[OS X TeX] MacOSX-TeX Digest, Vol 130, Issue 13
Brian Coleman
brian.coleman.velchron at gmail.com
Tue Aug 21 15:25:23 EDT 2018
Hi All,
\triangleq works fine for me.
See attached example.
Brian Coleman
On Tue, 21 Aug 2018 at 20:00, <macosx-tex-request at email.esm.psu.edu> wrote:
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> Today's Topics:
>
> 1. Re: Latex symbol for "define equal" (Markus Klyver)
> 2. Re: Latex symbol for "define equal" (Markus Klyver)
> 3. Re: Latex symbol for "define equal" (juan tolosa)
> 4. Re: Latex symbol for "define equal" (Josep Maria Font)
> 5. Re: Latex symbol for "define equal" (Markus Klyver)
> 6. Re: Latex symbol for "define equal" (Markus Klyver)
> 7. Re: Latex symbol for "define equal" (Josep Maria Font)
> 8. Re: Latex symbol for "define equal" (Ross Moore)
> 9. Re: Latex symbol for "define equal" (Markus Klyver)
> 10. Re: Latex symbol for "define equal" (Markus Klyver)
> 11. Re: Latex symbol for "define equal" (Martin Berggren)
> 12. Re: Latex symbol for "define equal" (N?stor E. Aguilera)
> 13. Re: Latex symbol for "define equal" (N?stor E. Aguilera)
> 14. Re: Latex symbol for "define equal" (Markus Klyver)
> 15. Re: Latex symbol for "define equal" (Markus Klyver)
> 16. Re: Latex symbol for "define equal" (Martin Berggren)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Mon, 20 Aug 2018 23:51:26 +0000
> From: Markus Klyver <markusklyver at hotmail.com>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID:
> <
> DB6PR0901MB120892CEA34A345E764C52C2C3320 at DB6PR0901MB1208.eurprd09.prod.outlook.com
> >
>
> Content-Type: text/plain; charset="utf-8"
>
> Really, there's no standardized notation for it. Just be consistent and
> use the most aesthetically pleasing and practical notation on your context.
> Maybe := looks ugly will your fonts. Use an other notation instead.
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> ------------------------------
>
> Message: 2
> Date: Mon, 20 Aug 2018 23:56:02 +0000
> From: Markus Klyver <markusklyver at hotmail.com>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID:
> <
> DB6PR0901MB12084C8135048CC9AAED626EC3320 at DB6PR0901MB1208.eurprd09.prod.outlook.com
> >
>
> Content-Type: text/plain; charset="utf-8"
>
> Really, there's no standardized notation for it. Just be consistent and
> use the most aesthetically pleasing and practical notation on your context.
> Maybe := looks ugly will your fonts. Use an other notation instead.
> -------------- next part --------------
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>
> ------------------------------
>
> Message: 3
> Date: Mon, 20 Aug 2018 23:40:59 -0400
> From: juan tolosa <juantolo at me.com>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID: <D3E510E8-28B1-487D-A2FA-5D46F6A7284B at me.com>
> Content-Type: text/plain; charset=utf-8
>
> This is tangential but, since we are at it, what really, really annoys me,
> is the tendency in some books to define concepts using ?if and only if?
> instead of plain ?if?, as in
>
> Definition. A sequence (a_n) converges to a real number p if and only if
> for every epsilon > 0 there is an N such that (etc.)
>
> (Is it my impression, or there is a growing number of such texts? And when
> did this nonsense begin?)
>
> One could simply agree that the ?if? in definitions is not the same as the
> ?if? in logical statements.
> If one is really fastidious, one could put a statement at the beginning of
> the book that ?if? in definitions can be reworded as, say,
> by ?(a_n) converges to p? we mean ? (etc)
> Or simply not use ?if? at all in definitions.
> What is even more annoying is that when the definition gets really
> involved, the ?if and only if? makes it even worse.
> And, invariably, in a really involved definition you will find that the
> author(s) abandon their own fastidiousness and just use ?if.?
> Question is, why not do it from the very beginning?
>
> Juan
>
> > On Aug 20, 2018, at 7:56 PM, Markus Klyver <markusklyver at hotmail.com>
> wrote:
> >
> > Really, there's no standardized notation for it. Just be consistent and
> use the most aesthetically pleasing and practical notation on your context.
> Maybe := looks ugly will your fonts. Use an other notation instead.
> > ----------- Please Consult the Following Before Posting -----------
> > TeX FAQ: http://www.tex.ac.uk/faq
> > List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> > List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
> > https://email.esm.psu.edu/pipermail/macosx-tex/
> > TeX on Mac OS X Website: http://mactex-wiki.tug.org/
> > List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
>
>
>
> ------------------------------
>
> Message: 4
> Date: Tue, 21 Aug 2018 10:15:24 +0000
> From: Josep Maria Font <jmfont at ub.edu>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID: <77496291-A014-4ACD-BB7F-6013F222CA83 at ub.edu>
> Content-Type: text/plain; charset="utf-8"
>
> That's why I prefer to use "when" for definitions: [...this NEW thing
> happens...] WHEN [...such-and-such known, understandable condition
> holds...].
>
>
> JMaF
>
>
> > El 21 /08/18, a les 5:40, juan tolosa <juantolo at me.com> va escriure:
> >
> > This is tangential but, since we are at it, what really, really annoys
> me, is the tendency in some books to define concepts using ?if and only if?
> instead of plain ?if?, as in
> >
> > Definition. A sequence (a_n) converges to a real number p if and only if
> for every epsilon > 0 there is an N such that (etc.)
> >
> > (Is it my impression, or there is a growing number of such texts? And
> when did this nonsense begin?)
> >
> > One could simply agree that the ?if? in definitions is not the same as
> the ?if? in logical statements.
> > If one is really fastidious, one could put a statement at the beginning
> of the book that ?if? in definitions can be reworded as, say,
> > by ?(a_n) converges to p? we mean ? (etc)
> > Or simply not use ?if? at all in definitions.
> > What is even more annoying is that when the definition gets really
> involved, the ?if and only if? makes it even worse.
> > And, invariably, in a really involved definition you will find that the
> author(s) abandon their own fastidiousness and just use ?if.?
> > Question is, why not do it from the very beginning?
> >
> > Juan
> >
> >> On Aug 20, 2018, at 7:56 PM, Markus Klyver <markusklyver at hotmail.com>
> wrote:
> >>
> >> Really, there's no standardized notation for it. Just be consistent and
> use the most aesthetically pleasing and practical notation on your context.
> Maybe := looks ugly will your fonts. Use an other notation instead.
> >> ----------- Please Consult the Following Before Posting -----------
> >> TeX FAQ: http://www.tex.ac.uk/faq
> >> List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> >> List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
> >> https://email.esm.psu.edu/pipermail/macosx-tex/
> >> TeX on Mac OS X Website: http://mactex-wiki.tug.org/
> >> List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
> >
> > ----------- Please Consult the Following Before Posting -----------
> > TeX FAQ: http://www.tex.ac.uk/faq
> > List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> > List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
> > https://email.esm.psu.edu/pipermail/macosx-tex/
> > TeX on Mac OS X Website: http://mactex-wiki.tug.org/
> > List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
>
>
>
> Aquest missatge, i els fitxers adjunts que hi pugui haver, pot contenir
> informaci? confidencial o protegida legalment i s?adre?a exclusivament a la
> persona o entitat destinat?ria. Si no consteu com a destinatari final o no
> teniu l?enc?rrec de rebre?l, no esteu autoritzat a llegir-lo, retenir-lo,
> modificar-lo, distribuir-lo, copiar-lo ni a revelar-ne el contingut. Si
> l?heu rebut per error, informeu-ne el remitent i elimineu del sistema tant
> el missatge com els fitxers adjunts que hi pugui haver.
>
> Este mensaje, y los ficheros adjuntos que pueda incluir, puede contener
> informaci?n confidencial o legalmente protegida y est? exclusivamente
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> remitente y elimine del sistema tanto el mensaje como los ficheros adjuntos
> que pueda contener.
>
> This email message and any attachments it carries may contain confidential
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> organization to whom they are addressed. If you are not the intended
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> ------------------------------
>
> Message: 5
> Date: Tue, 21 Aug 2018 10:40:21 +0000
> From: Markus Klyver <markusklyver at hotmail.com>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID:
> <
> HE1PR0901MB1209DAFA2D9115AE95FF1D57C3310 at HE1PR0901MB1209.eurprd09.prod.outlook.com
> >
>
> Content-Type: text/plain; charset="windows-1252"
>
> This will be a little rant, but I completely disagree with you. It annoys
> me when authors *don't* use "iff" in definitions, and here's why:
>
>
> P if Q means that Q implies P. But in the case of definitions, it's a fact
> of an equivalence. We mean P <---> Q, which also is the meaning of "iff".
> Using only "if" in definitions causes confusion. Consider the following
> example: x=0 if f(x)=0. Do we mean this as something we derived, that x=0
> implies f(x)=0? Or do we mean it as a definition of the function f, that f
> is defined to have only one root x=0? Since definitions are implications
> both ways, one should use "iff" and not "if".
>
>
> It all has to do with the fact that natural language is really bad to
> state mathematical facts in; and non-mathematicians (like engineers)
> totally fail to see this point because they are not trained in logic or how
> a logical statement looks like.
>
>
> ________________________________
> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> f?r juan tolosa <
> juantolo at me.com>
> Skickat: den 21 augusti 2018 05:40
> Till: TeX on Mac OS X Mailing List
> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
>
> This is tangential but, since we are at it, what really, really annoys me,
> is the tendency in some books to define concepts using ?if and only if?
> instead of plain ?if?, as in
>
> Definition. A sequence (a_n) converges to a real number p if and only if
> for every epsilon > 0 there is an N such that (etc.)
>
> (Is it my impression, or there is a growing number of such texts? And when
> did this nonsense begin?)
>
> One could simply agree that the ?if? in definitions is not the same as the
> ?if? in logical statements.
> If one is really fastidious, one could put a statement at the beginning of
> the book that ?if? in definitions can be reworded as, say,
> by ?(a_n) converges to p? we mean ? (etc)
> Or simply not use ?if? at all in definitions.
> What is even more annoying is that when the definition gets really
> involved, the ?if and only if? makes it even worse.
> And, invariably, in a really involved definition you will find that the
> author(s) abandon their own fastidiousness and just use ?if.?
> Question is, why not do it from the very beginning?
>
> Juan
>
> > On Aug 20, 2018, at 7:56 PM, Markus Klyver <markusklyver at hotmail.com>
> wrote:
> >
> > Really, there's no standardized notation for it. Just be consistent and
> use the most aesthetically pleasing and practical notation on your context.
> Maybe := looks ugly will your fonts. Use an other notation instead.
> > ----------- Please Consult the Following Before Posting -----------
> > TeX FAQ: http://www.tex.ac.uk/faq
> > List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> > List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
> > https://email.esm.psu.edu/pipermail/macosx-tex/
> > TeX on Mac OS X Website: http://mactex-wiki.tug.org/
> > List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
>
> ----------- Please Consult the Following Before Posting -----------
> TeX FAQ: http://www.tex.ac.uk/faq
> List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
> https://email.esm.psu.edu/pipermail/macosx-tex/
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> List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
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> ------------------------------
>
> Message: 6
> Date: Tue, 21 Aug 2018 11:02:03 +0000
> From: Markus Klyver <markusklyver at hotmail.com>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID:
> <
> HE1PR0901MB120961EFB979D77B61ABCB81C3310 at HE1PR0901MB1209.eurprd09.prod.outlook.com
> >
>
> Content-Type: text/plain; charset="windows-1252"
>
> But how would the reader know if it's a theorem or a definition? Consider
> the statement "f(x)=1 when x=2". Do we mean f(x)=1 <---> x=2 or just f(x)=1
> <--- x=2?
>
>
> ________________________________
> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> f?r Josep Maria
> Font <jmfont at ub.edu>
> Skickat: den 21 augusti 2018 12:15
> Till: TeX on Mac OS X Mailing List
> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
>
> That's why I prefer to use "when" for definitions: [...this NEW thing
> happens...] WHEN [...such-and-such known, understandable condition
> holds...].
>
>
> JMaF
>
>
> > El 21 /08/18, a les 5:40, juan tolosa <juantolo at me.com> va escriure:
> >
> > This is tangential but, since we are at it, what really, really annoys
> me, is the tendency in some books to define concepts using ?if and only if?
> instead of plain ?if?, as in
> >
> > Definition. A sequence (a_n) converges to a real number p if and only if
> for every epsilon > 0 there is an N such that (etc.)
> >
> > (Is it my impression, or there is a growing number of such texts? And
> when did this nonsense begin?)
> >
> > One could simply agree that the ?if? in definitions is not the same as
> the ?if? in logical statements.
> > If one is really fastidious, one could put a statement at the beginning
> of the book that ?if? in definitions can be reworded as, say,
> > by ?(a_n) converges to p? we mean ? (etc)
> > Or simply not use ?if? at all in definitions.
> > What is even more annoying is that when the definition gets really
> involved, the ?if and only if? makes it even worse.
> > And, invariably, in a really involved definition you will find that the
> author(s) abandon their own fastidiousness and just use ?if.?
> > Question is, why not do it from the very beginning?
> >
> > Juan
> >
> >> On Aug 20, 2018, at 7:56 PM, Markus Klyver <markusklyver at hotmail.com>
> wrote:
> >>
> >> Really, there's no standardized notation for it. Just be consistent and
> use the most aesthetically pleasing and practical notation on your context.
> Maybe := looks ugly will your fonts. Use an other notation instead.
> >> ----------- Please Consult the Following Before Posting -----------
> >> TeX FAQ: http://www.tex.ac.uk/faq
> >> List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> >> List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
> >> https://email.esm.psu.edu/pipermail/macosx-tex/
> >> TeX on Mac OS X Website: http://mactex-wiki.tug.org/
> >> List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
> >
> > ----------- Please Consult the Following Before Posting -----------
> > TeX FAQ: http://www.tex.ac.uk/faq
> > List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> > List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
> > https://email.esm.psu.edu/pipermail/macosx-tex/
> > TeX on Mac OS X Website: http://mactex-wiki.tug.org/
> > List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
>
>
>
> Aquest missatge, i els fitxers adjunts que hi pugui haver, pot contenir
> informaci? confidencial o protegida legalment i s?adre?a exclusivament a la
> persona o entitat destinat?ria. Si no consteu com a destinatari final o no
> teniu l?enc?rrec de rebre?l, no esteu autoritzat a llegir-lo, retenir-lo,
> modificar-lo, distribuir-lo, copiar-lo ni a revelar-ne el contingut. Si
> l?heu rebut per error, informeu-ne el remitent i elimineu del sistema tant
> el missatge com els fitxers adjunts que hi pugui haver.
>
> Este mensaje, y los ficheros adjuntos que pueda incluir, puede contener
> informaci?n confidencial o legalmente protegida y est? exclusivamente
> dirigido a la persona o entidad destinataria. Si usted no consta como
> destinatario final ni es la persona encargada de recibirlo, no est?
> autorizado a leerlo, retenerlo, modificarlo, distribuirlo o copiarlo, ni a
> revelar su contenido. Si lo ha recibido por error, informe de ello al
> remitente y elimine del sistema tanto el mensaje como los ficheros adjuntos
> que pueda contener.
>
> This email message and any attachments it carries may contain confidential
> or legally protected material and are intended solely for the individual or
> organization to whom they are addressed. If you are not the intended
> recipient of this message or the person responsible for processing it, then
> you are not authorized to read, save, modify, send, copy or disclose any
> part of it. If you have received the message by mistake, please inform the
> sender of this and eliminate the message and any attachments it carries
> from your account.
> ----------- Please Consult the Following Before Posting -----------
> TeX FAQ: http://www.tex.ac.uk/faq
> List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
> https://email.esm.psu.edu/pipermail/macosx-tex/
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> List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
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> ------------------------------
>
> Message: 7
> Date: Tue, 21 Aug 2018 11:12:27 +0000
> From: Josep Maria Font <jmfont at ub.edu>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID: <AABF4FED-3B93-4EAF-ADF1-C9647E712342 at ub.edu>
> Content-Type: text/plain; charset="utf-8"
>
> If the context does not make it clear whether you are reading a theorem or
> a definition, something is wrong with this book !
>
> Normally the reader is reading a statement marked as "Definition", or one
> marked as "Theorem" (Lemma, Proposition, etc.). And in the Preliminaries
> you can explain your usage of "when" in definitions as meaning "if and only
> if".
>
> In this way you keep the "if" family, including "if and only if", for real
> mathematical statements, that is, statements which you can (in theory)
> check whether they are true or false; notice that one of the main
> differences between a definition and a theorem is that it does not make
> sense to say that a definition is true or false!
>
>
> JMaF
>
>
> El 21 /08/18, a les 13:02, Markus Klyver <markusklyver at hotmail.com<mailto:
> markusklyver at hotmail.com>> va escriure:
>
> But how would the reader know if it's a theorem or a definition? Consider
> the statement "f(x)=1 when x=2". Do we mean f(x)=1 <---> x=2 or just f(x)=1
> <--- x=2?
>
>
> ________________________________
> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu<mailto:
> macosx-tex-bounces at email.esm.psu.edu>> f?r Josep Maria Font <jmfont at ub.edu
> <mailto:jmfont at ub.edu>>
> Skickat: den 21 augusti 2018 12:15
> Till: TeX on Mac OS X Mailing List
> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
>
> That's why I prefer to use "when" for definitions: [...this NEW thing
> happens...] WHEN [...such-and-such known, understandable condition
> holds...].
>
>
> JMaF
>
>
> > El 21 /08/18, a les 5:40, juan tolosa <juantolo at me.com<mailto:
> juantolo at me.com>> va escriure:
> >
> > This is tangential but, since we are at it, what really, really annoys
> me, is the tendency in some books to define concepts using ?if and only if?
> instead of plain ?if?, as in
> >
> > Definition. A sequence (a_n) converges to a real number p if and only if
> for every epsilon > 0 there is an N such that (etc.)
> >
> > (Is it my impression, or there is a growing number of such texts? And
> when did this nonsense begin?)
> >
> > One could simply agree that the ?if? in definitions is not the same as
> the ?if? in logical statements.
> > If one is really fastidious, one could put a statement at the beginning
> of the book that ?if? in definitions can be reworded as, say,
> > by ?(a_n) converges to p? we mean ? (etc)
> > Or simply not use ?if? at all in definitions.
> > What is even more annoying is that when the definition gets really
> involved, the ?if and only if? makes it even worse.
> > And, invariably, in a really involved definition you will find that the
> author(s) abandon their own fastidiousness and just use ?if.?
> > Question is, why not do it from the very beginning?
> >
> > Juan
> >
> >> On Aug 20, 2018, at 7:56 PM, Markus Klyver <markusklyver at hotmail.com
> <mailto:markusklyver at hotmail.com>> wrote:
> >>
> >> Really, there's no standardized notation for it. Just be consistent and
> use the most aesthetically pleasing and practical notation on your context.
> Maybe := looks ugly will your fonts. Use an other notation instead.
> >> ----------- Please Consult the Following Before Posting -----------
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> >
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>
>
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> informaci? confidencial o protegida legalment i s?adre?a exclusivament a la
> persona o entitat destinat?ria. Si no consteu com a destinatari final o no
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> l?heu rebut per error, informeu-ne el remitent i elimineu del sistema tant
> el missatge com els fitxers adjunts que hi pugui haver.
>
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> informaci?n confidencial o legalmente protegida y est? exclusivamente
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>
> Message: 8
> Date: Tue, 21 Aug 2018 11:21:34 +0000
> From: Ross Moore <ross.moore at mq.edu.au>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Cc: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID: <D6494BF8-D8D4-447A-8ADA-955E129E9BD5 at mq.edu.au>
> Content-Type: text/plain; charset="utf-8"
>
> Hi all.
>
> I agree with using "when", or better "precisely when", in definitions.
> Do not use these words in theorems. There are many alternative ways to
> express what you want.
>
> But didn't this thread start out asking for a symbolic notation?
> In that context, equivalence is *not* the word you want for a definition,
> as equivalence depends on a particular relation, so is weaker than a
> definition.
>
> There are nuances in mathematical concepts and notations that are not
> always correctly appreciated by those who work primarily in other (mostly
> applied) fields.
>
>
> On 21/08/2018, at 13:03, "Markus Klyver" <markusklyver at hotmail.com<mailto:
> markusklyver at hotmail.com>> wrote:
>
>
> But how would the reader know if it's a theorem or a definition? Consider
> the statement "f(x)=1 when x=2". Do we mean f(x)=1 <---> x=2 or just f(x)=1
> <--- x=2?
>
>
> ________________________________
> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu<mailto:
> macosx-tex-bounces at email.esm.psu.edu>> f?r Josep Maria Font <jmfont at ub.edu
> <mailto:jmfont at ub.edu>>
> Skickat: den 21 augusti 2018 12:15
> Till: TeX on Mac OS X Mailing List
> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
>
> That's why I prefer to use "when" for definitions: [...this NEW thing
> happens...] WHEN [...such-and-such known, understandable condition
> holds...].
>
>
> JMaF
>
>
> Cheers,
>
> Ross
> -------------- next part --------------
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>
> Message: 9
> Date: Tue, 21 Aug 2018 11:47:26 +0000
> From: Markus Klyver <markusklyver at hotmail.com>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID:
> <
> HE1PR0901MB120957F886843AB608FA5147C3310 at HE1PR0901MB1209.eurprd09.prod.outlook.com
> >
>
> Content-Type: text/plain; charset="windows-1252"
>
> The thing is that you can have several different definitions, all
> equivalent. Consider the definition "a matrix A \in \mathbb R^{n \times n}
> is invertiable iff A have a multiplicative inverse". It turns out that this
> is equivalent to a lot of things, among det(A) !=0, A having full rang, A
> having n linear independent eigenvectors, Ax=0 only having the trivial
> solution, Ax=b having a solution (which is unique) for every
> right-hand-side b, etc.
>
>
> In these cases it doesn't make much sense to use "if" and lose the
> importance that all these other statements, which could be taken as
> definitions. are equivalent. An other example: consider what it means for a
> complex to be holomorphic. It turns out that holomorphic and analytical is
> the same for complex functions, but it's not the same on R. You can have a
> real smooth non-analytical function, so clearly one should be careful to
> treat those concepts as the same. Which is why you keep "iff" in the
> definition of holomorphicy, not fooling students that entire and analytical
> functions are the same.
>
>
> ________________________________
> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> f?r Josep Maria
> Font <jmfont at ub.edu>
> Skickat: den 21 augusti 2018 13:12
> Till: TeX on Mac OS X Mailing List
> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
>
> If the context does not make it clear whether you are reading a theorem or
> a definition, something is wrong with this book !
>
> Normally the reader is reading a statement marked as "Definition", or one
> marked as "Theorem" (Lemma, Proposition, etc.). And in the Preliminaries
> you can explain your usage of "when" in definitions as meaning "if and only
> if".
>
> In this way you keep the "if" family, including "if and only if", for real
> mathematical statements, that is, statements which you can (in theory)
> check whether they are true or false; notice that one of the main
> differences between a definition and a theorem is that it does not make
> sense to say that a definition is true or false!
>
>
> JMaF
>
>
> El 21 /08/18, a les 13:02, Markus Klyver <markusklyver at hotmail.com<mailto:
> markusklyver at hotmail.com>> va escriure:
>
> But how would the reader know if it's a theorem or a definition? Consider
> the statement "f(x)=1 when x=2". Do we mean f(x)=1 <---> x=2 or just f(x)=1
> <--- x=2?
>
>
> ________________________________
> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu<mailto:
> macosx-tex-bounces at email.esm.psu.edu>> f?r Josep Maria Font <jmfont at ub.edu
> <mailto:jmfont at ub.edu>>
> Skickat: den 21 augusti 2018 12:15
> Till: TeX on Mac OS X Mailing List
> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
>
> That's why I prefer to use "when" for definitions: [...this NEW thing
> happens...] WHEN [...such-and-such known, understandable condition
> holds...].
>
>
> JMaF
>
>
> > El 21 /08/18, a les 5:40, juan tolosa <juantolo at me.com<mailto:
> juantolo at me.com>> va escriure:
> >
> > This is tangential but, since we are at it, what really, really annoys
> me, is the tendency in some books to define concepts using ?if and only if?
> instead of plain ?if?, as in
> >
> > Definition. A sequence (a_n) converges to a real number p if and only if
> for every epsilon > 0 there is an N such that (etc.)
> >
> > (Is it my impression, or there is a growing number of such texts? And
> when did this nonsense begin?)
> >
> > One could simply agree that the ?if? in definitions is not the same as
> the ?if? in logical statements.
> > If one is really fastidious, one could put a statement at the beginning
> of the book that ?if? in definitions can be reworded as, say,
> > by ?(a_n) converges to p? we mean ? (etc)
> > Or simply not use ?if? at all in definitions.
> > What is even more annoying is that when the definition gets really
> involved, the ?if and only if? makes it even worse.
> > And, invariably, in a really involved definition you will find that the
> author(s) abandon their own fastidiousness and just use ?if.?
> > Question is, why not do it from the very beginning?
> >
> > Juan
> >
> >> On Aug 20, 2018, at 7:56 PM, Markus Klyver <markusklyver at hotmail.com
> <mailto:markusklyver at hotmail.com>> wrote:
> >>
> >> Really, there's no standardized notation for it. Just be consistent and
> use the most aesthetically pleasing and practical notation on your context.
> Maybe := looks ugly will your fonts. Use an other notation instead.
> >> ----------- Please Consult the Following Before Posting -----------
> >> TeX FAQ: http://www.tex.ac.uk/faq
> >> List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> >> List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
> >> https://email.esm.psu.edu/pipermail/macosx-tex/
> >> TeX on Mac OS X Website: http://mactex-wiki.tug.org/
> >> List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
> >
> > ----------- Please Consult the Following Before Posting -----------
> > TeX FAQ: http://www.tex.ac.uk/faq
> > List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> > List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
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> > TeX on Mac OS X Website: http://mactex-wiki.tug.org/
> > List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
>
>
>
> Aquest missatge, i els fitxers adjunts que hi pugui haver, pot contenir
> informaci? confidencial o protegida legalment i s?adre?a exclusivament a la
> persona o entitat destinat?ria. Si no consteu com a destinatari final o no
> teniu l?enc?rrec de rebre?l, no esteu autoritzat a llegir-lo, retenir-lo,
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> l?heu rebut per error, informeu-ne el remitent i elimineu del sistema tant
> el missatge com els fitxers adjunts que hi pugui haver.
>
> Este mensaje, y los ficheros adjuntos que pueda incluir, puede contener
> informaci?n confidencial o legalmente protegida y est? exclusivamente
> dirigido a la persona o entidad destinataria. Si usted no consta como
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> revelar su contenido. Si lo ha recibido por error, informe de ello al
> remitente y elimine del sistema tanto el mensaje como los ficheros adjuntos
> que pueda contener.
>
> This email message and any attachments it carries may contain confidential
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> ----------- Please Consult the Following Before Posting -----------
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>
>
> Aquest missatge, i els fitxers adjunts que hi pugui haver, pot contenir
> informaci? confidencial o protegida legalment i s?adre?a exclusivament a la
> persona o entitat destinat?ria. Si no consteu com a destinatari final o no
> teniu l?enc?rrec de rebre?l, no esteu autoritzat a llegir-lo, retenir-lo,
> modificar-lo, distribuir-lo, copiar-lo ni a revelar-ne el contingut. Si
> l?heu rebut per error, informeu-ne el remitent i elimineu del sistema tant
> el missatge com els fitxers adjunts que hi pugui haver.
>
> Este mensaje, y los ficheros adjuntos que pueda incluir, puede contener
> informaci?n confidencial o legalmente protegida y est? exclusivamente
> dirigido a la persona o entidad destinataria. Si usted no consta como
> destinatario final ni es la persona encargada de recibirlo, no est?
> autorizado a leerlo, retenerlo, modificarlo, distribuirlo o copiarlo, ni a
> revelar su contenido. Si lo ha recibido por error, informe de ello al
> remitente y elimine del sistema tanto el mensaje como los ficheros adjuntos
> que pueda contener.
>
> This email message and any attachments it carries may contain confidential
> or legally protected material and are intended solely for the individual or
> organization to whom they are addressed. If you are not the intended
> recipient of this message or the person responsible for processing it, then
> you are not authorized to read, save, modify, send, copy or disclose any
> part of it. If you have received the message by mistake, please inform the
> sender of this and eliminate the message and any attachments it carries
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> ------------------------------
>
> Message: 10
> Date: Tue, 21 Aug 2018 11:49:09 +0000
> From: Markus Klyver <markusklyver at hotmail.com>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID:
> <
> HE1PR0901MB1209E67541A9F9A221589F4AC3310 at HE1PR0901MB1209.eurprd09.prod.outlook.com
> >
>
> Content-Type: text/plain; charset="iso-8859-1"
>
> Definitions *are* equivalences.
>
>
> ________________________________
> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> f?r Ross Moore <
> ross.moore at mq.edu.au>
> Skickat: den 21 augusti 2018 13:21
> Till: TeX on Mac OS X Mailing List
> Kopia: TeX on Mac OS X Mailing List
> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
>
> Hi all.
>
> I agree with using "when", or better "precisely when", in definitions.
> Do not use these words in theorems. There are many alternative ways to
> express what you want.
>
> But didn't this thread start out asking for a symbolic notation?
> In that context, equivalence is *not* the word you want for a definition,
> as equivalence depends on a particular relation, so is weaker than a
> definition.
>
> There are nuances in mathematical concepts and notations that are not
> always correctly appreciated by those who work primarily in other (mostly
> applied) fields.
>
>
> On 21/08/2018, at 13:03, "Markus Klyver" <markusklyver at hotmail.com<mailto:
> markusklyver at hotmail.com>> wrote:
>
>
> But how would the reader know if it's a theorem or a definition? Consider
> the statement "f(x)=1 when x=2". Do we mean f(x)=1 <---> x=2 or just f(x)=1
> <--- x=2?
>
>
> ________________________________
> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu<mailto:
> macosx-tex-bounces at email.esm.psu.edu>> f?r Josep Maria Font <jmfont at ub.edu
> <mailto:jmfont at ub.edu>>
> Skickat: den 21 augusti 2018 12:15
> Till: TeX on Mac OS X Mailing List
> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
>
> That's why I prefer to use "when" for definitions: [...this NEW thing
> happens...] WHEN [...such-and-such known, understandable condition
> holds...].
>
>
> JMaF
>
>
> Cheers,
>
> Ross
> -------------- next part --------------
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> https://email.esm.psu.edu/pipermail/macosx-tex/attachments/20180821/1ee9e03b/attachment-0001.html
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>
> ------------------------------
>
> Message: 11
> Date: Tue, 21 Aug 2018 13:59:18 +0200
> From: Martin Berggren <martin.berggren at cs.umu.se>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID: <59716C4D-A9A5-40D0-B16D-FA55623687DC at cs.umu.se>
> Content-Type: text/plain; charset="utf-8"
>
>
> > On 21 Aug 2018, at 13:47, Markus Klyver <markusklyver at hotmail.com>
> wrote:
> >
> > The thing is that you can have several different definitions, all
> equivalent. Consider the definition "a matrix A \in \mathbb R^{n \times n}
> is invertiable iff A have a multiplicative inverse". It turns out that this
> is equivalent to a lot of things, among det(A) !=0, A having full rang, A
> having n linearindependent eigenvectors, Ax=0 only having the trivial
> solution, Ax=b having a solution (which is unique) for every
> right-hand-side b, etc.
>
> I would save that this is a theorem, not a definition. I think of a
> definition as a ?macro?; that is, giving a short name to a mathematical
> property. Example: a matrix A is called positive semidefinite when x^T Ax
> \geq 0 for all vectors x. The point is that you in each instances when the
> name is used, it can be replaced by its definition.
>
> Sincerely,
>
>
> Martin Berggren
> -------------------------------------------------------------------
> Department of Computing Science,
> UMIT Research Lab
> Ume? Universitet
> Campustorget 5, S-901 87 Ume?, Sweden. Ph: +46-70-732 8111
> http://www.cs.umu.se/~martinb <http://www.cs.umu.se/~martinb>,
> Martin.Berggren at cs.umu.se <mailto:Martin.Berggren at cs.umu.se>
>
>
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> ------------------------------
>
> Message: 12
> Date: Tue, 21 Aug 2018 09:10:57 -0300
> From: "N?stor E. Aguilera" <nestoreaguilera at gmail.com>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID: <6A9FD85E-F6AD-4418-804F-F854FE87EDE1 at gmail.com>
> Content-Type: text/plain; charset=utf-8
>
> Hi everyone:
>
> It seems to me that when we use "=" or another symbol in a "definition" we
> are actually stablishin a "notation". When we say "A is invertible
> if/iff/when..." this looks more like a definition, but it is actually a
> naming convention (right?), that is, a variant of notation, where we use a
> name to describe a perhaps complex situation.
>
> Knuth "defines" (as stated in Appendix B) the "bracket notation" (ACP, Vol
> 1, ?1.2.3, equation (16)) using "=" and "if".
>
> Perhaps somebody can explain what are the differences, if any, between
> "definition" and "notation".
>
> Best,
>
> Nestor
>
> ===============================================
>
> > On 21 Aug 2018, at 08:49, Markus Klyver <markusklyver at hotmail.com>
> wrote:
> >
> > Definitions *are* equivalences.
> >
> >
> > Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> f?r Ross Moore <
> ross.moore at mq.edu.au>
> > Skickat: den 21 augusti 2018 13:21
> > Till: TeX on Mac OS X Mailing List
> > Kopia: TeX on Mac OS X Mailing List
> > ?mne: Re: [OS X TeX] Latex symbol for "define equal"
> >
> > Hi all.
> >
> > I agree with using "when", or better "precisely when", in definitions.
> > Do not use these words in theorems. There are many alternative ways to
> express what you want.
> >
> > But didn't this thread start out asking for a symbolic notation?
> > In that context, equivalence is *not* the word you want for a
> definition, as equivalence depends on a particular relation, so is weaker
> than a definition.
> >
> > There are nuances in mathematical concepts and notations that are not
> always correctly appreciated by those who work primarily in other (mostly
> applied) fields.
> >
> >
> > On 21/08/2018, at 13:03, "Markus Klyver" <markusklyver at hotmail.com>
> wrote:
> >
> >> But how would the reader know if it's a theorem or a definition?
> Consider the statement "f(x)=1 when x=2". Do we mean f(x)=1 <---> x=2 or
> just f(x)=1 <--- x=2?
> >>
> >>
> >> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> f?r Josep
> Maria Font <jmfont at ub.edu>
> >> Skickat: den 21 augusti 2018 12:15
> >> Till: TeX on Mac OS X Mailing List
> >> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
> >>
> >> That's why I prefer to use "when" for definitions: [...this NEW thing
> happens...] WHEN [...such-and-such known, understandable condition
> holds...].
> >>
> >>
> >> JMaF
> >
> >
> > Cheers,
> >
> > Ross
> > ----------- Please Consult the Following Before Posting -----------
> > TeX FAQ: http://www.tex.ac.uk/faq
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> > List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
>
>
>
> ------------------------------
>
> Message: 13
> Date: Tue, 21 Aug 2018 09:43:39 -0300
> From: "N?stor E. Aguilera" <nestoreaguilera at gmail.com>
> To: Ralph Martin <ralphrmartin at gmail.com>
> Cc: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID: <DB8137AC-4047-430F-A249-E12ED33D1F90 at gmail.com>
> Content-Type: text/plain; charset=utf-8
>
> Hi Ralph:
>
> > On 21 Aug 2018, at 09:30, Ralph Martin <ralphrmartin at gmail.com> wrote:
> >
> >> [...]
> >
> > Definitions give *names* for concepts:
> > - a "matrix" is a two-dimensional table ...
> > - an "invertible matrix" is a matrix which ...
> >
> > Notation gives *symbols* for concepts (or quantities):
> > - ? is used to denote the ratio of the circumference of any circle to
> its diameter
> > - t is used to denote time
> > - boldface letters are used to denote vectors
> >
> > I always have trouble getting computer science students not to use
> multi-letter variable *names* in maths, when they should be using *symbols*
> for variables.
>
> So, would you say that the equality sign (or some replacement of it) in a
> "definition" is actually describing a "notation"?
>
> Nestor
>
> ------------------------------
>
> Message: 14
> Date: Tue, 21 Aug 2018 12:48:48 +0000
> From: Markus Klyver <markusklyver at hotmail.com>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID:
> <
> HE1PR0901MB1209F6C8AA31316AC2E36D57C3310 at HE1PR0901MB1209.eurprd09.prod.outlook.com
> >
>
> Content-Type: text/plain; charset="windows-1252"
>
> Depends, literature tends to use the definitions interchangeably depending
> on what aspect is important to subject.
>
>
> ________________________________
> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> f?r Martin
> Berggren <martin.berggren at cs.umu.se>
> Skickat: den 21 augusti 2018 13:59
> Till: TeX on Mac OS X Mailing List
> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
>
>
> On 21 Aug 2018, at 13:47, Markus Klyver <markusklyver at hotmail.com<mailto:
> markusklyver at hotmail.com>> wrote:
>
> The thing is that you can have several different definitions, all
> equivalent. Consider the definition "a matrix A \in \mathbb R^{n \times n}
> is invertiable iff A have a multiplicative inverse". It turns out that this
> is equivalent to a lot of things, among det(A) !=0, A having full rang, A
> having n linearindependent eigenvectors, Ax=0 only having the trivial
> solution, Ax=b having a solution (which is unique) for every
> right-hand-side b, etc.
>
> I would save that this is a theorem, not a definition. I think of a
> definition as a ?macro?; that is, giving a short name to a mathematical
> property. Example: a matrix A is called positive semidefinite when x^T Ax
> \geq 0 for all vectors x. The point is that you in each instances when the
> name is used, it can be replaced by its definition.
>
> Sincerely,
>
>
> Martin Berggren
> -------------------------------------------------------------------
> Department of Computing Science,
> UMIT Research Lab
> Ume? Universitet
> Campustorget 5, S-901 87 Ume?, Sweden. Ph: +46-70-732 8111
> http://www.cs.umu.se/~martinb, Martin.Berggren at cs.umu.se<mailto:
> Martin.Berggren at cs.umu.se>
>
>
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>
> ------------------------------
>
> Message: 15
> Date: Tue, 21 Aug 2018 12:51:48 +0000
> From: Markus Klyver <markusklyver at hotmail.com>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID:
> <
> HE1PR0901MB12093547942945904F35ABABC3310 at HE1PR0901MB1209.eurprd09.prod.outlook.com
> >
>
> Content-Type: text/plain; charset="utf-8"
>
> Yes, the sign = usually denotes an equivalence relation between two sets
> of interest. So the equality sign is a matter of notation, denoting
> "equality" (in the sense of an equivalence relation).
>
>
> ________________________________
> Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> f?r "N?stor E.
> Aguilera" <nestoreaguilera at gmail.com>
> Skickat: den 21 augusti 2018 14:43
> Till: Ralph Martin
> Kopia: TeX on Mac OS X Mailing List
> ?mne: Re: [OS X TeX] Latex symbol for "define equal"
>
> Hi Ralph:
>
> > On 21 Aug 2018, at 09:30, Ralph Martin <ralphrmartin at gmail.com> wrote:
> >
> >> [...]
> >
> > Definitions give *names* for concepts:
> > - a "matrix" is a two-dimensional table ...
> > - an "invertible matrix" is a matrix which ...
> >
> > Notation gives *symbols* for concepts (or quantities):
> > - ? is used to denote the ratio of the circumference of any circle to
> its diameter
> > - t is used to denote time
> > - boldface letters are used to denote vectors
> >
> > I always have trouble getting computer science students not to use
> multi-letter variable *names* in maths, when they should be using *symbols*
> for variables.
>
> So, would you say that the equality sign (or some replacement of it) in a
> "definition" is actually describing a "notation"?
>
> Nestor
> ----------- Please Consult the Following Before Posting -----------
> TeX FAQ: http://www.tex.ac.uk/faq
> List Reminders and Etiquette: https://sites.esm.psu.edu/~gray/tex/
> List Archives: http://dir.gmane.org/gmane.comp.tex.macosx
> https://email.esm.psu.edu/pipermail/macosx-tex/
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> List Info: https://email.esm.psu.edu/mailman/listinfo/macosx-tex
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> ------------------------------
>
> Message: 16
> Date: Tue, 21 Aug 2018 14:58:46 +0200
> From: Martin Berggren <martin.berggren at cs.umu.se>
> To: TeX on Mac OS X Mailing List <macosx-tex at email.esm.psu.edu>
> Subject: Re: [OS X TeX] Latex symbol for "define equal"
> Message-ID: <774832C8-3EEC-4690-A783-37D99E269CD1 at cs.umu.se>
> Content-Type: text/plain; charset="utf-8"
>
> Regarding the difference between a theorem and a definition: In the words
> of my thesis advisor: You cannot argue with a definition. (It should be
> said with a french accent!) A definition can be nice, practical, useful, or
> impractical, unnecessary, confusing, and so on, but it cannot be true or
> false. A theorem, however, is true.
>
> Cheers,
>
> > On 21 Aug 2018, at 14:48, Markus Klyver <markusklyver at hotmail.com>
> wrote:
> >
> > Depends, literature tends to use the definitions interchangeably
> depending on what aspect is important to subject.
> >
> >
> > Fr?n: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu <mailto:
> macosx-tex-bounces at email.esm.psu.edu>> f?r Martin Berggren <
> martin.berggren at cs.umu.se <mailto:martin.berggren at cs.umu.se>>
> > Skickat: den 21 augusti 2018 13:59
> > Till: TeX on Mac OS X Mailing List
> > ?mne: Re: [OS X TeX] Latex symbol for "define equal"
> >
> >
> >> On 21 Aug 2018, at 13:47, Markus Klyver <markusklyver at hotmail.com
> <mailto:markusklyver at hotmail.com>> wrote:
> >>
> >> The thing is that you can have several different definitions, all
> equivalent. Consider the definition "a matrix A \in \mathbb R^{n \times n}
> is invertiable iff A have a multiplicative inverse". It turns out that this
> is equivalent to a lot of things, among det(A) !=0, A having full rang, A
> having n linearindependent eigenvectors, Ax=0 only having the trivial
> solution, Ax=b having a solution (which is unique) for every
> right-hand-side b, etc.
> >
> > I would save that this is a theorem, not a definition. I think of a
> definition as a ?macro?; that is, giving a short name to a mathematical
> property. Example: a matrix A is called positive semidefinite when x^T Ax
> \geq 0 for all vectors x. The point is that you in each instances when the
> name is used, it can be replaced by its definition.
> >
>
> Martin Berggren
> -------------------------------------------------------------------
> Department of Computing Science,
> UMIT Research Lab
> Ume? Universitet
> Campustorget 5, S-901 87 Ume?, Sweden. Ph: +46-70-732 8111
> http://www.cs.umu.se/~martinb <http://www.cs.umu.se/~martinb>,
> Martin.Berggren at cs.umu.se <mailto:Martin.Berggren at cs.umu.se>
>
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