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{SECT 0 {EXCHG {PARA 209 "" 0 "" }{PARA 210 "" 0 "" {TEXT 212 34 "Test
 suite for diffalg efficiency " }}{PARA 211 "" 0 "" {TEXT 213 22 "Crea
ted February, 1998" }}{PARA 209 "" 0 "" {TEXT 214 47 "Revised May 1998
, February 1999,  December 2000" }}{PARA 209 "" 0 "" {TEXT 214 14 "Eve
lyne Hubert" }}{PARA 212 "" 0 "" }}{EXCHG {PARA 209 "> " 0 "" 
{MPLTEXT 1 215 14 "with(diffalg);" }}{PARA 213 "" 1 "" {XPPMATH 20 "6#
7:I3Rosenfeld_GroebnerG6\"I-delta_leaderGF%I1delta_polynomialGF%I'deno
teGF%I,derivativesGF%I2differential_ringGF%I.differentiateGF%I*equatio
nsGF%I5essential_componentsGF%I0field_extensionGF%I(greaterGF%I,inequa
tionsGF%I(initialGF%I3initial_conditionsGF%I)is_primeGF%I'leaderGF%I6p
ower_series_solutionGF%I7preparation_polynomialGF%I.print_rankingGF%I%
rankGF%I'reduceGF%I.rewrite_rulesGF%I)separantGF%I(versionG6$I(_syslib
GF%F>" }{TEXT 216 1 " " }}}{SECT 1 {PARA 214 "" 0 "" {TEXT 217 23 "Pro
blem 0: the pendulum" }}{PARA 215 "" 0 "" {TEXT 218 177 "The equation \+
of the prendulum in cartesian coordinate, rewritten as a system of fir
st order equations, is known to be an index 3 problem. We wish to find
 the hidden constraints." }}{PARA 209 "" 0 "" }{EXCHG {PARA 209 "> " 0
 "" {MPLTEXT 1 215 55 "K0 := field_extension(transcendental_elements=[
m,l,g]);" }{MPLTEXT 1 215 88 "\nR0 := differential_ring(ranking=[[T,p,
q,x,y]], derivations=[t], field_of_constants=K0);" }}{PARA 213 "" 1 ""
 {XPPMATH 20 "6#>I#K0G6\"I-ground_fieldGF%" }{TEXT 216 1 " " }}{PARA 
213 "" 1 "" {XPPMATH 20 "6#>I#R0G6\"I)ODE_ringGF%" }{TEXT 216 1 " " }}
}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 75 "S0 := [m*q[t]+T[]*y[]+g
, y[t]-q[], m*p[t]+T[]*x[],x[t]-p[], x[]^2+y[]^2-1];" }}{PARA 213 "" 
1 "" {XPPMATH 20 "6#>I#S0G6\"7',(*&I\"mGF%\"\"\"&I\"qGF%6#I\"tGF%F*F**
&&I\"TGF%F%F*&I\"yGF%F%F*F*I\"gGF%F*,&&F3F-F*&F,F%!\"\",&*&F)F*&I\"pGF
%F-F*F**&F0F*&I\"xGF%F%F*F*,&&F?F-F*&F<F%F8,(*$F>\"\"#F**$F2FEF*F8F*" 
}{TEXT 216 1 " " }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 28 " Ros
enfeld_Groebner(S0,R0); " }{MPLTEXT 1 215 18 "\nrewrite_rules(%);" }
{MPLTEXT 1 215 1 "\n" }}{PARA 213 "" 1 "" {XPPMATH 20 "6#7$I0character
isableG6\"F$" }{TEXT 216 1 " " }}{PARA 213 "" 1 "" {XPPMATH 20 "6#7$7)
/&I\"qG6\"6#I\"tGF(*(,*I\"gGF(\"\"\"*&F-F.I\"yGF(\"\"#!\"#*&F0\"\"%F-F
.F.*(F0F.F'F1I\"mGF(F.F.F.F6!\"\",&F7F.*$F0F1F.F7/&F0F)F'/I\"TGF(,$*&,
(*&F0F.F-F.F7*&F0\"\"$F-F.F.*&F'F1F6F.F.F.F8F7F7/I\"pGF(**F0F.F'F.I\"x
GF(F.F8F7/*$FHF1,&F.F.F9F70F8\"\"!0FHFM7(/F=,$FAF7/FFFM/F'FM/FHFM/F9F.
0F0FM" }{TEXT 216 1 " " }}}}{SECT 1 {PARA 214 "" 0 "" {TEXT 217 73 "Pr
oblem 1 (submitted by E. Cheb Terrab, Instituto de Fisica-UERJ, Brazil
)" }}{PARA 209 "" 0 "" {TEXT 219 100 "\nProblem: under which condition
 on w  does a first order differential equation y'=w(x,y) does admit "
 }{TEXT 214 1 " " }{TEXT 219 31 "symmetries with infinitesimals " }
{TEXT 219 27 ". Find this infinitesimals." }}{PARA 209 "" 0 "" }
{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 71 "S1 := [diff(eta(x,y),x)-
diff(xi(x,y),y)*w(x,y)^2-xi(x,y)*diff(w(x,y),x)" }{MPLTEXT 1 215 72 "
\n           -eta(x,y)*diff(w(x,y),y),diff(xi(x,y),y), diff(eta(x,y),x
)];" }}{PARA 213 "" 1 "" {XPPMATH 20 "6#>I#S1G6\"7%,*-I%diffGI*protect
edGF*6$-I$etaGF%6$I\"xGF%I\"yGF%F/\"\"\"*&-F)6$-I#xiGF%F.F0F1-I\"wGF%F
.\"\"#!\"\"*&F5F1-F)6$F7F/F1F:*&F,F1-F)6$F7F0F1F:F3F(" }{TEXT 216 1 " \+
" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 81 "R1 := differential_
ring(notation=diff, derivations=[x,y], ranking=[[xi, eta],w]):" }}}
{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 28 "Rosenfeld_Groebner(S1, R
1); " }{MPLTEXT 1 215 18 "\nrewrite_rules(%);" }}{PARA 213 "" 1 "" 
{XPPMATH 20 "6#7'I0characterisableG6\"F$F$F$F$" }{TEXT 216 1 " " }}
{PARA 213 "" 1 "" {XPPMATH 20 "6#7'7$/-I#xiG6\"6$I\"xGF(I\"yGF(\"\"!/-
I$etaGF(F)F,7(/-I%diffGI*protectedGF46$F.F*F,/-F36$F.F+,$**F.\"\"\",&*
&-F36$-I\"wGF(F)F*F;-F36$F@-I\"$GF46$F+\"\"#F;F;*&-F36%F@F*F+F;-F36$F@
F+F;!\"\"F;FKFMF>FMFM/F&,$*(F.F;FKF;F>FMFM/-F36%F@-FE6$F*FGF+*(,(*(FKF
;F>FG-F36%F@F*FDF;F;*(-F36$F@FTF;FKFGFIF;F;*(F>FGFIF;FBF;FMF;FK!\"#F>F
M0F>F,0FKF,7%/-F36$F&F+F,F-/F>F,7&F]oF1F`o/FKF,7%F1F%Fbo" }{TEXT 216 
1 " " }}}}{SECT 1 {PARA 214 "" 0 "" {TEXT 217 67 "Problem 2 (submitted
 by T. Kolokolnikov, Univ. of British Columbia)" }}{PARA 209 "" 0 "" 
{TEXT 214 56 "At the crossing of diffalg and symmetry matching pattern
" }}{PARA 209 "" 0 "" }{PARA 209 "" 0 "" {TEXT 219 241 "Problem: does \+
there exist an equation y'' = A + y' B + y'^2 C that is non-self-adjoi
nt (i.e. does not come from a Lagrangian) and that, at the same time h
as a symmetry of the form [0, eta(x,y)] and an  integrating factor of \+
the form mu(x,y)." }{TEXT 214 1 "\n" }}{SECT 0 {PARA 216 "" 0 "" 
{TEXT 220 35 "The  equations (S1) for the problem" }}{PARA 209 "" 0 ""
 }{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 58 "S2 := \{-3*B[x]*B[y]*C
[x]-4*A[y]*C[]*C[x,x]-B[]*B[x,y]*B[y]" }{MPLTEXT 1 215 53 "\n-B[]*B[x,
y]*C[x]+3*A[y]*C[]*B[x,y]-2*A[y,y]*A[y]*C[]" }{MPLTEXT 1 215 53 "\n-4*
A[]*C[y]*C[x,x]-A[]*C[x,y]*B[y]+2*A[]*C[x,y]*C[x]" }{MPLTEXT 1 215 68 
"\n+3*A[]*C[y]*B[x,y]+2*B[]*C[x,x]*B[y]+A[]*C[]*B[y]^2-A[x,y]*C[]*B[y]
" }{MPLTEXT 1 215 66 "\n+4*A[]*C[]*C[x]^2+2*A[x,y]*C[]*C[x]+B[]^2*C[x]
*B[y]-2*A[y]*C[x]^2" }{MPLTEXT 1 215 71 "\n+B[x]*B[y]^2+2*B[x]*C[x]^2+
A[x,y,y]*B[y]-2*A[x,y,y]*C[x]-B[x,x,y]*B[y]" }{MPLTEXT 1 215 79 "\n+2*
B[x,x,y]*C[x]+C[x,x,x]*B[y]-2*C[x,x,x]*C[x]-3*A[y,y]*B[x,y]+4*A[y,y]*C
[x,x]" }{MPLTEXT 1 215 78 "\n-5*B[x,y]*C[x,x]-A[y]*B[y]^2+3*A[y]*B[y]*
C[x]-A[x]*C[y]*B[y]+2*A[x]*C[y]*C[x]" }{MPLTEXT 1 215 66 "\n-2*A[y,y]*
A[]*C[y]+2*B[x,y]^2+3*C[x,x]^2+A[y,y]^2+B[]*A[y,y]*B[y]" }{MPLTEXT 1 
215 76 "\n-B[]^2*C[x]^2+A[y]^2*C[]^2+A[]^2*C[y]^2-B[]*A[y]*C[]*B[y]-B[
]*A[]*C[y]*B[y]" }{MPLTEXT 1 215 43 "\n+2*A[y]*C[]*A[]*C[y]-4*A[]*C[]*
B[y]*C[x], " }{MPLTEXT 1 215 69 "\n   B[]*C[x,y]*B[y]+2*A[y,y]*C[]*C[x
]-A[]*C[y,y]*B[y]+A[y]*C[]*B[y,y]" }{MPLTEXT 1 215 70 "\n-2*A[y]*C[y]*
B[y]-2*A[]*C[y]*C[x,y]-2*A[y]*C[]*C[x,y]+A[]*C[y]*B[y,y]" }{MPLTEXT 1 
215 68 "\n-B[]*C[x]*B[y,y]+2*A[]*C[y,y]*C[x]-A[y,y]*C[]*B[y]+4*A[y]*C[
y]*C[x]" }{MPLTEXT 1 215 75 "\n+A[y,y,y]*B[y]-2*A[y,y,y]*C[x]-B[x,y,y]
*B[y]+2*B[x,y,y]*C[x]+2*B[y]^2*C[x]" }{MPLTEXT 1 215 75 "\n-6*B[y]*C[x
]^2+C[x,x,y]*B[y]-2*C[x,x,y]*C[x]-A[y,y]*B[y,y]+2*A[y,y]*C[x,y]" }
{MPLTEXT 1 215 71 "\n+B[x,y]*B[y,y]-2*B[x,y]*C[x,y]-C[x,x]*B[y,y]+2*C[
x,x]*C[x,y]+4*C[x]^3," }{MPLTEXT 1 215 77 "\n   32*C[x,y]*C[x,x]*B[x,y
,y]+32*B[y,y]*C[x,x]^2*C[]+16*C[x]^2*C[]*B[x,x,y,y]" }{MPLTEXT 1 215 
76 "\n   +8*B[y]^2*B[y,y]*B[x,y]-32*C[x,y]^2*C[x,x]*B[]-32*C[x,y]*C[x]
*C[x,x,x,y]" }{MPLTEXT 1 215 73 "\n+16*B[y,y]*C[x,y]*B[x,x,y]+4*B[y]^3
*C[x,y]*B[]+32*B[y,y]*C[x,x]*C[x,x,y]" }{MPLTEXT 1 215 50 "\n+8*B[y,y]
*B[x,y]*B[x,y,y]-8*B[y]^2*C[]*C[x,x,x,y]" }{MPLTEXT 1 215 77 "\n-16*B[
y,y]*C[x,x]*B[x,y,y]-64*C[x,y]*C[x,x]*C[x,x,y]-40*B[y]^2*C[x]*C[x,x,y]
" }{MPLTEXT 1 215 73 "\n-32*C[x]^2*C[]*C[x,x,x,y]+16*B[y,y]*C[x]*C[x,x
,x,y]-16*C[x]^3*B[x,y]*C[]" }{MPLTEXT 1 215 66 "\n-8*B[y,y]^2*C[x]*A[y
]+4*B[y,y]^2*B[x,y]*B[]+16*B[y]*C[x,y]^2*A[y]" }{MPLTEXT 1 215 75 "\n+
32*C[x,y]*B[x,y]*C[x,x,y]+16*C[x,y]^2*B[x,y]*B[]-8*B[y,y]*C[x]*B[x,x,y
,y]" }{MPLTEXT 1 215 76 "\n-64*C[x]*C[x,x,y]*B[x,y,y]-16*C[x,y]*B[x,y]
*B[x,y,y]-64*C[x,y]*C[x,x]^2*C[]" }{MPLTEXT 1 215 73 "\n-8*B[y,y]^2*C[
x,x]*B[]-16*B[y,y]*B[x,y]*C[x,x,y]-32*C[x,y]*C[x]^2*C[x,x]" }{MPLTEXT 
1 215 71 "\n+16*B[y,y]*C[x]^2*C[x,x]-12*B[y]^2*B[y,y]*C[x,x]-32*C[x,y]
^2*C[x]*A[y]" }{MPLTEXT 1 215 66 "\n-16*B[y]^2*C[x,y]*B[x,y]+24*B[y]^2
*C[x,y]*C[x,x]-2*B[y]^4*C[]*B[]" }{MPLTEXT 1 215 78 "\n-32*C[x]^3*C[]^
2*A[y]+48*B[y,y]*C[x,y]^2*A[]+4*B[y]^4*C[x]-16*B[x,y,y]*C[x]^3" }
{MPLTEXT 1 215 79 "\n+64*C[x]*C[x,x,y]^2+16*C[x]*B[x,y,y]^2-4*B[y,y]^2
*B[x,x,y]+8*B[y,y]^2*C[x,x,x]" }{MPLTEXT 1 215 72 "\n-8*B[y]*B[x,y,y]^
2-6*B[y]^3*B[x,y,y]+12*B[y]^3*C[x,x,y]-32*C[x,y]^3*A[]" }{MPLTEXT 1 
215 73 "\n-16*B[y]^2*C[x]^3+32*C[x,y]^2*C[x,x,x]+4*B[y,y]^3*A[]-32*B[y
]*C[x,x,y]^2" }{MPLTEXT 1 215 63 "\n-16*C[x,y]^2*B[x,x,y]+16*B[y]*C[x]
^4+32*C[x,x,y]*C[x]^3-B[y]^5" }{MPLTEXT 1 215 74 "\n+32*B[y]*C[x,x,y]*
B[x,y,y]-24*B[y,y]^2*C[x,y]*A[]+16*B[y]*C[x,x,y]*C[x]^2" }{MPLTEXT 1 
215 66 "\n+2*B[y]^3*C[]*B[x,y]-2*B[y]^3*B[y,y]*B[]-8*B[y]*B[y,y]*C[x,x
,x,y]" }{MPLTEXT 1 215 77 "\n+16*B[y]*C[x,y]*C[x,x,x,y]+4*B[y]*B[y,y]*
B[x,x,y,y]-8*B[y]*C[x,y]*B[x,x,y,y]" }{MPLTEXT 1 215 66 "\n+4*B[y]^2*C
[]*B[x,x,y,y]+4*B[y]^3*C[]^2*A[y]+4*B[y]*B[y,y]^2*A[y]" }{MPLTEXT 1 
215 48 "\n-8*B[y]*B[x,y,y]*C[x]^2+20*B[y]^2*C[x]*B[x,y,y]" }{MPLTEXT 
1 215 75 "\n-32*B[y,y]*C[x,y]*C[x,x,x]+16*C[x,y]*C[x]*B[x,x,y,y]+8*B[y
,y]*B[x,y]^2*C[]" }{MPLTEXT 1 215 78 "\n-16*C[x,y]*B[x,y]^2*C[]-32*B[y
]*C[x,y]*C[x]*C[x,x]+16*B[y]*B[y,y]*C[x]*C[x,x]" }{MPLTEXT 1 215 79 "
\n+32*B[y]*C[x,y]*C[x]*B[x,y]-16*B[y]*B[y,y]*C[x]*B[x,y]+8*B[y]*B[y,y]
^2*C[]*A[]" }{MPLTEXT 1 215 78 "\n+32*B[y]*C[x,y]^2*C[]*A[]+4*B[y]^2*B
[y,y]*C[]^2*A[]-24*B[y]^2*C[]^2*C[x]*A[y]" }{MPLTEXT 1 215 78 "\n-16*B
[y,y]^2*C[x]*C[]*A[]-32*C[x,y]*C[]^2*C[x]^2*A[]-64*C[x,y]^2*C[x]*C[]*A
[]" }{MPLTEXT 1 215 79 "\n+16*B[y,y]*C[]^2*C[x]^2*A[]+8*B[y]^2*B[y,y]*
C[]*A[y]-8*B[y]^2*C[x,y]*C[]^2*A[]" }{MPLTEXT 1 215 79 "\n+16*B[y]*B[x
,y,y]*C[x,x]*C[]+16*B[y]*C[x]^3*C[]*B[]-8*B[y]*B[x,y,y]*C[]*B[x,y]" }
{MPLTEXT 1 215 79 "\n+8*B[y]*C[]*C[x,y]*B[x,x,y]-24*B[y]^2*C[x]^2*C[]*
B[]-4*B[y]^2*B[x,y,y]*C[]*B[]" }{MPLTEXT 1 215 78 "\n-12*B[y]^2*C[x]*C
[]*B[x,y]-16*B[y]^2*C[x]*C[x,y]*B[]+8*B[y]^2*C[x]*B[y,y]*B[]" }
{MPLTEXT 1 215 55 "\n+8*B[y]^2*C[x,x,y]*C[]*B[]-16*B[y]*C[]*C[x]*B[x,x
,y,y]" }{MPLTEXT 1 215 55 "\n-16*B[y]*C[]*C[x,y]*C[x,x,x]+48*B[y]*C[]^
2*C[x]^2*A[y]" }{MPLTEXT 1 215 55 "\n+8*B[y]*C[x,x,y]*B[y,y]*B[]+8*B[y
]*B[x,y,y]*C[x,y]*B[]" }{MPLTEXT 1 215 55 "\n-4*B[y]*B[x,y,y]*B[y,y]*B
[]+8*C[x]*B[x,y,y]*B[y,y]*B[]" }{MPLTEXT 1 215 57 "\n-32*C[x]*B[x,y,y]
*C[x,x]*C[]-16*C[x]*C[]*C[x,y]*B[x,x,y]" }{MPLTEXT 1 215 55 "\n-64*C[x
,y]*C[x]^2*C[]*A[y]+32*C[x]*C[]*C[x,y]*C[x,x,x]" }{MPLTEXT 1 215 56 "
\n+8*C[x]*C[]*B[y,y]*B[x,x,y]-16*C[x]*C[]*B[y,y]*C[x,x,x]" }{MPLTEXT 
1 215 60 "\n+64*B[y]*C[x,y]*C[x]*C[]*A[y]+16*B[y,y]*C[x,x]*C[x]*C[]*B[
]" }{MPLTEXT 1 215 57 "\n-8*B[y,y]*B[x,y]*C[x]*C[]*B[]+32*B[y,y]*C[x]*
C[x,y]*A[y]" }{MPLTEXT 1 215 57 "\n+64*C[x,y]*C[x,x]*C[]*B[x,y]-16*C[x
,y]*B[x,y]*B[y,y]*B[]" }{MPLTEXT 1 215 55 "\n+32*B[y,y]*C[x]^2*C[]*A[y
]-32*B[y,y]*B[x,y]*C[x,x]*C[]" }{MPLTEXT 1 215 56 "\n+32*C[x]^2*C[x,x,
y]*C[]*B[]+16*C[x]*B[x,y,y]*C[]*B[x,y]" }{MPLTEXT 1 215 57 "\n+32*C[x]
*C[x,x,y]*C[x,y]*B[]-16*C[x]*C[x,x,y]*B[y,y]*B[]" }{MPLTEXT 1 215 57 "
\n+64*C[x]*C[x,x,y]*C[x,x]*C[]-32*C[x]*C[x,x,y]*C[]*B[x,y]" }{MPLTEXT 
1 215 56 "\n-16*C[x]^2*B[x,y,y]*C[]*B[]-16*C[x]*B[x,y,y]*C[x,y]*B[]" }
{MPLTEXT 1 215 59 "\n+32*C[x,y]*C[x,x]*B[y,y]*B[]-32*B[y]*C[x]*C[x,x,y
]*C[]*B[]" }{MPLTEXT 1 215 61 "\n+16*B[y]*C[x]*B[x,y,y]*C[]*B[]+16*B[y
]*C[x,y]*C[x,x]*C[]*B[]" }{MPLTEXT 1 215 57 "\n-32*B[y]*B[y,y]*C[x]*C[
]*A[y]-16*B[y]*B[y,y]*C[x,y]*A[y]" }{MPLTEXT 1 215 57 "\n-32*B[y]*C[x,
x,y]*C[x,x]*C[]+32*B[y]*C[]*C[x]*C[x,x,x,y]" }{MPLTEXT 1 215 55 "\n-4*
B[y]*C[]*B[y,y]*B[x,x,y]+8*B[y]*C[]*B[y,y]*C[x,x,x]" }{MPLTEXT 1 215 
55 "\n+16*B[y]*C[x,x,y]*C[]*B[x,y]+16*B[y]*C[x]^2*C[x,y]*B[]" }
{MPLTEXT 1 215 52 "\n-8*B[y]*C[x]^2*B[y,y]*B[]+24*B[y]*C[]*C[x]^2*B[x,
y]" }{MPLTEXT 1 215 51 "\n-16*B[y]^2*C[x,y]*C[]*A[y]+12*B[y]^3*C[x]*C[
]*B[]-" }{MPLTEXT 1 215 58 "\n16*B[y]*C[x,x,y]*C[x,y]*B[]-32*B[y]*B[y,
y]*C[x,y]*C[]*A[]" }{MPLTEXT 1 215 61 "\n-16*B[y]*B[y,y]*C[]^2*C[x]*A[
]+32*B[y]*C[x,y]*C[]^2*C[x]*A[]" }{MPLTEXT 1 215 59 "\n-8*B[y]*B[y,y]*
C[x,x]*C[]*B[]+4*B[y]*B[y,y]*B[x,y]*C[]*B[]" }{MPLTEXT 1 215 60 "\n+64
*B[y,y]*C[x,y]*C[x]*C[]*A[]-8*B[y]*C[x,y]*B[x,y]*C[]*B[]" }{MPLTEXT 1 
215 62 "\n+16*C[x,y]*B[x,y]*C[x]*C[]*B[]-32*C[x,y]*C[x,x]*C[x]*C[]*B[]
," }{MPLTEXT 1 215 70 "\n   -2*C[x]*B[y,y,y]-2*B[y,y]^2-8*C[x,y]^2+B[y
]^2*C[y]-B[y,y]*B[y]*C[]" }{MPLTEXT 1 215 61 "\n+4*C[x]^2*C[y]+8*B[y,y
]*C[x,y]+B[y]*B[y,y,y]-2*B[y]*C[x,y,y]" }{MPLTEXT 1 215 70 "\n+4*C[x]*
C[x,y,y]-4*B[y]*C[x]*C[y]+2*B[y,y]*C[x]*C[]+2*C[x,y]*B[y]*C[]" }
{MPLTEXT 1 215 21 "\n-4*C[x,y]*C[x]*C[]\}:" }{MPLTEXT 1 215 1 "\n" }
{MPLTEXT 1 215 1 "\n" }}}}{SECT 0 {PARA 216 "" 0 "" {TEXT 220 36 "The \+
inequation  (I1) for the problem" }}{PARA 209 "" 0 "" }{EXCHG {PARA 
209 "> " 0 "" {MPLTEXT 1 215 21 "I2:= [B[y] - 2*C[x]];" }}{PARA 213 ""
 1 "" {XPPMATH 20 "6#>I#I2G6\"7#,&&I\"BGF%6#I\"yGF%\"\"\"&I\"CGF%6#I\"
xGF%!\"#" }{TEXT 216 1 " " }}}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 
215 62 "R2 := differential_ring(ranking=[[A,B,C]], derivations=[x,y]);
" }}{PARA 213 "" 1 "" {XPPMATH 20 "6#>I#R2G6\"I)PDE_ringGF%" }{TEXT 
216 1 " " }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 34 "  Rosenfeld
_Groebner(S2, I2, R2); " }}{PARA 213 "" 1 "" {XPPMATH 20 "6#7\"" }
{TEXT 216 1 " " }}}{PARA 209 "" 0 "" }{PARA 209 "" 0 "" {TEXT 214 73 "
The answer is NO. An inconsistency is detected in the system S1=0, I1<
>0." }}}{SECT 1 {PARA 214 "" 0 "" {TEXT 217 74 "Problem 3 (submitted b
y E. Cheb Terrab, Instituto de Fisica-UERJ, Brazil))" }}{PARA 209 "" 0
 "" {TEXT 219 117 "Origin of the problem: Mapping an (up to now) unsol
vable Abel ODE to a solvable-to the end 2nd order  non-linear 0DE." }}
{PARA 209 "" 0 "" }{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 309 "S3 :
= [-diff(mu(x,y),y)-A(x,y)*mu(x,y),3*mu(x,y)*F(x,y)*y^2-diff(mu(x,y),x
)+ diff(mu(x,y),y)*H(x,y)+diff(mu(x,y),y)*F(x,y)*y^3-B(x,y)*mu(x,y),mu
(x,y)*diff(H(x,y),x)+diff(mu(x,y),x)*F(x,y)*y^3+ diff(mu(x,y),x)*H(x,y
)+mu(x,y)*diff(F(x,y),x)*y^3,diff(F(x,y),y),  diff(H(x,y),y),  diff(B(
x,y),y), diff(A(x,y),x)];" }{MPLTEXT 1 215 1 "\n" }}{PARA 213 "" 1 "" 
{XPPMATH 20 "6#>I#S3G6\"7),&-I%diffGI*protectedGF*6$-I#muGF%6$I\"xGF%I
\"yGF%F0!\"\"*&-I\"AGF%F.\"\"\"F,F5F1,,*(F,F5-I\"FGF%F.F5F0\"\"#\"\"$-
F)6$F,F/F1*&F(F5-I\"HGF%F.F5F5*(F(F5F8F5F0F;F5*&-I\"BGF%F.F5F,F5F1,**&
F,F5-F)6$F?F/F5F5*(F<F5F8F5F0F;F5*&F<F5F?F5F5*(F,F5-F)6$F8F/F5F0F;F5-F
)6$F8F0-F)6$F?F0-F)6$FCF0-F)6$F3F/" }{TEXT 216 1 " " }}}{EXCHG {PARA 
209 "> " 0 "" {MPLTEXT 1 215 85 "R3 := differential_ring(notation=diff
, derivations=[x,y], ranking=[[mu,A, B], F, H]);" }}{PARA 213 "" 1 "" 
{XPPMATH 20 "6#>I#R3G6\"I)PDE_ringGF%" }{TEXT 216 1 " " }}}{EXCHG 
{PARA 209 "> " 0 "" {MPLTEXT 1 215 28 "Rosenfeld_Groebner(S3, R3); " }
{MPLTEXT 1 215 18 "\nrewrite_rules(%);" }{MPLTEXT 1 215 1 "\n" }}
{PARA 213 "" 1 "" {XPPMATH 20 "6#7&I0characterisableG6\"F$F$F$" }
{TEXT 216 1 " " }}{PARA 213 "" 1 "" {XPPMATH 20 "6#7&7'/-I%diffGI*prot
ectedGF(6$-I\"AG6\"6$I\"xGF,I\"yGF,F.\"\"!/-F'6$-I\"BGF,F-F/F0/-I#muGF
,F-F0/-F'6$-I\"FGF,F-F/F0/-F'6$-I\"HGF,F-F/F07,/-F'6$F7F.,$*(F7\"\"\"-
F'6$FAF.FIFA!\"\"FL/-F'6$F7F/,$**F7FI,(FJFI*(FAFIF<FIF/\"\"#\"\"$*&FAF
IF4FIFLFIFAFL,&*&F<FIF/FUFIFAFIFLFL/-F'6$F4F.*&,(*&FAFI-F'6$FA-I\"$GF(
6$F.FTFIFI*(FAFIFJFIF4FIFI*$FJFT!\"#FIFAF`oF1/F**(FRFIFAFLFWFL/-F'6$F<
F.*(F<FIFJFIFAFLF9F>0FAF00FWF07*/FE,$*(FdoFIF7FIF<FLFL/FN,$**F7FI,(*&F
<FTF/FTFU*&F4FIF<FIFLFdoFIFIF<F`oF/!\"$FL/FZ*&,(*(F4FIFdoFIF<FIFI*$Fdo
FTF`o*&-F'6$F<F[oFIF<FIFIFIF<F`oF1/F**(F`pFIF<F`oF/FcpF9/FAF00F<F07(/F
E,$*&F4FIF7FIFL/FN,$*&F*FIF7FIFLF%F1/F<F0F^q" }{TEXT 216 1 " " }}}}
{SECT 1 {PARA 214 "" 0 "" {TEXT 217 30 "Problem 4: the double pendulum
" }}{PARA 217 "" 0 "" {TEXT 221 187 "Problem: the system of equations \+
describing a double pendulum in cartesian coordinates is an index 3 pr
oblem. We wish to determine the hidden algebraic constraints determini
ng the motion." }}{PARA 209 "" 0 "" }{PARA 218 "" 0 "" {TEXT 222 46 "N
ote : we look only for the non-singular case." }}{PARA 209 "" 0 "" }
{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 169 "S4 := [ -x[t,t] +2*x[]*
l[] - 2*X[]*L[],-y[t,t] +2*y[]*l[] - 2*Y[]*L[] - 1,-(X[t,t]+x[t,t]) +2
*X[]*L[], -(Y[t,t]+y[t,t]) +2*Y[]*L[]-1, -1 + x[]^2+y[]^2, -1 +X[]^2+Y
[]^2];" }}{PARA 213 "" 1 "" {XPPMATH 20 "6#>I#S4G6\"7(,(&I\"xGF%6$I\"t
GF%F+!\"\"*&&F)F%\"\"\"&I\"lGF%F%F/\"\"#*&&I\"XGF%F%F/&I\"LGF%F%F/!\"#
,*&I\"yGF%F*F,*&&F;F%F/F0F/F2*&&I\"YGF%F%F/F6F/F8F,F/,(&F5F*F,F(F,F3F2
,*&F@F*F,F:F,F>F2F,F/,(*$F.F2F/*$F=F2F/F,F/,(F,F/*$F4F2F/*$F?F2F/" }
{TEXT 216 1 " " }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 73 "R4 :=
 differential_ring(ranking=[[l, L], [X, x, Y, y]], derivations=[t]):" 
}}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 33 " G4:=Rosenfeld_Groebn
er(S4, R4); " }{MPLTEXT 1 215 19 "\nrewrite_rules(G4);" }}{PARA 213 ""
 1 "" {XPPMATH 20 "6#>I#G4G6\"7$I0characterisableGF%F'" }{TEXT 216 1 "
 " }}{PARA 213 "" 1 "" {XPPMATH 20 "6#7$7,/I\"lG6\",$*&,XI\"yGF'\"\"#*
$F+\"\"&!\"#*$&F+6#I\"tGF'F,F/*,I\"xGF'\"\"\"F+F6I\"XGF'F6I\"YGF'\"\"$
F1F,\"\"%*,F5F6F+F6F7F6F8F6F1F,!\"%**F5F6F7F6F8F,&F8F2F,!\"\"**F5F6F+F
:F7F6F8F9F:**F5F6F+F:F7F6F8F6F<**F5F6F+F:F>F,F7F6F6**F5F6F+F,F7F6F8F9F
<**F5F6F+F,F7F6F8F6F:*,F5F6F+F,F7F6F8F,F>F,F6*&F+F.F8F:F<*(F8F,F+F,F1F
,\"\"'*(F8F6F+F6F>F,!\"$*(F8F6F+F9F>F,F:*(F8F9F+F6F>F,F6*(F8F9F+F9F>F,
F?*(F5F6F>F,F7F6F?*(F8F6F+F.F>F,F?*(F8F:F1F,F+F,F<*&F8F:F+F9FH*&F+F9F8
F,!\"'*&F+F6F8F:F/*&F+F.F8F,FH*&F+F,F1F,F/*&F8F:F1F,F,F6,<F6F6*&F8FHF+
F,F6*&F+F:F8F:FS*&F+F,F8F:\"\"(*&F8F,F+F,!\"**&F+F:F8F,Ffn*&F+FHF8F,F6
*$F+FHF?*$F+F:F?*$F+F,F6*$F8FHF?*$F8F:F?*$F8F,F6F?#F6F,/I\"LGF',$*&,hn
**F7F6F5F6F8F:F1F,F6**F7F6F5F6F8F:F+F9F6*,F7F6F5F6F8F6F+F9F>F,F,*,F7F6
F5F6F8F6F+F6F>F,F/*&F+F:F8F.F?*&F+F:F>F,F6*&F+FHF8F6F6*&F+F:F8F9F.*,F7
F6F5F6F8F,F+F,F1F,F6*$F>F,F?**F7F6F5F6F+F.F8F,F6**F7F6F5F6F+F9F8F,F?**
F7F6F5F6F+F,F1F,F?**F7F6F5F6F+F6F8F:F?*(F5F6F7F6F1F,F?*&F8F,F>F,F?*(F7
F6F5F6F+F.F?*(F8F.F+F6F1F,F?*(F7F6F5F6F+F6F6*(F8F6F+F6F1F,FJ*(F8F9F+F6
F1F,F:*(F+F,F8F,F>F,F9*(F8F9F+F9F1F,F?*(F+F:F8F,F>F,F/*(F8F6F+F9F1F,F6
*&F+F:F8F6F<*&F8F6F+F,F9*&F+F,F8F9F<*&F+FHF8F9F?*&F+F,F8F.F6F6FXF?Fao/
&F86$F3F3,$*&,jo*(F8FHF+F6F1F,F,*(F8F:F+F6F1F,FS*(F8F,F+F6F1F,FH*(F+F:
F8F6F>F,F,*(F+F,F8F6F>F,FJ*(F8F,F+F9F1F,F:**F7F6F5F6F+F.F8F6F/**F5F6F7
F6F8F6F1F,F,*&F8F9F>F,F,*&F+FHF8F:F/*&F+F9F1F,F/*&F8F6F>F,F,*,F7F6F5F6
F8F,F+F9F>F,FJ*,F7F6F5F6F+F,F8F6F1F,F/*,F7F6F5F6F8F9F+F,F1F,F,*,F7F6F5
F6F8F,F+F6F>F,F9**F7F6F5F6F+F9F8F9F/**F7F6F5F6F8F.F1F,F/*(F+FHF8F6F>F,
F?FinF,FjnF:*&F+F:F8FHF,**F7F6F5F6F+F6F>F,F?**F7F6F5F6F+F9F8F.F/**F7F6
F5F6F+F.F8F9F,**F5F6F7F6F8F6F+F6F/*&F+F6F1F,F/FgnFS*(F8F:F+F9F1F,F/*(F
+F,F8F9F>F,!\"&*(F+F:F8F9F>F,F9FenFHFZF<F[oF/F]oF,FYF/**F7F6F5F6F+F.F>
F,F6**F7F6F5F6F8F.F+F6F,**F7F6F5F6F+F9F8F6F:F6FXF?F?/&F+Fgq*&,dpF?F6F[
rF6F\\rF/F]rF9F^rF9F_rFJF`rF.FarFJFbrF6FcrF6FdrFJFerF/FfrF6FgrF?FhrFJF
irF9FjrF6F[sFJF\\sF?F]sF?FinFhnFjnF:F^sF6F_sF?F`sF?FasF9FbsF?FcsF/FgnF
HFdsFJFesF/FgsF6FenFfsFZFfnF[oF?F\\oF6F]oF6F^oF6F_oF6F`oF?FYF/FhsF6Fis
F6FjsF:F6FXF?/*$F7F,,&F`oF?F6F6/*$F5F,,&F6F6F]oF?0,4F\\oF?FinF6Fgn\"\"
)F]oF/FenFSF`oF?F_oF6F?F6F^oF6\"\"!0F7Fht0F5Fht0,&F?F6F]oF6Fht7*/F&F+/
Fco,$F8Fao/F7Fht/F5Fht/F`oF6/F]oF60F+Fht0F8Fht" }{TEXT 216 1 " " }}}}
{SECT 1 {PARA 214 "" 0 "" {TEXT 217 37 "Problem 5: coupled tanks (K. F
orsman)" }}{PARA 219 "" 0 "" {TEXT 223 265 "This is a non-linear contr
ol problem. We study a system as follow: a tap pours water   in a funn
el that directs the water in a bucket having a small hole at the botto
m. We wish to determine the equation that determines the height of wat
er in the bucket (the output " }{TEXT 223 53 ") in function of the flo
w of water poured (the input " }{TEXT 223 105 "). In other words, we w
ish to deduce the input-output representation from the state-space rep
resentation." }}{PARA 209 "" 0 "" }{EXCHG {PARA 209 "> " 0 "" 
{MPLTEXT 1 215 69 "S5 := [ z[]^2*z[t]-u[]+rz[], y[t]-rz[]+ry[], ry[]^2
-y[], rz[]^2-z[]];" }}{PARA 213 "" 1 "" {XPPMATH 20 "6#>I#S5G6\"7&,(*&
&I\"zGF%F%\"\"#&F*6#I\"tGF%\"\"\"F/&I\"uGF%F%!\"\"&I#rzGF%F%F/,(&I\"yG
F%F-F/F3F2&I#ryGF%F%F/,&*$F8F+F/&F7F%F2,&*$F3F+F/F)F2" }{TEXT 216 1 " \+
" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 34 "I5 := \{u[], z[], r
z[], y[], ry[]\};" }}{PARA 213 "" 1 "" {XPPMATH 20 "6#>I#I5G6\"<'&I\"y
GF%F%&I\"zGF%F%&I\"uGF%F%&I#rzGF%F%&I#ryGF%F%" }{TEXT 216 1 " " }}}
{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 66 "R5 := differential_ring(
derivations=[t], ranking=[[rz,z,ry],y,u]):" }}}{EXCHG {PARA 209 "> " 0
 "" {MPLTEXT 1 215 67 "R5b := differential_ring(derivations=[t], ranki
ng=[[rz,ry,z,y,u]]):" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 38 
"G5b := Rosenfeld_Groebner(S5,I5,R5b); " }{MPLTEXT 1 215 40 "\nG5 :=Ro
senfeld_Groebner(S5,I5,R5,G5b); " }}{PARA 213 "" 1 "" {XPPMATH 20 "6#>
I$G5bG6\"7#I0characterisableGF%" }{TEXT 216 1 " " }}{PARA 220 "" 1 "" 
}}{EXCHG {PARA 209 "" 0 "" }{PARA 209 "" 0 "" {TEXT 224 7 "Answer:" }}
}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 49 "collect(op(-1, equation
s(G5[1])), [y[t,t],y[t]]);" }}}}{SECT 1 {PARA 214 "" 0 "" {TEXT 217 
83 "Problem 6: chemical oxydation of polymers (submitted by  Khaled Ab
deljaoued, ONERA)" }}{PARA 221 "" 0 "" {TEXT 225 70 "Problem: the syst
em describes a chemical system of polymers oxydation." }}{PARA 209 "" 
0 "" }{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 84 "a1 := -diff(Y1(t),
t)+alpha*k7*Q(t)+ri*X(t) - (k2*Z(t)+k5*Y2(t))*Y1(t)- k4*(Y1(t))^2:" }
{MPLTEXT 1 215 74 "\na2 := -diff(Y2(t),t)+k2*Y1(t)*Z(t)-(k3*X(t)+k5*Y1
(t))*Y2(t) - k6*Y2(t)^2:" }{MPLTEXT 1 215 46 "\na3 := -diff(Z(t),t)+k6
*Y2(t)^2-k2*Y1(t)*Z(t):" }{MPLTEXT 1 215 39 "\na5 := diff(Y1(t),t)+dif
f(Y2(t),t):    " }{MPLTEXT 1 215 42 "\na4 := -diff(X(t),t)  -(ri+k3*Y2
(t))*X(t):" }{MPLTEXT 1 215 44 "\na6 := -diff(Q(t),t)+ k3*X(t)*Y2(t)-k
7*Q(t):" }{MPLTEXT 1 215 34 "\na7 := -diff(V(t),t)+beta*k7*Q(t):" }
{MPLTEXT 1 215 61 "\na8 := -ro* diff(M(t),t)+32 * diff(Z(t),t)-Mv * di
ff(V(t),t):" }{MPLTEXT 1 215 27 "\nS6 := \{seq(a||i, i=1..8)\};" }}
{PARA 213 "" 1 "" {XPPMATH 20 "6#>I#S6G6\"<*,*-I%diffGI*protectedGF*6$
-I#Y2GF%6#I\"tGF%F/!\"\"*(I#k2GF%\"\"\"-I#Y1GF%F.F3-I\"ZGF%F.F3F3*&,&*
&I#k3GF%F3-I\"XGF%F.F3F3*&I#k5GF%F3F4F3F3F3F,F3F0*&I#k6GF%F3F,\"\"#F0,
(*&I#roGF%F3-F)6$-I\"MGF%F.F/F3F0-F)6$F6F/\"#K*&I#MvGF%F3-F)6$-I\"VGF%
F.F/F3F0,(FJF0F@F3F1F0,(-F)6$-I\"QGF%F.F/F0*(F;F3F<F3F,F3F3*&I#k7GF%F3
FWF3F0,,-F)6$F4F/F0*(I&alphaGF%F3FenF3FWF3F3*&I#riGF%F3F<F3F3*&,&*&F2F
3F6F3F3*&F?F3F,F3F3F3F4F3F0*&I#k4GF%F3F4FBF0,&FOF0*(I%betaGF%F3FenF3FW
F3F3,&-F)6$F<F/F0*&,&F\\oF3*&F;F3F,F3F3F3F<F3F0,&FgnF3F(F3" }{TEXT 
216 1 " " }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 97 "K6 := field
_extension(transcendental_elements=[alpha, ri, ro, beta, Mv, k2, k3, k
4, k5, k6, k7]):" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 215 112 "R6
 := differential_ring(ranking=[[M,V],[Q,Y2,Y1],[Z,X]], derivations=[t]
, field_of_constants=K6, notation=diff):" }}}{EXCHG {PARA 209 "> " 0 "
" {MPLTEXT 1 215 32 "G6:= Rosenfeld_Groebner(S6,R6); " }}{PARA 220 "" 
1 "" {TEXT 226 33 "Warning,  computation interrupted" }}}{EXCHG {PARA 
209 "> " 0 "" }}}{PARA 222 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 
}{PAGENUMBERS 0 1 2 33 1 1 }