Extracting Higher Order Critical Points and Topological Simplification of 3D Vector Fields

2006 ◽  
Author(s):  
T. Weinkauf ◽  
H. Theisel ◽  
Kuangyu Shi ◽  
H. Hege ◽  
H. Seidel
Keyword(s):  
Critical Points ◽  
Vector Fields ◽  
Higher Order

One variation of R. Miron's spray theory in OsckM

2003 ◽  
Vol 40 (4) ◽  
pp. 443-462
Author(s):  
Irena Čomić
Keyword(s):  
Vector Fields ◽  
Transformation Group ◽  
Higher Order

Lately a big attention has been payed on the higher order geometry. Some relevant papers are mentioned in the references. R. Miron and Gh. Atanasiu in [16], [17] studied the geometry of OsckM. R. Miron in [19] gave the comprehend theory of higher order geometry and its application. The whole theory of sprays in OsckM M is established. Here, using R. Miron's method, a variation of this theory is given. The transformation group is slightly different from that used in [19] and it will change the geometry. The adapted basis, the Liouville vector fields, the equation of sprays, will have different form. We give the relations between coefficients of S and the Liouville vector fields.

On the local solvability of vector fields with critical points

2011 ◽  
Vol 10 (3) ◽  
pp. 785-808
Author(s):  
François Treves
Keyword(s):  
Critical Points ◽  
Vector Fields ◽  
Local Solvability ◽  
Complex Vector ◽  
Base Manifold ◽  
Smooth Complex ◽  
Empty Subset

AbstractThe article discusses the local solvability (or lack thereof) of various classes of smooth, complex vector fields that vanish on some non-empty subset of the base manifold.

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Vector Fields ◽  
Higher Order ◽  
Beam Equation ◽  
Series Solutions ◽  
Local Conservation ◽  
Power Series Method ◽  
Potential System ◽  
Power Series Solutions ◽  
The Difference

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

Harmonicity of vector fields on a class of Lorentzian solvable Lie groups

2018 ◽  
Vol 18 (3) ◽  
pp. 337-344 ◽  
Author(s):  
Ju Tan ◽  
Shaoqiang Deng
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Algebras ◽  
Critical Points ◽  
Vector Fields ◽  
Energy Functional ◽  
Smooth Vector ◽  
Solvable Lie Groups ◽  
Invariant Vector ◽  
Invariant Vector Fields

AbstractIn this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

Phase diagrams and higher-order critical points

1975 ◽  
Vol 12 (1) ◽  
pp. 345-355 ◽  
Author(s):  
Robert B. Griffiths
Keyword(s):  
Phase Diagrams ◽  
Critical Points ◽  
Higher Order

Unique Normal Form and the Associated Coefficients for a Class of Three-Dimensional Nilpotent Vector Fields

2017 ◽  
Vol 27 (14) ◽  
pp. 1750224
Author(s):  
Jing Li ◽  
Liying Kou ◽  
Duo Wang ◽  
Wei Zhang
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
Three Dimensional ◽  
Recursive Formula ◽  
Higher Order ◽  
Dimensional Vector ◽  
Symbolic Calculations

In this paper, we mainly focus on the unique normal form for a class of three-dimensional vector fields via the method of transformation with parameters. A general explicit recursive formula is derived to compute the higher order normal form and the associated coefficients, which can be achieved easily by symbolic calculations. To illustrate the efficiency of the approach, a comparison of our result with others is also presented.

scholarly journals Modeling n -Symmetry Vector Fields using Higher-Order Energies

2018 ◽  
Vol 37 (2) ◽  
pp. 1-18 ◽  
Author(s):  
Christopher Brandt ◽  
Leonardo Scandolo ◽  
Elmar Eisemann ◽  
Klaus Hildebrandt
Keyword(s):  
Vector Fields ◽  
Higher Order
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