View Full Version : [SOLVED] What color is a domain wall?
Hannu Rjaniemi
Jun17-04, 06:15 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>If I lived in a universe with a domain wall, what would it look like?\n\nTo be more precise, suppose I have a universe where the Standard Model\nis not quite the same as ours, i.e. the gauge group is not SU(3) x\nSU(2) x U(1). Now introduce (via some magic) a bubble of our Standard\nModel into this universe. This region should, I believe, be separated\nfrom the rest of the universe by some sort of domain wall, a\ntopological defect.\n\nWhat would it look like? I realise that the answer may depend on what\nsort of particle states, confinement et cetera is allowed by the\n"alien" gauge group, but if I sat in my Standard Model bubble, what\nwould the sky look like? I suppose there might be some sort of\nscattering, since penetrating the wall might be energetically\nexpensive. But it might not be completely opaque.\n\nAny thoughts?\n\nthanks,\n\n- Hannu Rajaniemi\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>If I lived in a universe with a domain wall, what would it look like?
To be more precise, suppose I have a universe where the Standard Model
is not quite the same as ours, i.e. the gauge group is not SU(3) x
SU(2) x U(1). Now introduce (via some magic) a bubble of our Standard
Model into this universe. This region should, I believe, be separated
from the rest of the universe by some sort of domain wall, a
topological defect.
What would it look like? I realise that the answer may depend on what
sort of particle states, confinement et cetera is allowed by the
"alien" gauge group, but if I sat in my Standard Model bubble, what
would the sky look like? I suppose there might be some sort of
scattering, since penetrating the wall might be energetically
expensive. But it might not be completely opaque.
Any thoughts?
thanks,
- Hannu Rajaniemi
Mark Fergerson
Jun27-04, 05:55 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hannu Rjaniemi wrote:\n\n> If I lived in a universe with a domain wall, what would it look like?\n>\n> To be more precise, suppose I have a universe where the Standard Model\n> is not quite the same as ours, i.e. the gauge group is not SU(3) x\n> SU(2) x U(1). Now introduce (via some magic) a bubble of our Standard\n> Model into this universe. This region should, I believe, be separated\n> from the rest of the universe by some sort of domain wall, a\n> topological defect.\n\nSeems to me the boundary will be unstable.\n\n> What would it look like? I realise that the answer may depend on what\n> sort of particle states, confinement et cetera is allowed by the\n> "alien" gauge group, but if I sat in my Standard Model bubble, what\n> would the sky look like? I suppose there might be some sort of\n> scattering, since penetrating the wall might be energetically\n> expensive. But it might not be completely opaque.\n>\n> Any thoughts?\n\nEither light won\'t be able to propagate into it at all\n(mirror) or won\'t be able to get out (black).\n\nBut the stability issue keeps bugging me; you have\ndifferent versions of spacetime, and they will have\ndifferent zero-point energy levels and stuff like that. I\npersonally think the boundary will be bright as all hell,\nand racing one way or the other near (or at) c as the one\nkind of symmetry collapses into the other kind.\n\nIt\'d make a dandy doomsday weapon though...\n\nMark L. Fergerson\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hannu Rjaniemi wrote:
> If I lived in a universe with a domain wall, what would it look like?
>
> To be more precise, suppose I have a universe where the Standard Model
> is not quite the same as ours, i.e. the gauge group is not SU(3) x
> SU(2) x U(1). Now introduce (via some magic) a bubble of our Standard
> Model into this universe. This region should, I believe, be separated
> from the rest of the universe by some sort of domain wall, a
> topological defect.
Seems to me the boundary will be unstable.
> What would it look like? I realise that the answer may depend on what
> sort of particle states, confinement et cetera is allowed by the
> "alien" gauge group, but if I sat in my Standard Model bubble, what
> would the sky look like? I suppose there might be some sort of
> scattering, since penetrating the wall might be energetically
> expensive. But it might not be completely opaque.
>
> Any thoughts?
Either light won't be able to propagate into it at all
(mirror) or won't be able to get out (black).
But the stability issue keeps bugging me; you have
different versions of spacetime, and they will have
different zero-point energy levels and stuff like that. I
personally think the boundary will be bright as all hell,
and racing one way or the other near (or at) c as the one
kind of symmetry collapses into the other kind.
It'd make a dandy doomsday weapon though...
Mark L. Fergerson
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hannu Rjaniemi wrote:\n> To be more precise, suppose I have a universe where the Standard\nModel\n> is not quite the same as ours, i.e. the gauge group is not SU(3) x\n> SU(2) x U(1). Now introduce (via some magic) a bubble of our Standard\n> Model into this universe. This region should, I believe, be separated\n> from the rest of the universe by some sort of domain wall, a\n> topological defect.\n>\n> What would it look like? I realise that the answer may depend on what\n> sort of particle states, confinement et cetera is allowed by the\n> "alien" gauge group, but if I sat in my Standard Model bubble, what\n> would the sky look like? I suppose there might be some sort of\n> scattering, since penetrating the wall might be energetically\n> expensive. But it might not be completely opaque.\n\nFirst... Both the Q and A is probably "overly speculative" here :)\n\nSeems to me that _all_ of it depends on what particle states and groups\nare at action on the other side of the "wall". I mean the underlying\nassumption here is that there is at least a field connection between\nthe domains, meaning that an excitation in a field can propagate across\nthe wall. So some of the physics will be the same. But depending on the\n"mapping" between the fields in the two domains and on the common\nparticle states in the alien domain anything could appear here right ?\n\nA simple example would be an "uncoupling" I guess, where say two scalar\nreal fields in the alien universe have an U(1) symmetry, but appears in\nour domain as two separate, uncoupled scalars (with maybe different\nmass). So an incoming U(1) particle state would "scatter" into two\nseparate particle field excitations in this domain, provided that\nenergy conditions are met (another assumption about the "alien" domain\nof course).\n\nBut I have no idea how you would handle the case where separate fields\nin the alien domain could map into a more restricted group in our\ndomain. That could inject non-group-member states.. Like an SU(2)\nexcitation that is not one of the representations of the group :) (say\n4 scalar fields combining into the standard representation, which would\nbe bad since there are only 3 degrees of freedom in an SU(2) group).\nYou would probably have to disallow this kind of mapping and only allow\nmapping degrees of freedom.\n\nIn any case.. it would probably provide for an interesting fireworks\ndisplay at the edge :)\n\n/BW\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hannu Rjaniemi wrote:
> To be more precise, suppose I have a universe where the Standard
Model
> is not quite the same as ours, i.e. the gauge group is not SU(3) x
> SU(2) x U(1). Now introduce (via some magic) a bubble of our Standard
> Model into this universe. This region should, I believe, be separated
> from the rest of the universe by some sort of domain wall, a
> topological defect.
>
> What would it look like? I realise that the answer may depend on what
> sort of particle states, confinement et cetera is allowed by the
> "alien" gauge group, but if I sat in my Standard Model bubble, what
> would the sky look like? I suppose there might be some sort of
> scattering, since penetrating the wall might be energetically
> expensive. But it might not be completely opaque.
First... Both the Q and A is probably "overly speculative" here :)
Seems to me that _all_ of it depends on what particle states and groups
are at action on the other side of the "wall". I mean the underlying
assumption here is that there is at least a field connection between
the domains, meaning that an excitation in a field can propagate across
the wall. So some of the physics will be the same. But depending on the
"mapping" between the fields in the two domains and on the common
particle states in the alien domain anything could appear here right ?
A simple example would be an "uncoupling" I guess, where say two scalar
real fields in the alien universe have an U(1) symmetry, but appears in
our domain as two separate, uncoupled scalars (with maybe different
mass). So an incoming U(1) particle state would "scatter" into two
separate particle field excitations in this domain, provided that
energy conditions are met (another assumption about the "alien" domain
of course).
But I have no idea how you would handle the case where separate fields
in the alien domain could map into a more restricted group in our
domain. That could inject non-group-member states.. Like an SU(2)
excitation that is not one of the representations of the group :) (say
4 scalar fields combining into the standard representation, which would
be bad since there are only 3 degrees of freedom in an SU(2) group).
You would probably have to disallow this kind of mapping and only allow
mapping degrees of freedom.
In any case.. it would probably provide for an interesting fireworks
display at the edge :)
/BW
George W. Harris
Jun30-04, 05:35 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Mark Fergerson <nunya@biz.ness> wrote:\n\n: But the stability issue keeps bugging me; you have\n:different versions of spacetime, and they will have\n:different zero-point energy levels and stuff like that. I\n:personally think the boundary will be bright as all hell,\n:and racing one way or the other near (or at) c as the one\n:kind of symmetry collapses into the other kind.\n\nThat\'s the kind of thing that worries me about\npeople who want to tap zero-point energy. You lower\nthe zero-point energy in one spot, and it then\npropagates outward at c, destroying everything in its\n\'path\'.\n\n: Mark L. Fergerson\n\n--\ne^(i*pi)+1=0\n\nGeorge W. Harris For actual email address, replace each \'u\' with an \'i\'.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Mark Fergerson <nunya@biz.ness> wrote:
: But the stability issue keeps bugging me; you have
:different versions of spacetime, and they will have
:different zero-point energy levels and stuff like that. I
:personally think the boundary will be bright as all hell,
:and racing one way or the other near (or at) c as the one
:kind of symmetry collapses into the other kind.
That's the kind of thing that worries me about
people who want to tap zero-point energy. You lower
the zero-point energy in one spot, and it then
propagates outward at c, destroying everything in its
'path'.
: Mark L. Fergerson
--
e^(i*\pi)+1=0
George W. Harris For actual email address, replace each 'u' with an 'i'.
Brian Davis
Jun30-04, 05:36 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hannu Rjaniemi wrote:\n\n> If I lived in a universe with a domain wall, what would it look like?\n\nWell, who\'s to say you don\'t but the domain walls are simply not\nwithin our causal sphere? That said...\nWhat if during electroweak symmertry breaking different areas of\nthe universe had different masses for the photon (we got zero,\nsomeplace else got, well, a massive photon). The domain wall would be\nperfectly reflective, and what you would "see" is a mirror of the\nuniverse on your side. Note that for most such old domain walls, they\nwould be locally flat, so it would act very much like a flat mirror.\n\n--\nBrian Davis\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hannu Rjaniemi wrote:
> If I lived in a universe with a domain wall, what would it look like?
Well, who's to say you don't but the domain walls are simply not
within our causal sphere? That said...
What if during electroweak symmertry breaking different areas of
the universe had different masses for the photon (we got zero,
someplace else got, well, a massive photon). The domain wall would be
perfectly reflective, and what you would "see" is a mirror of the
universe on your side. Note that for most such old domain walls, they
would be locally flat, so it would act very much like a flat mirror.
--
Brian Davis
Dirk Bruere at Neopax
Jul1-04, 04:47 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Brian Davis wrote:\n\n> Hannu Rjaniemi wrote:\n>\n>\n>>If I lived in a universe with a domain wall, what would it look like?\n>\n>\n> Well, who\'s to say you don\'t but the domain walls are simply not\n> within our causal sphere? That said...\n> What if during electroweak symmertry breaking different areas of\n> the universe had different masses for the photon (we got zero,\n> someplace else got, well, a massive photon). The domain wall would be\n> perfectly reflective, and what you would "see" is a mirror of the\n> universe on your side. Note that for most such old domain walls, they\n> would be locally flat, so it would act very much like a flat mirror.\n\nAnd if one could walk up to it, what would it be like? Infinitely stiff?\nFlexible? Massive? Gravitational effects? etc\n\n--\nDirk\n\nThe Consensus:-\nThe political party for the new millenium\nhttp://www.theconsensus.org\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Brian Davis wrote:
> Hannu Rjaniemi wrote:
>
>
>>If I lived in a universe with a domain wall, what would it look like?
>
>
> Well, who's to say you don't but the domain walls are simply not
> within our causal sphere? That said...
> What if during electroweak symmertry breaking different areas of
> the universe had different masses for the photon (we got zero,
> someplace else got, well, a massive photon). The domain wall would be
> perfectly reflective, and what you would "see" is a mirror of the
> universe on your side. Note that for most such old domain walls, they
> would be locally flat, so it would act very much like a flat mirror.
And if one could walk up to it, what would it be like? Infinitely stiff?
Flexible? Massive? Gravitational effects? etc
--
Dirk
The Consensus:-
The political party for the new millenium
http://www.theconsensus.org
Bryan J. Maloney
Jul1-04, 04:47 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"George W. Harris" <gharrus@mundsprung.com> abagooba zoink larblortch\nnews:hdvud091ss0hm4851cmagl860fv2vov7r 7@4ax.com:\n\n> Mark Fergerson <nunya@biz.ness> wrote:\n>\n>: But the stability issue keeps bugging me; you have\n>:different versions of spacetime, and they will have\n>:different zero-point energy levels and stuff like that. I\n>:personally think the boundary will be bright as all hell,\n>:and racing one way or the other near (or at) c as the one\n>:kind of symmetry collapses into the other kind.\n>\n> That\'s the kind of thing that worries me about\n> people who want to tap zero-point energy. You lower\n> the zero-point energy in one spot, and it then\n> propagates outward at c, destroying everything in its\n> \'path\'.\n\nThe Zeepee Bomb.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"George W. Harris" <gharrus@mundsprung.com> abagooba zoink larblortch
news:hdvud091ss0hm4851cmagl860fv2vov7r7@4ax.com:
> Mark Fergerson <nunya@biz.ness> wrote:
>
>: But the stability issue keeps bugging me; you have
>:different versions of spacetime, and they will have
>:different zero-point energy levels and stuff like that. I
>:personally think the boundary will be bright as all hell,
>:and racing one way or the other near (or at) c as the one
>:kind of symmetry collapses into the other kind.
>
> That's the kind of thing that worries me about
> people who want to tap zero-point energy. You lower
> the zero-point energy in one spot, and it then
> propagates outward at c, destroying everything in its
> 'path'.
The Zeepee Bomb.
Brian Davis
Jul2-04, 09:08 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nDirk Bruere asked:\n\n>> The domain wall would be perfectly reflective, and what you\n>> would "see" is a mirror of the universe on your side. Note that\n>> for most such old domain walls, they would be locally flat, so it\n>> would act very much like a flat mirror.\n>\n> And if one could walk up to it, what would it be like? Infinitely stiff?\n> Flexible? Massive? Gravitational effects? etc\n\nUnderstand this is coming from my hazy memory of a cosomology class\nI had in grad school, nearly a decade ago. Domain walls like I\'m\ndescribing are gravitationally repulsive (they have a negative energy\ndensity... think of it as a surface tension), and because they are\nlarge-scale (old domain walls of this type would span the observable\nuniverse), gravitational repulsion would remain constant with distance\n(just like gravitational attraction to an infinate massive flat\nplain). I\'m not sure if it would correspond to the behavior of a\nnegative mass object (in a Newtonian sense), but I suspect it would be\nwell-nigh impenetrable - consider trying to stick your hand through\nit, you need to change the physics by raising the local energy through\nunification temperatures.\nI think.\nIt would be plenty flexible, and would behave somewhat like a soap\nfilm - the tension means it would try to minimize it\'s surface area\n(subject to c!), and so locally flatten out. Initially, such domain\nwalls are very complex almost fractal surfaces, but small variations\nsmooth out (via tension) very very rapidly, while moderate variations\n(moderate curvatures) flatten somewhat slower, and large-scale warps\n(areas with large curvature) flatten still slower, etc.\nDomain walls are pretty weird. And, for that matter, some types\n(like the one descibed above; remember, I\'m basing all this on *ONE*\ntype of domain wall, with a specific symmetry breaking during the\nelectroweak transition) are bloody obvious. And we don\'t see them.\nDraw your own conclusions, but keep in mind that such topological\ndefects, (including monopoles, strings, domain walls, domain walls\nbounded by strings, textures, etc.) are parts of almost any theory.\nThe lack of such observations is an interesting problem (inflation is\nsometimes invoked... after all, we made it up to solve other, similar\nproblems).\n\n--\nBrian Davis\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Dirk Bruere asked:
>> The domain wall would be perfectly reflective, and what you
>> would "see" is a mirror of the universe on your side. Note that
>> for most such old domain walls, they would be locally flat, so it
>> would act very much like a flat mirror.
>
> And if one could walk up to it, what would it be like? Infinitely stiff?
> Flexible? Massive? Gravitational effects? etc
Understand this is coming from my hazy memory of a cosomology class
I had in grad school, nearly a decade ago. Domain walls like I'm
describing are gravitationally repulsive (they have a negative energy
density... think of it as a surface tension), and because they are
large-scale (old domain walls of this type would span the observable
universe), gravitational repulsion would remain constant with distance
(just like gravitational attraction to an infinate massive flat
plain). I'm not sure if it would correspond to the behavior of a
negative mass object (in a Newtonian sense), but I suspect it would be
well-nigh impenetrable - consider trying to stick your hand through
it, you need to change the physics by raising the local energy through
unification temperatures.
I think.
It would be plenty flexible, and would behave somewhat like a soap
film - the tension means it would try to minimize it's surface area
(subject to c!), and so locally flatten out. Initially, such domain
walls are very complex almost fractal surfaces, but small variations
smooth out (via tension) very very rapidly, while moderate variations
(moderate curvatures) flatten somewhat slower, and large-scale warps
(areas with large curvature) flatten still slower, etc.
Domain walls are pretty weird. And, for that matter, some types
(like the one descibed above; remember, I'm basing all this on *ONE*
type of domain wall, with a specific symmetry breaking during the
electroweak transition) are bloody obvious. And we don't see them.
Draw your own conclusions, but keep in mind that such topological
defects, (including monopoles, strings, domain walls, domain walls
bounded by strings, textures, etc.) are parts of almost any theory.
The lack of such observations is an interesting problem (inflation is
sometimes invoked... after all, we made it up to solve other, similar
problems).
--
Brian Davis
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