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(Corrections (Draft))
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{| class="wikitable" cellspacing="0" style="margin: 1em auto 1em auto;
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!| Nr.
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!| Image of png File
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!| $\TeX$, 1st version
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!| $\TeX$, 2nd version
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!| Confidence, F?
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name of png
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|-
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!colspan="5" | [[Algebraic curve]]
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|-
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| 1.(23.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011450/a01145065.png || $g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$ ||  $$ g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n, } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n, } \end{array} \right.$$  || conf 0.698
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png = a01145065.png(65)
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|-
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!colspan="5" | [[Algebraic geometry]]
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|-
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| 2.(116.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150014.png || $\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ ||  $$\empty$$ || conf 0.997
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 +
png = a01150014.png(14)
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|-
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| 3.(133.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150021.png || $\omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ ||  $$\empty$$ || conf 0.973
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png = a01150021.png(21)
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|-
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| 4.(67.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150022.png || $\overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ || $$\widetilde{ w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } },$$ || conf 0.107
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png = a01150022.png(22)
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|-
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| 5.(105.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150044.png || $\theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v )$ ||  $$\empty$$ || conf 0.775
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 +
png = a01150044.png(44)
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|-
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| 6.(17.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150078.png || $\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ ||  $$\empty$$ || conf 0.440
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png = a01150078.png(78)
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|-
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!colspan="5" | [[Algebraic surface]]
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|-
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| 7.(144.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640132.png || $0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$ || $$0 \rightarrow {\cal O} _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$$  || conf 0.981
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png = a011640132.png(132)
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|-
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| 8.(73.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640137.png || $M = \operatorname { dim } \operatorname { Im } ( H ^ { 1 } ( V , E _ { \alpha } ) \rightarrow H ^ { 1 } ( V , T _ { V } ) )$ ||  $$\empty$$ || conf 0.997
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png = a011640137.png(137)
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|-
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| 9.(88.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640139.png || $\operatorname { dim } _ { k } H ^ { 2 } ( V , E _ { \alpha } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , T _ { V } )$ ||  $$\empty$$ || conf 0.996
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 +
png = a011640139.png(139)
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|-
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| 10.(117.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164027.png || $N _ { m } = \left( \begin{array} { c } { m + 3 } \\ { 3 } \end{array} \right) - d m + 2 t + \tau + p - 1$ ||  $$\empty$$ || conf 0.369
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png = a01164027.png(27)
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|-
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| 11.(72.) ||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164029.png || $p _ { \alpha } ( V ) = \left( \begin{array} { c } { n - 1 } \\ { 3 } \end{array} \right) - d ( n - 1 ) + 2 t + \tau + p - 1$ ||  $$\empty$$ || conf 0.396
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png = a01164029.png(29)
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|-
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| 12.(68.)*||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164047.png || $p _ { x } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , O _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , O _ { V } ) =$ ||  $$p _ { \alpha } ( V ) = - \operatorname { dim } _ { k } H _ { 1 } ( V , {\cal O} _ { V } ) + \operatorname { dim } _ { k } H ^ { 2 } ( V , {\cal O} _ { V } ) =$$ || conf 0.756  F
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png = a01164047.png(47)
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|-
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| 13.(93.)*||  https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164053.png || $1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$ ||  $$ 1 + p _ { \alpha } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 },$$|| conf 0.752  F
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png = a01164053.png(53)
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|-
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!colspan="5" | [[Cartan subalgebra]]
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|-
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| 14.(33.)*||  https://www.encyclopediaofmath.org/legacyimages/c/c020/c020550/c0205509.png || $\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$ ||  $$\empty$$ || conf 0.110  F
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png = c0205509.png(9)
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|-
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!colspan="5" | [[Cartan theorem]]
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|-
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| 15.(49.)*||  https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205704.png || $f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$ ||  $$\empty$$ || conf 0.149  F
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png = c0205704.png(4)
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|-
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| 16.(55.)*||  https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057064.png || $\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$ ||  $$\empty$$ || conf 0.853  F
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png = c02057064.png(64)
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|-
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!colspan="5" | [[Comitant]]
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|-
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| 17.(7.) ||  https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333033.png || $H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$ ||  $$\empty$$ || conf 0.956
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png = c02333033.png(33)
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|-
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| 18.(76.) ||  https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333034.png || $= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$ ||  $$\empty$$ || conf 0.549
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png = c02333034.png(34)
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|-
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| 19.(11.)*||  https://www.encyclopediaofmath.org/legacyimages/c/c023/c023330/c02333035.png || $( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$ ||  $$\empty$$ || conf 0.521  F
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png = c02333035.png(35)
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|-
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!colspan="5" | [[Deformation]]
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|-
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| 20.(26.) ||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700175.png || $\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$ ||  $$\empty$$ || conf 0.683
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png = d030700175.png(175)
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|-
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| 21.(27.) ||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700190.png || $\operatorname { dim } _ { k } H ^ { 1 } ( X _ { 0 } , T _ { X _ { 0 } } ) - \operatorname { dim } M _ { X _ { 0 } } \leq \operatorname { dim } _ { k } H ^ { 2 } ( X _ { 0 } , T _ { X _ { 0 } } )$ ||  $$\empty$$ || conf 0.944
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png = d030700190.png(190)
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|-
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| 22.(78.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700263.png || $\alpha \circ b = \alpha b + \sum _ { i = 1 } ^ { \infty } \phi _ { i } ( \alpha , b ) t ^ { i } , \quad \alpha , b \in V$ ||  $$\empty$$ || conf 0.097  F
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 +
png = d030700263.png(263)
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|-
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| 23.(96.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700270.png || $\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$ ||  $$\empty$$ || conf 0.873  F
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 +
png = d030700270.png(270)
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|-
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!colspan="5" | [[Differential algebra]]
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|-
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| 24.(106.) ||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830107.png || $S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$ ||  $$\empty$$ || conf 0.149
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png = d031830107.png(107)
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|-
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| 25.(146.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830141.png || $( \eta _ { 1 } , \ldots , \eta _ { k } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { k } )$ ||  $$\empty$$ || conf 0.562  F
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png = d031830141.png(141)
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|-
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| 26.(145.)$^F$*||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d031830150.png || $( \eta _ { 1 } , \ldots , \eta _ { n } ) \rightarrow F ( \zeta _ { 1 } , \ldots , \zeta _ { n } )$ ||  $$\empty$$ || conf 0.376  F
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png = d031830150.png(150)
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|-
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| 27.(57.) ||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d03183016.png || $\omega _ { V } = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ ||  $$\empty$$ || conf 0.780
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png = d03183016.png(16)
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|-
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| 28.(111.) ||  https://www.encyclopediaofmath.org/legacyimages/d/d031/d031830/d03183043.png || $e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$ ||  $$ e_ { i j } = \operatorname { ord } _  { { Y } _ { j } } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$$ || conf 0.187
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png = d03183043.png(43)
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|-
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!colspan="5" | [[Dimension polynomial]]
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|-
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| 29.(48.) ||  https://www.encyclopediaofmath.org/legacyimages/d/d032/d032490/d03249029.png || $\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ ||  $$\empty$$ || conf 0.968
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png = d03249029.png(29)
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|-
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!colspan="5" | [[Duality]]
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|-
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| 30.(118.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120173.png || $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$ ||  $$\empty$$ || conf 0.824  F
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png = d034120173.png(173)
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|-
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| 31.(59.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120175.png || $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow H _ { c } ^ { n } ( X , \Omega )$ ||  $$\empty$$ || conf 0.921  F
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png = d034120175.png(175)
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|-
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| 32.(124.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120184.png || $( H ^ { p } ( X , F ) ) ^ { \prime } \cong H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) )$ ||  $$\empty$$ || conf 0.829  F
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png = d034120184.png(184)
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|-
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| 33.(29.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120236.png || $\beta : \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X  F  , \Omega ) \rightarrow \operatorname { Ext } _ { c } ^ { n - p - 1 } ( X \backslash Y || F , \Omega )$ ||  $$\empty$$ || conf 0.634 || F
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 +
png = d034120236.png(236)
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|-
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| 34.(77.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120247.png || $\underset { n \rightarrow \infty } { \operatorname { lim } } | \alpha _ { n } | ^ { 1 / n } = \sigma < + \infty$ ||  $$\empty$$ || conf 0.521  F
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png = d034120247.png(247)
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|-
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| 35.(58.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120253.png || $h ( \phi ) = \operatorname { lim } _ { r \rightarrow \infty } \frac { \operatorname { ln } | A ( r e ^ { i \phi } ) | } { r }$ ||  $$\empty$$ || conf 0.861  F
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png = d034120253.png(253)
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|-
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| 36.(69.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120360.png || $\operatorname { sup } _ { l \in E ^ { \perp } } | l ( \omega ) | = \operatorname { inf } _ { x \in E } \| \omega - x \|$ ||  $$\empty$$ || conf 0.293  F
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png = d034120360.png(360)
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|-
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| 37.(15.) ||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120376.png || $\operatorname { sup } _ { f \in B ^ { 1 } } | \int _ { \partial G } f ( \zeta ) \omega ( \zeta ) d \zeta | = \operatorname { inf } _ { \phi \in E ^ { 1 } } \int _ { \partial G } | \omega ( \zeta ) - \phi ( \zeta ) \| d \zeta |$ ||  $$\empty$$ || conf 0.508
 +
 +
png = d034120376.png(376)
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|-
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| 38.(52.) ||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120509.png || $f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$ ||  $$\empty$$ || conf 0.491
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 +
png = d034120509.png(509)
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|-
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| 39.(140.) ||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120535.png || $f ^ { * } ( x ^ { * } ) = \operatorname { sup } _ { x \in X } ( \langle x ^ { * } , x \rangle - f ( x ) )$ ||  $$\empty$$ || conf 0.900
 +
 +
png = d034120535.png(535)
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|-
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| 40.(94.) ||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d034120555.png || $f _ { 0 } ( x ) \rightarrow \text { inf, } \quad f _ { i } ( x ) \leq 0 , \quad i = 1 , \ldots , m , \quad x \in B$ ||  $$\empty$$ || conf 0.810
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 +
png = d034120555.png(555)
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|-
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| 41.(74.)*||  https://www.encyclopediaofmath.org/legacyimages/d/d034/d034120/d03412079.png || $( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$ ||  $$\empty$$ || conf 0.117  F
 +
 +
png = d03412079.png(79)
 +
|-
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!colspan="5" | [[Extension of a differential field]]
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|-
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| 42.(63.) ||  https://www.encyclopediaofmath.org/legacyimages/e/e036/e036960/e03696024.png || $F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$ ||  $$\empty$$ || conf 0.628
 +
 +
png = e03696024.png(24)
 +
|-
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!colspan="5" | [[Formal group]]
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|-
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| 43.(120.)*||  https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f040820118.png || $\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$ ||  $$\empty$$ || conf 0.098  F
 +
 +
png = f040820118.png(118)
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|-
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| 44.(147.)*||  https://www.encyclopediaofmath.org/legacyimages/f/f040/f040820/f04082059.png || $( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$ ||  $$\empty$$ || conf 0.553  F
 +
 +
png = f04082059.png(59)
 +
|-
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!colspan="5" | [[Gel'fond-Schneider method]]
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|-
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| 45.(148.) ||  https://www.encyclopediaofmath.org/legacyimages/g/g130/g130020/g1300205.png || $\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$ ||  $$\empty$$ || conf 0.979
 +
 +
png = g1300205.png(5)
 +
|-
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!colspan="5" | [[Group]]
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|-
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| 46.(22.)*||  https://www.encyclopediaofmath.org/legacyimages/g/g045/g045210/g04521075.png || $\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$ ||  $$\empty$$ || conf 0.226  F
 +
 +
png = g04521075.png(75)
 +
|-
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!colspan="5" | [[Homogeneous space]]
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|-
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| 47.(89.) ||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047690/h04769069.png || $\mathfrak { g } = \mathfrak { f } + \mathfrak { m } , \quad \mathfrak { f } \cap \mathfrak { m } = \{ 0 \}$ ||  $$\empty$$ || conf 0.793
 +
 +
png = h04769069.png(69)
 +
|-
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!colspan="5" | [[Hopf algebra]]
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|-
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| 48.(103.) ||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970129.png || $m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$ ||  $$\empty$$ || conf 0.618
 +
 +
png = h047970129.png(129)
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|-
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| 49.(107.)*||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970139.png || $F _ { 1 } ( X || Y ) , \ldots , F _ { n } ( X || Y ) \in K [ X _ { 1 } , \ldots , X _ { n } || Y _ { 1 } , \ldots , Y _ { n } ] \}$ ||  $$\empty$$ || conf 0.353  F
 +
 +
png = h047970139.png(139)
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|-
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| 50.(97.) ||  https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797042.png || $\epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g }$ ||  $$\empty$$ || conf 0.213
 +
 +
png = h04797042.png(42)
 +
|-
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!colspan="5" | [[Invariants, theory of]]
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|-
 +
| 51.(149.)*||  https://www.encyclopediaofmath.org/legacyimages/i/i052/i052350/i05235015.png || $\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$ ||  $$\empty$$ || conf 0.142  F
 +
 +
png = i05235015.png(15)
 +
|-
 +
!colspan="5" | [[Jordan algebra]]
 +
|-
 +
| 52.(150.) ||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427030.png || $H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$ ||  $$\empty$$ || conf 0.651
 +
 +
png = j05427030.png(30)
 +
|-
 +
| 53.(42.) ||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427031.png || $\Gamma = \operatorname { diag } \{ \gamma _ { 1 } , \gamma _ { 2 } , \gamma _ { 3 } \} , \quad \gamma _ { i } \neq 0 , \quad \gamma _ { i } \in F$ ||  $$\empty$$ || conf 0.987
 +
 +
png = j05427031.png(31)
 +
|-
 +
| 54.(125.)*||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054270/j05427077.png || $\mathfrak { g } = \mathfrak { g } - 1 + \mathfrak { g } \mathfrak { d } + \mathfrak { g } _ { 1 }$ ||  $$\empty$$ || conf 0.598  F
 +
 +
png = j05427077.png(77)
 +
|-
 +
!colspan="5" | [[Jordan matrix]]
 +
|-
 +
| 55.(6.)*||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543403.png || $J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$ ||  $$J = \left\| \begin{array} { c c c c }
 +
J_{n_1}(\lambda_1)  &  0    &    0  & 0 \\
 +
      0            & \ddots  & \ddots & 0 \\
 +
      0            & \ddots  & \ddots & 0 \\
 +
      0            &  0    &    0  & J_{n_s}(\lambda_s)
 +
\end{array} \right\|,$$ || conf 0.072  F
 +
 +
png = j0543403.png(3)
 +
|-
 +
| 56.(64.) ||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j05434030.png || $C _ { m } ( \lambda ) = \operatorname { rk } ( A - \lambda E ) ^ { m - 1 } - 2 \operatorname { rk } ( A - \lambda E ) ^ { m } +$ ||  $$\empty$$ || conf 0.955
 +
 +
png = j05434030.png(30)
 +
|-
 +
| 57.(1.)*||  https://www.encyclopediaofmath.org/legacyimages/j/j054/j054340/j0543406.png || $J _ { m } ( \lambda ) = \| \begin{array} { c c c c c c } { \lambda } & { 1 } & { \square } & { \square } & { \square } & { \square } \\ { \square } & { \lambda } & { 1 } & { \square } & { 0 } & { \square } \\ { \square } & { \square } & { \cdots } & { \square } & { \square } & { \square } \\ { \square } & { \square } & { \square } & { \cdots } & { \square } & { \square } \\ { \square } & { 0 } & { \square } & { \square } & { \lambda } & { 1 } \\ { \square } & { \square } & { \square } & { \square } & { \square } & { \lambda } \end{array} ]$ || $$J_m(\lambda) = \left\| \begin{array} { c c c c c c }
 +
\lambda &    1    & \square &  \square &  \square &  \square \\
 +
\square & \lambda &    1    &  \square &  0      &  \square \\
 +
\square & \square & \ddots  &  \ddots  &  \square &  \square\\
 +
\square & \square & \square &  \ddots  &  \ddots  &  \square \\
 +
\square &  0    & \square &  \square & \lambda  &  1 \\
 +
\square & \square & \square &  \square &  \square & \lambda
 +
\end{array} \right\|,$$ || conf 0.098  F
 +
 +
png = j0543406.png(6)
 +
|-
 +
!colspan="5" | [[Lie algebra, semi-simple]]
 +
|-
 +
| 58.(5.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510127.png || $\left\| \begin{array} { r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 2 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } \end{array} \right\|$ || $$B_n: \quad \left\| \begin{array} { r r r r r r }
 +
{ 2 }  & { - 1 } & { 0 }  & { \dots } & { 0 } & { 0 }  \\
 +
{ - 1 } & { 2 }  & { - 1 } & { \dots } & { 0 } & { 0 }  \\
 +
{ 0 }  & { - 1 } & { 2 }  & { \dots } & { 0 } & { 0 }  \\
 +
\cdot  & \cdot  & \cdot  &  \dots  & \cdot & \cdot  \\
 +
{ 0 }  & { 0 }  & { 0 }  & { \dots } & { - 1 } & { 0 }\\
 +
{ 0 }  & { 0 }  & { 0 }  & { \dots } & { 2 } & { - 2 }\\
 +
{ 0 }  & { 0 }  & { 0 }  & { \dots } & { - 1 } & { 2 }
 +
\end{array} \right\|,$$|| conf 0.232
 +
 +
png = l058510127.png(127)
 +
|-
 +
| 59.(3.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510129.png || $\| \left. \begin{array} { r r r r r r r } { 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 } \end{array} \right. |$ ||  $$D_n: \quad
 +
\left\| \begin{array} { r r r r r r r }
 +
{ 2 } & { - 1 } & { 0 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 } \\
 +
{ - 1 } & { 2 } & { - 1 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 }\\
 +
{ 0 } & { - 1 } & { 2 } & { \dots } & { 0 } & { 0 } & { 0 } & { 0 }  \\
 +
\cdot  & \cdot  & \cdot  &  \dots  & \cdot & \cdot &\cdot & \cdot \\
 +
{ 0 } & { 0 } & { 0 } & { \dots } & { 2 } & { - 1 } & { 0 } & { 0 }  \\
 +
{ 0 } & { 0 } & { 0 } & { \dots } & { - 1 } & { 2 } & { - 1 } & { - 1 }\\
 +
{ 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 2 } & { 0 } \\
 +
{ 0 } & { 0 } & { 0 } & { \dots } & { 0 } & { - 1 } & { 0 } & { 2 }
 +
\end{array} \right\|,$$ || conf 0.055  F
  
−
$$A+B=C$$
+
png = l058510129.png(129)
 +
|-
 +
| 60.(8.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510130.png || $\left\| \begin{array} { r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ || $$E_6:
 +
\quad \left\| \begin{array} { r r r r r r }
 +
{ 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\
 +
{ 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } \\
 +
{ - 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } \\
 +
{ 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\
 +
{ 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\
 +
{ 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 }
 +
\end{array} \right\|,$$ || conf 0.628  F
  
−
 
+
png = l058510130.png(130)
−
{| class="wikitable" cellspacing="0" style="margin: 1em auto 1em auto;"
+
|-
−
! colspan="2" | Col 1
+
| 61.(4.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510131.png || $\left\| \begin{array} { r r r r r r r } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\|$ || $$E_7: \quad
−
! scope="col" | Col 2
+
\left\| \begin{array} { r r r r r r r }
 +
{ 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\
 +
{ 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } \\
 +
{-1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\
 +
{ 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\
 +
{ 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\
 +
{ 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\
 +
{ 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 }
 +
\end{array} \right\|,$$ || conf 0.278
 +
 +
png = l058510131.png(131)
 +
|-
 +
| 62.(2.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510132.png || $\left. \begin{array} { r l l l l l l l } { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right.$ || $$E_8: \quad
 +
\left\| \begin{array} { r r r r r r r r }
 +
{ 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } &
 +
{ 0 } \\ { 0 } & { 2 } & { 0 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\
 +
{-1 } & { 0 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\
 +
{ 0 } & { - 1 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } \\
 +
{ 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } & { 0 } \\
 +
{ 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } & { 0 } \\
 +
{ 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 } & { - 1 } \\
 +
{ 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { - 1 } & { 2 }
 +
\end{array} \right\|,$$ || conf 0.354  F
 +
 +
png = l058510132.png(132)
 +
|-
 +
| 63.(10.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l058510133.png || $\left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|$
 +
|| $$F_4: \quad \left\| \begin{array} { r r r r } { 2 } & { - 1 } & { 0 } & { 0 } \\ { - 1 } & { 2 } & { - 2 } & { 0 } \\ { 0 } & { - 1 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { - 1 } & { 2 } \end{array} \right\| , \quad G _ { 2 } : \quad \left\| \begin{array} { r r } { 2 } & { - 1 } \\ { - 3 } & { 2 } \end{array} \right\|.$$ || conf 0.374  F
 +
 +
png = l058510133.png(133)
 +
|-
 +
| 64.(98.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851030.png || $\mathfrak { g } _ { \alpha } = \{ X \in \mathfrak { g } : [ H , X ] = \alpha ( H ) X , H \in \mathfrak { h } \}$ ||  $$\empty$$ || conf 0.976
 +
 +
png = l05851030.png(30)
 +
|-
 +
| 65.(126.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851037.png || $\mathfrak { g } = \mathfrak { h } + \sum _ { \alpha \in \Sigma } \mathfrak { g } _ { \alpha }$ ||  $$\empty$$ || conf 0.945
 +
 +
png = l05851037.png(37)
 +
|-
 +
| 66.(61.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851044.png || $\mathfrak { g } _ { \alpha } = \operatorname { dim } [ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { - \alpha } ] = 1$ ||  $$\empty$$ || conf 0.520  F
 +
 +
png = l05851044.png(44)
 +
|-
 +
| 67.(65.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851050.png || $[ H _ { \alpha } , X _ { \alpha } ] = 2 X _ { \alpha } \quad \text { and } \quad [ H _ { \alpha } , Y _ { \alpha } ] = - 2 Y _ { 0 }$ ||  $$\empty$$ || conf 0.539  F
 +
 +
png = l05851050.png(50)
 +
|-
 +
| 68.(70.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851051.png || $\beta ( H _ { \alpha } ) = \frac { 2 ( \alpha , \beta ) } { ( \alpha , \alpha ) } , \quad \alpha , \beta \in \Sigma$ ||  $$\empty$$ || conf 0.997
 +
 +
png = l05851051.png(51)
 +
|-
 +
| 69.(112.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851057.png || $[ \mathfrak { g } _ { \alpha } , \mathfrak { g } _ { \beta } ] = \mathfrak { g } _ { \alpha + \beta }$ ||  $$\empty$$ || conf 0.917
 +
 +
png = l05851057.png(57)
 +
|-
 +
| 70.(127.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851064.png || $H _ { \alpha _ { 1 } } , \ldots , H _ { \alpha _ { k } } , X _ { \alpha } \quad ( \alpha \in \Sigma )$ ||  $$\empty$$ || conf 0.432
 +
 +
png = l05851064.png(64)
 +
|-
 +
| 71.(113.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851069.png || $[ [ X _ { \alpha _ { i } } , X _ { - } \alpha _ { i } ] , X _ { - \alpha _ { j } } ] = - n ( i , j ) X _ { \alpha _ { j } }$ ||  $$\empty$$ || conf 0.628  F
 +
 +
png = l05851069.png(69)
 +
|-
 +
| 72.(79.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851073.png || $n ( i , j ) = \alpha _ { j } ( H _ { i } ) = \frac { 2 ( \alpha _ { i } , \alpha _ { j } ) } { ( \alpha _ { j } , \alpha _ { j } ) }$ ||  $$\empty$$ || conf 0.992
 +
 +
png = l05851073.png(73)
 +
|-
 +
| 73.(13.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851074.png || $[ X _ { \alpha } , X _ { \beta } ] = \left\{ \begin{array} { l l } { N _ { \alpha , \beta } X _ { \alpha + \beta } } & { \text { if } \alpha + \beta \in \Sigma } \\ { 0 } & { \text { if } \alpha + \beta \notin \Sigma } \end{array} \right.$ ||  $$\empty$$ || conf 0.988
 +
 +
png = l05851074.png(74)
 +
|-
 +
| 74.(80.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851078.png || $N _ { \alpha , \beta } = - N _ { - \alpha , - \beta } \quad \text { and } \quad N _ { \alpha , \beta } = \pm ( p + 1 )$ ||  $$\empty$$ || conf 0.961
 +
 +
png = l05851078.png(78)
 +
|-
 +
| 75.(85.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058510/l05851085.png || $X _ { \alpha } - X _ { - \alpha } , \quad i ( X _ { \alpha } + X _ { - \alpha } ) \quad ( \alpha \in \Sigma _ { + } )$ ||  $$\empty$$ || conf 0.691  F
 +
 +
png = l05851085.png(85)
 +
|-
 +
!colspan="5" | [[Lie algebra, solvable]]
 +
|-
 +
| 76.(119.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852011.png || $[ \mathfrak { g } _ { i } , \mathfrak { g } _ { i } ] \subset \mathfrak { g } _ { \mathfrak { i } } + 1$ ||  $$\empty$$ || conf 0.276  F
 +
 +
png = l05852011.png(11)
 +
|-
 +
| 77.(141.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852046.png || $\operatorname { dim } \mathfrak { g } _ { i } = \operatorname { dim } \mathfrak { g } - i$ ||  $$\empty$$ || conf 0.901
 +
 +
png = l05852046.png(46)
 +
|-
 +
!colspan="5" | [[Lie group]]
 +
|-
 +
| 78.(62.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l058590115.png || $( G ) \cong \operatorname { Aut } ( L ( G ) ) \quad \text { and } \quad L ( \operatorname { Aut } ( G ) ) \cong D ( L ( G ) )$ ||  $$\empty$$ || conf 0.693  F
 +
 +
png = l058590115.png(115)
 +
|-
 +
| 79.(50.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058590/l05859086.png || $( X , Y ) \rightarrow \operatorname { exp } ^ { - 1 } ( \operatorname { exp } X \operatorname { exp } Y ) , \quad X , Y \in L ( G )$ ||  $$\empty$$ || conf 0.856
 +
 +
png = l05859086.png(86)
 +
|-
 +
!colspan="5" | [[Lie group, compact]]
 +
|-
 +
| 80.(121.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058610/l05861012.png || $J = \left\| \begin{array} { c c } { 0 } & { E _ { x } } \\ { - E _ { x } } & { 0 } \end{array} \right\|$ ||  $$\empty$$ || conf 0.364  F
 +
 +
png = l05861012.png(12)
 +
|-
 +
!colspan="5" | [[Lie group, nilpotent]]
 +
|-
 +
| 81.(83.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058660/l0586604.png || $N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \}$ ||  $$\empty$$ || conf 0.466
 +
 +
png = l0586604.png(4)
 +
|-
 +
!colspan="5" | [[Lie group, semi-simple]]
 +
|-
 +
| 82.(35.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l058680102.png || $L ( \mathfrak { g } ) \cong \Gamma _ { 0 } ( \mathfrak { u } ) \cap \mathfrak { h } ^ { \prime } / \Gamma _ { 0 } ( [ \mathfrak { k } , \mathfrak { k } ] )$ ||  $$\empty$$ || conf 0.659  F
 +
 +
png = l058680102.png(102)
 +
|-
 +
| 83.(81.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058680/l05868032.png || $\Gamma _ { 1 } = \Gamma _ { 1 } ( g ) = \{ X \in h : \alpha ( X ) \in 2 \pi i Z \text { for all } \alpha \in \Sigma \}$ ||  $$\empty$$ || conf 0.183  F
 +
 +
png = l05868032.png(32)
 +
|-
 +
!colspan="5" | [[Lie p-algebra]]
 +
|-
 +
| 84.(36.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872026.png || $( \operatorname { ad } x ) ^ { n } y = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \left( \begin{array} { c } { n } \\ { j } \end{array} \right) x ^ { n - j } y x ^ { j }$ ||  $$\empty$$ || conf 0.356
 +
 +
png = l05872026.png(26)
 +
|-
 +
| 85.(99.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058720/l05872078.png || $\pi ( x + y ) = \pi ( x ) + \pi ( y ) , \quad \pi ( \lambda x ) = \lambda ^ { p } \pi ( x ) , \quad \lambda \in k$ ||  $$\empty$$ || conf 0.964
 +
 +
png = l05872078.png(78)
 +
|-
 +
!colspan="5" | [[Lie theorem]]
 +
|-
 +
| 86.(134.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876010.png || $y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$ ||  $$\empty$$ || conf 0.276
 +
 +
png = l05876010.png(10)
 +
|-
 +
| 87.(86.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876016.png || $X _ { i } = \sum _ { j = 1 } ^ { n } \xi _ { i j } ( x ) \frac { \partial } { \partial x _ { j } } , \quad i = 1 , \ldots , r$ ||  $$\empty$$ || conf 0.656
 +
 +
png = l05876016.png(16)
 +
|-
 +
| 88.(66.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876030.png || $\frac { \partial f _ { j } } { \partial g _ { i } } ( g , x ) = \sum _ { k = 1 } ^ { r } \xi _ { k j } ( f ( g _ { s } x ) ) \psi _ { k i } ( g )$ ||  $$\empty$$ || conf 0.336  F
 +
 +
png = l05876030.png(30)
 +
|-
 +
| 89.(19.)*||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876037.png || $\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ ||  $$\empty$$ || conf 0.157  F
 +
 +
png = l05876037.png(37)
 +
|-
 +
| 90.(14.) ||  https://www.encyclopediaofmath.org/legacyimages/l/l058/l058760/l05876052.png || $\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$ ||  $$\empty$$ || conf 0.085
 +
 +
png = l05876052.png(52)
 +
|-
 +
!colspan="5" | [[Maximal torus]]
 +
|-
 +
| 91.(95.) ||  https://www.encyclopediaofmath.org/legacyimages/m/m063/m063010/m06301072.png || $F ( x _ { 1 } f _ { 1 } + \ldots + x _ { x } f _ { n } ) = x _ { 1 } x _ { n } + x _ { 2 } x _ { n } - 1 + \ldots + x _ { p } x _ { n } - p + 1$ ||  $$\empty$$ || conf 0.198
 +
 +
png = m06301072.png(72)
 +
|-
 +
!colspan="5" | [[Non-Abelian cohomology]]
 +
|-
 +
| 92.(114.)*||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900110.png || $\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ ||  $$\empty$$ || conf 0.443  F
 +
 +
png = n066900110.png(110)
 +
|-
 +
| 93.(90.)*||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n066900118.png || $( g _ { 1 } , g _ { 2 } ) = h ( g _ { 1 } ) ( \phi ( g _ { 1 } ) ( h ( g _ { 2 } ) ) ) m ( g _ { 1 } , g _ { 2 } ) h ( g _ { 1 } , g _ { 2 } ) ^ { - 1 }$ ||  $$\empty$$ || conf 0.764  F
 +
 +
png = n066900118.png(118)
 +
|-
 +
| 94.(44.) ||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690016.png || $\delta ( e ) = e \quad \text { and } \quad \delta ( \rho ( a ) b ) = \sigma ( a ) \delta ( b ) , \quad \alpha \in C ^ { 0 } , \quad b \in C ^ { 1 }$ ||  $$\empty$$ || conf 0.400
 +
 +
png = n06690016.png(16)
 +
|-
 +
| 95.(60.)*||  https://www.encyclopediaofmath.org/legacyimages/n/n066/n066900/n06690028.png || $C ^ { * } ( \mathfrak { U } , F ) = ( C ^ { 0 } ( \mathfrak { U } , F ) , C ^ { 1 } ( \mathfrak { U } , F ) , C ^ { 2 } ( \mathfrak { U } , F ) )$ ||  $$\empty$$ || conf 0.205  F
 +
 +
png = n06690028.png(28)
 +
|-
 +
!colspan="5" | [[Picard scheme]]
 +
|-
 +
| 96.(39.)*||  https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267025.png || $\operatorname { Pic } _ { X / k } ( S ^ { \prime } ) = \operatorname { Fic } ( X \times k S ^ { \prime } ) / \operatorname { Fic } ( S ^ { \prime } )$ ||  $$\empty$$ || conf 0.345  F +
 +
 +
png = p07267025.png(25)
 +
|-
 +
!colspan="5" | [[Principal analytic fibration]]
 +
|-
 +
| 97.(100.)*||  https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464025.png || $g j : U _ { i } \cap U _ { j } \rightarrow G , \quad i , j \in I , \quad U _ { i } \cap U _ { j } \neq \emptyset$ ||  $$\empty$$ || conf 0.184  F
 +
 +
png = p07464025.png(25)
 +
|-
 +
!colspan="5" | [[Quantum groups]]
 +
|-
 +
| 98.(101.) ||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631062.png || $\phi ^ { * } : \mathfrak { g } ^ { * } \otimes \mathfrak { g } ^ { * } \rightarrow \mathfrak { g } ^ { * }$ ||  $$\empty$$ || conf 0.837
 +
 +
png = q07631062.png(62)
 +
|-
 +
| 99.(108.) ||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631071.png || $\delta : U _ { \mathfrak { g } } \rightarrow U _ { \mathfrak { g } } \otimes U _ { \mathfrak { g } }$ ||  $$\empty$$ || conf 0.648
 +
 +
png = q07631071.png(71)
 +
|-
 +
| 100.(56.)*||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631072.png || $\delta ( \alpha ) = \operatorname { lim } _ { h \rightarrow 0 } h ^ { - 1 } ( \Delta ( a ) - \Delta ^ { \prime } ( \alpha ) )$ ||  $$\empty$$ || conf 0.304  F
 +
 +
png = q07631072.png(72)
 +
|-
 +
| 101.(129.)*||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631088.png || $[ \alpha , X _ { i } ^ { \pm } ] = \pm \alpha _ { i } ( \alpha ) X _ { i } ^ { \pm } \quad \text { for } a$ ||  $$\empty$$ || conf 0.544  F
 +
 +
png = q07631088.png(88)
 +
|-
 +
| 102.(128.) ||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631089.png || $[ X _ { i } ^ { + } , X _ { j } ^ { - } ] = 2 \delta _ { i j } h ^ { - 1 } \operatorname { sinh } ( h H _ { i } / 2 )$ ||  $$\empty$$ || conf 0.893
 +
 +
png = q07631089.png(89)
 +
|-
 +
| 103.(20.) ||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631092.png || $\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) q ^ { - k ( n - k ) / 2 } ( X _ { i } ^ { \pm } ) ^ { k } X _ { j } ^ { \pm } \cdot ( X _ { i } ^ { \pm } ) ^ { n - k } = 0$ ||  $$\empty$$ || conf 0.055
 +
 +
png = q07631092.png(92)
 +
|-
 +
| 104.(30.) ||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631095.png || $\left( \begin{array} { l } { n } \\ { k } \end{array} \right) _ { q } = \frac { ( q ^ { n } - 1 ) \ldots ( q ^ { n - k + 1 } - 1 ) } { ( q ^ { k } - 1 ) \ldots ( q - 1 ) }$ ||  $$\empty$$ || conf 0.443
 +
 +
png = q07631095.png(95)
 +
|-
 +
| 105.(21.)*||  https://www.encyclopediaofmath.org/legacyimages/q/q076/q076310/q07631099.png || $\Delta ( X _ { i } ^ { \pm } ) = X _ { i } ^ { \pm } \bigotimes \operatorname { exp } ( \frac { h H _ { i } } { 4 } ) + \operatorname { exp } ( \frac { - h H _ { i } } { 4 } ) \otimes x _ { i } ^ { \pm }$ ||  $$\empty$$ || conf 0.212  F
 +
 +
png = q07631099.png(99)
 +
|-
 +
!colspan="5" | [[Rational representation]]
 +
|-
 +
| 106.(91.) ||  https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630100.png || $0 \leq \frac { 2 ( \chi , \alpha ) } { ( \alpha , \alpha ) } < p \quad \text { for all } \alpha \in \Delta$ ||  $$\empty$$ || conf 0.879
 +
 +
png = r077630100.png(100)
 +
|-
 +
| 107.(135.) ||  https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r077630104.png || $\phi _ { 0 } \bigotimes \phi _ { 1 } ^ { Fr } \otimes \ldots \otimes \phi _ { d } ^ { FF ^ { d } }$ ||  $$\empty$$ || conf 0.136
 +
 +
png = r077630104.png(104)
 +
|-
 +
| 108.(45.)*||  https://www.encyclopediaofmath.org/legacyimages/r/r077/r077630/r07763055.png || $\chi = \delta _ { \phi } - \sum _ { \alpha \in \Delta } m _ { \alpha } \alpha , \quad m _ { \alpha } \in Z , \quad m _ { \alpha } \geq 0$ ||  $$\empty$$ || conf 0.862  F
 +
 +
png = r07763055.png(55)
 +
|-
 +
!colspan="5" | [[Singular point]]
 +
|-
 +
| 109.(31.) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590225.png || $\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ ||  $$\empty$$ || conf 0.324
 +
 +
png = s085590225.png(225)
 +
|-
 +
| 110.(46.) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590404.png || $\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$ ||  $$\empty$$ || conf 0.459
 +
 +
png = s085590404.png(404)
 +
|-
 +
| 111.(115.)*||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590429.png || $p ( Z ) = 1 - \operatorname { dim } H ^ { 0 } ( Z , O _ { Z } ) + \operatorname { dim } H ^ { 1 } ( Z , O _ { Z } )$ ||  $$\empty$$ || conf 0.997  F
 +
 +
png = s085590429.png(429)
 +
|-
 +
| 112.(136.)*||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590440.png || $X _ { \epsilon } = \{ ( x _ { 0 } , \ldots , x _ { x } ) : f ( x _ { 0 } , \ldots , x _ { x } ) = \epsilon \}$ ||  $$\empty$$ || conf 0.433  F
 +
 +
png = s085590440.png(440)
 +
|-
 +
| 113.(12.) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590458.png || $= \left\{ \begin{array} { l l } { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } } & { \text { if } \mu = 2 k } \\ { ( x + \lambda ) ^ { 2 } \ldots ( x + k \lambda ) ^ { 2 } ( x + ( k + 1 ) \lambda ) } & { \text { if } \mu = 2 k + 1 } \end{array} \right.$ ||  $$\empty$$ || conf 0.870
 +
 +
png = s085590458.png(458)
 +
|-
 +
| 114.(75.) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590482.png || $( \frac { \partial F ( x , y , \lambda ) } { \partial x } , \frac { \partial F ( x , y , \lambda ) } { \partial y } )$ ||  $$\empty$$ || conf 0.986
 +
 +
png = s085590482.png(482)
 +
|-
 +
| 115.(137.) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590515.png || $\frac { d x _ { i } } { d x _ { i _ { 0 } } } = f _ { i } ( x ) , \quad f _ { i } \in C ( U ) , \quad i \neq i _ { 0 }$ ||  $$\empty$$ || conf 0.594
 +
 +
png = s085590515.png(515)
 +
|-
 +
| 116.(142.)*||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590527.png || $A = \| \left. \begin{array} { l l } { \alpha } & { b } \\ { c } & { e } \end{array} \right. |$ ||  $$\empty$$ || conf 0.506  F
 +
 +
png = s085590527.png(527)
 +
|-
 +
| 117.(53.) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590634.png || $\Delta = ( F _ { x x } ^ { \prime \prime } ) _ { 0 } ( F _ { y y } ^ { \prime \prime } ) _ { 0 } - ( F _ { x y } ^ { \prime \prime } ) _ { 0 } ^ { 2 }$ ||  $$\empty$$ || conf 0.920
 +
 +
png = s085590634.png(634)
 +
|-
 +
| 118.(16.)*||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590645.png || $\left| \begin{array} { l l l } { F _ { X } ^ { \prime } } & { F _ { y } ^ { \prime } } & { F _ { z } ^ { \prime } } \\ { G _ { \chi } ^ { \prime } } & { G _ { y } ^ { \prime } } & { G _ { Z } ^ { \prime } } \end{array} \right|$ ||  $$\empty$$ || conf 0.230  F
 +
 +
png = s085590645.png(645)
 +
|-
 +
| 119.(92.) ||  https://www.encyclopediaofmath.org/legacyimages/s/s085/s085590/s085590653.png || $( F _ { X } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { y } ^ { \prime } ) _ { 0 } = 0 , \quad ( F _ { z } ^ { \prime } ) _ { 0 } = 0$ ||  $$\empty$$ || conf 0.300
 +
 +
png = s085590653.png(653)
 +
|-
 +
!colspan="5" | [[Solv manifold]]
 +
|-
 +
| 120.(138.) ||  https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610054.png || $\{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \}$ ||  $$\empty$$ || conf 0.972
 +
 +
png = s08610054.png(54)
 +
|-
 +
!colspan="5" | [[Stability theorems in algebraic K-theory]]
 +
|-
 +
| 121.(71.) ||  https://www.encyclopediaofmath.org/legacyimages/s/s087/s087060/s08706033.png || $\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$ ||  $$\empty$$ || conf 0.379
 +
 +
png = s08706033.png(33)
 +
|-
 +
!colspan="5" | [[Steinberg module]]
 +
|-
 +
| 122.(130.) ||  https://www.encyclopediaofmath.org/legacyimages/s/s130/s130530/s13053016.png || $e = \frac { | U | } { | G | } ( \sum _ { b \in B } b ) ( \sum _ { w \in W } \operatorname { sign } ( w ) w )$ ||  $$\empty$$ || conf 0.138
 +
 +
png = s13053016.png(16)
 +
|-
 +
!colspan="5" | [[Steinberg symbol]]
 +
|-
 +
| 123.(24.)*||  https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054017.png || $( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$ ||  $$\empty$$ || conf 0.381  F
 +
 +
png = s13054017.png(17)
 +
|-
 +
!colspan="5" | [[Tilting theory]]
 +
|-
 +
| 124.(84.) ||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130105.png || $0 \rightarrow \Lambda \rightarrow T _ { 1 } \rightarrow \ldots \rightarrow T _ { n } \rightarrow 0$ ||  $$\empty$$ || conf 0.946
 +
 +
png = t130130105.png(105)
 +
|-
 +
!colspan="5" | [[Tits quadratic form]]
 +
|-
 +
| 125.(18.) ||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140104.png || $q R ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { i } x _ { j } + \sum _ { \langle \beta : i \rightarrow j ) \in Q _ { 1 } } x _ { , j } x _ { i } x _ { j }$ ||  $$\empty$$ || conf 0.112
 +
 +
png = t130140104.png(104)
 +
|-
 +
| 126.(40.) ||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140118.png || $[ X ] \mapsto \chi _ { R } ( [ X ] ) = \sum _ { m = 0 } ^ { \infty } ( - 1 ) ^ { m } \operatorname { dim } _ { K } \operatorname { Ext } _ { R } ^ { m } ( X , X )$ ||  $$\empty$$ || conf 0.116
 +
 +
png = t130140118.png(118)
 +
|-
 +
| 127.(132.)*||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140119.png || $\operatorname { dim } _ { 1 } : K _ { 0 } ( \operatorname { mod } R ) \rightarrow Z ^ { Q _ { 0 } }$ ||  $$\empty$$ || conf 0.287 F
 +
 +
png = t130140119.png(119)
 +
|-
 +
| 128.(37.)*||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t130140140.png || $q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$ ||  $$\empty$$ || conf 0.197  F
 +
 +
png = t130140140.png(140)
 +
|-
 +
| 129.(131.)*||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014044.png || $X \mapsto \operatorname { dim } X = ( \operatorname { dim } _ { K } X _ { j } ) _ { j \in Q _ { 0 } }$ ||  $$\empty$$ || conf 0.819  F
 +
 +
png = t13014044.png(44)
 +
|-
 +
| 130.(25.) ||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014048.png || $[ X ] \mapsto \chi _ { Q } ( [ X ] ) = \operatorname { dim } _ { K } \operatorname { End } _ { Q } ( X ) - \operatorname { dim } _ { K } \operatorname { Ext } _ { Q } ^ { 1 } ( X , X )$ ||  $$\empty$$ || conf 0.661
 +
 +
png = t13014048.png(48)
 +
|-
 +
| 131.(38.)*||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t13014056.png || $A _ { Q } ( v ) = \prod _ { i , j \in Q _ { 0 } } \prod _ { \langle \beta : j \rightarrow i \rangle \in Q _ { 1 } } M _ { v _ { i } \times v _ { j } } ( K ) _ { \beta }$ ||  $$\empty$$ || conf 0.481  F
 +
 +
png = t13014056.png(56)
 +
|-
 +
| 132.(139.)*||  https://www.encyclopediaofmath.org/legacyimages/t/t130/t130140/t1301406.png || $\Phi ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }$ ||  $$\empty$$ || conf 0.648  F
 +
 +
png = t1301406.png(6)
 +
|-
 +
!colspan="5" | [[Torus]]
 +
|-
 +
| 133.(41.)*||  https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933502.png || $r = \alpha \operatorname { sin } u k + l ( 1 + \epsilon \operatorname { cos } u ) ( i \operatorname { cos } v + j \operatorname { sin } v )$ ||  $$\empty$$ || conf 0.585  F
 +
 +
png = t0933502.png(2)
 +
|-
 +
| 134.(122.)*||  https://www.encyclopediaofmath.org/legacyimages/t/t093/t093350/t0933507.png || $d s ^ { 2 } = \alpha ^ { 2 } d u ^ { 2 } + l ^ { 2 } ( 1 + \epsilon \operatorname { cos } u ) ^ { 2 } d v ^ { 2 }$ ||  $$\empty$$ || conf 0.696  F
 +
 +
png = t0933507.png(7)
 +
|-
 +
!colspan="5" | [[Uniform distribution]]
 +
|-
 +
| 135.(9.) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524027.png || $u _ { 3 } ( x ) = \left\{ \begin{array} { l l } { \frac { x ^ { 2 } } { 2 } , } & { 0 \leq x < 1 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } ] } { 2 } , } & { 1 \leq x < 2 } \\ { \frac { [ x ^ { 2 } - 3 ( x - 1 ) ^ { 2 } + 3 ( x - 2 ) ^ { 2 } ] } { 2 } , } & { 2 \leq x < 3 } \\ { 0 , } & { x \notin [ 0,3 ] } \end{array} \right.$ ||  $$\empty$$ || conf 0.733
 +
 +
png = u09524027.png(27)
 +
|-
 +
| 136.(32.)*||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952403.png || $p ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { b - \alpha } , } & { x \in [ \alpha , b ] } \\ { 0 , } & { x \notin [ \alpha , b ] } \end{array} \right.$ ||  $$\empty$$ || conf 0.681  F
 +
 +
png = u0952403.png(3)
 +
|-
 +
| 137.(34.) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524030.png || $u _ { n } ( x ) = \frac { 1 } { ( n - 1 ) ! } \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( x - k ) _ { + } ^ { n - 1 }$ ||  $$\empty$$ || conf 0.569
 +
 +
png = u09524030.png(30)
 +
|-
 +
| 138.(109.) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524034.png || $z _ { + } = \left\{ \begin{array} { l l } { z , } & { z > 0 } \\ { 0 , } & { z \leq 0 } \end{array} \right.$ ||  $$\empty$$ || conf 0.676
 +
 +
png = u09524034.png(34)
 +
|-
 +
| 139.(43.) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952407.png || $F ( x ) = \left\{ \begin{array} { l l } { 0 , } & { x \leq a } \\ { \frac { x - a } { b - a } , } & { a < x \leq b } \\ { 1 , } & { x > b } \end{array} \right.$ ||  $$\empty$$ || conf 0.468
 +
 +
png = u0952407.png(7)
 +
|-
 +
| 140.(47.) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524072.png || $p ( x _ { 1 } , \ldots , x _ { n } ) = \left\{ \begin{array} { l l } { C \neq 0 , } & { x \in D } \\ { 0 , } & { x \notin D } \end{array} \right.$ ||  $$\empty$$ || conf 0.705
 +
 +
png = u09524072.png(72)
 +
|-
 +
!colspan="5" | [[Unipotent group]]
 +
|-
 +
| 141.(143.) ||  https://www.encyclopediaofmath.org/legacyimages/u/u095/u095410/u0954106.png || $\{ g \in \operatorname { GL } ( V ) : ( 1 - g ) ^ { n } = 0 \} , \quad n = \operatorname { dim } V$ ||  $$\empty$$ || conf 0.287
 +
 +
png = u0954106.png(6)
 +
|-
 +
!colspan="5" | [[Weyl module]]
 +
|-
 +
| 142.(51.) ||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090122.png || $\operatorname { diag } ( t _ { 1 } , \ldots , t _ { n } ) \mapsto t _ { 1 } ^ { \lambda _ { 1 } } \ldots t _ { n } ^ { \lambda _ { n } } \in K$ ||  $$\empty$$ || conf 0.507
 +
 +
png = w120090122.png(122)
 +
|-
 +
| 143.(54.)*||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090135.png || $\chi _ { \lambda } = \sum _ { \mu \in \Lambda ( n ) } \operatorname { dim } _ { K } ( \Delta ( \lambda ) ^ { \mu } ) _ { e _ { \mu } }$ ||  $$\empty$$ || conf 0.461  F
 +
 +
png = w120090135.png(135)
 +
|-
 +
| 144.(110.) ||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090259.png || $\mathfrak { B } = \{ e _ { \pm } \alpha , h _ { \beta } : \alpha \in \Phi ^ { + } , \beta \in \Sigma \}$ ||  $$\empty$$ || conf 0.381
 +
 +
png = w120090259.png(259)
 +
|-
 +
| 145.(82.) ||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w120090342.png || $\left( \begin{array} { c } { h } \\ { i } \end{array} \right) = \frac { h ( h - 1 ) \ldots ( h - i + 1 ) } { i ! }$ ||  $$\empty$$ || conf 0.487
 +
 +
png = w120090342.png(342)
 +
|-
 +
| 146.(28.)*||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009095.png || $\mathfrak { S } _ { \{ 1 , \ldots , \lambda _ { 1 } \} } \times \mathfrak { S } _ { \{ \lambda _ { 1 } + 1 , \ldots , \lambda _ { 1 } + \lambda _ { 2 } \} } \times$ ||  $$\empty$$ || conf 0.312  F
 +
 +
png = w12009095.png(95)
 +
|-
 +
| 147.(104.) ||  https://www.encyclopediaofmath.org/legacyimages/w/w120/w120090/w12009096.png || $\ldots \times \mathfrak { S } _ { \{ \lambda _ { 1 } + \ldots + \lambda _ { n - 1 } + 1 , \ldots , r \} }$ ||  $$\empty$$ || conf 0.259
 +
 +
png = w12009096.png(96)
 +
|-
 +
!colspan="5" | [[Witt vector]]
 
|-
 
|-
−
| 1. || XXX || YYY
+
| 148.(87.)*||  https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100172.png || $\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$ || $$\empty$$ || conf 0.351  F
 +
 +
png = w098100172.png(172)
 
|-
 
|-
−
| 2. || ABC || DEFGH
+
| 149.(123.)*||  https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100177.png || $\langle \alpha + b \rangle = \sum _ { n = 1 } ^ { \infty } V _ { n } \langle r _ { n } ( \alpha , b ) f$ || $$\empty$$ || conf 0.143  F
 +
 +
png = w098100177.png(177)
 
|-
 
|-
−
| 23. || [[Algebraic curve]] || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011450/a01145065.png ||$$g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$$ || confidence 0.698 (dir = a011450, nr = 00065, png = a01145065.png)
+
| 150.(102.) || https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100190.png || $\sigma ( \alpha _ { 1 } , \alpha _ { 2 } , \ldots ) = ( \alpha _ { 1 } ^ { p } , \alpha _ { 2 } ^ { p } , \ldots )$ ||  $$\empty$$ || conf 0.771
 +
 +
png = w098100190.png(190)
 
|-
 
|-
−
| 116. || [[Algebraic geometry]] || https://www.encyclopediaofmath.org/legacyimages/a/a011/a011500/a01150014.png || $$\theta = \displaystyle\int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$$ || confidence 0.997 (dir = a011500, nr = 00014, png = a01150014.png)
 
 
|}
 
|}
−
 
−
$$\theta = \int_0^\lambda \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$$
 
−
 
−
$$A = \sum_{i=0}^k a_i$$
 
−
 
−
$A = \sum_{i=0}^k a_i$
 

Revision as of 21:24, 2 November 2019

Nr. Image of png File $\TeX$, 1st version $\TeX$, 2nd version Confidence, F?

name of png

Algebraic curve
1.(23.) a01145065.png $g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n } \end{array} \right.$ $$ g \leq \left\{ \begin{array} { l l } { \frac { ( n - 2 ) ^ { 2 } } { 4 } } & { \text { for even } n, } \\ { \frac { ( n - 1 ) ( n - 3 ) } { 4 } } & { \text { for odd } n, } \end{array} \right.$$ conf 0.698

png = a01145065.png(65)

Algebraic geometry
2.(116.) a01150014.png $\theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } }$ $$\empty$$ conf 0.997

png = a01150014.png(14)

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

png = a01150021.png(21)

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

png = a01150022.png(22)

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

png = a01150044.png(44)

6.(17.) a01150078.png $\left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 )$ $$\empty$$ conf 0.440

png = a01150078.png(78)

Algebraic surface
7.(144.) a011640132.png $0 \rightarrow O _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$ $$0 \rightarrow {\cal O} _ { V } \rightarrow E _ { \alpha } \rightarrow T _ { V } \rightarrow 0$$ conf 0.981

png = a011640132.png(132)

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

png = a011640137.png(137)

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

png = a011640139.png(139)

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

png = a01164027.png(27)

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

png = a01164029.png(29)

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

png = a01164047.png(47)

13.(93.)* a01164053.png $1 + p _ { x } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 }$ $$ 1 + p _ { \alpha } ( V ) = \frac { \operatorname { deg } ( c _ { 1 } ^ { 2 } ) + \operatorname { deg } ( c _ { 2 } ) } { 12 },$$ conf 0.752 F

png = a01164053.png(53)

Cartan subalgebra
14.(33.)* c0205509.png $\mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \}$ $$\empty$$ conf 0.110 F

png = c0205509.png(9)

Cartan theorem
15.(49.)* c0205704.png $f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j }$ $$\empty$$ conf 0.149 F

png = c0205704.png(4)

16.(55.)* c02057064.png $\rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow$ $$\empty$$ conf 0.853 F

png = c02057064.png(64)

Comitant
17.(7.) c02333033.png $H = \frac { 1 } { 36 } \left| \begin{array} { c c } { \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } } & { \frac { \partial ^ { 2 } f } { \partial x \partial y } } \\ { \frac { \partial ^ { 2 } f } { \partial x \partial y } } & { \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } } \end{array} \right| =$ $$\empty$$ conf 0.956

png = c02333033.png(33)

18.(76.) c02333034.png $= ( a _ { 0 } a _ { 2 } - a _ { 1 } ^ { 2 } ) x ^ { 2 } + ( a _ { 0 } a _ { 3 } - a _ { 1 } a _ { 2 } ) x y + ( a _ { 1 } a _ { 3 } - a _ { 2 } ^ { 2 } ) y ^ { 2 }$ $$\empty$$ conf 0.549

png = c02333034.png(34)

19.(11.)* c02333035.png $( \alpha _ { 0 } , \alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } ) \mapsto ( \alpha _ { 0 } \alpha _ { 2 } - \alpha _ { 1 } ^ { 2 } , \frac { 1 } { 2 } ( \alpha _ { 0 } \alpha _ { 3 } - \alpha _ { 1 } \alpha _ { 2 } ) , \alpha _ { 1 } \alpha _ { 3 } - \alpha _ { 2 } ^ { 2 } )$ $$\empty$$ conf 0.521 F

png = c02333035.png(35)

Deformation
20.(26.) d030700175.png $\operatorname { Aut } _ { R ^ { \prime } } ( X ^ { \prime } | X _ { 0 } ) \rightarrow \operatorname { Aut } _ { R } ( X _ { R ^ { \prime } } ^ { \prime } \otimes R | X _ { 0 } )$ $$\empty$$ conf 0.683

png = d030700175.png(175)

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

png = d030700190.png(190)

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

png = d030700263.png(263)

23.(96.)* d030700270.png $\Phi ( \alpha ) = \alpha + \sum _ { i = 1 } ^ { \infty } t ^ { i } \phi _ { i } ( \alpha ) , \quad \alpha \in V$ $$\empty$$ conf 0.873 F

png = d030700270.png(270)

Differential algebra
24.(106.) d031830107.png $S ^ { t } F = \sum _ { j = 1 } ^ { r } c _ { j } A ^ { p _ { j } } A _ { 1 } ^ { i _ { 1 j } } \dots A _ { m - l } ^ { i _ { m - l } , j }$ $$\empty$$ conf 0.149

png = d031830107.png(107)

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

png = d031830141.png(141)

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

png = d031830150.png(150)

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

png = d03183016.png(16)

28.(111.) d03183043.png $e _ { i j } = \operatorname { ord } _ { Y } _ { j } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$ $$ e_ { i j } = \operatorname { ord } _ { { Y } _ { j } } F _ { i } , \quad 1 \leq i \leq n , \quad i \leq j \leq n$$ conf 0.187

png = d03183043.png(43)

Dimension polynomial
29.(48.) d03249029.png $\omega _ { \eta / F } ( x ) = \sum _ { 0 \leq i \leq m } \alpha _ { i } \left( \begin{array} { c } { x + i } \\ { i } \end{array} \right)$ $$\empty$$ conf 0.968

png = d03249029.png(29)

Duality
30.(118.)* d034120173.png $H ^ { p } ( X , F ) \times H _ { c } ^ { n - p } ( X , \operatorname { Hom } ( F , \Omega ) ) \rightarrow C$ $$\empty$$ conf 0.824 F

png = d034120173.png(173)

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

png = d034120175.png(175)

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

png = d034120184.png(184)

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

png = d034120236.png(236)

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

png = d034120247.png(247)

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

png = d034120253.png(253)

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

png = d034120360.png(360)

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

png = d034120376.png(376)

38.(52.) d034120509.png $f = \{ f _ { \alpha } \} \in \prod _ { \alpha } F _ { \alpha } , \quad g = \{ g _ { \alpha } \} \in \oplus _ { \alpha } G _ { \alpha }$ $$\empty$$ conf 0.491

png = d034120509.png(509)

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

png = d034120535.png(535)

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

png = d034120555.png(555)

41.(74.)* d03412079.png $( c _ { \gamma } , c ^ { r } ) = \sum _ { t ^ { r } \in K } c _ { r } ( t ^ { \prime } ) c ^ { r } ( t ^ { r } ) \operatorname { mod } 1$ $$\empty$$ conf 0.117 F

png = d03412079.png(79)

Extension of a differential field
42.(63.) e03696024.png $F _ { 1 } F _ { 2 } = F _ { 1 } \langle F _ { 2 } \rangle = F _ { 1 } ( F _ { 2 } ) = F _ { 2 } ( F _ { 1 } ) = F _ { 2 } \langle F _ { 1 } \rangle$ $$\empty$$ conf 0.628

png = e03696024.png(24)

Formal group
43.(120.)* f040820118.png $\operatorname { og } F _ { MU } ( X ) = \sum _ { i = 1 } ^ { \infty } i ^ { - 1 } [ C ^ { - } P ^ { - 1 } ] X ^ { i }$ $$\empty$$ conf 0.098 F

png = f040820118.png(118)

44.(147.)* f04082059.png $( x _ { 1 } , \ldots , x _ { x } ) \circ ( y _ { 1 } , \ldots , y _ { n } ) = ( z _ { 1 } , \ldots , z _ { x } )$ $$\empty$$ conf 0.553 F

png = f04082059.png(59)

Gel'fond-Schneider method
45.(148.) g1300205.png $\alpha ^ { \beta } = \operatorname { exp } \{ \beta \operatorname { log } \alpha \}$ $$\empty$$ conf 0.979

png = g1300205.png(5)

Group
46.(22.)* g04521075.png $\left. \begin{array} { l l l } { A } & { \rightarrow Y } & { \square } \\ { \downarrow } & { \square } & { } & { \square } \\ { X } & { \square } & { } & { A } \end{array} \right.$ $$\empty$$ conf 0.226 F

png = g04521075.png(75)

Homogeneous space
47.(89.) h04769069.png $\mathfrak { g } = \mathfrak { f } + \mathfrak { m } , \quad \mathfrak { f } \cap \mathfrak { m } = \{ 0 \}$ $$\empty$$ conf 0.793

png = h04769069.png(69)

Hopf algebra
48.(103.) h047970129.png $m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon$ $$\empty$$ conf 0.618

png = h047970129.png(129)

49.(107.)* h047970139.png $F _ { 1 } ( X || Y ) , \ldots , F _ { n } ( X || Y ) \in K [ X _ { 1 } , \ldots , X _ { n } || Y _ { 1 } , \ldots , Y _ { n } ] \}$ $$\empty$$ conf 0.353 F

png = h047970139.png(139)

50.(97.) h04797042.png $\epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g }$ $$\empty$$ conf 0.213

png = h04797042.png(42)

Invariants, theory of
51.(149.)* i05235015.png $\alpha _ { 1 } , \ldots , i _ { R } \rightarrow \alpha _ { 2 } ^ { \prime } , \ldots , i _ { R }$ $$\empty$$ conf 0.142 F

png = i05235015.png(15)

Jordan algebra
52.(150.) j05427030.png $H ( C _ { 3 } , \Gamma ) = \{ X \in C _ { 3 } : X = \Gamma ^ { - 1 } X \square ^ { \prime } \Gamma \}$ $$\empty$$ conf 0.651

png = j05427030.png(30)

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

png = j05427031.png(31)

54.(125.)* j05427077.png $\mathfrak { g } = \mathfrak { g } - 1 + \mathfrak { g } \mathfrak { d } + \mathfrak { g } _ { 1 }$ $$\empty$$ conf 0.598 F

png = j05427077.png(77)

Jordan matrix
55.(6.)* j0543403.png $J = \left| \begin{array} { c c c c } { J _ { n _ { 1 } } ( \lambda _ { 1 } ) } & { \square } & { \square } & { \square } \\ { \square } & { \ldots } & { \square } & { 0 } \\ { 0 } & { \square } & { \ldots } & { \square } \\ { \square } & { \square } & { \square } & { J _ { n _ { S } } ( \lambda _ { s } ) } \end{array} \right|$ $$J = \left\| \begin{array} { c c c c } J_{n_1}(\lambda_1) & 0 & 0 & 0 \\ 0 & \ddots & \ddots & 0 \\ 0 & \ddots & \ddots & 0 \\ 0 & 0 & 0 & J_{n_s}(\lambda_s) \end{array} \right\|,$$ conf 0.072 F

png = j0543403.png(3)

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

png = j05434030.png(30)

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

png = j0543406.png(6)

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

png = l058510127.png(127)

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

png = l058510129.png(129)

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

png = l058510130.png(130)

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

png = l058510131.png(131)

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

png = l058510132.png(132)

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

png = l058510133.png(133)

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

png = l05851030.png(30)

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

png = l05851037.png(37)

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

png = l05851044.png(44)

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

png = l05851050.png(50)

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

png = l05851051.png(51)

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

png = l05851057.png(57)

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

png = l05851064.png(64)

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

png = l05851069.png(69)

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

png = l05851073.png(73)

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

png = l05851074.png(74)

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

png = l05851078.png(78)

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

png = l05851085.png(85)

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

png = l05852011.png(11)

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

png = l05852046.png(46)

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

png = l058590115.png(115)

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

png = l05859086.png(86)

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

png = l05861012.png(12)

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

png = l0586604.png(4)

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

png = l058680102.png(102)

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

png = l05868032.png(32)

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

png = l05872026.png(26)

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

png = l05872078.png(78)

Lie theorem
86.(134.) l05876010.png $y _ { i } = f _ { i } ( g _ { 1 } , \ldots , g _ { i } || x _ { 1 } , \ldots , x _ { n } ) , \quad i = 1 , \ldots , n$ $$\empty$$ conf 0.276

png = l05876010.png(10)

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

png = l05876016.png(16)

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

png = l05876030.png(30)

89.(19.)* l05876037.png $\sum _ { k = 1 } ^ { N } ( \xi _ { i k } \frac { \partial \xi _ { j l } } { \partial x _ { k } } - \xi _ { j k } \frac { \partial \xi _ { i l } } { \partial x _ { k } } ) = \sum _ { k = 1 } ^ { r } c _ { i j } ^ { k } \xi _ { k l }$ $$\empty$$ conf 0.157 F

png = l05876037.png(37)

90.(14.) l05876052.png $\left. \begin{array} { c } { c _ { i j } ^ { k } = - c _ { j i } ^ { k } } \\ { \sum _ { l = 1 } ^ { r } ( c _ { i l } ^ { m } c _ { j k } ^ { l } + c _ { k l } ^ { m } c _ { i j } ^ { l } + c _ { j l } ^ { m } c _ { k i } ^ { l } ) = 0 , \quad 1 \leq i , j , k , l , m \leq r } \end{array} \right.$ $$\empty$$ conf 0.085

png = l05876052.png(52)

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

png = m06301072.png(72)

Non-Abelian cohomology
92.(114.)* n066900110.png $\phi ( g _ { 1 } ) \phi ( g ) \phi ( g _ { 1 } g _ { 2 } ) ^ { - 1 } = \operatorname { Int } m ( g _ { 1 } , g _ { 2 } )$ $$\empty$$ conf 0.443 F

png = n066900110.png(110)

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

png = n066900118.png(118)

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

png = n06690016.png(16)

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

png = n06690028.png(28)

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

png = p07267025.png(25)

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

png = p07464025.png(25)

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

png = q07631062.png(62)

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

png = q07631071.png(71)

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

png = q07631072.png(72)

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

png = q07631088.png(88)

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

png = q07631089.png(89)

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

png = q07631092.png(92)

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

png = q07631095.png(95)

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

png = q07631099.png(99)

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

png = r077630100.png(100)

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

png = r077630104.png(104)

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

png = r07763055.png(55)

Singular point
109.(31.) s085590225.png $\sum _ { k _ { 1 } , \ldots , k _ { n } = 0 } ^ { \infty } c _ { k _ { 1 } \cdots k _ { n } } ( z _ { 1 } - \zeta _ { 1 } ) ^ { k _ { 1 } } \ldots ( z _ { n } - \zeta _ { n } ) ^ { k _ { n } }$ $$\empty$$ conf 0.324

png = s085590225.png(225)

110.(46.) s085590404.png $\frac { m _ { 1 } } { n _ { 1 } } < \frac { m _ { 2 } } { n _ { 1 } n _ { 2 } } < \ldots < \frac { m _ { g } } { n _ { 1 } \ldots n _ { g } } = \frac { m _ { g } } { n }$ $$\empty$$ conf 0.459

png = s085590404.png(404)

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

png = s085590429.png(429)

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

png = s085590440.png(440)

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

png = s085590458.png(458)

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

png = s085590482.png(482)

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

png = s085590515.png(515)

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

png = s085590527.png(527)

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

png = s085590634.png(634)

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

png = s085590645.png(645)

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

png = s085590653.png(653)

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

png = s08610054.png(54)

Stability theorems in algebraic K-theory
121.(71.) s08706033.png $\psi _ { t _ { 1 } , \ldots , t _ { R } } ^ { \prime } : S K _ { 1 } ( R ) \rightarrow S K _ { 1 } ( R ( t _ { 1 } , \ldots , t _ { n } ) )$ $$\empty$$ conf 0.379

png = s08706033.png(33)

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

png = s13053016.png(16)

Steinberg symbol
123.(24.)* s13054017.png $( x _ { i j } ( a ) , x _ { k l } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq 1 , j \neq k } \\ { x _ { 1 } ( a b ) } & { \text { if } i \neq 1 , j = k } \end{array} \right.$ $$\empty$$ conf 0.381 F

png = s13054017.png(17)

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

png = t130130105.png(105)

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

png = t130140104.png(104)

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

png = t130140118.png(118)

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

png = t130140119.png(119)

128.(37.)* t130140140.png $q ( x ) = \sum _ { i \in I } x _ { i } ^ { 2 } + \sum _ { i \prec j } x _ { i } x _ { j } - \sum _ { p \in \operatorname { max } l } ( \sum _ { i \prec p } x _ { i } ) x _ { p }$ $$\empty$$ conf 0.197 F

png = t130140140.png(140)

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

png = t13014044.png(44)

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

png = t13014048.png(48)

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

png = t13014056.png(56)

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

png = t1301406.png(6)

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

png = t0933502.png(2)

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

png = t0933507.png(7)

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

png = u09524027.png(27)

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

png = u0952403.png(3)

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

png = u09524030.png(30)

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

png = u09524034.png(34)

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

png = u0952407.png(7)

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

png = u09524072.png(72)

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

png = u0954106.png(6)

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

png = w120090122.png(122)

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

png = w120090135.png(135)

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

png = w120090259.png(259)

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

png = w120090342.png(342)

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

png = w12009095.png(95)

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

png = w12009096.png(96)

Witt vector
148.(87.)* w098100172.png $\langle \alpha > < b \rangle = \langle \alpha b \rangle , \quad \langle 1 \rangle = f _ { 1 } = V _ { 1 } =$ $$\empty$$ conf 0.351 F

png = w098100172.png(172)

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

png = w098100177.png(177)

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

png = w098100190.png(190)

How to Cite This Entry:
Ulf Rehmann/Table of automatically generated TeX code. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ulf_Rehmann/Table_of_automatically_generated_TeX_code&oldid=44161
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