On the first Betti number of spacetimes with parallel lightlike vector field

2020 ◽  
Vol 72 ◽  
pp. 101648
Author(s):  
Raymond Hounnonkpe
Keyword(s):  
Vector Field ◽  
Betti Number ◽  
Lightlike Vector

Transverse Kähler holonomy in Sasaki Geometry and S-Stability

2021 ◽  
Vol 8 (1) ◽  
pp. 336-353
Author(s):  
Charles P. Boyer ◽  
Hongnian Huang ◽  
Christina W. Tønnesen-Friedman
Keyword(s):  
Vector Field ◽  
Betti Number ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Stability Properties ◽  
Hodge Number ◽  
Sasakian Structure ◽  
Characteristic Foliation ◽  
Holonomy Groups ◽  
Sasaki Manifolds

Abstract We study the transverse Kähler holonomy groups on Sasaki manifolds (M, S) and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number b 1(M) and the basic Hodge number h 0,2 B(S) vanish, then S is stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure is S-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is S-stable when dim M ≥ 7. Finally, we prove that the standard Sasaki join operation (transverse holonomy U(n 1) × U(n 2)) as well as the fiber join operation preserve S-stability.

scholarly journals Tri-partitions and Bases of an Ordered Complex

2020 ◽  
Vol 64 (3) ◽  
pp. 759-775
Author(s):  
Herbert Edelsbrunner ◽  
Katharina Ölsböck
Keyword(s):  
Vector Field ◽  
Planar Graph ◽  
Betti Number ◽  
Hodge Decomposition ◽  
Reduction Algorithm ◽  
Smooth Vector ◽  
Polyhedral Complex ◽  
Canonical Bases ◽  
Smooth Vector Field ◽  
Connected Planar Graph

Abstract Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.

scholarly journals On maximal hypersurfaces in Lorentz manifolds admitting a parallel lightlike vector field

2016 ◽  
Vol 33 (5) ◽  
pp. 055003 ◽  
Author(s):  
José A S Pelegrín ◽  
Alfonso Romero ◽  
Rafael M Rubio
Keyword(s):  
Vector Field ◽  
Lorentz Manifolds ◽  
Lightlike Vector

Mathematical Model of Flat Surface Machining Inaccuracies due to Elastic Strains of Technological System

2018 ◽  
pp. 1-14 ◽  
Author(s):  
I. I. Kravchenko
Keyword(s):  
Mathematical Model ◽  
Vector Field ◽  
Model Development ◽  
Technological System ◽  
Mutual Arrangement ◽  
Real Surface ◽  
Main Bearing ◽  
Flat Surfaces ◽  
Internal Surfaces ◽  
Elastic Strains

The paper considers the mathematical model development technique to build a vector field of the shape deviations when machining flat surfaces of shell parts on multi-operational machines under conditions of anisotropic rigidity in technological system (TS). The technological system has an anisotropic rigidity, as its elastic strains do not obey the accepted concepts, i.e. the rigidity towards the coordinate axes of the machine is the same, and they occur only towards the external force. The record shows that the diagrams of elastic strains of machine units are substantially different from the circumference. The issues to ensure the specified accuracy require that there should be mathematical models describing kinematic models and physical processes of mechanical machining under conditions of the specific TS. There are such models for external and internal surfaces of rotation [2,3], which are successfully implemented in practice. Flat surfaces (FS) of shell parts (SP) are both assembly and processing datum surfaces. Therefore, on them special stipulations are made regarding deviations of shape and mutual arrangement. The axes of the main bearing holes are coordinated with respect to them. The joints that ensure leak tightness and distributed load on the product part are closed on these surfaces. The paper deals with the analytical construction of the vector field F, which describes with appropriate approximation the real surface obtained as a result of modeling the process of machining flat surfaces (MFS) through face milling under conditions of anisotropic properties.

On the geometry of Killing vector field

2019 ◽  
Vol 2019 (2) ◽  
pp. 62-67
Author(s):  
R.A. Ilyasova
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Killing Vector Field

Some Properties of Para-Sasakian Manifolds

2017 ◽  
Vol 5 (2) ◽  
pp. 73-78
Author(s):  
Jay Prakash Singh ◽  
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Sasakian Manifold ◽  
Subject Classification ◽  
Killing Vector Field ◽  
Sasakian Manifolds ◽  
Geometric Properties ◽  
Paper Author

In this paper author present an investigation of some differential geometric properties of Para-Sasakian manifolds. Condition for a vector field to be Killing vector field in Para-Sasakian manifold is obtained. Mathematics Subject Classification (2010). 53B20, 53C15.

scholarly journals Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Martin de Borbon
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Projective Line ◽  
Tangent Cones ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Complex Curves ◽  
Complex Dimensions ◽  
Irregular Case ◽  
Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

Vector field path following of a full-wing solar-powered Unmanned Aerial Vehicle (UAV) landing based on Dubins path: a lesson from multiple landing failures

2021 ◽  
pp. 1-30
Author(s):  
A. Guo ◽  
Z. Zhou ◽  
R. Wang ◽  
X. Zhao ◽  
X. Zhu
Keyword(s):  
Vector Field ◽  
Path Following ◽  
Endurance Performance ◽  
Electrical Energy ◽  
Lateral Control ◽  
Special Configuration ◽  
Straight Line ◽  
Lateral Distance ◽  
Solar Powered ◽  
Line Path

Abstract The full-wing solar-powered UAV has a large aspect ratio, special configuration, and excellent aerodynamic performance. This UAV converts solar energy into electrical energy for level flight and storage to improve endurance performance. The UAV only uses a differential throttle for lateral control, and the insufficient control capability during crosswind landing results in a large lateral distance bias and leads to multiple landing failures. This paper analyzes 11 landing failures and finds that a large lateral distance bias at the beginning of the approach and the coupling of base and differential throttle control is the main reason for multiple landing failures. To improve the landing performance, a heading angle-based vector field (VF) method is applied to the straight-line and orbit paths following and two novel 3D Dubins landing paths are proposed to reduce the initial lateral control bias. The results show that the straight-line path simulation exhibits similar phenomenon with the practical failure; the single helical path has the highest lateral control accuracy; the left-arc to left-arc (L-L) path avoids the saturation of the differential throttle; and both paths effectively improve the probability of successful landing.

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