Difference between revisions of "User:Ulf Rehmann/TEST7"
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Ulf Rehmann (talk | contribs) (Created page with "** Nr 1. Cf. A priori and a posteriori bounds in matrix computations <pre> Markup: .BDIS {A15} p \eql {fnnme max} from {1 \...") |
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\$\$ \tag{A15} | \$\$ \tag{A15} | ||
p = \displaystyle\max _ {1 <= i <= n} | p = \displaystyle\max _ {1 <= i <= n} | ||
| − | {{\left | + | {{\left | {b _ i - \sum _ {j = 1} ^ n a _ {ij} {\hat x} _ j} \right |} |
| − | \over {BN + AN \cdot \sum _ {j = 1} ^ n \left | + | \over {BN + AN \cdot \sum _ {j = 1} ^ n \left | {{\hat x} _ j} \right |}} , |
\$\$ | \$\$ | ||
</pre> | </pre> | ||
| − | TeX Code displayed: | + | TeX Code displayed [https://www.encyclopediaofmath.org/legacyimages/a/a110/a110010/a110010108.png from this a110010108.png ] |
$$ \tag{A15} | $$ \tag{A15} | ||
p = \displaystyle\max _ {1 <= i <= n} | p = \displaystyle\max _ {1 <= i <= n} | ||
| − | {{\left | + | {{\left | {b _ i - \sum _ {j = 1} ^ n a _ {ij} {\hat x} _ j} \right |} |
| − | \over {BN + AN \cdot \sum _ {j = 1} ^ n \left | + | \over {BN + AN \cdot \sum _ {j = 1} ^ n \left | {{\hat x} _ j} \right |}} , |
$$ | $$ | ||
| Line 36: | Line 36: | ||
TeX rendered (inline) | TeX rendered (inline) | ||
$ H _ {\mathfrak A / \mathfrak A _ 1} (A,\, B) $ | $ H _ {\mathfrak A / \mathfrak A _ 1} (A,\, B) $ | ||
| − | + | [https://www.encyclopediaofmath.org/legacyimages/a/a010/a010200/a01020061.png from this a01020061.png] | |
---------------------------------------------------------- | ---------------------------------------------------------- | ||
| Line 50: | Line 50: | ||
TeX rendered (inline) | TeX rendered (inline) | ||
$ {\operatorname\Spec} \, \mathbf Z [ 1 / n ,\, \xi _ n ] $ | $ {\operatorname\Spec} \, \mathbf Z [ 1 / n ,\, \xi _ n ] $ | ||
| − | + | [https://www.encyclopediaofmath.org/legacyimages/m/m064/m064510/m06451074.png from this m06451074.png] | |
----------------------------------------------------------------- | ----------------------------------------------------------------- | ||
| Line 69: | Line 69: | ||
\$\S | \$\S | ||
\beta ( \mathcal A, \mathcal B ) = \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B} | \beta ( \mathcal A, \mathcal B ) = \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B} | ||
| − | \left | + | \left | {\mathrm P _ {\mathcal A \otimes \mathcal B} ( C ) - |
| − | ( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right | + | ( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right | |
= | = | ||
\$\$ | \$\$ | ||
| Line 77: | Line 77: | ||
= | = | ||
{1 \over 2} \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J | {1 \over 2} \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J | ||
| − | \left | + | \left | {\mathrm P ( A _ i \cap B _ j ) - \mathrm P ( A _ i ) \mathrm P ( B _ j )} \right | , |
\$\$ | \$\$ | ||
</pre> | </pre> | ||
| − | TeX Code displayed: | + | TeX Code displayed [https://www.encyclopediaofmath.org/legacyimages/a/a110/a110060/a11006022.png from this a11006022.png] and |
| + | [https://www.encyclopediaofmath.org/legacyimages/a/a110/a110060/a11006023.png this a11006023.png] | ||
$$ | $$ | ||
\beta ( \mathcal A, \mathcal B ) = \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B} | \beta ( \mathcal A, \mathcal B ) = \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B} | ||
| − | \left | + | \left | {\mathrm P _ {\mathcal A \otimes \mathcal B} ( C ) - |
| − | ( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right | + | ( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right | = |
$$ | $$ | ||
$$ | $$ | ||
| − | = {1 \over 2} \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J \left | + | = {1 \over 2} \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J \left | {\mathrm P ( A _ i \cap B _ j ) - \mathrm P ( A _ i ) \mathrm P ( B _ j )} \right | , |
$$ | $$ | ||
Latest revision as of 22:16, 21 November 2019
Markup:
.BDIS {A15}
p \eql {fnnme max} from {1 \leq i \leq n}
{{left \Lmi {b sub i \mns \sum from {j \eql 1} to n a sub {ij} {x hat} sub j} right \Rmi}
over {BN \pls AN \cdt \sum from {j \eql 1} to n left \Lmi {{x hat} sub j} right \Rmi}} ,
.EDIS
TeX Code:
\$\$ \tag{A15}
p = \displaystyle\max _ {1 <= i <= n}
{{\left | {b _ i - \sum _ {j = 1} ^ n a _ {ij} {\hat x} _ j} \right |}
\over {BN + AN \cdot \sum _ {j = 1} ^ n \left | {{\hat x} _ j} \right |}} ,
\$\$
TeX Code displayed from this a110010108.png $$ \tag{A15} p = \displaystyle\max _ {1 <= i <= n} {{\left | {b _ i - \sum _ {j = 1} ^ n a _ {ij} {\hat x} _ j} \right |} \over {BN + AN \cdot \sum _ {j = 1} ^ n \left | {{\hat x} _ j} \right |}} , $$
Markup:
\BMI H sub {\FgA / \FgA sub 1} \LpaA,\sph B\Rpa \EMI
TeX Code:
\$ H _ {\mathfrak A / \mathfrak A _ 1} (A,\, B) \$
TeX rendered (inline) $ H _ {\mathfrak A / \mathfrak A _ 1} (A,\, B) $ from this a01020061.png
Markup:
\BMI fnnme Spec \sph \FbZ \Lbk 1 / n ,\sph \Gxi sub n \Rbk\EMI
TeX Code:
\$ {\operatorname\Spec} \, \mathbf Z [ 1 / n ,\, \xi _ n ] \$
TeX rendered (inline) $ {\operatorname\Spec} \, \mathbf Z [ 1 / n ,\, \xi _ n ] $ from this m06451074.png
Markup:
.BDIS
\Gba \Lpa \FfA, \FfB \Rpa \eql {fnnme sup} from {C \seo \FfA \otm \FfB}
left \Lmi {\FsP sub {\FfA \otm \FfB} \Lpa
C \Rpa \mns \Lpa \FsP sub \FfA \tms \FsP sub \FfB \Rpa \Lpa C \Rpa} right \Rmi
.CDIS {@\eql@}
{1 over 2} {fnnme sup} \sum from {i \eql 1} to I \sum from {j \eql 1} to J
left \Lmi {\FsP \Lpa A sub i \cap B sub j \Rpa \mns
\FsP \Lpa A sub i \Rpa \FsP \Lpa B sub j \Rpa} right \Rmi ,
.EDIS
TeX Code:
\$\S
\beta ( \mathcal A, \mathcal B ) = \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B}
\left | {\mathrm P _ {\mathcal A \otimes \mathcal B} ( C ) -
( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right |
=
\$\$
\$\$
=
{1 \over 2} \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J
\left | {\mathrm P ( A _ i \cap B _ j ) - \mathrm P ( A _ i ) \mathrm P ( B _ j )} \right | ,
\$\$
TeX Code displayed from this a11006022.png and this a11006023.png $$ \beta ( \mathcal A, \mathcal B ) = \displaystyle\sup _ {C \in \mathcal A \otimes \mathcal B} \left | {\mathrm P _ {\mathcal A \otimes \mathcal B} ( C ) - ( \mathrm P _ \mathcal A - \mathrm P _ \mathcal B ) ( C )} \right | = $$
$$ = {1 \over 2} \displaystyle\sup \sum _ {i = 1} ^ I \sum _ {j = 1} ^ J \left | {\mathrm P ( A _ i \cap B _ j ) - \mathrm P ( A _ i ) \mathrm P ( B _ j )} \right | , $$
How to Cite This Entry:
Ulf Rehmann/TEST7. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ulf_Rehmann/TEST7&oldid=44221
Ulf Rehmann/TEST7. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ulf_Rehmann/TEST7&oldid=44221