<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\nI\'ve always seen the line element expressed as:\n\nds^2 = g_ab dx^a dx^b\n\nThis, apparently, holds for rectilinear as well as curvilinear\nand skew coordinates... It doesn\'t seem immediately\nunreasonable, to me, to construct a space with line element:\n\nds^2 = g_ab dx^a dx^b + h_abcd dx^a dx^b dx^c dx^d + ...\n\nexcept, I suppose, that the units seem a little odd. I have\nencountered a system which might be neatly solved if: (1) such\nspaces exist; and (2) certain things are known about transforming\nto such a space.\n\nIs there a name for such a beast? A place I might look for an\nintroduction to it? Thanks in advance.\n\n-----------------------------------------------------------------\n. . . Except when they don\'t,\nBecause sometimes they won\'t. - Dr. Seuss\n-----------------------------------------------------------------\nJason Cooper jcooper@acs.ucalgary.ca\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>I've always seen the line element expressed as:
ds^2 = g_{ab} dx^a dx^b
This, apparently, holds for rectilinear as well as curvilinear
and skew coordinates... It doesn't seem immediately
unreasonable, to me, to construct a space with line element:
except, I suppose, that the units seem a little odd. I have
encountered a system which might be neatly solved if: (1) such
spaces exist; and (2) certain things are known about transforming
to such a space.
Is there a name for such a beast? A place I might look for an
introduction to it? Thanks in advance.
-----------------------------------------------------------------
. . . Except when they don't,
Because sometimes they won't. - Dr. Seuss
-----------------------------------------------------------------
Jason Cooper jcooper@acs.ucalgary.ca
tessel@tum.bot
Jul28-04, 04:58 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>On Tue, 27 Jul 2004, jason cooper wrote:\n\n> I\'ve always seen the line element expressed as:\n>\n> ds^2 = g_ab dx^a dx^b\n>\n> This, apparently, holds for rectilinear as well as curvilinear\n> and skew coordinates...\n\nSure, but note that rectilinear coordinates only exist in flat\nspaces/spacetimes. Also, if you mean by "skew coordinates" what I think\nyou mean, these are a special case of rectilinear coordinates.\n\n> It doesn\'t seem immediately unreasonable, to me, to construct a space\n> with line element:\n>\n> ds^2 = g_ab dx^a dx^b + h_abcd dx^a dx^b dx^c dx^d + ...\n>\n> except, I suppose, that the units seem a little odd.\n\nMaybe write\n\nds^4 = + (g_ab dx^a dx^b)^2 + h_abcd dx^a dx^b dx^c dx^d\n\nand reexpress the first term as a trace? If you continue playing along\nthese lines, I think you will eventually be led to a more general notion:\nRiemann-Finsler geometry.\n\nNext, assuming your question is "Why not cubic or quartic forms instead of\nquadratic forms?", then, IIRC, this is discussed somewhere in volume two\nof classic treatise:\n\nauthor = {Michael Spivak},\ntitle = {A Comprehensive Introduction to Differential Geometry},\nnote = {five volumes},\npublisher = {Publish or Perish},\naddress = {Berkeley, CA}\nyear = 1979}\n\n> Is there a name for such a beast? A place I might look for an\n> introduction to it?\n\nTry this:\n\nauthor = {D. Bao and S.-S. Chern and Z. Shen},\ntitle = {An introduction to {R}iemann-{F}insler geometry},\npublisher = {Springer},\nyear = 2000}\n\nP.S.: if your problem involves ordinary differential equations, note that\na currently hot topic in pure math (see the ArXiV!) amounts to picking up\na thread dropped long ago by Elie Cartan (and his most prominent\nsuccessor, S.-S. Chern!), namely associating a Cartan geometry with a\nsolution space. (Cartan geometry is the simplest common generalization of\nRiemannian and Kleinian geometry; Lie groups and "homogeneous spaces"\n[e.g. projective spaces, Grassmannians, spheres, etc.] are classic\nexamples of Kleinian geometries.) I have mentioned this fascinating\nphenomenon several times recently.\n\nAt some point I hope to explain a related notion: studying the symmetries\nof certain ODEs leads naturally to the Bianchi groups, the Lorentz group,\netc.; turning this around, we can study the Kleinian geometries associated\nto the Bianchi groups, etc. Thus for example we have the classic\n-euclidean- differential geometry of plane curves associated with E(2),\nbut also the differential geometry of curves associated with SA(2), or\nwith the projective group, etc.\n\nP.P.S.: I guess the theory of elasticity would be one place where one\nmeets quartic forms. But in such cases, standard tricks (e.g. treat\nbivectors as vectors) should reduce this to ordinary semiriemannian\ngeometry.\n\n"T. Essel" (hiding somewhere in cyberspace)\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>On Tue, 27 Jul 2004, jason cooper wrote:
> I've always seen the line element expressed as:
>
> ds^2 = g_{ab} dx^a dx^b
>
> This, apparently, holds for rectilinear as well as curvilinear
> and skew coordinates...
Sure, but note that rectilinear coordinates only exist in flat
spaces/spacetimes. Also, if you mean by "skew coordinates" what I think
you mean, these are a special case of rectilinear coordinates.
> It doesn't seem immediately unreasonable, to me, to construct a space
> with line element:
>
> ds^2 = g_{ab} dx^a dx^b + h_{abcd} dx^a dx^b dx^c dx^d + ...
>
> except, I suppose, that the units seem a little odd.
and reexpress the first term as a trace? If you continue playing along
these lines, I think you will eventually be led to a more general notion:
Riemann-Finsler geometry.
Next, assuming your question is "Why not cubic or quartic forms instead of
quadratic forms?", then, IIRC, this is discussed somewhere in volume two
of classic treatise:
author = {Michael Spivak},
title = {A Comprehensive Introduction to Differential Geometry},
note = {five volumes},
publisher = {Publish or Perish},
address = {Berkeley, CA}
year = 1979}
> Is there a name for such a beast? A place I might look for an
> introduction to it?
Try this:
author = {D. Bao and S.-S. Chern and Z. Shen},
title = {An introduction to {R}iemann-{F}insler geometry},
publisher = {Springer},
year = 2000}
P.S.: if your problem involves ordinary differential equations, note that
a currently hot topic in pure math (see the ArXiV!) amounts to picking up
a thread dropped long ago by Elie Cartan (and his most prominent
successor, S.-S. Chern!), namely associating a Cartan geometry with a
solution space. (Cartan geometry is the simplest common generalization of
Riemannian and Kleinian geometry; Lie groups and "homogeneous spaces"
[e.g. projective spaces, Grassmannians, spheres, etc.] are classic
examples of Kleinian geometries.) I have mentioned this fascinating
phenomenon several times recently.
At some point I hope to explain a related notion: studying the symmetries
of certain ODEs leads naturally to the Bianchi groups, the Lorentz group,
etc.; turning this around, we can study the Kleinian geometries associated
to the Bianchi groups, etc. Thus for example we have the classic
-euclidean- differential geometry of plane curves associated with E(2),
but also the differential geometry of curves associated with SA(2), or
with the projective group, etc.
P.P.S.: I guess the theory of elasticity would be one place where one
meets quartic forms. But in such cases, standard tricks (e.g. treat
bivectors as vectors) should reduce this to ordinary semiriemannian
geometry.
"T. Essel" (hiding somewhere in cyberspace)
jason cooper
Jul28-04, 04:58 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\njason cooper (jcooper@acs4.acs.ucalgary.ca) wrote:\n\n: ds^2 = g_ab dx^a dx^b + h_abcd dx^a dx^b dx^c dx^d + ...\n\nIt occurs to me that I could be clearer. The problem I\'m working\non, in its basic form, gives a kinetic energy of the form:\n\n2T = g^ab p_a p_b\n\nwhere g^ab is usually interpreted as a metric. To a higher order\napproximation, the kinetic energy is given by:\n\n2T = g^ab p_a p_b + h^abcd p_a p_b p_c p_d\n\nMy question, I suppose, is can the combination of \'g\' and \'h\' be\ninterpreted similarly as a metric on a different sort of space?\n\n-----------------------------------------------------------------\n. . . Except when they don\'t,\nBecause sometimes they won\'t. - Dr. Seuss\n-----------------------------------------------------------------\nJason Cooper jcooper@acs.ucalgary.ca\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>jason cooper (jcooper@acs4.acs.ucalgary.ca) wrote:
It occurs to me that I could be clearer. The problem I'm working
on, in its basic form, gives a kinetic energy of the form:
2T = g^{ab} p_a p_b
where g^{ab} is usually interpreted as a metric. To a higher order
approximation, the kinetic energy is given by:
2T = g^{ab} p_a p_b + h^{abcd} p_a p_b p_c p_d
My question, I suppose, is can the combination of 'g' and 'h' be
interpreted similarly as a metric on a different sort of space?
-----------------------------------------------------------------
. . . Except when they don't,
Because sometimes they won't. - Dr. Seuss
-----------------------------------------------------------------
Jason Cooper jcooper@acs.ucalgary.ca
Igor
Jul29-04, 05:59 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\njcooper@acs4.acs.ucalgary.ca (jason cooper) wrote in message news:<ce6g3t\\$u97\\$1@news.ucalgary.ca>...\n> jason cooper (jcooper@acs4.acs.ucalgary.ca) wrote:\n>\n> : ds^2 = g_ab dx^a dx^b + h_abcd dx^a dx^b dx^c dx^d + ...\n>\n> It occurs to me that I could be clearer. The problem I\'m working\n> on, in its basic form, gives a kinetic energy of the form:\n>\n> 2T = g^ab p_a p_b\n>\n> where g^ab is usually interpreted as a metric. To a higher order\n> approximation, the kinetic energy is given by:\n>\n> 2T = g^ab p_a p_b + h^abcd p_a p_b p_c p_d\n>\n> My question, I suppose, is can the combination of \'g\' and \'h\' be\n> interpreted similarly as a metric on a different sort of space?\n>\n> -----------------------------------------------------------------\n> . . . Except when they don\'t,\n> Because sometimes they won\'t. - Dr. Seuss\n> -----------------------------------------------------------------\n> Jason Cooper jcooper@acs.ucalgary.ca\n\n\n\nYes, you can do it that way, but it can get very involved, not the\nmention tricky. In order for T to have the proper units, h would have\nto depend upon the components of p. Finsler geometry can cover this\ntype of problem, but only if the metric generating function (in this\ncase T) is homogeneous to second order in vector p. In other words,\nT(cp) = c^2 T(p), where c is some arbitrary non-zero constant. The\nfirst term will be covered in that regard, provided that it is\nReimannian. From the second term, we deduce that all the components\nof h must then be homogeneous to negative second order. So we\'ll need\nh(cp) = c^(-2)h(p).\n\nIf the Finsler condition is not met, the problem may still be\nworkable, but may be much more complicated to deal with. The\nchallenge is to set up and solve the variational problem associated\nwith the prescribed metric form. AFAIK, there is no general treatment\nfor proposed metrics of this type. You might want to look up "areal\nmetrics" or "areal spaces". These are the general headings under\nwhich this type of material tends to be classified. Good luck.\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>jcooper@acs4.acs.ucalgary.ca (jason cooper) wrote in message news:<ce6g3t$u97$1@news.ucalgary.ca>...
> jason cooper (jcooper@acs4.acs.ucalgary.ca) wrote:
>
> : ds^2 = g_{ab} dx^a dx^b + h_{abcd} dx^a dx^b dx^c dx^d + ...
>
> It occurs to me that I could be clearer. The problem I'm working
> on, in its basic form, gives a kinetic energy of the form:
>
> 2T = g^{ab} p_a p_b
>
> where g^{ab} is usually interpreted as a metric. To a higher order
> approximation, the kinetic energy is given by:
>
> 2T = g^{ab} p_a p_b + h^{abcd} p_a p_b p_c p_d
>
> My question, I suppose, is can the combination of 'g' and 'h' be
> interpreted similarly as a metric on a different sort of space?
>
> -----------------------------------------------------------------
> . . . Except when they don't,
> Because sometimes they won't. - Dr. Seuss
> -----------------------------------------------------------------
> Jason Cooper jcooper@acs.ucalgary.ca
Yes, you can do it that way, but it can get very involved, not the
mention tricky. In order for T to have the proper units, h would have
to depend upon the components of p. Finsler geometry can cover this
type of problem, but only if the metric generating function (in this
case T) is homogeneous to second order in vector p. In other words,
T(cp) = c^2 T(p), where c is some arbitrary non-zero constant. The
first term will be covered in that regard, provided that it is
Reimannian. From the second term, we deduce that all the components
of h must then be homogeneous to negative second order. So we'll need
h(cp) = c^(-2)h(p).
If the Finsler condition is not met, the problem may still be
workable, but may be much more complicated to deal with. The
challenge is to set up and solve the variational problem associated
with the prescribed metric form. AFAIK, there is no general treatment
for proposed metrics of this type. You might want to look up "areal
metrics" or "areal spaces". These are the general headings under
which this type of material tends to be classified. Good luck.
tessel@tum.bot
Jul30-04, 04:21 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>On Thu, 29 Jul 2004, Igor wrote:\n\n> AFAIK, there is no general treatment for proposed metrics of this type.\n\n[ i.e. ds^2 = g_ab dx^a dx^b + h_abcd dx^a dx^b dx^c dx^d + ...]\n\n> You might want to look up "areal metrics" or "areal spaces". These are\n> the general headings under which this type of material tends to be\n> classified.\n\nInteresting--- am I correct in guessing that a phrase like "the areal\ngeometry of plane curves" refers to the Kleinian geometry associated with\nSA(2)? (I.e., the special affine group acting on the real plane.) If so,\nmy guesses concerning a possible connection with Cartanian geometry and a\nreformulation in terms of bivectors would both be right on the money! In\nthat case, if you want to pursue this, Jason, I can say more about this\ninteresting geometry, and give some references.\n\n"T. Essel" (hiding somewhere in cyberspace)\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>On Thu, 29 Jul 2004, Igor wrote:
> AFAIK, there is no general treatment for proposed metrics of this type.
> You might want to look up "areal metrics" or "areal spaces". These are
> the general headings under which this type of material tends to be
> classified.
Interesting--- am I correct in guessing that a phrase like "the areal
geometry of plane curves" refers to the Kleinian geometry associated with
SA(2)? (I.e., the special affine group acting on the real plane.) If so,
my guesses concerning a possible connection with Cartanian geometry and a
reformulation in terms of bivectors would both be right on the money! In
that case, if you want to pursue this, Jason, I can say more about this
interesting geometry, and give some references.
"T. Essel" (hiding somewhere in cyberspace)
tessel@tum.bot
Jul30-04, 04:21 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>On Wed, 28 Jul 2004, jason cooper wrote:\n\n> The problem I\'m working on, in its basic form, gives a kinetic energy of\n> the form:\n>\n> 2T = g^ab p_a p_b\n>\n> where g^ab is usually interpreted as a metric. To a higher order\n> approximation, the kinetic energy is given by:\n>\n> 2T = g^ab p_a p_b + h^abcd p_a p_b p_c p_d\n\nDid this arise as the "kinetic energy part" of a Lagrangian, by any\nchance? (It could be considered a Lagrangian as it stands, but perhaps\nyou have an additional "potential energy term" in mind!)\n\nIf so, do you know how to obtain the Euler-Lagrange equation for a fourth\norder Lagrangian, and how to use a variational symmetry analysis to obtain\nconserved quantitites, i.e. invariants partially characterizing solutions\nof the EL equation ("field equation")? These computational skills happen\nto be one of the topics I have been discussing in the "Solitons in One\nPost" thread.\n\n> can the combination of \'g\' and \'h\' be interpreted similarly as a metric\n> on a different sort of space?\n\nGood question--- but right now, I don\'t know the answer.\n\n"T. Essel" (hiding somewhere in cyberspace)\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>On Wed, 28 Jul 2004, jason cooper wrote:
> The problem I'm working on, in its basic form, gives a kinetic energy of
> the form:
>
> 2T = g^{ab} p_a p_b
>
> where g^{ab} is usually interpreted as a metric. To a higher order
> approximation, the kinetic energy is given by:
>
> 2T = g^{ab} p_a p_b + h^{abcd} p_a p_b p_c p_d
Did this arise as the "kinetic energy part" of a Lagrangian, by any
chance? (It could be considered a Lagrangian as it stands, but perhaps
you have an additional "potential energy term" in mind!)
If so, do you know how to obtain the Euler-Lagrange equation for a fourth
order Lagrangian, and how to use a variational symmetry analysis to obtain
conserved quantitites, i.e. invariants partially characterizing solutions
of the EL equation ("field equation")? These computational skills happen
to be one of the topics I have been discussing in the "Solitons in One
Post" thread.
> can the combination of 'g' and 'h' be interpreted similarly as a metric
> on a different sort of space?
Good question--- but right now, I don't know the answer.
"T. Essel" (hiding somewhere in cyberspace)
jason cooper
Jul30-04, 12:43 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>\ntessel@tum.bot wrote:\n\n: > 2T = g^ab p_a p_b + h^abcd p_a p_b p_c p_d\n\n: Did this arise as the "kinetic energy part" of a Lagrangian, by any\n: chance? (It could be considered a Lagrangian as it stands, but perhaps\n: you have an additional "potential energy term" in mind!)\n\nIt absolutely did. And there is, in fact, a potential energy\nterm as well; I had omitted it because it wasn\'t really part of\nthe original question.\n\n: If so, do you know how to obtain the Euler-Lagrange equation for a fourth\n: order Lagrangian, and how to use a variational symmetry analysis to obtain\n: conserved quantitites, i.e. invariants partially characterizing solutions\n: of the EL equation ("field equation")? These computational skills happen\n: to be one of the topics I have been discussing in the "Solitons in One\n: Post" thread.\n\nI hadn\'t given any thought to the idea that the EL equation might\ndiffer for the above. I\'ll look into that thread...\n\n-----------------------------------------------------------------\n. . . Except when they don\'t,\nBecause sometimes they won\'t. - Dr. Seuss\n-----------------------------------------------------------------\nJason Cooper jcooper@acs.ucalgary.ca\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>tessel@tum.bot wrote:
: Did this arise as the "kinetic energy part" of a Lagrangian, by any
: chance? (It could be considered a Lagrangian as it stands, but perhaps
: you have an additional "potential energy term" in mind!)
It absolutely did. And there is, in fact, a potential energy
term as well; I had omitted it because it wasn't really part of
the original question.
: If so, do you know how to obtain the Euler-Lagrange equation for a fourth
: order Lagrangian, and how to use a variational symmetry analysis to obtain
: conserved quantitites, i.e. invariants partially characterizing solutions
: of the EL equation ("field equation")? These computational skills happen
: to be one of the topics I have been discussing in the "Solitons in One
: Post" thread.
I hadn't given any thought to the idea that the EL equation might
differ for the above. I'll look into that thread...
-----------------------------------------------------------------
. . . Except when they don't,
Because sometimes they won't. - Dr. Seuss
-----------------------------------------------------------------
Jason Cooper jcooper@acs.ucalgary.ca