scholarly journals 3D topological models and Heegaard splitting. II. Pontryagin duality and observables

2020 ◽  
Vol 61 (11) ◽  
pp. 112302
Author(s):  
F. Thuillier
Keyword(s):  
Heegaard Splitting ◽  
Pontryagin Duality ◽  
Topological Models

scholarly journals A Kinematical Analysis of the Flap and Wing Mechanism of a Light Sport Aircraft Using Topological Models

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1243
Author(s):  
Sorin Vlase ◽  
Ion-Marius Ghiţescu ◽  
Marius Paun
Keyword(s):  
Graph Theory ◽  
Mechanical System ◽  
Linear Systems ◽  
Kinematic Analysis ◽  
Control Mechanism ◽  
Practical Application ◽  
Dynamic Level ◽  
New Formulation ◽  
Topological Models ◽  
Independent Loops

In this, paper, we propose a method of kinematic analysis of a planar mechanism with application to the flap and wing mechanism of a light sport aircraft. A topological model is used to describe a mechanical system, which is a model that allows the study of the maneuverability of the system. The proposed algorithm is applied to determine the velocity and acceleration field of this multibody mechanical system. The graph associated with the mechanical system is generated in a new formulation and based on it, the fundamental loops of the graph are identified (corresponding to the independent loops of the mechanism), the equations for closing vectorial contours are written, and the kinematic conditions for determining velocities and accelerations and the associated linear systems are solved, which provides the field of speeds and accelerations. Graph Theory is applied at a kinematic level and not at a dynamic level, as in previous studies. A practical application for the kinematic analysis of the control mechanism of a light aircraft illustrates the proposed method.

scholarly journals DETECTING TORSION IN SKEIN MODULES USING HOCHSCHILD HOMOLOGY

2006 ◽  
Vol 15 (02) ◽  
pp. 259-277 ◽  
Author(s):  
MICHAEL McLENDON
Keyword(s):  
Tensor Product ◽  
Spectral Sequence ◽  
Boundary Surface ◽  
Hochschild Homology ◽  
Heegaard Splitting ◽  
Common Boundary ◽  
Skein Modules ◽  
Skein Module ◽  
Hochschild Complex

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.

Local uniqueness of certain geodesics related to Heegaard splittings

2018 ◽  
Vol 27 (09) ◽  
pp. 1842003
Author(s):  
Liang Liang ◽  
Fengling Li ◽  
Fengchun Lei ◽  
Jie Wu
Keyword(s):  
Heegaard Splitting ◽  
Sufficient Condition ◽  
Heegaard Splittings ◽  
Local Uniqueness ◽  
Uniqueness Property

Suppose [Formula: see text] is a Heegaard splitting and [Formula: see text] is an essential separating disk in [Formula: see text] such that a component of [Formula: see text] is homeomorphic to [Formula: see text], [Formula: see text]. In this paper, we prove that if there is a locally complicated simplicial path in [Formula: see text] connecting [Formula: see text] to [Formula: see text], then the geodesic connecting [Formula: see text] to [Formula: see text] is unique. Moreover, we give a sufficient condition such that [Formula: see text] is keen and the geodesic between any pair of essential disks on the opposite sides has local uniqueness property.

scholarly journals Groups of quasi-invariance and the Pontryagin duality

2010 ◽  
Vol 157 (18) ◽  
pp. 2786-2802 ◽  
Author(s):  
S.S. Gabriyelyan
Keyword(s):  
Pontryagin Duality

One-relator surface groups

1990 ◽  
Vol 108 (3) ◽  
pp. 467-474 ◽  
Author(s):  
John Hempel
Keyword(s):  
Normal Subgroup ◽  
Single Point ◽  
Surface Group ◽  
Compact Surface ◽  
Heegaard Splitting ◽  
Normal Closure ◽  
Single Element ◽  
Homotopy Classes ◽  
Simple Loop ◽  
Proper Power

For X a subset of a group G, the smallest normal subgroup of G which contains X is called the normal closure of X and is denoted by ngp (X; G) or simply by ngp (X) if there is no possibility of ambiguity. By a surface group we mean the fundamental group of a compact surface. We are interested in determining when a normal subgroup of a surface group contains a simple loop – the homotopy class of an embedding of S1 in the surface, or more generally, a power of a simple loop. This is significant to the study of 3-manifolds since a Heegaard splitting of a 3-manifold is reducible (cf. [2]) if and only if the kernel of the corresponding splitting homomorphism contains a simple loop. We give an answer in the case that the normal subgroup is the normal closure ngp (α) of a single element α: if ngp (α) contains a (power of a) simple loop β then α is homotopic to a (power of a) simple loop and β±1 is homotopic either to (a power of) α or to the commutator [α, γ] of a with some simple loop γ meeting a transversely in a single point. This implies that if a is not homotopic to a power of a simple loop, then the quotient map π1(S) → π1(S)/ngp (α) does not factor through a group with more than one end. In the process we show that π1(S)/ngp (α) is locally indicable if and only if α is not a proper power and that α always lifts to a simple loop in the covering space Sα of S corresponding to ngp (α). We also obtain some estimates on the minimal number of double points in certain homotopy classes of loops.

Object Extraction Using Topological Models from Complex Scene Image

2017 ◽  
pp. 335-357
Author(s):  
Uday Pratap Singh ◽  
Sanjeev Jain
Keyword(s):  
Object Recognition ◽  
Object Segmentation ◽  
Multimedia Data ◽  
Region Merging ◽  
Complex Scene ◽  
Object Contour ◽  
Merging Process ◽  
Feasible Solutions ◽  
Neighbourhood Graph ◽  
Topological Models

Efficient and effective object recognition from a multimedia data are very complex. Automatic object segmentation is usually very hard for natural images; interactive schemes with a few simple markers provide feasible solutions. In this chapter, we propose topological model based region merging. In this work, we will focus on topological models like, Relative Neighbourhood Graph (RNG) and Gabriel graph (GG), etc. From the Initial segmented image, we constructed a neighbourhood graph represented different regions as the node of graph and weight of the edges are the value of dissimilarity measures function for their colour histogram vectors. A method of similarity based region merging mechanism (supervised and unsupervised) is proposed to guide the merging process with the help of markers. The region merging process is adaptive to the image content and it does not need to set the similarity threshold in advance. To the validation of proposed method extensive experiments are performed and the result shows that the proposed method extracts the object contour from the complex background.

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