Algebraic and Differential Invariants Evelyne Hubert INRIA, M´ editerran´ ee www.inria.fr/members/Evelyne.Hubert Budapest - FoCM 2011 Joint work and shared thoughts with: Elizabeth Mansfield, Irina Kogan, Gloria Mari-Beffa Peter Olver, Michael Singer,George Labahn, Peter van der Kamp, Mark Hickman. Contents 1 Scaling symmetries 2 2 Rational and algebraic invariants 4 3 Differential Invariants 8 4 Generalized Differential Algebra 12 References a M. Fels and P. J. Olver. Moving coframes: II. Regularization and theoretical foundations. Acta Applicandae Mathematicae, 55(2) (1999). a P. J. Olver. Generating differential invariants. Journal of Mathematical Analysis and Applications 333 (2007). a E. Mansfield,P. H. van der Kamp. Evolution of curvature invariants and lifting integrability J. Geom. Phys. 56:8 (2006). a E. Mansfield. Algorithms for symmetric differential systems. Foundations of Computational Mathematics, 1:4 (2001). a E. Mansfield. A Practical Guide to the Invariant Calculus. Cambridge Monographs on Applied and Computational Mathematics 26. Cambridge University Press (2010). a E. Hubert and I. A. Kogan. Rational invariants of a group action. Construction and rewriting. J. of Symbolic Computation, 42:1-2 (2007). a E. Hubert and I. A. Kogan. Smooth and algebraic invariants of a group action. Local and global constructions. Foundations of Computational Mathematics, 7:4 (2007). a E. Hubert. Differential invariants of a Lie group action: syzygies on a generating set. J. of Symbolic Computation , 44:3 (2009). a E. Hubert. Differential algebra for derivations with nontrivial commutation rules. J. of Pure and Applied Algebra, 200:1-2 (2005). a E. Hubert and P. J. Olver. Differential invariants of conformal and projective surfaces. Symmetry Integrability and Geometry: Methods and Applications, 3 (2007), Art. 097. 4 E. Hubert. Generation properties of Maurer-Cartan invariants. (Preliminary version) http://hal.inria.fr/inria- 00194528/en. ‘ I. Anderson DifferentialGeometry . Maple 11-15. ‘ E. Hubert. diffalg: extension to non commuting derivations. INRIA, 2005. www.inria.fr/cafe/Evelyne.Hubert/diffalg. O E. Hubert. aida - Algebraic Invariants and their Differential Algebra. INRIA, 2007. www.inria.fr/cafe/Evelyne.Hubert/aida. 1
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Algebraic and Di erential Invariants3 Di erential Invariants 8 4 Generalized Di erential Algebra 12 References a M. Fels and P. J. Olver. Moving coframes: II. Regularization and theoretical
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Algebraic and Differential Invariants
Evelyne HubertINRIA, Mediterranee
www.inria.fr/members/Evelyne.Hubert
Budapest - FoCM 2011
Joint work and shared thoughts with:Elizabeth Mansfield, Irina Kogan, Gloria Mari-BeffaPeter Olver, Michael Singer,George Labahn,Peter van der Kamp, Mark Hickman.
Contents
1 Scaling symmetries 2
2 Rational and algebraic invariants 4
3 Differential Invariants 8
4 Generalized Differential Algebra 12
Referencesa
M. Fels and P. J. Olver. Moving coframes: II. Regularization and theoretical foundations. Acta Applicandae Mathematicae,55(2) (1999).
aP. J. Olver. Generating differential invariants. Journal of Mathematical Analysis and Applications 333 (2007).
aE. Mansfield,P. H. van der Kamp. Evolution of curvature invariants and lifting integrability J. Geom. Phys. 56:8 (2006).
aE. Mansfield. Algorithms for symmetric differential systems. Foundations of Computational Mathematics, 1:4 (2001).
aE. Mansfield. A Practical Guide to the Invariant Calculus. Cambridge Monographs on Applied and ComputationalMathematics 26. Cambridge University Press (2010).
aE. Hubert and I. A. Kogan. Rational invariants of a group action. Construction and rewriting. J. of Symbolic Computation,42:1-2 (2007).
aE. Hubert and I. A. Kogan. Smooth and algebraic invariants of a group action. Local and global constructions. Foundationsof Computational Mathematics, 7:4 (2007).
aE. Hubert. Differential invariants of a Lie group action: syzygies on a generating set. J. of Symbolic Computation , 44:3(2009).
aE. Hubert. Differential algebra for derivations with nontrivial commutation rules. J. of Pure and Applied Algebra, 200:1-2(2005).
aE. Hubert and P. J. Olver. Differential invariants of conformal and projective surfaces. Symmetry Integrability and Geometry:Methods and Applications, 3 (2007), Art. 097.
4 E. Hubert. Generation properties of Maurer-Cartan invariants. (Preliminary version) http://hal.inria.fr/inria-00194528/en.
`I. Anderson DifferentialGeometry . Maple 11-15.
`E. Hubert. diffalg: extension to non commuting derivations. INRIA, 2005. www.inria.fr/cafe/Evelyne.Hubert/diffalg.
O E. Hubert. aida - Algebraic Invariants and their Differential Algebra. INRIA, 2007. www.inria.fr/cafe/Evelyne.Hubert/aida.
1
1 Scaling symmetries
Dimensional analysisPrey-predator model [Murray, Mathematical Biology (2002)]{
n =((
1− nk1
)r − k2
pn+e
)n,
p = s(1−h pn
)p.
r, s, e, h, k1, k2 parameters.{n =
(1− n
k − h pn+1
)n,
p = s(1− p
n
)p.
s, h, k parameters
t → λ−1 t,n → µn,p → µν−1 p,
r → λ r,s → λ s,e → µ e,
h → ν h,k1 → µk1,k2 → λν k2
t = r t, n =n
e, p =
h p
e, s =
s
r, h =
k2
rh, k =
k1
e.
Symmetry reduction [Mansfield 01]
• Determine the scaling symmetry
• Compute a generating set of monomial invariants
• Rewrite the system in terms of those
It’s all linear algebra in the case of scalings!
Recipe for scaling reduction [H., Labahn 2012]
• Fill the matrix A determining the scaling
t → λ−1 t,n → µn,p → µν−1 p,
r → λ r,s → λ s,e → µ e,
h → ν h,k1 → µk1,k2 → λν k2
s r e h k1 k2 n p tν 0 0 0 1 0 1 0 −1 0µ 0 0 1 0 1 0 1 1 0λ 1 1 0 0 0 1 0 0 −1
Maurer-Cartan inv. & Serret-Frenet [Mansfield & van der Kamp 06]
d
ds
tnbx
=
0 κ(s) 0 0
−κ(s) 0 τ(s) 00 −τ(s) 0 01 0 0 0
tnbx
E3 =
{(R 0at 1
)∣∣∣∣ R ∈ SO(3), a ∈ R3
}
κ
0 1 0 01 0 0 00 0 0 00 0 0 0
+ τ
0 0 0 00 0 −1 00 1 0 00 0 0 0
+ 1
0 0 0 00 0 0 00 0 0 01 0 0 0
+ 0
0 0 0 00 0 0 00 0 0 00 1 0 0
+ 0. . . .
K =(ιx2ss −
ιx3sss
ιx2ss
1 0 0)
=(κ τ 1 0 0
)
Syzygies on M-C Invariants [H. 2012]
P (Kia) = 0
Di(Kjc)−Dj(Kic) =∑
1≤a<b≤r
Cabc(KiaKjb −KjaKib) +
m∑k=1
ΛijkKkc = 0
where
[vi, vj ] =
r∑k=1
Cijkvk[Di,Dj
]=
m∑k=1
Λijk Dk.
11
4 Generalized Differential Algebra
Differential Polynomial Rings
F = Q(x, y)
∂1 =∂
∂x, ∂2 =
∂
∂y
Y = {φ, ψ}
F Jφ, ψK = F[φ, φx, φy, . . . , ψ . . .]
φxxy ; φx2y ; φ(2,1)
∂
∂x(φxxy) = φxxxy ; ∂1
(φ(2,1)
)= φ(3,1)
∂
∂x
∂
∂y=
∂
∂y
∂
∂x
F a field
∂ = {∂1, . . . , ∂m} derivations on F
Y = {y1, . . . , yn}
F[ yα | α ∈ Nm, y ∈ Y ] = F JYK
∂i (yα) = yα+εi
εi = (0, . . . , 1ith
, . . . , 0)
∂i∂j = ∂j∂i
Derivations with nontrivial commutations
Y = {y1, . . . , yn}
D = {D1, . . . ,Dm}
DiDj −Dj Di =
m∑l=1
ΛijlDl Λijl ∈ K JYK
K JYK?
Mononotone derivatives: yα = Dα11 . . .Dαmm y
Differential polynomial ring K JYK with non commuting derivations [H. 05]
Y = {y1, . . . , yn}
D = {D1, . . . ,Dm}
K[yα | α ∈ Nm, y ∈ Y]
Di (yα) =
yα+εiifα1 = . . . = αi−1 = 0
DjDi
(yα−εj
)+
m∑l=1
cijl Dl
(yα−εj
)where j < i is s.t. αj > 0
while α1 = . . . = αj−1 = 0
If the cijl satisfy
- cijl = −cjil
- Dk(cijl) + Di(cjkl) + Dj(ckil) =
m∑µ=1
cijµcµkl + cjkµcµil + ckiµcµjl
& there exists an admissible ranking ≺
- |α| < |β| ⇒ yα ≺ yβ ,
- yα ≺ zβ ⇒ yα+γ ≺ zβ+γ ,
-∑l∈Nm
cijlDl(yα) ≺ yα+εi+εj
then DiDj(p)−DjDi(p) =
m∑l=1
cijlDl(p)
∀p ∈ K[ yα |α ∈ Nm ] = K JYK
12
Results obtained with the software developed
[aida,diffalg]
Conformal & Projective Surfaces
A differential invariants for surfaces in R3 under the conformal/projective group canbe effectively written in terms of a one (two) differential invariants of order 3/4, andtheir invariant (monotone) derivatives. [H. & Olver 07]
Generalized Orthogonal Group O(3− l, l) nR3 acting on R3 × R:
Differential invariants of all orders can be effectively written in terms of 3 secondorder differential invariants. [H. 09]