1 Introduction
2 Preliminaries
2.1 The Tensor Algebra
\(h\)
|
\(u\)
|
\(v\)
|
\(b_h\)
|
---|---|---|---|
\({\mathtt {1}}\)
| – | – |
\({\mathtt {1}}\)
|
\({\mathtt {2}}\)
| – | – |
\({\mathtt {2}}\)
|
\({\mathtt {12}}\)
|
\({\mathtt {1}}\)
|
\({\mathtt {2}}\)
|
\([{\mathtt {1}},{\mathtt {2}}]\)
|
\({\mathtt {112}}\)
|
\({\mathtt {1}}\)
|
\({\mathtt {12}}\)
|
\([{\mathtt {1}},[{\mathtt {1}},{\mathtt {2}}]]\)
|
\({\mathtt {122}}\)
|
\({\mathtt {12}}\)
|
\({\mathtt {2}}\)
|
\([[{\mathtt {1}},{\mathtt {2}}],{\mathtt {2}}]\)
|
\({\mathtt {1112}}\)
|
\({\mathtt {1}}\)
|
\({\mathtt {112}}\)
|
\([{\mathtt {1}},[{\mathtt {1}},[{\mathtt {1}},{\mathtt {2}}]]]\)
|
\({\mathtt {1122}}\)
|
\({\mathtt {1}}\)
|
\({\mathtt {122}}\)
|
\([{\mathtt {1}},[[{\mathtt {1}},{\mathtt {2}}],{\mathtt {2}}]]\)
|
\({\mathtt {1222}}\)
|
\({\mathtt {122}}\)
|
\({\mathtt {2}}\)
|
\([[[{\mathtt {1}},{\mathtt {2}}],{\mathtt {2}}],{\mathtt {2}}]\)
|
2.2 The Iterated-Integrals Signature
2.3 Invariants
2.4 Moving-Frame Method
2.5 Algebraic Groups and Invariants
-
There exists a non-empty, G-invariant, and Zariski-open subset \(U\subset X\), such that S intersects each orbit that is contained in U. In other words, we have that \(\overline{\varPhi (G\times S)} = X\), where closure is taken in the Zariski topology.
-
One has \(N = \{ n\in G\, |\, nS=S\}\).
3 Rigid-Motion Invariant Iterated-Integrals Signature in Low Dimensions
3.1 Planar Curves
3.2 Spatial Curves
\(\varepsilon \)
|
\(p_1\)
|
\(p_2^2\)
|
\(p_3\)
|
\(c_{12}^2+c_{13}^2+c_{23}^2\)
|
---|---|---|---|---|
\(0.001\)
|
\(40.478457\pm 0.031897\)
|
\(388.852598\pm 0.719423\)
|
\(410.1601\pm 1.451587\)
|
\(19.73920\pm 0.042489\)
|
\(0.002\)
|
\(40.478609\pm 0.063788\)
|
\(388.851188\pm 1.439533\)
|
\(410.1662\pm 2.904117\)
|
\(19.73923\pm 0.084997\)
|
\(0.003\)
|
\(40.478613\pm 0.095669\)
|
\(388.855217\pm 2.157932\)
|
\(410.1829\pm 4.361125\)
|
\(19.73970\pm 0.127661\)
|
\(0.005\)
|
\(40.478961\pm 0.159355\)
|
\(388.863724\pm 3.599937\)
|
\(410.2244\pm 7.260393\)
|
\(19.74067\pm 0.212676\)
|
\(0.005\)
|
\(40.478944\pm 0.159579\)
|
\(388.850491\pm 3.596824\)
|
\(410.2100\pm 7.250626\)
|
\(19.74000\pm 0.212337\)
|
\(0.006\)
|
\(40.479522\pm 0.191205\)
|
\(388.869176\pm 4.322178\)
|
\(410.2567\pm 8.712330\)
|
\(19.74127\pm 0.255310\)
|
\(0.007\)
|
\(40.479154\pm 0.222931\)
|
\(388.857348\pm 5.038757\)
|
\(410.2875\pm 10.16106\)
|
\(19.74183\pm 0.297526\)
|
\(0.008\)
|
\(40.479987\pm 0.255116\)
|
\(388.875886\pm 5.761005\)
|
\(410.3340\pm 11.62311\)
|
\(19.74294\pm 0.340253\)
|
\(0.009\)
|
\(40.479755\pm 0.286871\)
|
\(388.870847\pm 6.484605\)
|
\(410.3469\pm 13.07437\)
|
\(19.74314\pm 0.382924\)
|
\(0.01\)
|
\(40.480158\pm 0.318563\)
|
\(388.868822\pm 7.194929\)
|
\(410.3979\pm 14.50687\)
|
\(19.74404\pm 0.424583\)
|
\(0.1\)
|
\(40.666276\pm 3.192057\)
|
\(392.299953\pm 72.90078\)
|
\(436.6014\pm 151.5948\)
|
\(20.31963\pm 4.325157\)
|
\(0.2\)
|
\(41.230255\pm 6.405785\)
|
\(402.754860\pm 151.8107\)
|
\(518.9669\pm 343.5288\)
|
\(22.07039\pm 9.104890\)
|
\(0.3\)
|
\(42.179230\pm 9.658141\)
|
\(420.782060\pm 241.9202\)
|
\(670.0478\pm 622.4717\)
|
\(25.11996\pm 14.80631\)
|
\(0.4\)
|
\(43.487670\pm 13.000592\)
|
\(447.571714\pm 349.8711\)
|
\(908.8922\pm 1046.932\)
|
\(29.61702\pm 21.91037\)
|
\(0.5\)
|
\(45.180001\pm 16.397537\)
|
\(486.859950\pm 486.2594\)
|
\(1266.963\pm 1699.083\)
|
\(35.85023\pm 30.92696\)
|
\(0.6\)
|
\(47.250322\pm 19.924696\)
|
\(539.227263\pm 659.2860\)
|
\(1779.548\pm 2665.387\)
|
\(44.00729\pm 42.37409\)
|
\(0.7\)
|
\(49.737706\pm 23.587601\)
|
\(611.273555\pm 876.6830\)
|
\(2519.318\pm 4123.652\)
|
\(54.69984\pm 57.03797\)
|
\(0.8\)
|
\(52.540981\pm 27.312587\)
|
\(709.388988\pm 1163.299\)
|
\(3550.598\pm 6231.497\)
|
\(68.39113\pm 75.64911\)
|
\(0.9\)
|
\(55.768387\pm 31.331033\)
|
\(833.369659\pm 1526.606\)
|
\(4961.398\pm 9284.847\)
|
\(85.19148\pm 98.74314\)
|
\(1\)
|
\(59.282290\pm 35.392961\)
|
\(1001.783641\pm 1989.893\)
|
\(6865.127\pm 1341.885\)
|
\(106.2537\pm 126.9262\)
|
4 Orthogonal Invariants on \({\mathfrak {g}}_{\le 2}(\!({\mathbb {R}}^{d})\!)\)
-
\(f_1 = c_1^2+\cdots +c_d^2\),
-
\(f_i|_{{L_{d}^{(i-1)}}} = c_{1(d-i+2)}^2+\cdots +c_{(d-i+1)(d-i+2)}^2\) for \(2\le i <d\).
-
\(f_d|_{{L_{d}^{(d-1)}}} = c_{12}^2\).