| Full text | |
| Author(s): |
Grebenev, Vladimir
;
Grishkov, Alexandre
Total Authors: 2
|
| Document type: | Journal article |
| Source: | SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES; v. 16, n. 2, p. 17-pg., 2022-03-24. |
| Abstract | |
Random curves (produced by the Schramm-Loewner evolution SLE kappa) of the fractal dimension d(kappa) = 1 + kappa/8, kappa < 8 given on a surface D subset of R-3 and the conformal group (CG) that acts on D are considered. We study the action integral L[X(t)], X(t) is an element of SLE kappa (the fractal variation of length of a random curve (Kennedy in J Stat Phys 128(6):1263-1277, 2006)) together with the conformal group extension CGE of CG which invariant transforms SLE kappa and L[X(t)]. We calculate the second-order universal differential invariant J(2) (or the multiscale representation of invariants) of the GCE and show that L[X(t)] generates all second-order differential invariants of CGE by the operators of invariant differentiation. The differential invariants look like invariant quantities of different scales wherein L[X(t)] plays a role of "the fractal length scale". The method of calculations of differential invariants is a kind of modern multiscale analysis (Olver and Pohjanpelto in Adv Math 222(5):1746-1792, 2009) based on the theory of symmetry group. This investigation is also motivated by Cartan's point of view (in: Cartan, La Theoric des Groupes Finis et Continus et la Geometric Differentielle traittee par le Methode du Repere Mobile, Gauthier-Villars, Paris, 1937) that the local geometry properties are entirely governed by differential invariants of the group admitted. (AU) | |
| FAPESP's process: | 21/09845-0 - Conformal invariance of the probability density function for passive scalar fields (including arbitrary virus) in 2D turbulence: the proof of Polyakov conjecture |
| Grantee: | Alexandre Grichkov |
| Support Opportunities: | Research Grants - Visiting Researcher Grant - International |