ i}(g,x)=\sum _ {k=1}^r\xi _ {kj}(f(g_sx))\psi _ {ki}(g),$$ conf 0.336 F

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89.(19.)* l05876037.png $\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ $$\sum _ {k=1}^N(\xi _ {ik}\frac {\partial \xi _ {jl}}{\partial x _ k}-\xi _ {jk}\frac {\partial \xi _ {il}}{\partial x _ k})=\sum _ {k=1}^rc_{ij}^k\xi _ {kl},$$ conf 0.157 F

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90.(14.) l05876052.png $\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$ $$\left.\begin {array}c{c_{ij}^k=-c_{ji}^k},\\ {\displaystyle\sum _ {l=1}^r(c_{il}^mc_{jk}^l+c_{kl}^mc_{ij}^l+c_{jl}^mc_{ki}^l)=0,\quad 1 \leq i ,j,k,l,m\leq r,} \end {array} \right\}$$ conf 0.085

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Maximal torus

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

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Non-Abelian cohomology

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92.(114.)* n066900110.png $\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ $$\phi (g_1)\phi (g_2)\phi (g_1g_2)^{-1}=\operatorname {Int}m(g_1,g_2),$$ conf 0.443 F

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93.(90.)* n066900118.png $( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$ $$m'(g_1,g_2)=h(g_1)(\phi (g_1)(h(g_2)))m(g_1,g_2)h(g_1,g_2)^{-1}.$$ conf 0.764 F

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94.(44.) n06690016.png $\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$ $$\delta (e)=e\quad \;\text{and }\quad \delta (\rho (a)b)=\sigma (a)\delta (b),\quad \alpha \in C ^0,\quad b \in C ^1,$$ conf 0.400

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95.(60.)* n06690028.png $C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$ $$C^{*}(\mathfrak U,{\cal F})=(C^0(\mathfrak U,{\cal F}),C^1(\mathfrak U,{\cal F}),C^2(\mathfrak U,{\cal F})),$$ conf 0.205 F

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Picard scheme

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

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Principal analytic fibration

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

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Quantum groups

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98.(101.) q07631062.png $\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$ $$\phi ^{*}:\mathfrak g^{*}\otimes \mathfrak g^{*}\rightarrow \mathfrak g^{*}$$ conf 0.837

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99.(108.) q07631071.png $\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$ $$\delta :U_{\mathfrak g}\rightarrow U _ {\mathfrak g}\otimes U _ {\mathfrak g}$$ conf 0.648

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100.(56.)* q07631072.png $\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$ $$\delta (\alpha )=\operatorname {lim}_{h\rightarrow 0 }h^{-1}(\Delta (a)-\Delta ^{\prime }(\alpha ))$$ conf 0.304 F

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101.(129.)* q07631088.png $[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$ $$[\alpha ,X_i^{\pm }]=\pm \alpha _ i(\alpha )X_i^{\pm }\quad \text{for }a\in \mathfrak h;$$ conf 0.544 F

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102.(128.) q07631089.png $[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$ $$[X_i^{+},X_j^{-}]=2\delta _ {ij}h^{-1}\operatorname {sinh}(hH_i/2).$$ conf 0.893

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103.(20.) q07631092.png $\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$ $$\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\ k \end {array} \right)q^{-k(n-k)/2}(X_i^{\pm })^kX_j^{\pm }\cdot (X_i^{\pm })^{n-k}=0.$$ conf 0.055

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104.(30.) q07631095.png $\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$ $$\left( \begin {array}ln\\ k \end {array} \right)_q=\frac {(q^n-1)\ldots (q^{n-k+1}-1)}{(q^k-1)\ldots (q-1)} .$$ conf 0.443

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105.(21.)* q07631099.png $\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$ $$\Delta (X_i^{\pm })=X_i^{\pm }\otimes \operatorname {exp}(\frac {hH_i}4)+\operatorname {exp}(\frac {-hH_i}4)\otimes X _ i^{\pm }.$$ conf 0.212 F

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Rational representation

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106.(91.) r077630100.png $0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$ $$0\leq \frac {2(\chi ,\alpha )}{(\alpha ,\alpha )}<p\quad \text{for all }\alpha \in \Delta.$$ conf 0.879

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107.(135.) r077630104.png $\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$ $$\phi _ 0\otimes \phi _ 1^{Fr}\otimes \ldots \otimes \phi _ d^{{Fr}^d},$$ conf 0.136

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108.(45.)* r07763055.png $\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$ $$\chi =\delta _ {\phi }-\sum _ {\alpha \in \Delta }m_{\alpha }\alpha ,\quad m _ {\alpha }\in Z ,\quad m _ {\alpha }\geq 0.$$ conf 0.862 F

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Singular point

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109.(31.) s085590225.png $\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ $$\sum _ {k_1,\ldots ,k_n=0}^{\infty }c_{k_1\cdots k _ n}(z_1-\zeta _ 1)^{k_1}\ldots (z_n-\zeta _ n)^{k_n}$$ conf 0.324

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110.(46.) s085590404.png $\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$ $$\frac {m_1}{n_1}<\frac {m_2}{n_1n_2}<\ldots <\frac {m_g}{n_1\ldots n _ g}=\frac {m_g}n$$ conf 0.459

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111.(115.)* s085590429.png $p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$ $$p(Z)=1-\operatorname {dim}H^0({\mathbf Z},{\cal O}_{\mathbf Z })+\operatorname {dim}H^1({\mathbf Z},{\cal O}_{\mathbf Z })$$ conf 0.997 F

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112.(136.)* s085590440.png $X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$ $$X_{\epsilon }=\{(x_0,\ldots ,x_x):f(x_0,\ldots ,x_x)=\epsilon \}$$ conf 0.433 F

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113.(12.) s085590458.png $= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$ $$=\left\{ \begin {array}{ll} {(x+\lambda )^2\ldots (x+k\lambda )^2} &{\text{ if }\mu =2k,}\\ {(x+\lambda )^2\ldots (x+k\lambda )^2(x+(k+1)\lambda )} &{\text{ if }\mu =2k+1,} \end {array} \right.$$ conf 0.870

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114.(75.) s085590482.png $( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$ $$\big(\frac {\partial F (x,y,\lambda )}{\partial x },\frac {\partial F (x,y,\lambda )}{\partial y }\big)$$ conf 0.986

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115.(137.) s085590515.png $\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$ $$\frac {dx_i}{dx_{i_0}}=f_i(x),\quad f _ i\in C (U),\quad i \neq i _ 0.$$ conf 0.594

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116.(142.)* s085590527.png $A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$ $$A=\left\| \begin {array}{ll} {\alpha } &b\\ c &e \end {array} \right\|$$ conf 0.506 F

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117.(53.) s085590634.png $\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$ $$\Delta =(F_{xx}^{\prime \prime })_0(F_{yy}^{\prime \prime })_0-(F_{xy}^{\prime \prime })_0^2$$ conf 0.920

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118.(16.)* s085590645.png $\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$ $$\left\| \begin {array}{lll} {F_x^{\prime }} &{F_y^{\prime }} &{F_z^{\prime }}\\ {G_x^{\prime }} &{G_y^{\prime }} &{G_Z^{\prime }} \end {array} \right\|$$ conf 0.230 F

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119.(92.) s085590653.png $( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$ $$(F_x^{\prime })_0=0,\quad (F_y^{\prime })_0=0,\quad (F_z^{\prime })_0=0.$$ conf 0.300

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Solv manifold

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

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Stability theorems in algebraic K-theory

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

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Steinberg module

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

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Steinberg symbol

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

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Tilting theory

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

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Tits quadratic form

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125.(18.) t130140104.png $q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$ $$q_R(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}x_ix_j+\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}r_{i,j}x_ix_j,$$ conf 0.112

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126.(40.) t130140118.png $[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$ $$[X]\mapsto \chi _ R([X])=\sum _ {m=0}^{\infty }(-1)^m\operatorname {dim}_K\operatorname {Ext}_R^m(X,X)$$ conf 0.116

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127.(132.)* t130140119.png $\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$ $$\underline {\dim }:K_0(\operatorname {mod}R)\rightarrow {\mathbf Z}^{Q_0}$$ conf 0.287 F

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128.(37.)* t130140140.png $q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$ $$q_I(x)=\sum _ {i\in I }x_i^2+\sum _ {i\prec j \atop j\in I\setminus {\rm max}I}x_ix_j-\sum _ {p\in \operatorname {max}I}\big(\sum _ {i\prec p }x_i\big)x_p$$ conf 0.197 F

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129.(131.)* t13014044.png $X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ $$X\mapsto \underline {\dim }X=(\operatorname {dim}_KX_j)_{j\in Q _ 0}$$ conf 0.819 F

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130.(25. t13014048.png $[ X ] \mapsto \chi _ { Q } ( [ X ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( X ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( X , X )$ $$[X]\mapsto \chi _ Q([X])=\operatorname {dim}_K\operatorname {End}_Q(X)-\operatorname {dim}_K\operatorname {Ext}_Q^1(X,X)$$ conf 0.661

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131.(38.)* t13014056.png $A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$ $$A_Q(v)=\prod _ {i,j\in Q _ 0}\prod _ {\langle \beta :j\rightarrow i \rangle \in Q _ 1}M_{v_i\times v _ j}(K)_{\beta }$$ conf 0.481 F

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132.(139.)* t1301406.png $\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$ $$q_Q(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {i,j\in Q _ 0}d_{ij}x_ix_j,$$ conf 0.648 F

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Torus

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133.(41.)* t0933502.png $r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$ $$r=\alpha \operatorname {sin}u{\bf k}+l(1+\epsilon \operatorname {cos}u)({\bf i}\operatorname {cos}v+{\bf j}\operatorname {sin}v)$$ conf 0.585 F

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134.(122.)* t0933507.png $d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$ $$ds^2=\alpha ^2du^2+l^2(1+\epsilon \operatorname {cos}u)^2dv^2,$$ conf 0.696 F

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Uniform distribution

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135.(9.) u09524027.png $u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$ $$u_3(x)=\left\{ \begin {array}{ll} {\frac {x^2}2,} &{0\leq x <1,}\\ {\frac {[x^2-3(x-1)^2]}2,} &{1\leq x <2,}\\ {\frac {[x^2-3(x-1)^2+3(x-2)^2]}2,} &{2\leq x <3,}\\ {0,} &{x\notin [0,3].} \end {array} \right.$$ conf 0.733

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136.(32.)* u0952403.png $p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$ $$p(x)=\left\{ \begin {array}{ll} {\frac 1{b-\alpha },} &{x\in [\alpha ,b],}\\ {0,} &{x\notin [\alpha ,b].} \end {array} \right.$$ conf 0.681 F

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137.(34.) u09524030.png $u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$ $$u_n(x)=\frac 1{(n-1)!}\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\ k \end {array} \right)(x-k)_{+}^{n-1}$$ conf 0.569

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138.(109.) u09524034.png $z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$ $$z_{+}=\left\{ \begin {array}{ll} {z,} &{z>0}.\\ {0,} &{z\leq 0 }. \end {array} \right.$$ conf 0.676

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139.(43.) u0952407.png $F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$ $$F(x)=\left\{ \begin {array}{ll} {0,} &{x\leq a },\\ {\frac {x-a}{b-a},} &{a<x\leq b },\\ {1,} &{x>b}, \end {array} \right.$$ conf 0.468

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140.(47.) u09524072.png $p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$ $$p(x_1,\ldots ,x_n)=\left\{ \begin {array}{ll} {C\neq 0 ,} &{x\in D },\\ {0,} &{x\notin D }, \end {array} \right.$$ conf 0.705

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Unipotent group

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

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Weyl module

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142.(51.) w120090122.png $\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$ $$\operatorname {diag}(t_1,\ldots ,t_n)\mapsto t _ 1^{\lambda _ 1}\ldots t _ n^{\lambda _ n}\in K,$$ conf 0.507

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143.(54.)* w120090135.png $\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$ $$\chi _ {\lambda }=\sum _ {\mu \in \Lambda (n)}\operatorname {dim}_K(\Delta (\lambda )^{\mu })_{e_{\mu }},$$ conf 0.461 F

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144.(110.) w120090259.png $\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$ $$\mathfrak B=\{e_{\pm }\alpha ,h_{\beta }:\alpha \in \Phi ^{+},\beta \in \Sigma \}.$$ conf 0.381

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145.(82.) w120090342.png $\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$ $$\left( \begin {array}ch\\ i \end {array} \right)=\frac {h(h-1)\ldots (h-i+1)}{i!} $$ conf 0.487

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146.(28.)* w12009095.png $\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$ $$\mathfrak S_{\{1,\ldots ,\lambda _ 1\}}\times \mathfrak S_{\{\lambda _ 1+1,\ldots ,\lambda _ 1+\lambda _ 2\}}\times \dots $$ conf 0.312 F

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147.(104.) w12009096.png $\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$ $$\ldots \times \mathfrak S_{\{\lambda _ 1+\ldots +\lambda _ {n-1}+1,\ldots ,r\}},$$ conf 0.259

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Witt vector

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148.(87.)* w098100172.png $\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$ $$\langle \alpha ><b\rangle =\langle \alpha b \rangle ,\quad \langle {\bf 1}\rangle ={\bf f}_1={\bf V}_1=\text{ unit element}1,$$ conf 0.351 F

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149.(123.)* w098100177.png $\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$ $$\langle \alpha +b\rangle =\sum _ {n=1}^{\infty }{\bf V}_n\langle r _ n(\alpha ,b){\bf f}_n.$$ conf 0.143 F

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150.(102.) w098100190.png $\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$ $$\sigma (\alpha _ 1,\alpha _ 2,\ldots )=(\alpha _ 1^p,\alpha _ 2^p,\ldots )$$ conf 0.771

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How to Cite This Entry:
Ulf Rehmann/Table of automatically generated TeX code. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ulf_Rehmann/Table_of_automatically_generated_TeX_code&oldid=44218