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− | {| class="wikitable | + | This page gives an analysis of [[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|the code here]], [[User:Maximilian Janisch/latexlist|generated automatically from some png files underlying our old wiki pages]]. |
− | !| Nr. | + | As this page does contain a lot of $\TeX$ code, it loads slowly. |
− | !| Image of png File | + | |
− | !| $\TeX$, | + | 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). |
− | !| $\TeX$, | + | The first column gives the running number in this table, followed (in parentheses) by the number used [[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E| here]]. |
− | !| Confidence, F? | + | 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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 1.(23.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011450/a01145065.png || $g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$ || | + | | 1.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|23.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011450/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 2.(116.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150014.png || $\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ || | + | | 2.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|116.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/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.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150021.png || $\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ || | + | | 3.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|133.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/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.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150022.png || $\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ || $$\widetilde | + | | 4.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|67.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150022.png | ||
+ | || $\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ | ||
+ | ||$$\widetilde w=2\int\limits _ 0^{1/\varepsilon }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ | ||
+ | || conf 0.107 | ||
− | + | a01150022.png (22) | |
|- | |- | ||
− | | 5.(105.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150044.png || $\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$ || | + | | 5.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|105.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150044.png | ||
+ | || $\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$ | ||
+ | ||$$\theta (v+\pi i r )=\theta (r),\quad \theta (v+\alpha _ j)=e^{L_j(v)}\theta (v),$$ | ||
+ | || conf 0.775 | ||
− | + | a01150044.png (44) | |
|- | |- | ||
− | | 6.(17.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150078.png || $\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ || | + | | 6.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|17.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150078.png | ||
+ | || $\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ | ||
+ | ||$$\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).$$ | ||
+ | || conf 0.440 | ||
− | + | a01150078.png (78) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Algebraic surface]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 7.(144.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640132.png || $0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$ || $$0 \rightarrow {\cal O} | + | | 7.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|144.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/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.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640137.png || $M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$ || | + | | 8.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|73.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/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.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640139.png || $\operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } )$ || | + | | 9.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|88.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/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.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164027.png || $N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$ || | + | | 10.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|117.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/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.) || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164029.png || $p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$ || | + | | 11.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|72.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/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.)*|| https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164047.png || $p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$ || | + | | 12.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|68.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/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.)*|| https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164053.png || $1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$ || | + | | 13.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|93.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 14.(33.)*|| https://www.encyclopediaofmath.org/legacyimages/c/c020/c020550/c0205509.png || $\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$ || | + | | 14.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|33.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/c/c020/c020550/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 15.(49.)*|| https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205704.png || $f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$ || | + | | 15.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|49.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/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.)*|| https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057064.png || $\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$ || | + | | 16.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|55.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 17.(7.) || https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333033.png || $H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$ || | + | | 17.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|7.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/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.) || https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333034.png || $= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$ || | + | | 18.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|76.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/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.)*|| https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333035.png || $( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$ || | + | | 19.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|11.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 20.(26.) || https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700175.png || $\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$ || | + | | 20.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|26.]]) |
− | + | || https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/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.) || https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700190.png || $\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$ || | + | | 21.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|27.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/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.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700263.png || $\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$ || | + | | 22.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|78.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/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.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700270.png || $\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$ || | + | | 23.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|96.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 24.(106.) || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830107.png || $S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$ || | + | | 24.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|106.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/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.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830141.png || $( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$ || | + | | 25.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|146.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/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.)$^F$*|| https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830150.png || $( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$ || | + | | 26.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|145.]])$^F$* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/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.) || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d03183016.png || $\omega _ { V } = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ || | + | | 27.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|57.]]) |
+ | |||
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/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.) || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d03183043.png || $e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$ || | + | | 28.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|111.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 29.(48.) || https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249029.png || $\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ || | + | | 29.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|48.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 30.(118.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120173.png || $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$ || | + | | 30.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|118.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120175.png || $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow H _ { c } ^ { n } ( X , \Omega )$ || | + | | 31.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|59.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120184.png || $( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$ || | + | | 32.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|124.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120236.png || $\beta : \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X F , \Omega ) \rightarrow \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X \backslash Y || F , \Omega )$ || | + | | 33.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|29.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120247.png || $\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$ || | + | | 34.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|77.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120253.png || $h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$ || | + | | 35.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|58.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120360.png || $\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$ || | + | | 36.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|69.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.) || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120376.png || $\operatorname { sup } _ { f \in B ^ { 1 } } | \int _ { \partial G } f ( \zeta ) \omega ( \zeta ) d \zeta | = \operatorname { inf } _ { \phi \in E ^ { 1 } } \int _ { \partial G } | \omega ( \zeta ) - \phi ( \zeta ) \| d \zeta |$ || | + | | 37.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|15.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.) || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120509.png || $f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$ || | + | | 38.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|52.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.) || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120535.png || $f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$ || | + | | 39.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|140.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.) || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120555.png || $f _ { 0 } ( x ) \rightarrow \text { inf, } \quad f _ { i } ( x ) \leq 0 , \quad i = 1 , \ldots , m , \quad x \in B$ || | + | | 40.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|94.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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.)*|| https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d03412079.png || $( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$ || | + | | 41.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|74.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 42.(63.) || https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696024.png || $F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$ || | + | | 42.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|63.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 43.(120.)*|| https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f040820118.png || $\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$ || | + | | 43.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|120.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/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.)*|| https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f04082059.png || $( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$ || | + | | 44.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|147.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 45.(148.) || https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300205.png || $\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$ || | + | | 45.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|148.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 46.(22.)*|| https://www.encyclopediaofmath.org/legacyimages/g/g045/g045210/g04521075.png || $\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$ || | + | | 46.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|22.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/g/g045/g045210/g04521075.png | ||
+ | || $\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$ | ||
+ | | style="text-align:center;"| source incomplete | ||
+ | || conf 0.226 F | ||
− | + | g04521075.png (75) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Homogeneous space]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 47.(89.) || https://www.encyclopediaofmath.org/legacyimages/h/h047/h047690/h04769069.png || $\mathfrak { g } = \mathfrak { f } + \mathfrak { m } , \quad \mathfrak { f } \cap \mathfrak { m } = \{ 0 \}$ || | + | | 47.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|89.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/h/h047/h047690/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 48.(103.) || https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970129.png || $m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$ || | + | | 48.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|103.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/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.)*|| https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970139.png || $F _ { 1 } ( X || Y ) , \ldots , F _ { n } ( X || Y ) \in K [ X _ { 1 } , \ldots , X _ { n } || Y _ { 1 } , \ldots , Y _ { n } ] \}$ || | + | | 49.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|107.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/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.) || https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797042.png || $\epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g }$ || | + | | 50.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|97.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 51.(149.)*|| https://www.encyclopediaofmath.org/legacyimages/i/i052/i052350/i05235015.png || $\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$ || | + | | 51.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|149.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/i/i052/i052350/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 52.(150.) || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427030.png || $H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$ || | + | | 52.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|150.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/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.) || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427031.png || $\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$ || | + | | 53.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|42.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/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.)*|| https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427077.png || $\mathfrak { g } = \mathfrak { g } - 1 + \mathfrak { g } \mathfrak { d } + \mathfrak { g } _ { 1 }$ || | + | | 54.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|125.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 55.(6.)*|| https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543403.png || $J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$ || | + | | 55.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|6.]])* |
− | J_{n_1}(\lambda_1) | + | || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543403.png |
− | + | || $J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$ | |
− | + | ||$$J=\left\| | |
− | + | \begin {array}{cccc} | |
− | \end{array} \right\|,$$ || conf 0.072 F | + | |
+ | 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\|,$$ | ||
+ | || conf 0.072 F | ||
− | + | j0543403.png (3) | |
|- | |- | ||
− | | 56.(64.) || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434030.png || $C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$ || | + | | 56.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|64.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434030.png | ||
+ | || $C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$ | ||
+ | ||$$C_m(\lambda )=\operatorname {rk}(A-\lambda E )^{m-1}-2\operatorname {rk}(A-\lambda E )^m+$$ | ||
+ | || conf 0.955 | ||
− | + | j05434030.png (30) | |
|- | |- | ||
− | | 57.(1.)*|| https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543406.png || $J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$ || $$J_m(\lambda) = \left\| \begin{array} { | + | | 57.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|1.]])* |
− | + | || https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 58.(5.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510127.png || $\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$ || $$B_n: \quad \left\| \begin{array} { | + | | 58.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|5.]]) |
− | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510127.png | |
− | { - 1 } & | + | || $\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$ |
− | + | ||$$B_n:\quad \left\| | |
− | \cdot | + | \begin {array}{rrrrrr} |
− | + | ||
− | + | 2 &{-1} &0 &{\dots } &0 &0\\ | |
− | + | ||
− | \end{array} \right\|,$$|| conf 0.232 | + | {-1} &2 &{-1} &{\dots } &0 &0\\ |
+ | |||
+ | 0 &{-1} &2 &{\dots } &0 &0\\ | ||
+ | |||
+ | \cdot &\cdot &\cdot &\dots &\cdot &\cdot \\ | ||
+ | |||
+ | 0 &0 &0 &{\dots } &{-1} &0\\ | ||
+ | |||
+ | 0 &0 &0 &{\dots } &2 &{-2}\\ | ||
+ | |||
+ | 0 &0 &0 &{\dots } &{-1} &2 | ||
+ | |||
+ | \end {array} | ||
+ | \right\|,$$ | ||
+ | || conf 0.232 | ||
− | + | l058510127.png (127) | |
|- | |- | ||
− | | 59.(3.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510129.png || $\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$ || | + | | 59.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|3.]])* |
− | \left\| \begin{array} { | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510129.png |
− | + | || $\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$ | |
− | { - 1 } & | + | ||$$D_n:\quad \left\| |
− | + | \begin {array}{rrrrrrr} | |
− | \cdot | + | |
− | + | 2 &{-1} &0 &{\dots } &0 &0 &0 &0\\ | |
− | + | ||
− | + | {-1} &2 &{-1} &{\dots } &0 &0 &0 &0\\ | |
− | + | ||
− | \end{array} \right\|,$$ || conf 0.055 F | + | 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\|,$$ | ||
+ | || conf 0.055 F | ||
− | + | l058510129.png (129) | |
|- | |- | ||
− | | 60.(8.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510130.png || $\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ || $$E_6: | + | | 60.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|8.]])* |
− | \quad \left\| \begin{array} { | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510130.png |
− | + | || $\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ | |
− | + | ||$$E_6: | |
− | { - 1 } & | + | \quad \left\| |
− | + | \begin {array}{rrrrrr} | |
− | + | ||
− | + | 2 &0 &{-1} &0 &0 &0\\ | |
− | \end{array} \right\|,$$ || conf 0.628 F | + | |
+ | 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\|,$$ | ||
+ | || conf 0.628 F | ||
− | + | l058510130.png (130) | |
|- | |- | ||
− | | 61.(4.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510131.png || $\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ || $$E_7: \quad | + | | 61.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|4.]]) |
− | \left\| \begin{array} { | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510131.png |
− | + | || $\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ | |
− | + | ||$$E_7:\quad \left\| | |
− | {-1 } & | + | \begin {array}{rrrrrrr} |
− | + | ||
− | + | 2 &0 &{-1} &0 &0 &0 &0\\ | |
− | + | ||
− | + | 0 &2 &0 &{-1} &0 &0 &0\\ | |
− | + | ||
+ | {-1} &0 &2 &{-1} &0 &0 &0\\ | ||
+ | |||
+ | 0 &{-1} &{-1} &2 &{-1} &0 &0\\ | ||
+ | |||
+ | 0 &0 &0 &{-1} &2 &{-1} &0\\ | ||
+ | |||
+ | 0 &0 &0 &0 &{-1} &2 &{-1}\\ | ||
+ | |||
+ | 0 &0 &0 &0 &0 &{-1} &2 | ||
− | + | \end {array} | |
+ | \right\|,$$ | ||
+ | || conf 0.278 | ||
+ | |||
+ | l058510131.png (131) | ||
|- | |- | ||
− | | 62.(2.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510132.png || $\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$ || $$E_8: \quad | + | | 62.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|2.]])* |
− | \left\| \begin{array} { | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510132.png |
− | + | || $\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$ | |
− | + | ||$$E_8:\quad \left\| | |
− | {-1 } & | + | \begin {array}{rrrrrrrr} |
− | + | ||
− | + | 2 &0 &{-1} &0 &0 &0 &0 & | |
− | + | 0\\ | |
− | + | 0 &2 &0 &{-1} &0 &0 &0 &0\\ | |
− | + | ||
− | \end{array} \right\|,$$ || conf 0.354 F | + | {-1} &0 &2 &{-1} &0 &0 &0 &0\\ |
+ | |||
+ | 0 &{-1} &{-1} &2 &{-1} &0 &0 &0\\ | ||
+ | |||
+ | 0 &0 &0 &{-1} &2 &{-1} &0 &0\\ | ||
+ | |||
+ | 0 &0 &0 &0 &{-1} &2 &{-1} &0\\ | ||
+ | |||
+ | 0 &0 &0 &0 &0 &{-1} &2 &{-1}\\ | ||
+ | |||
+ | 0 &0 &0 &0 &0 &0 &{-1} &2 | ||
+ | |||
+ | \end {array} | ||
+ | \right\|,$$ | ||
+ | || conf 0.354 F | ||
− | + | l058510132.png (132) | |
|- | |- | ||
− | | 63.(10.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510133.png || $\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$ | + | | 63.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|10.]])* |
− | || $$F_4: \quad \left\| \begin{array} { | + | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510133.png |
+ | || $\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$ | ||
+ | |||
+ | ||$$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\|.$$ | ||
+ | || conf 0.374 F | ||
− | + | l058510133.png (133) | |
|- | |- | ||
− | | 64.(98.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851030.png || $\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$ || | + | | 64.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|98.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851030.png | ||
+ | || $\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$ | ||
+ | ||$$\mathfrak g_{\alpha }=\{X\in \mathfrak g:[H,X]=\alpha (H)X,H\in \mathfrak h\}.$$ | ||
+ | || conf 0.976 | ||
− | + | l05851030.png (30) | |
|- | |- | ||
− | | 65.(126.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851037.png || $\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$ || | + | | 65.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|126.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851037.png | ||
+ | || $\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$ | ||
+ | ||$$\mathfrak g=\mathfrak h+\sum _ {\alpha \in \Sigma }\mathfrak g_{\alpha }.$$ | ||
+ | || conf 0.945 | ||
− | + | l05851037.png (37) | |
|- | |- | ||
− | | 66.(61.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851044.png || $\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$ || | + | | 66.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|61.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851044.png | ||
+ | || $\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$ | ||
+ | ||$$\mathfrak g_{\alpha }=\operatorname {dim}[\mathfrak g_{\alpha },\mathfrak g_{-\alpha }]=1.$$ | ||
+ | || conf 0.520 F | ||
− | + | l05851044.png (44) | |
|- | |- | ||
− | | 67.(65.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851050.png || $[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$ || | + | | 67.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|65.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851050.png | ||
+ | || $[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$ | ||
+ | ||$$[H_{\alpha },X_{\alpha }]=2X_{\alpha }\quad {\rm and }\quad [H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }.$$ | ||
+ | || conf 0.539 F | ||
− | + | l05851050.png (50) | |
|- | |- | ||
− | | 68.(70.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851051.png || $\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$ || | + | | 68.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|70.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851051.png | ||
+ | || $\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$ | ||
+ | ||$$\beta (H_{\alpha })=\frac {2(\alpha ,\beta )}{(\alpha ,\alpha )},\quad \alpha ,\beta \in \Sigma,$$ | ||
+ | || conf 0.997 | ||
− | + | l05851051.png (51) | |
|- | |- | ||
− | | 69.(112.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851057.png || $[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$ || | + | | 69.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|112.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851057.png | ||
+ | || $[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$ | ||
+ | ||$$[\mathfrak g_{\alpha },\mathfrak g_{\beta }]=\mathfrak g_{\alpha +\beta }$$ | ||
+ | || conf 0.917 | ||
− | + | l05851057.png (57) | |
|- | |- | ||
− | | 70.(127.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851064.png || $H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$ || | + | | 70.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|127.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851064.png | ||
+ | || $H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$ | ||
+ | ||$$H_{\alpha _ 1},\ldots ,H_{\alpha _ k},X_{\alpha }\quad (\alpha \in \Sigma )$$ | ||
+ | || conf 0.432 | ||
− | + | l05851064.png (64) | |
|- | |- | ||
− | | 71.(113.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851069.png || $[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$ || | + | | 71.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|113.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851069.png | ||
+ | || $[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$ | ||
+ | ||$$[[X_{\alpha _ i},X_{-}\alpha _ i],X_{-\alpha _ j}]=-n(i,j)X_{\alpha _ j},$$ | ||
+ | || conf 0.628 F | ||
− | + | l05851069.png (69) | |
|- | |- | ||
− | | 72.(79.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851073.png || $n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$ || | + | | 72.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|79.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851073.png | ||
+ | || $n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$ | ||
+ | ||$$n(i,j)=\alpha _ j(H_i)=\frac {2(\alpha _ i,\alpha _ j)}{(\alpha _ j,\alpha _ j)}.$$ | ||
+ | || conf 0.992 | ||
− | + | l05851073.png (73) | |
|- | |- | ||
− | | 73.(13.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851074.png || $[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$ || | + | | 73.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|13.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851074.png | ||
+ | || $[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$ | ||
+ | ||$$[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.$$ | ||
+ | || conf 0.988 | ||
− | + | l05851074.png (74) | |
|- | |- | ||
− | | 74.(80.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851078.png || $N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$ || | + | | 74.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|80.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851078.png | ||
+ | || $N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$ | ||
+ | ||$$N_{\alpha ,\beta }=-N_{-\alpha ,-\beta }\quad {\rm and }\quad N _ {\alpha ,\beta }=\pm (p+1),$$ | ||
+ | || conf 0.961 | ||
− | + | l05851078.png (78) | |
|- | |- | ||
− | | 75.(85.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851085.png || $X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$ || | + | | 75.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|85.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851085.png | ||
+ | || $X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$ | ||
+ | ||$$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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 76.(119.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852011.png || $[ \mathfrak { g } _ { i } , \mathfrak { g } _ { i } ] \subset \mathfrak { g } _ { \mathfrak { i } } + 1$ || | + | | 76.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|119.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/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.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852046.png || $\operatorname { dim } \mathfrak { g } _ { i } = \operatorname { dim } \mathfrak { g } - i$ || | + | | 77.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|141.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/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]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 78.(62.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l058590115.png || $( G ) \cong \operatorname { Aut } ( L ( G ) ) \quad \text { and } \quad L ( \operatorname { Aut } ( G ) ) \cong D ( L ( G ) )$ || | + | | 78.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|62.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/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)),$$ | ||
+ | || conf 0.693 F | ||
− | + | l058590115.png (115) | |
|- | |- | ||
− | | 79.(50.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l05859086.png || $( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$ || | + | | 79.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|50.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l05859086.png | ||
+ | || $( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$ | ||
+ | ||$$(X,Y)\rightarrow \operatorname {exp}^{-1}(\operatorname {exp}X\operatorname {exp}Y),\quad X ,Y\in L (G),$$ | ||
+ | || conf 0.856 | ||
− | + | l05859086.png (86) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Lie group, compact]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 80.(121.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058610/l05861012.png || $J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$ || | + | | 80.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|121.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058610/l05861012.png | ||
+ | || $J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$ | ||
+ | ||$$J=\left\| | ||
+ | \begin {array}{cc} | ||
+ | 0 &{E_x}\\ | ||
+ | {-E_x} &0 | ||
+ | \end {array} | ||
+ | \right\|,$$ | ||
+ | || conf 0.364 F | ||
− | + | l05861012.png (12) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Lie group, nilpotent]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 81.(83.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586604.png || $N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$ || | + | | 81.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|83.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586604.png | ||
+ | || $N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$ | ||
+ | ||$$N(F)=\{g\in GL (V):gv\equiv v \operatorname {mod}V_i\;\text{for all }v\in V _ i,\;i\geq 1 \}$$ | ||
+ | || conf 0.466 | ||
− | + | l0586604.png (4) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Lie group, semi-simple]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 82.(35.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680102.png || $L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$ || | + | | 82.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|35.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680102.png | ||
+ | || $L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$ | ||
+ | ||$$L(\mathfrak g)\cong \Gamma _ 0(\mathfrak u)\cap \mathfrak h^{\prime }/\Gamma _ 0([\mathfrak k,\mathfrak k])$$ | ||
+ | || conf 0.659 F | ||
− | + | l058680102.png (102) | |
|- | |- | ||
− | | 83.(81.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868032.png || $\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$ || | + | | 83.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|81.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868032.png | ||
+ | || $\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$ | ||
+ | ||$$\Gamma _ 1=\Gamma _ 1(g)=\{X\in h :\alpha (X)\in 2 \pi i {\mathbf Z}\;\text{for all }\alpha \in \Sigma \}.$$ | ||
+ | || conf 0.183 F | ||
− | + | l05868032.png (32) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Lie p-algebra]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 84.(36.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872026.png || $( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$ || | + | | 84.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|36.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872026.png | ||
+ | || $( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$ | ||
+ | ||$$(\operatorname {ad}x)^ny=\sum _ {j=1}^n(-1)^j\left(\begin {array}cn\\ | ||
+ | j | ||
+ | \end {array} | ||
+ | \right)x^{n-j}yx^j$$ | ||
+ | || conf 0.356 | ||
− | + | l05872026.png (26) | |
|- | |- | ||
− | | 85.(99.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872078.png || $\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$ || | + | | 85.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|99.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872078.png | ||
+ | || $\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$ | ||
+ | ||$$\pi (x+y)=\pi (x)+\pi (y),\quad \pi (\lambda x )=\lambda ^p\pi (x),\quad \lambda \in k .$$ | ||
+ | || conf 0.964 | ||
− | + | l05872078.png (78) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Lie theorem]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 86.(134.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876010.png || $y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$ || | + | | 86.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|134.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876010.png | ||
+ | || $y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } | ||
+ | || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$ | ||
+ | ||$$y_i=f_i(g_1,\ldots ,g_i;x_1,\ldots ,x_n),\quad i =1,\ldots ,n$$ | ||
+ | || conf 0.276 | ||
− | + | l05876010.png (10) | |
|- | |- | ||
− | | 87.(86.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876016.png || $X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$ || | + | | 87.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|86.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876016.png | ||
+ | || $X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$ | ||
+ | ||$$X_i=\sum _ {j=1}^n\xi _ {ij}(x)\frac {\partial }{\partial x _ j},\quad i =1,\ldots ,r,$$ | ||
+ | || conf 0.656 | ||
− | + | l05876016.png (16) | |
|- | |- | ||
− | | 88.(66.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876030.png || $\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$ || | + | | 88.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|66.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876030.png | ||
+ | || $\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$ | ||
+ | ||$$\frac {\partial f _ j}{\partial g _ i}(g,x)=\sum _ {k=1}^r\xi _ {kj}(f(g_sx))\psi _ {ki}(g),$$ | ||
+ | || conf 0.336 F | ||
− | + | l05876030.png (30) | |
|- | |- | ||
− | | 89.(19.)*|| https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876037.png || $\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ || | + | | 89.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|19.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876037.png | ||
+ | || $\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ | ||
+ | ||$$\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},$$ | ||
+ | || conf 0.157 F | ||
− | + | l05876037.png (37) | |
|- | |- | ||
− | | 90.(14.) || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876052.png || $\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$ || | + | | 90.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|14.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876052.png | ||
+ | || $\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$ | ||
+ | ||$$\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\}$$ | ||
+ | || conf 0.085 | ||
− | + | l05876052.png (52) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Maximal torus]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 91.(95.) || https://www.encyclopediaofmath.org/legacyimages/m/m063/m063010/m06301072.png || $F ( x _ { 1 } f _ { 1 } + \ldots + x _ { x } f _ { n } ) = x _ { 1 } x _ { n } + x _ { 2 } x _ { n } - 1 + \ldots + x _ { p } x _ { n } - p + 1$ || | + | | 91.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|95.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/m/m063/m063010/m06301072.png | ||
+ | || $F ( x _ { 1 } f _ { 1 } + \ldots + x _ { x } f _ { n } ) = x _ { 1 } x _ { n } + x _ { 2 } x _ { n } - 1 + \ldots + x _ { p } x _ { n } - p + 1$ | ||
+ | ||$$F(x_1f_1+\ldots +x_xf_n)=x_1x_n+x_2x_{n-1}+\ldots +x_px_{n-p+1},$$ | ||
+ | || conf 0.198 | ||
− | + | m06301072.png (72) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Non-Abelian cohomology]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 92.(114.)*|| https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900110.png || $\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ || | + | | 92.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|114.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900110.png | ||
+ | || $\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ | ||
+ | ||$$\phi (g_1)\phi (g_2)\phi (g_1g_2)^{-1}=\operatorname {Int}m(g_1,g_2),$$ | ||
+ | || conf 0.443 F | ||
− | + | n066900110.png (110) | |
|- | |- | ||
− | | 93.(90.)*|| https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900118.png || $( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$ || | + | | 93.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|90.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900118.png | ||
+ | || $( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$ | ||
+ | ||$$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}.$$ | ||
+ | || conf 0.764 F | ||
− | + | n066900118.png (118) | |
|- | |- | ||
− | | 94.(44.) || https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690016.png || $\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$ || | + | | 94.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|44.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690016.png | ||
+ | || $\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$ | ||
+ | ||$$\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,$$ | ||
+ | || conf 0.400 | ||
− | + | n06690016.png (16) | |
|- | |- | ||
− | | 95.(60.)*|| https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690028.png || $C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$ || | + | | 95.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|60.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690028.png | ||
+ | || $C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$ | ||
+ | ||$$C^{*}(\mathfrak U,{\cal F})=(C^0(\mathfrak U,{\cal F}),C^1(\mathfrak U,{\cal F}),C^2(\mathfrak U,{\cal F})),$$ | ||
+ | || conf 0.205 F | ||
− | + | n06690028.png (28) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Picard scheme]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 96.(39.)*|| https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267025.png || $\operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Fic } ( X \times k S ^ { \prime } ) / \operatorname { Fic } ( S ^ { \prime } )$ || | + | | 96.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|39.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267025.png | ||
+ | || $\operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Fic } ( X \times k S ^ { \prime } ) / \operatorname { Fic } ( S ^ { \prime } )$ | ||
+ | ||$$\operatorname {Pic}_{X/k}(S^{\prime })=\operatorname {Pic}(X\times_k S ^{\prime })/\operatorname {Pic}(S^{\prime })$$ | ||
+ | || conf 0.345 F + | ||
− | + | p07267025.png (25) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Principal analytic fibration]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 97.(100.)*|| https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464025.png || $g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$ || | + | | 97.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|100.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464025.png | ||
+ | || $g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$ | ||
+ | ||$$g_j:U_i\cap U _ j\rightarrow G ,\quad i ,j\in I ,\quad U _ i\cap U _ j\neq \emptyset,$$ | ||
+ | || conf 0.184 F | ||
− | + | p07464025.png (25) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Quantum groups]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 98.(101.) || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631062.png || $\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$ || | + | | 98.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|101.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631062.png | ||
+ | || $\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$ | ||
+ | ||$$\phi ^{*}:\mathfrak g^{*}\otimes \mathfrak g^{*}\rightarrow \mathfrak g^{*}$$ | ||
+ | || conf 0.837 | ||
− | + | q07631062.png (62) | |
|- | |- | ||
− | | 99.(108.) || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631071.png || $\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$ || | + | | 99.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|108.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631071.png | ||
+ | || $\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$ | ||
+ | ||$$\delta :U_{\mathfrak g}\rightarrow U _ {\mathfrak g}\otimes U _ {\mathfrak g}$$ | ||
+ | || conf 0.648 | ||
− | + | q07631071.png (71) | |
|- | |- | ||
− | | 100.(56.)*|| https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631072.png || $\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$ || | + | | 100.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|56.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631072.png | ||
+ | || $\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$ | ||
+ | ||$$\delta (\alpha )=\operatorname {lim}_{h\rightarrow 0 }h^{-1}(\Delta (a)-\Delta ^{\prime }(\alpha ))$$ | ||
+ | || conf 0.304 F | ||
− | + | q07631072.png (72) | |
|- | |- | ||
− | | 101.(129.)*|| https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631088.png || $[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$ || | + | | 101.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|129.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631088.png | ||
+ | || $[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$ | ||
+ | ||$$[\alpha ,X_i^{\pm }]=\pm \alpha _ i(\alpha )X_i^{\pm }\quad \text{for }a\in \mathfrak h;$$ | ||
+ | || conf 0.544 F | ||
− | + | q07631088.png (88) | |
|- | |- | ||
− | | 102.(128.) || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631089.png || $[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$ || | + | | 102.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|128.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631089.png | ||
+ | || $[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$ | ||
+ | ||$$[X_i^{+},X_j^{-}]=2\delta _ {ij}h^{-1}\operatorname {sinh}(hH_i/2).$$ | ||
+ | || conf 0.893 | ||
− | + | q07631089.png (89) | |
|- | |- | ||
− | | 103.(20.) || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631092.png || $\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$ || | + | | 103.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|20.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631092.png | ||
+ | || $\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$ | ||
+ | ||$$\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.$$ | ||
+ | || conf 0.055 | ||
− | + | q07631092.png (92) | |
|- | |- | ||
− | | 104.(30.) || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631095.png || $\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$ || | + | | 104.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|30.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631095.png | ||
+ | || $\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$ | ||
+ | ||$$\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)} | ||
+ | .$$ | ||
+ | || conf 0.443 | ||
− | + | q07631095.png (95) | |
|- | |- | ||
− | | 105.(21.)*|| https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631099.png || $\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$ || | + | | 105.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|21.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631099.png | ||
+ | || $\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$ | ||
+ | ||$$\Delta (X_i^{\pm })=X_i^{\pm }\otimes \operatorname {exp}(\frac {hH_i}4)+\operatorname {exp}(\frac {-hH_i}4)\otimes X _ i^{\pm }.$$ | ||
+ | || conf 0.212 F | ||
− | + | q07631099.png (99) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Rational representation]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 106.(91.) || https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630100.png || $0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$ || | + | | 106.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|91.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630100.png | ||
+ | || $0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$ | ||
+ | ||$$0\leq \frac {2(\chi ,\alpha )}{(\alpha ,\alpha )}<p\quad \text{for all }\alpha \in \Delta.$$ | ||
+ | || conf 0.879 | ||
− | + | r077630100.png (100) | |
|- | |- | ||
− | | 107.(135.) || https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630104.png || $\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$ || | + | | 107.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|135.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630104.png | ||
+ | || $\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$ | ||
+ | ||$$\phi _ 0\otimes \phi _ 1^{Fr}\otimes \ldots \otimes \phi _ d^{{Fr}^d},$$ | ||
+ | || conf 0.136 | ||
− | + | r077630104.png (104) | |
|- | |- | ||
− | | 108.(45.)*|| https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763055.png || $\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$ || | + | | 108.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|45.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763055.png | ||
+ | || $\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$ | ||
+ | ||$$\chi =\delta _ {\phi }-\sum _ {\alpha \in \Delta }m_{\alpha }\alpha ,\quad m _ {\alpha }\in Z ,\quad m _ {\alpha }\geq 0.$$ | ||
+ | || conf 0.862 F | ||
− | + | r07763055.png (55) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Singular point]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 109.(31.) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590225.png || $\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ || | + | | 109.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|31.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590225.png | ||
+ | || $\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ | ||
+ | ||$$\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}$$ | ||
+ | || conf 0.324 | ||
− | + | s085590225.png (225) | |
|- | |- | ||
− | | 110.(46.) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590404.png || $\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$ || | + | | 110.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|46.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590404.png | ||
+ | || $\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$ | ||
+ | ||$$\frac {m_1}{n_1}<\frac {m_2}{n_1n_2}<\ldots <\frac {m_g}{n_1\ldots n _ g}=\frac {m_g}n$$ | ||
+ | || conf 0.459 | ||
− | + | s085590404.png (404) | |
|- | |- | ||
− | | 111.(115.)*|| https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590429.png || $p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$ || | + | | 111.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|115.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590429.png | ||
+ | || $p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$ | ||
+ | ||$$p(Z)=1-\operatorname {dim}H^0({\mathbf Z},{\cal O}_{\mathbf Z })+\operatorname {dim}H^1({\mathbf Z},{\cal O}_{\mathbf Z })$$ | ||
+ | || conf 0.997 F | ||
− | + | s085590429.png (429) | |
|- | |- | ||
− | | 112.(136.)*|| https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590440.png || $X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$ || | + | | 112.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|136.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590440.png | ||
+ | || $X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$ | ||
+ | ||$$X_{\epsilon }=\{(x_0,\ldots ,x_x):f(x_0,\ldots ,x_x)=\epsilon \}$$ | ||
+ | || conf 0.433 F | ||
− | + | s085590440.png (440) | |
|- | |- | ||
− | | 113.(12.) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590458.png || $= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$ || | + | | 113.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|12.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590458.png | ||
+ | || $= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$ | ||
+ | ||$$=\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.$$ | ||
+ | || conf 0.870 | ||
− | + | s085590458.png (458) | |
|- | |- | ||
− | | 114.(75.) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590482.png || $( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$ || | + | | 114.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|75.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590482.png | ||
+ | || $( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$ | ||
+ | ||$$\big(\frac {\partial F (x,y,\lambda )}{\partial x },\frac {\partial F (x,y,\lambda )}{\partial y }\big)$$ | ||
+ | || conf 0.986 | ||
− | + | s085590482.png (482) | |
|- | |- | ||
− | | 115.(137.) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590515.png || $\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$ || | + | | 115.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|137.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590515.png | ||
+ | || $\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$ | ||
+ | ||$$\frac {dx_i}{dx_{i_0}}=f_i(x),\quad f _ i\in C (U),\quad i \neq i _ 0.$$ | ||
+ | || conf 0.594 | ||
− | + | s085590515.png (515) | |
|- | |- | ||
− | | 116.(142.)*|| https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590527.png || $A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$ || | + | | 116.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|142.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590527.png | ||
+ | || $A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$ | ||
+ | ||$$A=\left\| | ||
+ | \begin {array}{ll} | ||
+ | {\alpha } &b\\ | ||
+ | c &e | ||
+ | \end {array} | ||
+ | \right\|$$ | ||
+ | || conf 0.506 F | ||
− | + | s085590527.png (527) | |
|- | |- | ||
− | | 117.(53.) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590634.png || $\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$ || | + | | 117.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|53.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590634.png | ||
+ | || $\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$ | ||
+ | ||$$\Delta =(F_{xx}^{\prime \prime })_0(F_{yy}^{\prime \prime })_0-(F_{xy}^{\prime \prime })_0^2$$ | ||
+ | || conf 0.920 | ||
− | + | s085590634.png (634) | |
|- | |- | ||
− | | 118.(16.)*|| https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590645.png || $\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$ || | + | | 118.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|16.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590645.png | ||
+ | || $\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$ | ||
+ | ||$$\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\|$$ | ||
+ | || conf 0.230 F | ||
− | + | s085590645.png (645) | |
|- | |- | ||
− | | 119.(92.) || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590653.png || $( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$ || | + | | 119.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|92.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590653.png | ||
+ | || $( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$ | ||
+ | ||$$(F_x^{\prime })_0=0,\quad (F_y^{\prime })_0=0,\quad (F_z^{\prime })_0=0.$$ | ||
+ | || conf 0.300 | ||
− | + | s085590653.png (653) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Solv manifold]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 120.(138.) || https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610054.png || $\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$ || | + | | 120.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|138.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610054.png | ||
+ | || $\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$ | ||
+ | ||$$\{e\}\rightarrow \Delta \rightarrow \pi \rightarrow {\mathbf Z}^s\rightarrow \{e\}$$ | ||
+ | || conf 0.972 | ||
− | + | s08610054.png (54) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Stability theorems in algebraic K-theory]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 121.(71.) || https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706033.png || $\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$ || | + | | 121.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|71.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706033.png | ||
+ | || $\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$ | ||
+ | ||$$\psi _ {t_1,\ldots ,t_n}^{\prime }:SK_1(R)\rightarrow S K _ 1(R(t_1,\ldots ,t_n)).$$ | ||
+ | || conf 0.379 | ||
− | + | s08706033.png (33) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Steinberg module]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 122.(130.) || https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053016.png || $e = \frac { | U | } { | G | } ( \sum _ { b \in B } b ) ( \sum _ { w \in W } \operatorname { sign } ( w ) w )$ || | + | | 122.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|130.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053016.png | ||
+ | || $e = \frac { | U | } { | G | } ( \sum _ { b \in B } b ) ( \sum _ { w \in W } \operatorname { sign } ( w ) w )$ | ||
+ | ||$$e=\frac {|U|}{|G|}\big(\sum _ {b\in B }b\big)\big(\sum _ {w\in W }\operatorname {sign}(w)w\big)$$ | ||
+ | || conf 0.138 | ||
− | + | s13053016.png (16) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Steinberg symbol]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 123.(24.)*|| https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054017.png || $( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$ || | + | | 123.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|24.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054017.png | ||
+ | || $( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$ | ||
+ | ||$$(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.$$ | ||
+ | || conf 0.381 F | ||
− | + | s13054017.png (17) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Tilting theory]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 124.(84.) || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130105.png || $0 \rightarrow \Lambda \rightarrow T _ { 1 } \rightarrow \ldots \rightarrow T _ { n } \rightarrow 0$ || | + | | 124.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|84.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130105.png | ||
+ | || $0 \rightarrow \Lambda \rightarrow T _ { 1 } \rightarrow \ldots \rightarrow T _ { n } \rightarrow 0$ | ||
+ | ||$$0\rightarrow \Lambda \rightarrow T _ 1\rightarrow \ldots \rightarrow T _ n\rightarrow 0 $$ | ||
+ | || conf 0.946 | ||
− | + | t130130105.png (105) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Tits quadratic form]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 125.(18.) || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140104.png || $q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$ || | + | | 125.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|18.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140104.png | ||
+ | || $q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$ | ||
+ | ||$$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,$$ | ||
+ | || conf 0.112 | ||
− | + | t130140104.png (104) | |
|- | |- | ||
− | | 126.(40.) || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140118.png || $[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$ || | + | | 126.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|40.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140118.png | ||
+ | || $[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$ | ||
+ | ||$$[X]\mapsto \chi _ R([X])=\sum _ {m=0}^{\infty }(-1)^m\operatorname {dim}_K\operatorname {Ext}_R^m(X,X)$$ | ||
+ | || conf 0.116 | ||
− | + | t130140118.png (118) | |
|- | |- | ||
− | | 127.(132.)*|| https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140119.png || $\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$ || | + | | 127.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|132.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140119.png | ||
+ | || $\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$ | ||
+ | ||$$\underline {\dim }:K_0(\operatorname {mod}R)\rightarrow {\mathbf Z}^{Q_0}$$ | ||
+ | || conf 0.287 F | ||
− | + | t130140119.png (119) | |
|- | |- | ||
− | | 128.(37.)*|| https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140140.png || $q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$ || | + | | 128.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|37.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140140.png | ||
+ | || $q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$ | ||
+ | ||$$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$$ | ||
+ | || conf 0.197 F | ||
− | + | t130140140.png (140) | |
|- | |- | ||
− | | 129.(131.)*|| https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014044.png || $X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ || | + | | 129.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|131.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014044.png | ||
+ | || $X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ | ||
+ | ||$$X\mapsto \underline {\dim }X=(\operatorname {dim}_KX_j)_{j\in Q _ 0}$$ | ||
+ | || conf 0.819 F | ||
− | + | t13014044.png (44) | |
|- | |- | ||
− | | 130.(25. | + | | 130.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|25.]] |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014048.png | ||
+ | || $[ X ] \mapsto \chi _ { Q } ( [ X ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( X ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( X , X )$ | ||
+ | ||$$[X]\mapsto \chi _ Q([X])=\operatorname {dim}_K\operatorname {End}_Q(X)-\operatorname {dim}_K\operatorname {Ext}_Q^1(X,X)$$ | ||
+ | || conf 0.661 | ||
− | + | t13014048.png (48) | |
|- | |- | ||
− | | 131.(38.)*|| https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014056.png || $A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$ || | + | | 131.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|38.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014056.png | ||
+ | || $A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$ | ||
+ | ||$$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 }$$ | ||
+ | || conf 0.481 F | ||
− | + | t13014056.png (56) | |
|- | |- | ||
− | | 132.(139.)*|| https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301406.png || $\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$ || | + | | 132.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|139.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301406.png | ||
+ | || $\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$ | ||
+ | ||$$q_Q(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {i,j\in Q _ 0}d_{ij}x_ix_j,$$ | ||
+ | || conf 0.648 F | ||
− | + | t1301406.png (6) | |
|- | |- | ||
− | + | |} | |
+ | |||
+ | ==[[Torus]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 133.(41.)*|| https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933502.png || $r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$ || | + | | 133.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|41.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933502.png | ||
+ | || $r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$ | ||
+ | ||$$r=\alpha \operatorname {sin}u{\bf k}+l(1+\epsilon \operatorname {cos}u)({\bf i}\operatorname {cos}v+{\bf j}\operatorname {sin}v)$$ | ||
+ | || conf 0.585 F | ||
− | + | t0933502.png (2) | |
|- | |- | ||
− | | 134.(122.)*|| https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933507.png || $d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$ || | + | | 134.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|122.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933507.png | ||
+ | || $d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$ | ||
+ | ||$$ds^2=\alpha ^2du^2+l^2(1+\epsilon \operatorname {cos}u)^2dv^2,$$ | ||
+ | || conf 0.696 F | ||
− | + | t0933507.png (7) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Uniform distribution]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 135.(9.) || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524027.png || $u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$ || | + | | 135.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|9.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524027.png | ||
+ | || $u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$ | ||
+ | ||$$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.$$ | ||
+ | || conf 0.733 | ||
− | + | u09524027.png (27) | |
|- | |- | ||
− | | 136.(32.)*|| https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952403.png || $p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$ || | + | | 136.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|32.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952403.png | ||
+ | || $p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$ | ||
+ | ||$$p(x)=\left\{ | ||
+ | \begin {array}{ll} | ||
+ | {\frac 1{b-\alpha },} &{x\in [\alpha ,b],}\\ | ||
+ | {0,} &{x\notin [\alpha ,b].} | ||
+ | \end {array} | ||
+ | \right.$$ | ||
+ | || conf 0.681 F | ||
− | + | u0952403.png (3) | |
|- | |- | ||
− | | 137.(34.) || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524030.png || $u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$ || | + | | 137.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|34.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524030.png | ||
+ | || $u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$ | ||
+ | ||$$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}$$ | ||
+ | || conf 0.569 | ||
− | + | u09524030.png (30) | |
|- | |- | ||
− | | 138.(109.) || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524034.png || $z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$ || | + | | 138.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|109.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524034.png | ||
+ | || $z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$ | ||
+ | ||$$z_{+}=\left\{ | ||
+ | \begin {array}{ll} | ||
+ | {z,} &{z>0}.\\ | ||
+ | {0,} &{z\leq 0 }. | ||
+ | \end {array} | ||
+ | \right.$$ | ||
+ | || conf 0.676 | ||
− | + | u09524034.png (34) | |
|- | |- | ||
− | | 139.(43.) || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952407.png || $F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$ || | + | | 139.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|43.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952407.png | ||
+ | || $F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$ | ||
+ | ||$$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.$$ | ||
+ | || conf 0.468 | ||
− | + | u0952407.png (7) | |
|- | |- | ||
− | | 140.(47.) || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524072.png || $p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$ || | + | | 140.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|47.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524072.png | ||
+ | || $p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$ | ||
+ | ||$$p(x_1,\ldots ,x_n)=\left\{ | ||
+ | \begin {array}{ll} | ||
+ | {C\neq 0 ,} &{x\in D },\\ | ||
+ | {0,} &{x\notin D }, | ||
+ | \end {array} | ||
+ | \right.$$ | ||
+ | || conf 0.705 | ||
− | + | u09524072.png (72) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Unipotent group]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 141.(143.) || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954106.png || $\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$ || | + | | 141.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|143.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954106.png | ||
+ | || $\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$ | ||
+ | ||$$\{g\in \operatorname {GL}(V):(1-g)^n=0\},\quad n =\operatorname {dim}V,$$ | ||
+ | || conf 0.287 | ||
− | + | u0954106.png (6) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Weyl module]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 142.(51.) || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090122.png || $\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$ || | + | | 142.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|51.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090122.png | ||
+ | || $\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$ | ||
+ | ||$$\operatorname {diag}(t_1,\ldots ,t_n)\mapsto t _ 1^{\lambda _ 1}\ldots t _ n^{\lambda _ n}\in K,$$ | ||
+ | || conf 0.507 | ||
− | + | w120090122.png (122) | |
|- | |- | ||
− | | 143.(54.)*|| https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090135.png || $\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$ || | + | | 143.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|54.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090135.png | ||
+ | || $\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$ | ||
+ | ||$$\chi _ {\lambda }=\sum _ {\mu \in \Lambda (n)}\operatorname {dim}_K(\Delta (\lambda )^{\mu })_{e_{\mu }},$$ | ||
+ | || conf 0.461 F | ||
− | + | w120090135.png (135) | |
|- | |- | ||
− | | 144.(110.) || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090259.png || $\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$ || | + | | 144.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|110.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090259.png | ||
+ | || $\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$ | ||
+ | ||$$\mathfrak B=\{e_{\pm }\alpha ,h_{\beta }:\alpha \in \Phi ^{+},\beta \in \Sigma \}.$$ | ||
+ | || conf 0.381 | ||
− | + | w120090259.png (259) | |
|- | |- | ||
− | | 145.(82.) || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090342.png || $\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$ || | + | | 145.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|82.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090342.png | ||
+ | || $\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$ | ||
+ | ||$$\left( | ||
+ | \begin {array}ch\\ | ||
+ | i | ||
+ | \end {array} | ||
+ | \right)=\frac {h(h-1)\ldots (h-i+1)}{i!} | ||
+ | $$ | ||
+ | || conf 0.487 | ||
− | + | w120090342.png (342) | |
|- | |- | ||
− | | 146.(28.)*|| https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009095.png || $\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$ || | + | | 146.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|28.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009095.png | ||
+ | || $\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$ | ||
+ | ||$$\mathfrak S_{\{1,\ldots ,\lambda _ 1\}}\times \mathfrak S_{\{\lambda _ 1+1,\ldots ,\lambda _ 1+\lambda _ 2\}}\times \dots $$ | ||
+ | || conf 0.312 F | ||
− | + | w12009095.png (95) | |
|- | |- | ||
− | | 147.(104.) || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009096.png || $\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$ || | + | | 147.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|104.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009096.png | ||
+ | || $\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$ | ||
+ | ||$$\ldots \times \mathfrak S_{\{\lambda _ 1+\ldots +\lambda _ {n-1}+1,\ldots ,r\}},$$ | ||
+ | || conf 0.259 | ||
− | + | w12009096.png (96) | |
|- | |- | ||
− | + | |} | |
+ | ==[[Witt vector]]== | ||
+ | {| class="wikitable" style="text-align: left; width: 1740px;" | ||
+ | !style=width: 3%| Nr. | ||
+ | !style=width: 30%| Image of png File | ||
+ | !style=width: 30%| $\TeX$, automatically generated version | ||
+ | !style=width: 30%| $\TeX$, manually corrected version | ||
+ | !style=width: 7%| Confidence, F? | ||
+ | png file | ||
|- | |- | ||
− | | 148.(87.)*|| https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100172.png || $\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$ || | + | | 148.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|87.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100172.png | ||
+ | || $\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$ | ||
+ | ||$$\langle \alpha ><b\rangle =\langle \alpha b \rangle ,\quad \langle {\bf 1}\rangle ={\bf f}_1={\bf V}_1=\text{ unit element}1,$$ | ||
+ | || conf 0.351 F | ||
− | + | w098100172.png (172) | |
|- | |- | ||
− | | 149.(123.)*|| https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100177.png || $\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$ || | + | | 149.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|123.]])* |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100177.png | ||
+ | || $\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$ | ||
+ | ||$$\langle \alpha +b\rangle =\sum _ {n=1}^{\infty }{\bf V}_n\langle r _ n(\alpha ,b){\bf f}_n.$$ | ||
+ | || conf 0.143 F | ||
− | + | w098100177.png (177) | |
|- | |- | ||
− | | 150.(102.) || https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100190.png || $\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$ || | + | | 150.([[User:Maximilian Janisch/latexlist/latex/Algebraic Groups/E|102.]]) |
+ | || https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100190.png | ||
+ | || $\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$ | ||
+ | ||$$\sigma (\alpha _ 1,\alpha _ 2,\ldots )=(\alpha _ 1^p,\alpha _ 2^p,\ldots )$$ | ||
+ | || conf 0.771 | ||
− | + | w098100190.png (190) | |
|- | |- | ||
|} | |} |
Latest revision as of 17:27, 11 November 2019
This page gives an analysis of the code here, generated automatically from some png files underlying our old wiki pages. As this page does contain a lot of $\TeX$ code, it loads slowly.
Under the name of some of our EoM-pages the table below lists some png files, displaying their image and their $TeX$ rendering (automatically retrieved and corrected by hand). The first column gives the running number in this table, followed (in parentheses) by the number used here. The last column gives the confidence and the name of the png file, followed (in parentheses) by the number it has in the sequence of all png files called by its calling EoM-page.
Here is a short survey of the more systematic errors which seem to occur:
- 1. Trailing punctuation is dismissed.
- [concerns almost all images] ; technically: pixels in sparse last pixel columns of bit images are suppressed/ignored?
- 2. "Displayed" images are not recognized as such.
- [concerns almost all images]
- Therefore these are displayed too small, and like "inline" $\TeX$ format.
- Remark: This cannot be discovered from the png file, it has to be retrieved from the html markup in the calling file: Displayed images are embedded in some html <table> markup.
- 3. Sparse initial column pixels of the bit image are dismissed
- (in parts this affects essential symbols), [see nr. 15,16,36,43,58,59,60,61,62,63,97,109]
- 4. Some fonts are not recognized
- \cal: [7.12.25.26,30,31,32,33,95,111] \mathbf: [30,83,111,127] \bf:[ 133,148,149]
- 5. Semi-colon is interpreted as double pipe = "||"
- [33,49,86,101]
- 6. Some code is not displayed at all.
- (This seems to be a bug of our MathJax TeX interpreter.) [67,74,78,81,83,94,101,106]
- This seems to happen when a string "\text {" is involved, can apparently be fixed by using "\text{", but still unclear.
- 7. Questions
- The different interpretation of the matrix delimiters in [56-63] is a bit surprising. Should be checked!
- Also, the vanishing of some '-' signs in the first column of some matrices, maybe that is related to 3.?
Algebraic curve
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
---|---|---|---|---|
1.(23.) | ![]() |
$g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$ | $$g\leq \left\{ \begin {array}{ll} {\frac {(n-2)^2}4} &{\text{ for even }n,}\\ {\frac {(n-1)(n-3)}4} &{\text{ for odd }n,} \end {array} \right.$$ | conf 0.698
a01145065.png (65) |
Algebraic geometry
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
---|---|---|---|---|
2.(116.) | ![]() |
$\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ | $$\theta =\int\limits _ 0^{\lambda }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ | conf 0.997
a01150014.png (14) |
3.(133.) | ![]() |
$\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ | $$\omega =2\int\limits _ 0^{1/c}\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ | conf 0.973
a01150021.png (21) |
4.(67.) | ![]() |
$\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ | $$\widetilde w=2\int\limits _ 0^{1/\varepsilon }\frac {dx}{\sqrt {(1-c^2x^2)(1-e^2x^2)}},$$ | conf 0.107
a01150022.png (22) |
5.(105.) | ![]() |
$\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$ | $$\theta (v+\pi i r )=\theta (r),\quad \theta (v+\alpha _ j)=e^{L_j(v)}\theta (v),$$ | conf 0.775
a01150044.png (44) |
6.(17.) | ![]() |
$\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ | $$\left( \begin {array}{ll} {\alpha } &b\\ c &d \end {array} \right)\equiv \left( \begin {array}{ll} 1&0\\ 0&1 \end {array} \right)(\operatorname {mod}7).$$ | conf 0.440
a01150078.png (78) |
Algebraic surface
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
---|---|---|---|---|
7.(144.) | ![]() |
$0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$ | $$0\rightarrow {\cal O}_V\rightarrow E _ {\alpha }\rightarrow T _ V\rightarrow 0$$ | conf 0.981
a011640132.png (132) |
8.(73.) | ![]() |
$M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$ | $$M=\operatorname {dim}\operatorname {Im}(H^1(V,E_{\alpha })\rightarrow H ^1(V,T_V)).$$ | conf 0.997
a011640137.png (137) |
9.(88.) | ![]() |
$\operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } )$ | $$\operatorname {dim}_kH^2(V,E_{\alpha })+\operatorname {dim}_kH^2(V,T_V).$$ | conf 0.996
a011640139.png (139) |
10.(117.) | ![]() |
$N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$ | $$N_m=\left(\begin {array}c{m+3}\\ 3 \end {array} \right)-dm+2t+\tau +p-1.$$ | conf 0.369
a01164027.png (27) |
11.(72.) | ![]() |
$p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$ | $$p_{\alpha }(V)=\left(\begin {array}c{n-1}\\ 3 \end {array} \right)-d(n-1)+2t+\tau +p-1$$ | conf 0.396
a01164029.png (29) |
12.(68.)* | ![]() |
$p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$ | $$p_{\alpha }(V)=-\operatorname {dim}_kH_1(V,{\cal O}_V)+\operatorname {dim}_kH^2(V,{\cal O}_V)=$$ | conf 0.756 F
a01164047.png (47) |
13.(93.)* | ![]() |
$1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$ | $$1+p_{\alpha }(V)=\frac {\operatorname {deg}(c_1^2)+\operatorname {deg}(c_2)}{12},$$ | conf 0.752 F
a01164053.png (53) |
Cartan subalgebra
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14.(33.)* | ![]() |
$\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$ | $$\mathfrak g_0=\big\{X\in \mathfrak g:\forall H \in \mathfrak t\exists n_{X,H}\in {\mathbb Z}((\text{ ad }H)^{n_{X,H}}(X)=0)\big\},$$ | conf 0.110 F
c0205509.png (9) |
Cartan theorem
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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15.(49.)* | ![]() |
$f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$ | $$[e_i,f_j]=\delta _ {ij}h_i,\quad [h_i,e_j]=\alpha _ {ij}e_j,\quad [h_i,f_j]=-\alpha _ {ij}f_j,$$ | conf 0.149 F
c0205704.png (4) |
16.(55.)* | ![]() |
$\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$ | $$\dots \rightarrow H ^p(X,S)\rightarrow H ^p(X,F)\stackrel {\phi_p }{\rightarrow }H^p(X,G)\rightarrow $$ | conf 0.853 F
c02057064.png (64) |
Comitant
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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17.(7.) | ![]() |
$H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$ | $$H=\frac 1{36}\left| \begin {array}{cc} {\frac {\partial ^2f}{\partial x ^2}} &{\frac {\partial ^2f}{\partial x \partial y }}\\ {\frac {\partial ^2f}{\partial x \partial y }} &{\frac {\partial ^2f}{\partial y ^2}} \end {array} \right|=$$ | conf 0.956
c02333033.png (33) |
18.(76.) | ![]() |
$= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$ | $$=(a_0a_2-a_1^2)x^2+(a_0a_3-a_1a_2)xy+(a_1a_3-a_2^2)y^2$$ | conf 0.549
c02333034.png (34) |
19.(11.)* | ![]() |
$( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$ | $$(\alpha _ 0,\alpha _ 1,\alpha _ 2,\alpha _ 3)\mapsto (\alpha _ 0\alpha _ 2-\alpha _ 1^2,\frac 12(\alpha _ 0\alpha _ 3-\alpha _ 1\alpha _ 2),\alpha _ 1\alpha _ 3-\alpha _ 2^2)$$ | conf 0.521 F
c02333035.png (35) |
Deformation
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
---|---|---|---|---|
20.(26.) | ![]() |
$\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$ | $$\operatorname {Aut}_{R^{\prime }}(X^{\prime }|X_0)\rightarrow \operatorname {Aut}_R(X_{R^{\prime }}^{\prime }\otimes R |X_0)$$ | conf 0.683
\ d030700175.png (175) |
21.(27.) | ![]() |
$\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$ | $$\operatorname {dim}_kH^1(X_0,T_{X_0})-\operatorname {dim}M_{X_0}\leq \operatorname {dim}_kH^2(X_0,T_{X_0}).$$ | conf 0.944
d030700190.png (190) |
22.(78.)* | ![]() |
$\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$ | $$\alpha \circ b =\alpha b +\sum _ {i=1}^{\infty }\phi _ i(\alpha ,b)t^i,\quad \alpha ,b\in V,$$ | conf 0.097 F
d030700263.png (263) |
23.(96.)* | ![]() |
$\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$ | $$\Phi (\alpha )=\alpha +\sum _ {i=1}^{\infty }t^i\phi _ i(\alpha ),\quad \alpha \in V,$$ | conf 0.873 F
d030700270.png (270) |
Differential algebra
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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24.(106.) | ![]() |
$S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$ | $$S^tF=\sum _ {j=1}^rc_jA^{p_j}A_1^{i_{1j}}\dots A _ {m-l}^{i_{{m-l},j}},$$ | conf 0.149
d031830107.png (107) |
25.(146.)* | ![]() |
$( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$ | $(\eta _ 1,\ldots ,\eta _ k)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ k)$ | conf 0.562 F
d031830141.png (141) |
26.(145.)$^F$* | ![]() |
$( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$ | $(\eta _ 1,\ldots ,\eta _ n)\rightarrow {}_{\cal F}(\zeta _ 1,\ldots ,\zeta _ n)$ | conf 0.376 F
d031830150.png (150) |
27.(57.) | ![]() |
$\omega _ { V } = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ | $$\omega _ V=\sum _ {0\leq i \leq m }\alpha _ i\left( \begin {array}c{x+i}\\ i \end {array} \right),$$ | conf 0.780
d03183016.png (16) |
28.(111.) | ![]() |
$e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$ | $$e_{ij}=\operatorname {ord}_{Y_j}F_i,\quad 1 \leq i \leq n ,\quad i \leq j \leq n,$$ | conf 0.187
d03183043.png (43) |
Dimension polynomial
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
29.(48.) | ![]() |
$\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ | $$\omega _ {\eta /F}(x)=\sum _ {0\leq i \leq m }\alpha _ i\left(\begin {array}c{x+i}\\ i \end {array} \right),$$ | conf 0.968
d03249029.png (29) |
Duality
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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30.(118.)* | ![]() |
$H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$ | $$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow {\mathbf C},$$ | conf 0.824 F
d034120173.png (173) | |
31.(59.)* | ![]() |
$H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow H _ { c } ^ { n } ( X , \Omega )$ | $$H^p(X,{\cal F})\times H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega ))\rightarrow H _ c^n(X,\Omega )$$ | conf 0.921 F
d034120175.png (175) | |
32.(124.)* | ![]() |
$( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$ | $$(H^p(X,{\cal F}))^{\prime }\cong H _ c^{n-p}(X,\operatorname {Hom}({\cal F},\Omega )).$$ | conf 0.829 F
d034120184.png (184) | |
33.(29.)* | ![]() |
$\beta : \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X F , \Omega ) \rightarrow \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X \backslash Y || F , \Omega )$ | $$\beta :\operatorname {Ext}_c^{n-p-1}(X;{\cal F},\Omega )\rightarrow \operatorname {Ext}_c^{n-p-1}(X\backslash Y ;{\cal F},\Omega ).$$ | conf 0.634 | F
d034120236.png (236) |
34.(77.)* | ![]() |
$\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$ | $$\underset {n\rightarrow \infty }{\overline {\lim }}|\alpha _ n|^{1/n}=\sigma <+\infty.$$ | conf 0.521 F
d034120247.png (247) | |
35.(58.)* | ![]() |
$h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$ | $$h(\phi )=\underset {n\rightarrow \infty }{\overline {\lim }}\frac {\operatorname {ln}|A(re^{i\phi })|}r$$ | conf 0.861 F
d034120253.png (253) | |
36.(69.)* | ![]() |
$\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$ | $$\operatorname*{sup}_{l\in E^\perp \atop \|l\|\le 1 }|l(\omega )|=\operatorname*{inf}_{x\in E }\|\omega -x\|,$$ | conf 0.293 F
d034120360.png (360) | |
37.(15.) | ![]() |
$\operatorname { sup } _ { f \in B ^ { 1 } } | \int _ { \partial G } f ( \zeta ) \omega ( \zeta ) d \zeta | = \operatorname { inf } _ { \phi \in E ^ { 1 } } \int _ { \partial G } | \omega ( \zeta ) - \phi ( \zeta ) \| d \zeta |$ | $$\operatorname*{sup}_{f\in B ^1}\big|\int\limits _ {\partial G }f(\zeta )\omega (\zeta )d\zeta \big|=\operatorname*{inf}_{\phi \in E ^1}\int\limits _ {\partial G }|\omega (\zeta )-\phi (\zeta ) ||d\zeta |.$$ | conf 0.508
d034120376.png (376) | |
38.(52.) | ![]() |
$f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$ | $$f=\{f_{\alpha }\}\in \prod _ {\alpha }F_{\alpha },\quad g =\{g_{\alpha }\}\in \operatorname*\oplus _ {\alpha }G_{\alpha }.$$ | conf 0.491
d034120509.png (509) | |
39.(140.) | ![]() |
$f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$ | $$f^{*}(x^{*})=\operatorname*{sup}_{x\in X }(\langle x ^{*},x\rangle -f(x))$$ | conf 0.900
d034120535.png (535) | |
40.(94.) | ![]() |
$f _ { 0 } ( x ) \rightarrow \text { inf, } \quad f _ { i } ( x ) \leq 0 , \quad i = 1 , \ldots , m , \quad x \in B$ | $$f_0(x)\rightarrow \text{ inf, }\quad f _ i(x)\leq 0 ,\quad i =1,\ldots ,m,\quad x \in B,$$ | conf 0.810
d034120555.png (555) | |
41.(74.)* | ![]() |
$( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$ | $$(c_{\gamma },c^r)=\sum _ {t^r\in K }c_r(t^{\prime })c^r(t^r)\operatorname {mod}1$$ | conf 0.117 F
d03412079.png (79) |
Extension of a differential field
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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42.(63.) | ![]() |
$F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$ | $$F_1F_2=F_1\langle F _ 2\rangle =F_1(F_2)=F_2(F_1)=F_2\langle F _ 1\rangle,$$ | conf 0.628
e03696024.png (24) |
Formal group
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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43.(120.)* | ![]() |
$\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$ | $$\operatorname {log}F_{\rm MU }(X)=\sum _ {i=1}^{\infty }i^{-1}[{\rm CP}^{i-1}]X^i,$$ | conf 0.098 F
f040820118.png (118) |
44.(147.)* | ![]() |
$( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$ | $$(x_1,\ldots ,x_n)\circ (y_1,\ldots ,y_n)=(z_1,\ldots ,z_n),$$ | conf 0.553 F
f04082059.png (59) |
Gel'fond-Schneider method
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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45.(148.) | ![]() |
$\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$ | $\alpha ^{\beta }=\operatorname {exp}\{\beta \operatorname {log}\alpha \}$ | conf 0.979
g1300205.png (5) |
Group
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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46.(22.)* | ![]() |
$\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$ | source incomplete | conf 0.226 F
g04521075.png (75) |
Homogeneous space
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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47.(89.) | ![]() |
$\mathfrak { g } = \mathfrak { f } + \mathfrak { m } , \quad \mathfrak { f } \cap \mathfrak { m } = \{ 0 \}$ | $$\mathfrak g=\mathfrak f+\mathfrak m,\quad \mathfrak f\cap \mathfrak m=\{0\},$$ | conf 0.793
h04769069.png (69) |
Hopf algebra
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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48.(103.) | ![]() |
$m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$ | $m\circ (\iota \otimes 1 )\circ \mu =m\circ (1\otimes \iota )\circ \mu =e\circ \epsilon$ | conf 0.618
h047970129.png (129) |
49.(107.)* | ![]() |
$F _ { 1 } ( X || Y ) , \ldots , F _ { n } ( X || Y ) \in K [ X _ { 1 } , \ldots , X _ { n } || Y _ { 1 } , \ldots , Y _ { n } ] \}$ | $F_1(X;Y),\ldots ,F_n(X;Y)\in K [X_1,\ldots ,X_n;Y_1,\ldots ,Y_n]\}$ | conf 0.353 F
h047970139.png (139) |
50.(97.) | ![]() |
$\epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g }$ | $$\epsilon (x)=0,\quad \delta (x)=x\otimes 1 +1\otimes x ,\quad x \in \mathfrak g.$$ | conf 0.213
h04797042.png (42) |
Invariants, theory of
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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51.(149.)* | ![]() |
$\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$ | $$\alpha _ {i_1,\dots,i_n}\rightarrow \alpha _ {i_1,\dots,i_n}^{\prime }.$$ | conf 0.142 F
i05235015.png (15) |
Jordan algebra
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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52.(150.) | ![]() |
$H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$ | $$(C_3,\Gamma )=\big\{X\in C _ 3:X=\Gamma ^{-1}X\square ^{\prime }\Gamma \big\},$$ | conf 0.651
j05427030.png (30) |
53.(42.) | ![]() |
$\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$ | $$\Gamma =\operatorname {diag}\{\gamma _ 1,\gamma _ 2,\gamma _ 3\},\quad \gamma _ i\neq 0 ,\quad \gamma _ i\in F,$$ | conf 0.987
j05427031.png (31) |
54.(125.)* | ![]() |
$\mathfrak { g } = \mathfrak { g } - 1 + \mathfrak { g } \mathfrak { d } + \mathfrak { g } _ { 1 }$ | $\mathfrak g=\mathfrak g_{-1}+\mathfrak g_0+\mathfrak g_1$ | conf 0.598 F
j05427077.png (77) |
Jordan matrix
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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55.(6.)* | ![]() |
$J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$ | $$J=\left\| \begin {array}{cccc} J_{n_1}(\lambda_1) &0 &0 &0\\ 0 &\ddots &\ddots &0\\ 0 &\ddots &\ddots &0\\ 0 &0 &0 &J_{n_s}(\lambda_s) \end {array} \right\|,$$ | conf 0.072 F
j0543403.png (3) |
56.(64.) | ![]() |
$C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$ | $$C_m(\lambda )=\operatorname {rk}(A-\lambda E )^{m-1}-2\operatorname {rk}(A-\lambda E )^m+$$ | conf 0.955
j05434030.png (30) |
57.(1.)* | ![]() |
$J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$ | $$J_m(\lambda)=\left\| \begin {array}{cccccc} \lambda &1 &\square &\square &\square &\square \\ \square &\lambda &1 &\square &0 &\square \\ \square &\square &\ddots &\ddots &\square &\square\\ \square &\square &\square &\ddots &\ddots &\square \\ \square &0 &\square &\square &\lambda &1\\ \square &\square &\square &\square &\square &\lambda \end {array} \right\|,$$ | conf 0.098 F
j0543406.png (6) |
Lie algebra, semi-simple
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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58.(5.) | ![]() |
$\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$ | $$B_n:\quad \left\| \begin {array}{rrrrrr} 2 &{-1} &0 &{\dots } &0 &0\\ {-1} &2 &{-1} &{\dots } &0 &0\\ 0 &{-1} &2 &{\dots } &0 &0\\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot \\ 0 &0 &0 &{\dots } &{-1} &0\\ 0 &0 &0 &{\dots } &2 &{-2}\\ 0 &0 &0 &{\dots } &{-1} &2 \end {array} \right\|,$$ | conf 0.232
l058510127.png (127) |
59.(3.)* | ![]() |
$\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$ | $$D_n:\quad \left\| \begin {array}{rrrrrrr} 2 &{-1} &0 &{\dots } &0 &0 &0 &0\\ {-1} &2 &{-1} &{\dots } &0 &0 &0 &0\\ 0 &{-1} &2 &{\dots } &0 &0 &0 &0\\ \cdot &\cdot &\cdot &\dots &\cdot &\cdot &\cdot &\cdot \\ 0 &0 &0 &{\dots } &2 &{-1} &0 &0\\ 0 &0 &0 &{\dots } &{-1} &2 &{-1} &{-1}\\ 0 &0 &0 &{\dots } &0 &{-1} &2 &0\\ 0 &0 &0 &{\dots } &0 &{-1} &0 &2 \end {array} \right\|,$$ | conf 0.055 F
l058510129.png (129) |
60.(8.)* | ![]() |
$\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ | $$E_6: \quad \left\| \begin {array}{rrrrrr} 2 &0 &{-1} &0 &0 &0\\ 0 &2 &0 &{-1} &0 &0\\ {-1} &0 &2 &{-1} &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0\\ 0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ | conf 0.628 F
l058510130.png (130) |
61.(4.) | ![]() |
$\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ | $$E_7:\quad \left\| \begin {array}{rrrrrrr} 2 &0 &{-1} &0 &0 &0 &0\\ 0 &2 &0 &{-1} &0 &0 &0\\ {-1} &0 &2 &{-1} &0 &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0 &0\\ 0 &0 &0 &{-1} &2 &{-1} &0\\ 0 &0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ | conf 0.278
l058510131.png (131) |
62.(2.)* | ![]() |
$\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$ | $$E_8:\quad \left\| \begin {array}{rrrrrrrr} 2 &0 &{-1} &0 &0 &0 &0 & 0\\ 0 &2 &0 &{-1} &0 &0 &0 &0\\ {-1} &0 &2 &{-1} &0 &0 &0 &0\\ 0 &{-1} &{-1} &2 &{-1} &0 &0 &0\\ 0 &0 &0 &{-1} &2 &{-1} &0 &0\\ 0 &0 &0 &0 &{-1} &2 &{-1} &0\\ 0 &0 &0 &0 &0 &{-1} &2 &{-1}\\ 0 &0 &0 &0 &0 &0 &{-1} &2 \end {array} \right\|,$$ | conf 0.354 F
l058510132.png (132) |
63.(10.)* | ![]() |
$\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$ | $$F_4:\quad \left\| \begin {array}{rrrr} 2 &{-1} &0 &0\\ {-1} &2 &{-2} &0\\ 0 &{-1} &2 &{-1}\\ 0 &0 &{-1} &2 \end {array} \right\|,\quad G _ 2:\quad \left\| \begin {array}{rr} 2&{-1}\\ {-3}&2 \end {array} \right\|.$$ | conf 0.374 F
l058510133.png (133) |
64.(98.) | ![]() |
$\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$ | $$\mathfrak g_{\alpha }=\{X\in \mathfrak g:[H,X]=\alpha (H)X,H\in \mathfrak h\}.$$ | conf 0.976
l05851030.png (30) |
65.(126.) | ![]() |
$\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$ | $$\mathfrak g=\mathfrak h+\sum _ {\alpha \in \Sigma }\mathfrak g_{\alpha }.$$ | conf 0.945
l05851037.png (37) |
66.(61.)* | ![]() |
$\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$ | $$\mathfrak g_{\alpha }=\operatorname {dim}[\mathfrak g_{\alpha },\mathfrak g_{-\alpha }]=1.$$ | conf 0.520 F
l05851044.png (44) |
67.(65.)* | ![]() |
$[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$ | $$[H_{\alpha },X_{\alpha }]=2X_{\alpha }\quad {\rm and }\quad [H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }.$$ | conf 0.539 F
l05851050.png (50) |
68.(70.) | ![]() |
$\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$ | $$\beta (H_{\alpha })=\frac {2(\alpha ,\beta )}{(\alpha ,\alpha )},\quad \alpha ,\beta \in \Sigma,$$ | conf 0.997
l05851051.png (51) |
69.(112.) | ![]() |
$[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$ | $$[\mathfrak g_{\alpha },\mathfrak g_{\beta }]=\mathfrak g_{\alpha +\beta }$$ | conf 0.917
l05851057.png (57) |
70.(127.) | ![]() |
$H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$ | $$H_{\alpha _ 1},\ldots ,H_{\alpha _ k},X_{\alpha }\quad (\alpha \in \Sigma )$$ | conf 0.432
l05851064.png (64) |
71.(113.)* | ![]() |
$[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$ | $$[[X_{\alpha _ i},X_{-}\alpha _ i],X_{-\alpha _ j}]=-n(i,j)X_{\alpha _ j},$$ | conf 0.628 F
l05851069.png (69) |
72.(79.) | ![]() |
$n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$ | $$n(i,j)=\alpha _ j(H_i)=\frac {2(\alpha _ i,\alpha _ j)}{(\alpha _ j,\alpha _ j)}.$$ | conf 0.992
l05851073.png (73) |
73.(13.) | ![]() |
$[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$ | $$[X_{\alpha },X_{\beta }]=\left\{ \begin {array}{ll} {N_{\alpha ,\beta }X_{\alpha +\beta }} &{\text{ if }\alpha +\beta \in \Sigma,}\\ 0 &{\text{ if }\alpha +\beta \notin \Sigma,} \end {array} \right.$$ | conf 0.988
l05851074.png (74) |
74.(80.) | ![]() |
$N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$ | $$N_{\alpha ,\beta }=-N_{-\alpha ,-\beta }\quad {\rm and }\quad N _ {\alpha ,\beta }=\pm (p+1),$$ | conf 0.961
l05851078.png (78) |
75.(85.)* | ![]() |
$X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$ | $$iH_\alpha,X_{\alpha }-X_{-\alpha },\quad i (X_{\alpha }+X_{-\alpha })\quad (\alpha \in \Sigma _ {+})$$ | conf 0.691 F
l05851085.png (85) |
Lie algebra, solvable
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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76.(119.)* | ![]() |
$[ \mathfrak { g } _ { i } , \mathfrak { g } _ { i } ] \subset \mathfrak { g } _ { \mathfrak { i } } + 1$ | $[\mathfrak g_i,\mathfrak g_i]\subset \mathfrak g_{i+1}$ | conf 0.276 F
l05852011.png (11) |
77.(141.) | ![]() |
$\operatorname { dim } \mathfrak { g } _ { i } = \operatorname { dim } \mathfrak { g } - i$ | $\operatorname {dim}\mathfrak g_i=\operatorname {dim}\mathfrak g-i$ | conf 0.901
l05852046.png (46) |
Lie group
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
78.(62.)* | ![]() |
$( G ) \cong \operatorname { Aut } ( L ( G ) ) \quad \text { and } \quad L ( \operatorname { Aut } ( G ) ) \cong D ( L ( G ) )$ | $$\operatorname {Aut}(G)\cong \operatorname {Aut}(L(G))\quad {\rm and }\quad L (\operatorname {Aut}(G))\cong D (L(G)),$$ | conf 0.693 F
l058590115.png (115) |
79.(50.) | ![]() |
$( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$ | $$(X,Y)\rightarrow \operatorname {exp}^{-1}(\operatorname {exp}X\operatorname {exp}Y),\quad X ,Y\in L (G),$$ | conf 0.856
l05859086.png (86) |
Lie group, compact
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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80.(121.)* | ![]() |
$J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$ | $$J=\left\| \begin {array}{cc} 0 &{E_x}\\ {-E_x} &0 \end {array} \right\|,$$ | conf 0.364 F
l05861012.png (12) |
Lie group, nilpotent
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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81.(83.) | ![]() |
$N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$ | $$N(F)=\{g\in GL (V):gv\equiv v \operatorname {mod}V_i\;\text{for all }v\in V _ i,\;i\geq 1 \}$$ | conf 0.466
l0586604.png (4) |
Lie group, semi-simple
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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82.(35.)* | ![]() |
$L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$ | $$L(\mathfrak g)\cong \Gamma _ 0(\mathfrak u)\cap \mathfrak h^{\prime }/\Gamma _ 0([\mathfrak k,\mathfrak k])$$ | conf 0.659 F
l058680102.png (102) |
83.(81.)* | ![]() |
$\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$ | $$\Gamma _ 1=\Gamma _ 1(g)=\{X\in h :\alpha (X)\in 2 \pi i {\mathbf Z}\;\text{for all }\alpha \in \Sigma \}.$$ | conf 0.183 F
l05868032.png (32) |
Lie p-algebra
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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84.(36.) | ![]() |
$( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$ | $$(\operatorname {ad}x)^ny=\sum _ {j=1}^n(-1)^j\left(\begin {array}cn\\ j \end {array} \right)x^{n-j}yx^j$$ | conf 0.356
l05872026.png (26) |
85.(99.) | ![]() |
$\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$ | $$\pi (x+y)=\pi (x)+\pi (y),\quad \pi (\lambda x )=\lambda ^p\pi (x),\quad \lambda \in k .$$ | conf 0.964
l05872078.png (78) |
Lie theorem
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
---|---|---|---|---|
86.(134.) | ![]() |
$y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$ | $$y_i=f_i(g_1,\ldots ,g_i;x_1,\ldots ,x_n),\quad i =1,\ldots ,n$$ | conf 0.276
l05876010.png (10) |
87.(86.) | ![]() |
$X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$ | $$X_i=\sum _ {j=1}^n\xi _ {ij}(x)\frac {\partial }{\partial x _ j},\quad i =1,\ldots ,r,$$ | conf 0.656
l05876016.png (16) |
88.(66.)* | ![]() |
$\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$ | $$\frac {\partial f _ j}{\partial g _ i}(g,x)=\sum _ {k=1}^r\xi _ {kj}(f(g_sx))\psi _ {ki}(g),$$ | conf 0.336 F
l05876030.png (30) |
89.(19.)* | ![]() |
$\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ | $$\sum _ {k=1}^N(\xi _ {ik}\frac {\partial \xi _ {jl}}{\partial x _ k}-\xi _ {jk}\frac {\partial \xi _ {il}}{\partial x _ k})=\sum _ {k=1}^rc_{ij}^k\xi _ {kl},$$ | conf 0.157 F
l05876037.png (37) |
90.(14.) | ![]() |
$\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$ | $$\left.\begin {array}c{c_{ij}^k=-c_{ji}^k},\\ {\displaystyle\sum _ {l=1}^r(c_{il}^mc_{jk}^l+c_{kl}^mc_{ij}^l+c_{jl}^mc_{ki}^l)=0,\quad 1 \leq i ,j,k,l,m\leq r,} \end {array} \right\}$$ | conf 0.085
l05876052.png (52) |
Maximal torus
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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---|---|---|---|---|
91.(95.) | ![]() |
$F ( x _ { 1 } f _ { 1 } + \ldots + x _ { x } f _ { n } ) = x _ { 1 } x _ { n } + x _ { 2 } x _ { n } - 1 + \ldots + x _ { p } x _ { n } - p + 1$ | $$F(x_1f_1+\ldots +x_xf_n)=x_1x_n+x_2x_{n-1}+\ldots +x_px_{n-p+1},$$ | conf 0.198
m06301072.png (72) |
Non-Abelian cohomology
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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92.(114.)* | ![]() |
$\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ | $$\phi (g_1)\phi (g_2)\phi (g_1g_2)^{-1}=\operatorname {Int}m(g_1,g_2),$$ | conf 0.443 F
n066900110.png (110) |
93.(90.)* | ![]() |
$( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$ | $$m'(g_1,g_2)=h(g_1)(\phi (g_1)(h(g_2)))m(g_1,g_2)h(g_1,g_2)^{-1}.$$ | conf 0.764 F
n066900118.png (118) |
94.(44.) | ![]() |
$\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$ | $$\delta (e)=e\quad \;\text{and }\quad \delta (\rho (a)b)=\sigma (a)\delta (b),\quad \alpha \in C ^0,\quad b \in C ^1,$$ | conf 0.400
n06690016.png (16) |
95.(60.)* | ![]() |
$C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$ | $$C^{*}(\mathfrak U,{\cal F})=(C^0(\mathfrak U,{\cal F}),C^1(\mathfrak U,{\cal F}),C^2(\mathfrak U,{\cal F})),$$ | conf 0.205 F
n06690028.png (28) |
Picard scheme
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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96.(39.)* | ![]() |
$\operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Fic } ( X \times k S ^ { \prime } ) / \operatorname { Fic } ( S ^ { \prime } )$ | $$\operatorname {Pic}_{X/k}(S^{\prime })=\operatorname {Pic}(X\times_k S ^{\prime })/\operatorname {Pic}(S^{\prime })$$ | conf 0.345 F +
p07267025.png (25) |
Principal analytic fibration
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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97.(100.)* | ![]() |
$g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$ | $$g_j:U_i\cap U _ j\rightarrow G ,\quad i ,j\in I ,\quad U _ i\cap U _ j\neq \emptyset,$$ | conf 0.184 F
p07464025.png (25) |
Quantum groups
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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98.(101.) | ![]() |
$\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$ | $$\phi ^{*}:\mathfrak g^{*}\otimes \mathfrak g^{*}\rightarrow \mathfrak g^{*}$$ | conf 0.837
q07631062.png (62) |
99.(108.) | ![]() |
$\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$ | $$\delta :U_{\mathfrak g}\rightarrow U _ {\mathfrak g}\otimes U _ {\mathfrak g}$$ | conf 0.648
q07631071.png (71) |
100.(56.)* | ![]() |
$\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$ | $$\delta (\alpha )=\operatorname {lim}_{h\rightarrow 0 }h^{-1}(\Delta (a)-\Delta ^{\prime }(\alpha ))$$ | conf 0.304 F
q07631072.png (72) |
101.(129.)* | ![]() |
$[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$ | $$[\alpha ,X_i^{\pm }]=\pm \alpha _ i(\alpha )X_i^{\pm }\quad \text{for }a\in \mathfrak h;$$ | conf 0.544 F
q07631088.png (88) |
102.(128.) | ![]() |
$[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$ | $$[X_i^{+},X_j^{-}]=2\delta _ {ij}h^{-1}\operatorname {sinh}(hH_i/2).$$ | conf 0.893
q07631089.png (89) |
103.(20.) | ![]() |
$\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$ | $$\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\ k \end {array} \right)q^{-k(n-k)/2}(X_i^{\pm })^kX_j^{\pm }\cdot (X_i^{\pm })^{n-k}=0.$$ | conf 0.055
q07631092.png (92) |
104.(30.) | ![]() |
$\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$ | $$\left( \begin {array}ln\\ k \end {array} \right)_q=\frac {(q^n-1)\ldots (q^{n-k+1}-1)}{(q^k-1)\ldots (q-1)} .$$ | conf 0.443
q07631095.png (95) |
105.(21.)* | ![]() |
$\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$ | $$\Delta (X_i^{\pm })=X_i^{\pm }\otimes \operatorname {exp}(\frac {hH_i}4)+\operatorname {exp}(\frac {-hH_i}4)\otimes X _ i^{\pm }.$$ | conf 0.212 F
q07631099.png (99) |
Rational representation
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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106.(91.) | ![]() |
$0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$ | $$0\leq \frac {2(\chi ,\alpha )}{(\alpha ,\alpha )}<p\quad \text{for all }\alpha \in \Delta.$$ | conf 0.879
r077630100.png (100) |
107.(135.) | ![]() |
$\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$ | $$\phi _ 0\otimes \phi _ 1^{Fr}\otimes \ldots \otimes \phi _ d^{{Fr}^d},$$ | conf 0.136
r077630104.png (104) |
108.(45.)* | ![]() |
$\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$ | $$\chi =\delta _ {\phi }-\sum _ {\alpha \in \Delta }m_{\alpha }\alpha ,\quad m _ {\alpha }\in Z ,\quad m _ {\alpha }\geq 0.$$ | conf 0.862 F
r07763055.png (55) |
Singular point
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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109.(31.) | ![]() |
$\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ | $$\sum _ {k_1,\ldots ,k_n=0}^{\infty }c_{k_1\cdots k _ n}(z_1-\zeta _ 1)^{k_1}\ldots (z_n-\zeta _ n)^{k_n}$$ | conf 0.324
s085590225.png (225) |
110.(46.) | ![]() |
$\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$ | $$\frac {m_1}{n_1}<\frac {m_2}{n_1n_2}<\ldots <\frac {m_g}{n_1\ldots n _ g}=\frac {m_g}n$$ | conf 0.459
s085590404.png (404) |
111.(115.)* | ![]() |
$p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$ | $$p(Z)=1-\operatorname {dim}H^0({\mathbf Z},{\cal O}_{\mathbf Z })+\operatorname {dim}H^1({\mathbf Z},{\cal O}_{\mathbf Z })$$ | conf 0.997 F
s085590429.png (429) |
112.(136.)* | ![]() |
$X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$ | $$X_{\epsilon }=\{(x_0,\ldots ,x_x):f(x_0,\ldots ,x_x)=\epsilon \}$$ | conf 0.433 F
s085590440.png (440) |
113.(12.) | ![]() |
$= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$ | $$=\left\{ \begin {array}{ll} {(x+\lambda )^2\ldots (x+k\lambda )^2} &{\text{ if }\mu =2k,}\\ {(x+\lambda )^2\ldots (x+k\lambda )^2(x+(k+1)\lambda )} &{\text{ if }\mu =2k+1,} \end {array} \right.$$ | conf 0.870
s085590458.png (458) |
114.(75.) | ![]() |
$( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$ | $$\big(\frac {\partial F (x,y,\lambda )}{\partial x },\frac {\partial F (x,y,\lambda )}{\partial y }\big)$$ | conf 0.986
s085590482.png (482) |
115.(137.) | ![]() |
$\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$ | $$\frac {dx_i}{dx_{i_0}}=f_i(x),\quad f _ i\in C (U),\quad i \neq i _ 0.$$ | conf 0.594
s085590515.png (515) |
116.(142.)* | ![]() |
$A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$ | $$A=\left\| \begin {array}{ll} {\alpha } &b\\ c &e \end {array} \right\|$$ | conf 0.506 F
s085590527.png (527) |
117.(53.) | ![]() |
$\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$ | $$\Delta =(F_{xx}^{\prime \prime })_0(F_{yy}^{\prime \prime })_0-(F_{xy}^{\prime \prime })_0^2$$ | conf 0.920
s085590634.png (634) |
118.(16.)* | ![]() |
$\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$ | $$\left\| \begin {array}{lll} {F_x^{\prime }} &{F_y^{\prime }} &{F_z^{\prime }}\\ {G_x^{\prime }} &{G_y^{\prime }} &{G_Z^{\prime }} \end {array} \right\|$$ | conf 0.230 F
s085590645.png (645) |
119.(92.) | ![]() |
$( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$ | $$(F_x^{\prime })_0=0,\quad (F_y^{\prime })_0=0,\quad (F_z^{\prime })_0=0.$$ | conf 0.300
s085590653.png (653) |
Solv manifold
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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120.(138.) | ![]() |
$\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$ | $$\{e\}\rightarrow \Delta \rightarrow \pi \rightarrow {\mathbf Z}^s\rightarrow \{e\}$$ | conf 0.972
s08610054.png (54) |
Stability theorems in algebraic K-theory
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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121.(71.) | ![]() |
$\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$ | $$\psi _ {t_1,\ldots ,t_n}^{\prime }:SK_1(R)\rightarrow S K _ 1(R(t_1,\ldots ,t_n)).$$ | conf 0.379
s08706033.png (33) |
Steinberg module
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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122.(130.) | ![]() |
$e = \frac { | U | } { | G | } ( \sum _ { b \in B } b ) ( \sum _ { w \in W } \operatorname { sign } ( w ) w )$ | $$e=\frac {|U|}{|G|}\big(\sum _ {b\in B }b\big)\big(\sum _ {w\in W }\operatorname {sign}(w)w\big)$$ | conf 0.138
s13053016.png (16) |
Steinberg symbol
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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123.(24.)* | ![]() |
$( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$ | $$(x_{ij}(a),x_{kl}(b))=\left\{ \begin {array}{ll} 1 &{\text{ if }i\neq l ,j\neq k },\\ {x_{il}(ab)} &{\text{ if }i\neq l ,j=k}. \end {array} \right.$$ | conf 0.381 F
s13054017.png (17) |
Tilting theory
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
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124.(84.) | ![]() |
$0 \rightarrow \Lambda \rightarrow T _ { 1 } \rightarrow \ldots \rightarrow T _ { n } \rightarrow 0$ | $$0\rightarrow \Lambda \rightarrow T _ 1\rightarrow \ldots \rightarrow T _ n\rightarrow 0 $$ | conf 0.946
t130130105.png (105) |
Tits quadratic form
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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125.(18.) | ![]() |
$q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$ | $$q_R(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}x_ix_j+\sum _ {\langle \beta :i\rightarrow j )\in Q _ 1}r_{i,j}x_ix_j,$$ | conf 0.112
t130140104.png (104) |
126.(40.) | ![]() |
$[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$ | $$[X]\mapsto \chi _ R([X])=\sum _ {m=0}^{\infty }(-1)^m\operatorname {dim}_K\operatorname {Ext}_R^m(X,X)$$ | conf 0.116
t130140118.png (118) |
127.(132.)* | ![]() |
$\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$ | $$\underline {\dim }:K_0(\operatorname {mod}R)\rightarrow {\mathbf Z}^{Q_0}$$ | conf 0.287 F
t130140119.png (119) |
128.(37.)* | ![]() |
$q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$ | $$q_I(x)=\sum _ {i\in I }x_i^2+\sum _ {i\prec j \atop j\in I\setminus {\rm max}I}x_ix_j-\sum _ {p\in \operatorname {max}I}\big(\sum _ {i\prec p }x_i\big)x_p$$ | conf 0.197 F
t130140140.png (140) |
129.(131.)* | ![]() |
$X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ | $$X\mapsto \underline {\dim }X=(\operatorname {dim}_KX_j)_{j\in Q _ 0}$$ | conf 0.819 F
t13014044.png (44) |
130.(25. | ![]() |
$[ X ] \mapsto \chi _ { Q } ( [ X ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( X ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( X , X )$ | $$[X]\mapsto \chi _ Q([X])=\operatorname {dim}_K\operatorname {End}_Q(X)-\operatorname {dim}_K\operatorname {Ext}_Q^1(X,X)$$ | conf 0.661
t13014048.png (48) |
131.(38.)* | ![]() |
$A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$ | $$A_Q(v)=\prod _ {i,j\in Q _ 0}\prod _ {\langle \beta :j\rightarrow i \rangle \in Q _ 1}M_{v_i\times v _ j}(K)_{\beta }$$ | conf 0.481 F
t13014056.png (56) |
132.(139.)* | ![]() |
$\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$ | $$q_Q(x)=\sum _ {j\in Q _ 0}x_j^2-\sum _ {i,j\in Q _ 0}d_{ij}x_ix_j,$$ | conf 0.648 F
t1301406.png (6) |
Torus
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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133.(41.)* | ![]() |
$r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$ | $$r=\alpha \operatorname {sin}u{\bf k}+l(1+\epsilon \operatorname {cos}u)({\bf i}\operatorname {cos}v+{\bf j}\operatorname {sin}v)$$ | conf 0.585 F
t0933502.png (2) |
134.(122.)* | ![]() |
$d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$ | $$ds^2=\alpha ^2du^2+l^2(1+\epsilon \operatorname {cos}u)^2dv^2,$$ | conf 0.696 F
t0933507.png (7) |
Uniform distribution
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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135.(9.) | ![]() |
$u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$ | $$u_3(x)=\left\{ \begin {array}{ll} {\frac {x^2}2,} &{0\leq x <1,}\\ {\frac {[x^2-3(x-1)^2]}2,} &{1\leq x <2,}\\ {\frac {[x^2-3(x-1)^2+3(x-2)^2]}2,} &{2\leq x <3,}\\ {0,} &{x\notin [0,3].} \end {array} \right.$$ | conf 0.733
u09524027.png (27) |
136.(32.)* | ![]() |
$p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$ | $$p(x)=\left\{ \begin {array}{ll} {\frac 1{b-\alpha },} &{x\in [\alpha ,b],}\\ {0,} &{x\notin [\alpha ,b].} \end {array} \right.$$ | conf 0.681 F
u0952403.png (3) |
137.(34.) | ![]() |
$u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$ | $$u_n(x)=\frac 1{(n-1)!}\sum _ {k=0}^n(-1)^k\left(\begin {array}ln\\ k \end {array} \right)(x-k)_{+}^{n-1}$$ | conf 0.569
u09524030.png (30) |
138.(109.) | ![]() |
$z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$ | $$z_{+}=\left\{ \begin {array}{ll} {z,} &{z>0}.\\ {0,} &{z\leq 0 }. \end {array} \right.$$ | conf 0.676
u09524034.png (34) |
139.(43.) | ![]() |
$F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$ | $$F(x)=\left\{ \begin {array}{ll} {0,} &{x\leq a },\\ {\frac {x-a}{b-a},} &{a<x\leq b },\\ {1,} &{x>b}, \end {array} \right.$$ | conf 0.468
u0952407.png (7) |
140.(47.) | ![]() |
$p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$ | $$p(x_1,\ldots ,x_n)=\left\{ \begin {array}{ll} {C\neq 0 ,} &{x\in D },\\ {0,} &{x\notin D }, \end {array} \right.$$ | conf 0.705
u09524072.png (72) |
Unipotent group
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
---|---|---|---|---|
141.(143.) | ![]() |
$\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$ | $$\{g\in \operatorname {GL}(V):(1-g)^n=0\},\quad n =\operatorname {dim}V,$$ | conf 0.287
u0954106.png (6) |
Weyl module
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
---|---|---|---|---|
142.(51.) | ![]() |
$\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$ | $$\operatorname {diag}(t_1,\ldots ,t_n)\mapsto t _ 1^{\lambda _ 1}\ldots t _ n^{\lambda _ n}\in K,$$ | conf 0.507
w120090122.png (122) |
143.(54.)* | ![]() |
$\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$ | $$\chi _ {\lambda }=\sum _ {\mu \in \Lambda (n)}\operatorname {dim}_K(\Delta (\lambda )^{\mu })_{e_{\mu }},$$ | conf 0.461 F
w120090135.png (135) |
144.(110.) | ![]() |
$\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$ | $$\mathfrak B=\{e_{\pm }\alpha ,h_{\beta }:\alpha \in \Phi ^{+},\beta \in \Sigma \}.$$ | conf 0.381
w120090259.png (259) |
145.(82.) | ![]() |
$\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$ | $$\left( \begin {array}ch\\ i \end {array} \right)=\frac {h(h-1)\ldots (h-i+1)}{i!} $$ | conf 0.487
w120090342.png (342) |
146.(28.)* | ![]() |
$\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$ | $$\mathfrak S_{\{1,\ldots ,\lambda _ 1\}}\times \mathfrak S_{\{\lambda _ 1+1,\ldots ,\lambda _ 1+\lambda _ 2\}}\times \dots $$ | conf 0.312 F
w12009095.png (95) |
147.(104.) | ![]() |
$\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$ | $$\ldots \times \mathfrak S_{\{\lambda _ 1+\ldots +\lambda _ {n-1}+1,\ldots ,r\}},$$ | conf 0.259
w12009096.png (96) |
Witt vector
Nr. | Image of png File | $\TeX$, automatically generated version | $\TeX$, manually corrected version | Confidence, F?
png file |
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148.(87.)* | ![]() |
$\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$ | $$\langle \alpha ><b\rangle =\langle \alpha b \rangle ,\quad \langle {\bf 1}\rangle ={\bf f}_1={\bf V}_1=\text{ unit element}1,$$ | conf 0.351 F
w098100172.png (172) |
149.(123.)* | ![]() |
$\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$ | $$\langle \alpha +b\rangle =\sum _ {n=1}^{\infty }{\bf V}_n\langle r _ n(\alpha ,b){\bf f}_n.$$ | conf 0.143 F
w098100177.png (177) |
150.(102.) | ![]() |
$\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$ | $$\sigma (\alpha _ 1,\alpha _ 2,\ldots )=(\alpha _ 1^p,\alpha _ 2^p,\ldots )$$ | conf 0.771
w098100190.png (190) |
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=44162