Lie Group Modeling of Nonlinear Point Set Shape Variability

2000 ◽  
pp. 259-268
Author(s):  
Niels Holm Olsen ◽  
Mads Nielsen
Keyword(s):  
Lie Group ◽  
Group Modeling ◽  
Point Set

The Topology of a Group Action

2020 ◽  
pp. 205-212
Author(s):  
Loring W. Tu
Keyword(s):  
Fixed Point ◽  
Vector Bundle ◽  
Group Action ◽  
Lie Group ◽  
Tubular Neighborhood ◽  
Compact Lie Group ◽  
Fixed Point Set ◽  
Smooth Action ◽  
Point Set ◽  
Neighborhood Theorem

This chapter describes the topology of a group action. It proves some topological facts about the fixed point set and the stabilizers of a continuous or a smooth action. The chapter also introduces the equivariant tubular neighborhood theorem and the equivariant Mayer–Vietoris sequence. A tubular neighborhood of a submanifold S in a manifold M is a neighborhood that has the structure of a vector bundle over S. Because the total space of a vector bundle has the same homotopy type as the base space, in calculating cohomology one may replace a submanifold by a tubular neighborhood. The tubular neighborhood theorem guarantees the existence of a tubular neighborhood for a compact regular submanifold. The Mayer–Vietoris sequence is a powerful tool for calculating the cohomology of a union of two open subsets. Both the tubular neighborhood theorem and the Mayer–Vietoris sequence have equivariant counterparts for a G-manifold where G is a compact Lie group.

scholarly journals Equivariant basic cohomology of Riemannian foliations

2018 ◽  
Vol 2018 (745) ◽  
pp. 1-40 ◽  
Author(s):  
Oliver Goertsches ◽  
Dirk Töben
Keyword(s):  
Fixed Point ◽  
Lie Group ◽  
Betti Number ◽  
Group Actions ◽  
Fixed Point Set ◽  
Foliated Manifold ◽  
Riemannian Foliations ◽  
Basic Cohomology ◽  
Lie Group Actions ◽  
Point Set

Abstract The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C, we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino’s observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting.

scholarly journals Three-dimensional connected groups of automorphisms of toroidal circle planes

2020 ◽  
Vol 20 (4) ◽  
pp. 585-594
Author(s):  
Brendan Creutz ◽  
Duy Ho ◽  
Günter F. Steinke
Keyword(s):  
Lie Group ◽  
Three Dimensional ◽  
Minkowski Planes ◽  
Point Set ◽  
Groups Of Automorphisms ◽  
Full Classification

AbstractWe contribute to the classification of toroidal circle planes and flat Minkowski planes possessing three-dimensional connected groups of automorphisms. When such a group is an almost simple Lie group, we show that it is isomorphic to PSL(2, ℝ). Using this result, we describe a framework for the full classification based on the action of the group on the point set.

Lie Group Modeling of Nonlinear Helical Beam Elements

2014 ◽  
Author(s):  
Ralph Jödicke ◽  
Uwe Jungnickel ◽  
Andreas Müller
Keyword(s):  
Lie Group ◽  
Beam Model ◽  
Deflection Curve ◽  
Finite Frames ◽  
Beam Elements ◽  
Group Modeling ◽  
Stiffness Matrices ◽  
Torsion Energy ◽  
Shearing Energy ◽  
Helical Beam

A viscoelastic beam model is presented based on SE(3) group theory. We discretize a rod with beams between finite frames on the rod and regard the configurations of these frames as elements of the SE(3) Lie group. Two subsequent frames are connected by a beam. The curvatures and strains are assumed to be constant on the trajectory between them. If the deflection curve of the beam is modeled as a helix, the resulting beam model is geometrical exact for large bending deformations. The stiffness matrices of the discrete beam elements result from the potential extensional and shearing energy as well as from the potential bending and torsion energy. The benefit of this SE(3) modeling for translational elastic coordinates and for translational forces in comparison to an SO(3) × ℝ3 variant is demonstrated.

scholarly journals The homotopy fixed point set of Lie group actions on elliptic spaces

2015 ◽  
Vol 110 (5) ◽  
pp. 1135-1156 ◽  
Author(s):  
Urtzi Buijs ◽  
Yves Félix ◽  
Sergio Huerta ◽  
Aniceto Murillo
Keyword(s):  
Fixed Point ◽  
Lie Group ◽  
Group Actions ◽  
Fixed Point Set ◽  
Lie Group Actions ◽  
Point Set ◽  
Elliptic Spaces

scholarly journals Lie-Group Modeling and Numerical Simulation of a Helicopter

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2682
Author(s):  
Alessandro Tarsi ◽  
Simone Fiori
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Group Theory ◽  
Lie Group ◽  
Technical Literature ◽  
Lie Group Theory ◽  
Pontryagin Principle ◽  
Group Modeling ◽  
The Many ◽  
The Mathematical Model

Helicopters are extraordinarily complex mechanisms. Such complexity makes it difficult to model, simulate and pilot a helicopter. The present paper proposes a mathematical model of a fantail helicopter type based on Lie-group theory. The present paper first recalls the Lagrange–d’Alembert–Pontryagin principle to describe the dynamics of a multi-part object, and subsequently applies such principle to describe the motion of a helicopter in space. A good part of the paper is devoted to the numerical simulation of the motion of a helicopter, which was obtained through a dedicated numerical method. Numerical simulation was based on a series of values for the many parameters involved in the mathematical model carefully inferred from the available technical literature.

scholarly journals Almost free actions on manifolds

1974 ◽  
Vol 10 (2) ◽  
pp. 177-196 ◽  
Author(s):  
Philip T. Church ◽  
Klaus Lamotke
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lie Group ◽  
Orbit Space ◽  
Fixed Point Set ◽  
Point Set ◽  
Free Actions ◽  
Connected Lie Group

Let X be a compact, connected, oriented topological G-manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U(1) = S1 and dimX =4 or G = Sp(1) = S3 and dimX = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, Sp(1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and(2) the (even) number of fixed points.Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U(1) is also proved.

Computational Micrograph Registration with Sieve Processes

1996 ◽  
Vol 54 ◽  
pp. 440-441
Author(s):  
P.J. Phillips ◽  
J. Huang ◽  
S. M. Dunn
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Spatial Organization ◽  
Spatial Arrangement ◽  
Stereo Image ◽  
Image Model ◽  
Geometric Invariants ◽  
Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

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