de Lucas, Javier Gràcia, Xavier Rivas, Xavier Román-Roy, Narciso Vilariño, Silvia
Published in
Journal of Physics A: Mathematical and Theoretical

A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra. In this work, multisymplectic forms are applied to the study of the reduction of Lie...

Mahony, Robert van Goor, Pieter Hamel, Tarek
Published in
Annual Review of Control, Robotics, and Autonomous Systems

Equivariance is a common and natural property of many nonlinear control systems, especially those associated with models of mechatronic and navigation systems. Such systems admit a symmetry, associated with the equivariance, that provides structure enabling the design of robust and high-performance observers. A key insight is to pose the observer s...

Hayashi, Masahito Shigemoto, Kazuyasu Tsukioka, Takuya
Published in
Journal of Physics Communications

By using the first and second flows of the Kowalevski top, we can recreate the Kowalevski top into two−flows Kowalevski top, which has two−time variables. Then, we demonstrate that equations of the two−flows Kowalevski top become those of the full genus two Jacobi inversion problem. In addition to the Lax pair for the first flow, we construct a Lax...

Thanwerdas, Yann Pennec, Xavier

In contrast to SPD matrices, few tools exist to perform Riemannian statistics on the open elliptope of full-rank correlation matrices. The quotient-affine metric was recently built as the quotient of the affine-invariant metric by the congruence action of positive diagonal matrices. The space of SPD matrices had always been thought of as a Riemanni...

bruno, t. m., peloso m. vallarino, m.

In this paper we estimate the Sobolev embedding constant on general noncompact Lie groups, for sub-Riemannian inhomogeneous Sobolev spaces endowed with a left invariant measure. The bound that we obtain, up to a constant depending only on the group and its sub-Riemannian structure, reduces to the best known bound for the classical inhomogeneous Sob...

Bourdon, Marc RÉMY, Bertrand

We obtain non-vanishing of group L^p-cohomology of Lie groups for p large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromov's question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings a...

Klepikov, P. N. Rodionov, E. D. Khromova, O. P.
Published in
Russian Mathematics

In this paper, we investigate invariant Ricci solitons, an important subclass in the class of homogeneous Ricci solitons. A classification of invariant Ricci solitons on three-dimensional Lie groups with left-invariant Riemannian metric and semi-symmetric connection different from the Levi-Civita connection is obtained. It is proved that in this ca...

Molla, Arman Nicolay, Samuel

peer reviewed / We study a generalization of the Fréchet functional equation in the setting of Lie groups. We provide theorems that give a description of the solutions and their regularity. These properties are applied to obtain new results about the Fréchet functional equations generalized to homogeneous spaces. In the end, we present a few exampl...

Bourdon, Marc Remy, Bertrand Emy, Bertrand

We show that the continuous L p-cohomology of locally compact second countable groups is a quasi-isometric invariant. As an application, we prove partial results supporting a positive answer to a question asked by M. Gromov, suggesting a classical behaviour of continuous L p-cohomology of simple real Lie groups. In addition to quasi-isometric invar...

Brossard, Martin Barrau, Axel Chauchat, Paul Bonnabel, Silvère

The recently introduced matrix group SE2(3) provides a 5×5 matrix representation for the orientation, velocity and position of an object in the 3-D space, a triplet we call "extended pose". In this paper we build on this group to develop a theory to associate uncertainty with extended poses represented by 5×5 matrices. Our approach is particularly ...