A few questions about aligned maths

A few questions about aligned maths

[Morgan Brassel]

Hi everyone,

I read carefully the two 'My Way' from Aditya regarding math alignment 
(thank you for those, they were extremely useful to me, and should maybe get even 
more visibility on the wiki), and I have some questions about it:

1) Is there a way to modify interline spacing in 'cases' environment? I 
find it a bit short when using the displaystyle option.

2) I noticed that the interline spacing is not the same in the 'align' and 
'aligned' environment (it's larger in 'align'). Is there a way to make 
'aligned' use the spacing of align? It looks better with displaystyle 
maths.

3) Aditya, I saw your remark in your 'My Way' concerning the 'multline' 
environment. I must admit I would really be glad to see it appear in 
ConTeXt! I'm afraid I'm not able to implement it myself... I use it a 
lot with amsmath: when a formula is just too long for one line, I put it 
on two with multline. The first part of the formula is left aligned on the 
first line, and the second part is right aligned on the second line. It seems 
impossible to get the same result with only 'align'. Would it be difficult 
to make it available in ConTeXt?

Best regards,
Morgan
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Re: A few questions about aligned maths

[Aditya Mahajan]

On Thu, 13 Mar 2008, Morgan Brassel wrote:

>
> Hi everyone,
>
> I read carefully the two 'My Way' from Aditya regarding math alignment 
> (thank you for those, they were extremely useful to me, and should maybe 
> get even more visibility on the wiki), and I have some questions about 
> it:
>
>
> 1) Is there a way to modify interline spacing in 'cases' environment? I
> find it a bit short when using the displaystyle option.

There is no clean way to do this at the moment. You can force a looser 
interline by adding \noalign{\vskip ...} after \NR.

> 2) I noticed that the interline spacing is not the same in the 'align' and
> 'aligned' environment (it's larger in 'align'). Is there a way to make
> 'aligned' use the spacing of align? It looks better with displaystyle
> maths.

I do not like the definition of aligned using matrices that I have 
presented in the 'My Way'. Matrices try very hard to have a compact 
interline spacing, while in a aligned environment you need the opposite. 
At some point, I played with some of the internals of core-mat, and had 
a working definition of aligned. I cannot find it at the moment :(

> 3) Aditya, I saw your remark in your 'My Way' concerning the 'multline'
> environment. I must admit I would really be glad to see it appear in
> ConTeXt! I'm afraid I'm not able to implement it myself... I use it a
> lot with amsmath: when a formula is just too long for one line, I put it
> on two with multline. The first part of the formula is left aligned on the
> first line, and the second part is right aligned on the second line. It seems
> impossible to get the same result with only 'align'. Would it be difficult
> to make it available in ConTeXt?

No, multline is probably the simplest of all math environments. I do not 
really understand what all features it should have. If you can explain 
everything you want from a multiline environment, I can give a shot at 
trying to implement that, and Hans and Taco could polish it up.

Aditya
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___________________________________________________________________________________

Re: A few questions about aligned maths

[Morgan Brassel]

Le Thursday 13 March 2008 13:23:06 Aditya Mahajan, vous avez écrit :
> On Thu, 13 Mar 2008, Morgan Brassel wrote:
> > Hi everyone,
> >
> > I read carefully the two 'My Way' from Aditya regarding math alignment
> > (thank you for those, they were extremely useful to me, and should maybe
> > get even more visibility on the wiki), and I have some questions about
> > it:
> >
> >
> > 1) Is there a way to modify interline spacing in 'cases' environment? I
> > find it a bit short when using the displaystyle option.
>
> There is no clean way to do this at the moment. You can force a looser
> interline by adding \noalign{\vskip ...} after \NR.

Ok, I'll do that. This is the \needspace command described in your 'My Way', 
isn't it? I should have remembered it!

> > 2) I noticed that the interline spacing is not the same in the 'align'
> > and 'aligned' environment (it's larger in 'align'). Is there a way to
> > make 'aligned' use the spacing of align? It looks better with
> > displaystyle maths.
>
> I do not like the definition of aligned using matrices that I have
> presented in the 'My Way'. Matrices try very hard to have a compact
> interline spacing, while in a aligned environment you need the opposite.
> At some point, I played with some of the internals of core-mat, and had
> a working definition of aligned. I cannot find it at the moment :(

Maybe another solution would be to add left and right options directly to the 
align environment. But of course I have no idea if it is possible... Anyway, 
if you find your new definition back, I would be glad to test it!

> > 3) Aditya, I saw your remark in your 'My Way' concerning the 'multline'
> > environment. I must admit I would really be glad to see it appear in
> > ConTeXt! I'm afraid I'm not able to implement it myself... I use it a
> > lot with amsmath: when a formula is just too long for one line, I put it
> > on two with multline. The first part of the formula is left aligned on
> > the first line, and the second part is right aligned on the second line.
> > It seems impossible to get the same result with only 'align'. Would it be
> > difficult to make it available in ConTeXt?
>
> No, multline is probably the simplest of all math environments. I do not
> really understand what all features it should have. If you can explain
> everything you want from a multiline environment, I can give a shot at
> trying to implement that, and Hans and Taco could polish it up.
>
> Aditya

Concerning multline, I only read the specification given in amsldoc.pdf:
"3.3 Split equations without alignment". Multline does not support alignment, 
so you can't put '&' inside it, only '\\'. All it does is cut the equation 
into several lines: the first is left aligned, the last is right aligned, and 
all the intermediate are centered. I don't if this is sufficient as an 
explanation. Please let me know if I can help or test. And thank you for your 
time!

Morgan
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Re: A few questions about aligned maths

[Aditya Mahajan]

On Thu, 13 Mar 2008, Morgan Brassel wrote:

> Concerning multline, I only read the specification given in amsldoc.pdf:
> "3.3 Split equations without alignment". Multline does not support alignment,
> so you can't put '&' inside it, only '\\'. All it does is cut the equation
> into several lines: the first is left aligned, the last is right aligned, and
> all the intermediate are centered. I don't if this is sufficient as an
> explanation. Please let me know if I can help or test. And thank you for your
> time!

So, will this user interface be enough:

\startformula \startmultline
   \NC line 1 \FR
   \NC line 2 \MR
   \NC line 3 \LR
\stopmultline \stopformula

Lines with \FR are flush left, lines with \MR are centered, and lines with 
\LR are right flushed. (First row, middle row, last row; as in tables). 
multline also has \pushleft and \pushright with go against the ConTeXt way 
of things.

It is also possible to just have \NR at each row, and let ConTeXt figure 
out the correct flushing. That will involve a two-pass algorithm, and I 
will give that a try.

The other thing is placement of equation numbers. It is easy to implement 
multline so that equation number is always centered. I am not sure if the 
current ConTeXt mechanism allows for equation number to be placed on the 
bottom (similar to tbtags options).

Aditya
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___________________________________________________________________________________

Re: A few questions about aligned maths

[Morgan Brassel]

Le Friday 14 March 2008 00:48:55 Aditya Mahajan, vous avez écrit :
> So, will this user interface be enough:
>
> \startformula \startmultline
>    \NC line 1 \FR
>    \NC line 2 \MR
>    \NC line 3 \LR
> \stopmultline \stopformula
>
> Lines with \FR are flush left, lines with \MR are centered, and lines with
> \LR are right flushed. (First row, middle row, last row; as in tables).
> multline also has \pushleft and \pushright with go against the ConTeXt way
> of things.

It seems great to me. Would it be possible to use \FR ou \LR for intermediate 
lines? It could be a way to get the same result as \pushleft and \pushright 
in amsmath. Anyway, I'm not sure this is a very useful feature: centered 
lines in the middle is clearly the best choice (at least for me!).

> It is also possible to just have \NR at each row, and let ConTeXt figure
> out the correct flushing. That will involve a two-pass algorithm, and I
> will give that a try.

If the use of \FR and \LR can address the problem of \pushleft and \pushright, 
the first solution would be more powerful... And maybe it's more coherent 
with ConTeXt tables.

> The other thing is placement of equation numbers. It is easy to implement
> multline so that equation number is always centered. I am not sure if the
> current ConTeXt mechanism allows for equation number to be placed on the
> bottom (similar to tbtags options).
>
> Aditya

As multline is supposed to be used to typeset one only equation, I guess the 
choice made in amsmath is the best for tag placement: last line if placed 
right and first line if placed left. But if it's not possible currently, a 
centered tag would not hurt me at all...

Morgan
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Re: A few questions about aligned maths

[Wolfgang Schuster]

On Thu, 13 Mar 2008 19:48:55 -0400 (EDT)
Aditya Mahajan <adityam@umich.edu> wrote:

> On Thu, 13 Mar 2008, Morgan Brassel wrote:
> 
> > Concerning multline, I only read the specification given in amsldoc.pdf:
> > "3.3 Split equations without alignment". Multline does not support alignment,
> > so you can't put '&' inside it, only '\\'. All it does is cut the equation
> > into several lines: the first is left aligned, the last is right aligned, and
> > all the intermediate are centered. I don't if this is sufficient as an
> > explanation. Please let me know if I can help or test. And thank you for your
> > time!
> 
> So, will this user interface be enough:
> 
> \startformula \startmultline
>    \NC line 1 \FR
>    \NC line 2 \MR
>    \NC line 3 \LR
> \stopmultline \stopformula

I would change myself the name for the environment from multline to
multiline, we don't have to rely on the short command names as old TeX
packages did for many macros.

> Lines with \FR are flush left, lines with \MR are centered, and lines with 
> \LR are right flushed. (First row, middle row, last row; as in tables). 
> multline also has \pushleft and \pushright with go against the ConTeXt way 
> of things.
> 
> It is also possible to just have \NR at each row, and let ConTeXt figure 
> out the correct flushing. That will involve a two-pass algorithm, and I 
> will give that a try.

I think you should use NR at the end of a line and not different
commands for the first, the last and all other lines. You could change
the alignment of each line with \NC[align=...], this will fit to the
normal commands for tables.

> The other thing is placement of equation numbers. It is easy to implement 
> multline so that equation number is always centered. I am not sure if the 
> current ConTeXt mechanism allows for equation number to be placed on the 
> bottom (similar to tbtags options).

Wolfgang
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___________________________________________________________________________________

Re: A few questions about aligned maths

[Aditya Mahajan]

On Fri, 14 Mar 2008, Wolfgang Schuster wrote:

> On Thu, 13 Mar 2008 19:48:55 -0400 (EDT)
> Aditya Mahajan <adityam@umich.edu> wrote:
>
>> On Thu, 13 Mar 2008, Morgan Brassel wrote:
>>
>>> Concerning multline, I only read the specification given in amsldoc.pdf:
>>> "3.3 Split equations without alignment". Multline does not support alignment,
>>> so you can't put '&' inside it, only '\\'. All it does is cut the equation
>>> into several lines: the first is left aligned, the last is right aligned, and
>>> all the intermediate are centered. I don't if this is sufficient as an
>>> explanation. Please let me know if I can help or test. And thank you for your
>>> time!
>>
>> So, will this user interface be enough:
>>
>> \startformula \startmultline
>>    \NC line 1 \FR
>>    \NC line 2 \MR
>>    \NC line 3 \LR
>> \stopmultline \stopformula
>
> I would change myself the name for the environment from multline to
> multiline, we don't have to rely on the short command names as old TeX
> packages did for many macros.

Yes, of course. I never really understood the stange naming conventions of 
amstex.

>> Lines with \FR are flush left, lines with \MR are centered, and lines with
>> \LR are right flushed. (First row, middle row, last row; as in tables).
>> multline also has \pushleft and \pushright with go against the ConTeXt way
>> of things.
>>
>> It is also possible to just have \NR at each row, and let ConTeXt figure
>> out the correct flushing. That will involve a two-pass algorithm, and I
>> will give that a try.
>
> I think you should use NR at the end of a line and not different
> commands for the first, the last and all other lines. You could change
> the alignment of each line with \NC[align=...], this will fit to the
> normal commands for tables.

This will also make it easier to implement ;)


Aditya
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___________________________________________________________________________________

Re: A few questions about aligned maths

[Aditya Mahajan]

On Thu, 13 Mar 2008, Morgan Brassel wrote:
> 3) Aditya, I saw your remark in your 'My Way' concerning the 'multline'
> environment. I must admit I would really be glad to see it appear in
> ConTeXt! I'm afraid I'm not able to implement it myself... I use it a
> lot with amsmath: when a formula is just too long for one line, I put it
> on two with multline. The first part of the formula is left aligned on the
> first line, and the second part is right aligned on the second line. It seems
> impossible to get the same result with only 'align'. Would it be difficult
> to make it available in ConTeXt?

Here is my first attempt. Completely bare-bones right now, but short 
enough to show what needs to be done.

\unprotect
\def\startmathlines
   {\def\stop@math{$}
    \def\NC{}
    \def\NR{\def\NC{\cr}}
    % Hack for number. Should be done by measuring the width of the number
    \scratchdimen\dimexpr\displaywidth-4em \relax
    \vbox \bgroup
    \halign\bgroup
     \hbox to \scratchdimen
       {\hfil \strut
         $\mathsurround\zeropoint\displaystyle{}## \stop@math
        \hfil}%
        \crcr
        \hfilneg}

\def\stopmathlines
   {\hfilneg
    \crcr
    \egroup
    \egroup}
\protect

\starttext

\placeformula \startformula \startmathlines
   \NC f(x) = (a+b)^n \NR
   \NC = a^n + n \cdot a^{n-1} b + \cdots + n \cdot a b^{n-1} + b^n \NR
   \NC = a^n + n \cdot a^{n-1} b + \cdots + n \cdot a b^{n-1} + b^n \NR
\stopmathlines \stopformula

\stoptext

Taking care of [align=left|right] should be easy. This version will not 
break across pages. I cannot figure out how to get rid of the outer \vbox. 
Any suggestions?

Aditya
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___________________________________________________________________________________

Re: A few questions about aligned maths

[Morgan Brassel]

On Sun, 23 Mar 2008, Aditya Mahajan wrote:

> On Thu, 13 Mar 2008, Morgan Brassel wrote:
>> 3) Aditya, I saw your remark in your 'My Way' concerning the 'multline'
>> environment. I must admit I would really be glad to see it appear in
>> ConTeXt! I'm afraid I'm not able to implement it myself... I use it a
>> lot with amsmath: when a formula is just too long for one line, I put it
>> on two with multline. The first part of the formula is left aligned on the
>> first line, and the second part is right aligned on the second line. It seems
>> impossible to get the same result with only 'align'. Would it be difficult
>> to make it available in ConTeXt?
>
> Here is my first attempt. Completely bare-bones right now, but short
> enough to show what needs to be done.
>
> \unprotect
> \def\startmathlines
>   {\def\stop@math{$}
>    \def\NC{}
>    \def\NR{\def\NC{\cr}}
>    % Hack for number. Should be done by measuring the width of the number
>    \scratchdimen\dimexpr\displaywidth-4em \relax
>    \vbox \bgroup
>    \halign\bgroup
>     \hbox to \scratchdimen
>       {\hfil \strut
>         $\mathsurround\zeropoint\displaystyle{}## \stop@math
>        \hfil}%
>        \crcr
>        \hfilneg}
>
> \def\stopmathlines
>   {\hfilneg
>    \crcr
>    \egroup
>    \egroup}
> \protect
>
> \starttext
>
> \placeformula \startformula \startmathlines
>   \NC f(x) = (a+b)^n \NR
>   \NC = a^n + n \cdot a^{n-1} b + \cdots + n \cdot a b^{n-1} + b^n \NR
>   \NC = a^n + n \cdot a^{n-1} b + \cdots + n \cdot a b^{n-1} + b^n \NR
> \stopmathlines \stopformula
>
> \stoptext
>
> Taking care of [align=left|right] should be easy. This version will not
> break across pages. I cannot figure out how to get rid of the outer \vbox.
> Any suggestions?
>
> Aditya

Sorry that I can't help you with the implementation, Aditya... I just want 
to confirm that your new macro perfectly fits my needs. I hope it will 
make it into the core, it really is useful!

As always, a big thanks for your time and patience!
Morgan
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___________________________________________________________________________________

Re: A few questions about aligned maths

[Hans Hagen]

Morgan Brassel wrote:

> Sorry that I can't help you with the implementation, Aditya... I just want 
> to confirm that your new macro perfectly fits my needs. I hope it will 
> make it into the core, it really is useful!

once aditya is confident that it makes sense and works as expected it 
will probably end up in the core, but as usual i'll wait till he sends 
me an updated core file

Hans

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Re: A few questions about aligned maths

[Aditya Mahajan]

On Mon, 24 Mar 2008, Hans Hagen wrote:

> Morgan Brassel wrote:
>
>> Sorry that I can't help you with the implementation, Aditya... I just want
>> to confirm that your new macro perfectly fits my needs. I hope it will
>> make it into the core, it really is useful!
>
> once aditya is confident that it makes sense and works as expected it
> will probably end up in the core, but as usual i'll wait till he sends
> me an updated core file

It will be a while before I am able to get all the bells and whistles 
(take care of where the formula number is to be placed, and how how the 
formula is to be aligned).

Aditya
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Re: A few questions about aligned maths

[Dave]

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Is there any progress on this?  I notice it is not yet in the core.  I'm in
need of an AMSL split-like environment but the above doesn't work entirely
as expected.  Below is an example.
Any ideas?
Thanks, Dave

\placeformula
\startformula \startalign
  \NC \sigma_{L_s}^2
      \NC= E\left\{ \eps_L^2 \right\} \NR
  \NC \NC= \startmathlines
       \NC E\left\{ \left( \frac{1}{L_{\ref}^2(\nu)}

 \Re{\eps_{\pth}+\eps_{\off}}\,\cos\left(\phi(\nu)\right) +

 \Im{\eps_{\pth}+\eps_{\off}}\,\sin\left(\phi(\nu)\right)
                    \right)^2 \right\} - \NR
       \NC E\left\{ \frac{2}{L_{\ref}^2(\nu)} \,
\frac{L_{\pth}(\nu)}{L_{\ref}(\nu)}
                    \left(
                      \Re{\eps_{\pth} +
\eps_{\off}}\,\cos\left(\phi(\nu)\right) +
                      \Im{\eps_{\pth} +
\eps_{\off}}\,\sin\left(\phi(\nu)\right)
                    \right) \NR
                \NC \left(
                      \Re{\eps_{\ref} +
\eps_{\off}}\,\cos\left(\phi(\nu)\right) +
                      \Im{\eps_{\ref} +
\eps_{\off}}\,\sin\left(\phi(\nu)\right) +
                    \right)
                    \right\} + \NR
       \NC E\left\{ \frac{1}{L_{\ref}^2(\nu)} \,
\frac{L_{\pth}^2(\nu)}{L_{\ref}^2(\nu)}
                    \left(
                      \Re{\eps_{\ref} +
\eps_{\off}}\,\cos\left(\phi(\nu)\right) +
                      \Im{\eps_{\ref} +
\eps_{\off}}\,\sin\left(\phi(\nu)\right)
                    \right)^2 \right\} \NR
            \stopmathlines \NR
  \NC \NC= \frac{1}{L_{\ref}^2} \, \left( \sigma_{\pth}^2 +
           \left(\frac{L_{\pth}}{L_{\ref}} - 1\right)^2 \sigma_{\off}^2 +
           \left(\frac{L_{\pth}}{L_{\ref}}\right)^2 \sigma_{\ref}^2 \right)
           \NR[eq:SpcTransVarC2]
\stopalign \stopformula


On Mon, Mar 24, 2008 at 7:17 AM, Aditya Mahajan <adityam@umich.edu> wrote:

> On Mon, 24 Mar 2008, Hans Hagen wrote:
>
> > Morgan Brassel wrote:
> >
> >> Sorry that I can't help you with the implementation, Aditya... I just
> want
> >> to confirm that your new macro perfectly fits my needs. I hope it will
> >> make it into the core, it really is useful!
> >
> > once aditya is confident that it makes sense and works as expected it
> > will probably end up in the core, but as usual i'll wait till he sends
> > me an updated core file
>
> It will be a while before I am able to get all the bells and whistles
> (take care of where the formula number is to be placed, and how how the
> formula is to be aligned).
>
> Aditya
>
> ___________________________________________________________________________________
> If your question is of interest to others as well, please add an entry to
> the Wiki!
>
> maillist : ntg-context@ntg.nl /
> http://www.ntg.nl/mailman/listinfo/ntg-context
> webpage  : http://www.pragma-ade.nl / http://tex.aanhet.net
> archive  : https://foundry.supelec.fr/projects/contextrev/
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>
> ___________________________________________________________________________________
>

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Re: A few questions about aligned maths

[Aditya Mahajan]

This can be corrected by changing the \vbox in the definition of 
\startmathlines to \vtop. This can be made configurable by adding a 
location key. Here is a complete example

\unprotect
\def\startmathlines
{\def\stop@math{$}
\def\NC{}
\def\NR{\def\NC{\cr}}
% Hack for number. Should be done by measuring the width of the number
\scratchdimen\dimexpr\displaywidth-4em \relax
\vbox \bgroup
\halign\bgroup
\hbox to \scratchdimen
{\hfil \strut
$\mathsurround\zeropoint\displaystyle{}## \stop@math
\hfil}%
\crcr
\hfilneg}

\def\stopmathlines
{\hfilneg
\crcr
\egroup
\egroup}
\protect

Once I understand the new experimental code, I will start playing around 
with aligned math support.

Aditya

On Sat, 17 Jan 2009, Dave wrote:

> Is there any progress on this?  I notice it is not yet in the core.  I'm in
> need of an AMSL split-like environment but the above doesn't work entirely
> as expected.  Below is an example.
> Any ideas?
> Thanks, Dave
>
> \placeformula
> \startformula \startalign
>  \NC \sigma_{L_s}^2
>      \NC= E\left\{ \eps_L^2 \right\} \NR
>  \NC \NC= \startmathlines
>       \NC E\left\{ \left( \frac{1}{L_{\ref}^2(\nu)}
>
> \Re{\eps_{\pth}+\eps_{\off}}\,\cos\left(\phi(\nu)\right) +
>
> \Im{\eps_{\pth}+\eps_{\off}}\,\sin\left(\phi(\nu)\right)
>                    \right)^2 \right\} - \NR
>       \NC E\left\{ \frac{2}{L_{\ref}^2(\nu)} \,
> \frac{L_{\pth}(\nu)}{L_{\ref}(\nu)}
>                    \left(
>                      \Re{\eps_{\pth} +
> \eps_{\off}}\,\cos\left(\phi(\nu)\right) +
>                      \Im{\eps_{\pth} +
> \eps_{\off}}\,\sin\left(\phi(\nu)\right)
>                    \right) \NR
>                \NC \left(
>                      \Re{\eps_{\ref} +
> \eps_{\off}}\,\cos\left(\phi(\nu)\right) +
>                      \Im{\eps_{\ref} +
> \eps_{\off}}\,\sin\left(\phi(\nu)\right) +
>                    \right)
>                    \right\} + \NR
>       \NC E\left\{ \frac{1}{L_{\ref}^2(\nu)} \,
> \frac{L_{\pth}^2(\nu)}{L_{\ref}^2(\nu)}
>                    \left(
>                      \Re{\eps_{\ref} +
> \eps_{\off}}\,\cos\left(\phi(\nu)\right) +
>                      \Im{\eps_{\ref} +
> \eps_{\off}}\,\sin\left(\phi(\nu)\right)
>                    \right)^2 \right\} \NR
>            \stopmathlines \NR
>  \NC \NC= \frac{1}{L_{\ref}^2} \, \left( \sigma_{\pth}^2 +
>           \left(\frac{L_{\pth}}{L_{\ref}} - 1\right)^2 \sigma_{\off}^2 +
>           \left(\frac{L_{\pth}}{L_{\ref}}\right)^2 \sigma_{\ref}^2 \right)
>           \NR[eq:SpcTransVarC2]
> \stopalign \stopformula
>
>
> On Mon, Mar 24, 2008 at 7:17 AM, Aditya Mahajan <adityam@umich.edu> wrote:
>
>> On Mon, 24 Mar 2008, Hans Hagen wrote:
>>
>>> Morgan Brassel wrote:
>>>
>>>> Sorry that I can't help you with the implementation, Aditya... I just
>> want
>>>> to confirm that your new macro perfectly fits my needs. I hope it will
>>>> make it into the core, it really is useful!
>>>
>>> once aditya is confident that it makes sense and works as expected it
>>> will probably end up in the core, but as usual i'll wait till he sends
>>> me an updated core file
>>
>> It will be a while before I am able to get all the bells and whistles
>> (take care of where the formula number is to be placed, and how how the
>> formula is to be aligned).
>>
>> Aditya
>>
>> ___________________________________________________________________________________
>> If your question is of interest to others as well, please add an entry to
>> the Wiki!
>>
>> maillist : ntg-context@ntg.nl /
>> http://www.ntg.nl/mailman/listinfo/ntg-context
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>>
>> ___________________________________________________________________________________
>>
>
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