Namespaces
Variants
Actions

User:Ulf Rehmann/Table of automatically generated TeX code

From Encyclopedia of Mathematics
Jump to: navigation, search

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

j0543403.png (3)

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

j05434030.png (30)

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

j0543406.png (6)

Lie algebra, semi-simple

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

l058510127.png (127)

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

l058510129.png (129)

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

l058510130.png (130)

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

l058510131.png (131)

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

l058510132.png (132)

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

l058510133.png (133)

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

l05851037.png (37)

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

l05851044.png (44)

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

l05851050.png (50)

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

l05851051.png (51)

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

l05851057.png (57)

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

l05851064.png (64)

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

l05851085.png (85)

Lie algebra, solvable

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

l05852011.png (11)

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

l05852046.png (46)

Lie group

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

l058590115.png (115)

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

l05859086.png (86)

Lie group, compact

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

l05868032.png (32)

Lie p-algebra

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

l05872078.png (78)

Lie theorem

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

l05876010.png (10)

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

l05876016.png (16)

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

l05876030.png (30)

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

l05876037.png (37)

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

l05876052.png (52)

Maximal torus

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

m06301072.png (72)

Non-Abelian cohomology

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

n066900110.png (110)

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

n066900118.png (118)

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

n06690016.png (16)

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

n06690028.png (28)

Picard scheme

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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 +

p07267025.png (25)

Principal analytic fibration

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

p07464025.png (25)

Quantum groups

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

q07631062.png (62)

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

q07631071.png (71)

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

q07631072.png (72)

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

q07631088.png (88)

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

q07631089.png (89)

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

q07631092.png (92)

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

q07631095.png (95)

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

q07631099.png (99)

Rational representation

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

r077630100.png (100)

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

r077630104.png (104)

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

r07763055.png (55)

Singular point

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

s085590225.png (225)

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

s085590404.png (404)

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

s085590429.png (429)

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

s085590440.png (440)

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

s085590458.png (458)

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

s085590482.png (482)

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

s085590515.png (515)

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

s085590527.png (527)

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

s085590634.png (634)

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

s085590645.png (645)

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

s085590653.png (653)

Solv manifold

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

s08610054.png (54)

Stability theorems in algebraic K-theory

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

s08706033.png (33)

Steinberg module

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

s13053016.png (16)

Steinberg symbol

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

s13054017.png (17)

Tilting theory

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

t130130105.png (105)

Tits quadratic form

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

t130140104.png (104)

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

t130140118.png (118)

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

t130140119.png (119)

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

t130140140.png (140)

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

t13014044.png (44)

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

t13014048.png (48)

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

t13014056.png (56)

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

t1301406.png (6)

Torus

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

t0933502.png (2)

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

t0933507.png (7)

Uniform distribution

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

u09524027.png (27)

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

u0952403.png (3)

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

u09524030.png (30)

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

u09524034.png (34)

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

u0952407.png (7)

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

u09524072.png (72)

Unipotent group

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

u0954106.png (6)

Weyl module

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

w120090122.png (122)

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

w120090135.png (135)

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

w120090259.png (259)

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

w120090342.png (342)

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

w12009095.png (95)

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

w12009096.png (96)

Witt vector

Nr. Image of png File $\TeX$, automatically generated version $\TeX$, manually corrected version Confidence, F?

png file

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

w098100172.png (172)

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

w098100177.png (177)

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

w098100190.png (190)

How to Cite This Entry:
Ulf Rehmann/Table of automatically generated TeX code. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Ulf_Rehmann/Table_of_automatically_generated_TeX_code&oldid=44219