The Experts below are selected from a list of 9 Experts worldwide ranked by ideXlab platform
Sergei Merenkov - One of the best experts on this subject based on the ideXlab platform.
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quasisymmetric rigidity of square sierpinski carpets
Annals of Mathematics, 2013Co-Authors: Mario Bonk, Sergei MerenkovAbstract:We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpinski carpet S3 is a Euclidean Isometry. For carpets in a more general family, the standard 1/p-Sierpinski carpets Sp, p ≥ 3 odd, we show that the groups of quasisymmetric self-maps are finite dihedral. We also establish that Sp and Sq are quasisymmetrically equivalent only if p = q. The main tool in the proof for these facts is a new invariant—a certain discrete modulus of a path family—that is preserved under quasisymmetric
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quasisymmetric rigidity of square sierpinski carpets
arXiv: Complex Variables, 2011Co-Authors: Mario Bonk, Sergei MerenkovAbstract:We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpi\'nski carpet $S_3$ is a Euclidean Isometry. For carpets in a more general family, the standard $1/p$-Sierpi\'nski carpets $S_p$, $p\ge 3$ odd, we show that the groups of quasisymmetric self-maps are finite dihedral. We also establish that $S_p$ and $S_q$ are quasisymmetrically equivalent only if $p=q$. The main tool in the proof for these facts is a new invariant---a certain discrete modulus of a path family---that is preserved under quasisymmetric maps of carpets.
Mario Bonk - One of the best experts on this subject based on the ideXlab platform.
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quasisymmetric rigidity of square sierpinski carpets
Annals of Mathematics, 2013Co-Authors: Mario Bonk, Sergei MerenkovAbstract:We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpinski carpet S3 is a Euclidean Isometry. For carpets in a more general family, the standard 1/p-Sierpinski carpets Sp, p ≥ 3 odd, we show that the groups of quasisymmetric self-maps are finite dihedral. We also establish that Sp and Sq are quasisymmetrically equivalent only if p = q. The main tool in the proof for these facts is a new invariant—a certain discrete modulus of a path family—that is preserved under quasisymmetric
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quasisymmetric rigidity of square sierpinski carpets
arXiv: Complex Variables, 2011Co-Authors: Mario Bonk, Sergei MerenkovAbstract:We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpi\'nski carpet $S_3$ is a Euclidean Isometry. For carpets in a more general family, the standard $1/p$-Sierpi\'nski carpets $S_p$, $p\ge 3$ odd, we show that the groups of quasisymmetric self-maps are finite dihedral. We also establish that $S_p$ and $S_q$ are quasisymmetrically equivalent only if $p=q$. The main tool in the proof for these facts is a new invariant---a certain discrete modulus of a path family---that is preserved under quasisymmetric maps of carpets.
Peter Donelan - One of the best experts on this subject based on the ideXlab platform.
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invariants of the k fold adjoint action of the Euclidean Isometry group
Journal of Geometry, 2016Co-Authors: Mohammed Daher, Peter DonelanAbstract:A non-zero element of the Lie algebra \({\mathfrak{se}(3)}\) of the special Euclidean spatial Isometry group SE(3) is known as a twist and the corresponding element of the projective Lie algebra is termed a screw. Either can be used to describe a one-degree-of-freedom joint between rigid components in a mechanical device or robot manipulator. This leads to a practical interest in multiple twists or screws, describing the overall instantaneous motion of such a device. In this paper, invariants of multiple twists under the action induced by the adjoint action of the group are determined. The ring of the polynomial invariants for the adjoint action of SE(3) acting on a single twist is well known to be finitely generated by the Klein and Killing forms, while a theorem of Panyushev (Publ. Res. Inst. Math. Sci. 4:1199–1257, 2007) gives finite generation for the real invariants of the induced action on two twists. However we are not aware of a corresponding theorem for k twists, where \({k\geq3}\). Following Study, Geometrie der Dynamen, (1903), we use the principle of transference to determine fundamental algebraic invariants and their syzygies. We prove that the ring of invariants for triple twists is rationally finitely generated by 13 of these invariants.
Mohammed Daher - One of the best experts on this subject based on the ideXlab platform.
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invariants of the k fold adjoint action of the Euclidean Isometry group
Journal of Geometry, 2016Co-Authors: Mohammed Daher, Peter DonelanAbstract:A non-zero element of the Lie algebra \({\mathfrak{se}(3)}\) of the special Euclidean spatial Isometry group SE(3) is known as a twist and the corresponding element of the projective Lie algebra is termed a screw. Either can be used to describe a one-degree-of-freedom joint between rigid components in a mechanical device or robot manipulator. This leads to a practical interest in multiple twists or screws, describing the overall instantaneous motion of such a device. In this paper, invariants of multiple twists under the action induced by the adjoint action of the group are determined. The ring of the polynomial invariants for the adjoint action of SE(3) acting on a single twist is well known to be finitely generated by the Klein and Killing forms, while a theorem of Panyushev (Publ. Res. Inst. Math. Sci. 4:1199–1257, 2007) gives finite generation for the real invariants of the induced action on two twists. However we are not aware of a corresponding theorem for k twists, where \({k\geq3}\). Following Study, Geometrie der Dynamen, (1903), we use the principle of transference to determine fundamental algebraic invariants and their syzygies. We prove that the ring of invariants for triple twists is rationally finitely generated by 13 of these invariants.