About Gordan’s Algorithm for Binary Forms

Olive, Marc

doi:10.1007/s10208-016-9324-x

Cited by 32 publications

(53 citation statements)

References 56 publications

(81 reference statements)

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“…Both of them rely on the resolution of a Diophantine equation such as (6). It is the second version that has been used to produce the tables of section 7 and that we shortly outline next (a more detailed treatment of these algorithms is provided in [64]).…”

Section: 2mentioning

confidence: 99%

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A Minimal Integrity Basis for the Elasticity Tensor

Olive

Kolev

Auffray

2017

Arch Rational Mech Anal

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We definitively solve the old problem of finding a minimal integrity basis of polynomial invariants of the fourth-order elasticity tensor C. Decomposing C into its SO(3)-irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are second-order harmonic tensors, and D is a fourth-order harmonic tensor. Combining theorems of classical invariant theory and formal computations, a minimal integrity basis of 297 polynomial invariants for the elasticity tensor is obtained for the first time.

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Section: 2mentioning

confidence: 99%

“…Indeed, a degree 11 joint invariant in Inv(S 8 ⊕ S 4 ) (which needs to be counted twice for our purpose) was superfluous. This mistake has been corrected in [64].…”

Section: Explicit Computationsmentioning

confidence: 99%

A Minimal Integrity Basis for the Elasticity Tensor

Olive

Kolev

Auffray

2017

Arch Rational Mech Anal

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“…In continuous and discrete mathematics, group actions are widespread. They also arise in various fields of applied science, especially in classical mechanics [2,7,29]. Invariants may be used e.g.…”

Section: Introductionmentioning

confidence: 99%

On differential invariants of parabolic surfaces

Chen

Merker

2021

Dissertationes Math.

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We study equivalence classes of local graphed analytic surfaces {u = F (x, y)} in R 3 under the action of the special affine group SA3(R), assuming that their Hessian matrices Fxx Fxy Fyx Fyy have rank 1 at every point (x, y). Such parabolic surfaces have identically zero Gaussian curvature, hence are developable. After the treatment of the rank 2 case by Olver [Differential Geom. Appl. 27 (2007)], we determine the structures of various algebras of differential invariants in all possible branches, and we employ the power series method in order to compute all incoming relative or absolute differential invariants. Starting with our rank 1 root hypothesis Fxx = 0 ≡ FxxFyy − F 2 xy , we quickly encounter the first relative differential invariant S := FxxFxxy − FxyFxxx F 2 xx. A surface {u = F (x, y)} is SA3(R)-equivalent to a curve {u = F (x)} times Ry (a cylinder) if and only if S ≡ 0. This branch S ≡ 0 amounts to the (well-known) A2(R)-equivalence problem for planar curves. In the more interesting branch S = 0, we find the first absolute differential invariant W := F 2 xx Fxxxy − Fxx Fxy Fxxxx + 2 Fxy F 2 xxx − 2 Fxx Fxxx Fxxy (Fxx) 2 Fxx Fxxy − Fxy Fxxx 2/3. When W ≡ 0, the surface is conical, and we establish that two differential invariants, X of order 5 and Y of order 7, generate the full algebra of differential invariants. In the thickest branch W = 0 (= S), we find another differential invariant M of order 5 whose numerator has 57 differential monomials, and we show that M, W are generators. Mainly, we set up the celebrated Fels-Olver recurrence formulas for differential invariants under the assumptions that one or two (relative) differential invariants vanish identically. These degenerate cases, apparently, have not been studied before in the literature, and will be developed further.

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“…In 2014, an integrity basis with thirteen isotropic invariants of a symmetric third order three-dimensional tensor was presented by Olive and Auffray [15]. It was noted that the Olive-Auffray integrity basis is actually a minimal integrity basis [14]. In 2017, Olive, Kolev and Auffray [16] gave a minimal integrity basis of the elasticity tensors, with 297 invariants.…”

Section: Introductionmentioning

confidence: 99%

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

Zhang

Ming

Chen

2019

Appl. Math. Mech.-Engl. Ed.

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Representation theorems for both isotropic and anisotropic functions are of prime importance in both theoretical and applied mechanics. In this article, we discuss about two-dimensional Eshelby tensors (denoted as M (2) ), which have wide applications in many fields of mechanics. Based upon the complex variable method, we obtain an integrity basis of ten isotropic invariants of M (2) . Since an integrity basis is always a polynomial functional basis, we further confirm that this integrity basis is also an irreducible polynomial functional basis of M (2) .(4.14)H, L, K, θ 1 , θ 2 and θ 3 are independent to each other. Moreover, we introduce six scalarvalued functions of H, L, K, θ 1 , θ 2 and θ 3 as below:(4.15) By some calculations, we have Thus, J 11 , . . . , J 16 are polynomials in J 1 , . . . , J 7 . In view of this, they are also polynomial invariants of M (2) and we only need to testify that any polynomial invariant of M (2) is polynomial in J 1 , . . . , J 16 .Recalling that each non-zero monomial

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About Gordan’s Algorithm for Binary Forms

Cited by 32 publications

References 56 publications

A Minimal Integrity Basis for the Elasticity Tensor

A Minimal Integrity Basis for the Elasticity Tensor

On differential invariants of parabolic surfaces

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

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