Characterization of integral curves of a linear vector field in Lorentz 3-space

2016 ◽  
Vol 13 (06) ◽  
pp. 1650073
Author(s):  
Tunahan Turhan ◽  
Nihat Ayyıldız
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Symmetric Matrix ◽  
Order System ◽  
Linear Vector ◽  
Integral Curves ◽  
The Matrix ◽  
Vector Formula

We propose a detail study of integral curves or flow lines of a linear vector field in Lorentz [Formula: see text]-space. We construct the matrix [Formula: see text] depending on the causal characters of the vector [Formula: see text] by analyzing the non-zero solutions of the equation [Formula: see text], [Formula: see text] in such a space, where [Formula: see text] is the skew-symmetric matrix corresponding to the linear map [Formula: see text]. Considering the structure of a linear vector field, we obtain the linear first-order system of differential equations. The solutions of this system of equations give rise to integral curves of linear vector fields from which we provide a classification of such curves.

scholarly journals Flows of measures generated by vector fields

2018 ◽  
Vol 148 (4) ◽  
pp. 773-818
Author(s):  
Emanuele Paolini ◽  
Eugene Stepanov
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Borel Measure ◽  
Weak Sense ◽  
Positive Borel Measure ◽  
Smooth Structure ◽  
Smooth Vector ◽  
Integral Curves ◽  
The Given ◽  
Measurable Vector

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

scholarly journals Clifford algebra-based structure filtering analysis for geophysical vector fields

2013 ◽  
Vol 20 (4) ◽  
pp. 563-570 ◽  
Author(s):  
Z. Yu ◽  
W. Luo ◽  
L. Yi ◽  
Y. Hu ◽  
L. Yuan
Keyword(s):  
Vector Field ◽  
Clifford Algebra ◽  
El Niño ◽  
Vector Fields ◽  
El Nino ◽  
Field Analysis ◽  
Directional Information ◽  
Filtering Method ◽  
Additional Information

Abstract. A new Clifford algebra-based vector field filtering method, which combines amplitude similarity and direction difference synchronously, is proposed. Firstly, a modified correlation product is defined by combining the amplitude similarity and direction difference. Then, a structure filtering algorithm is constructed based on the modified correlation product. With custom template and thresholds applied to the modulus and directional fields independently, our approach can reveal not only the modulus similarities but also the classification of the angular distribution. Experiments on exploring the tempo-spatial evolution of the 2002–2003 El Niño from the global wind data field are used to test the algorithm. The results suggest that both the modulus similarity and directional information given by our approach can reveal the different stages and dominate factors of the process of the El Niño evolution. Additional information such as the directional stability of the El Niño can also be extracted. All the above suggest our method can provide a new powerful and applicable tool for geophysical vector field analysis.

scholarly journals Finitely curved orbits of complex polynomial vector fields

2007 ◽  
Vol 79 (1) ◽  
pp. 13-16
Author(s):  
Albetã C. Mafra
Keyword(s):  
Vector Field ◽  
Gaussian Curvature ◽  
Algebraic Curve ◽  
Vector Fields ◽  
Complex Polynomial ◽  
Holomorphic Foliations ◽  
Holomorphic Curve ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

This note is about the geometry of holomorphic foliations. Let X be a polynomial vector field with isolated singularities on C². We announce some results regarding two problems: 1. Given a finitely curved orbit L of X, under which conditions is L algebraic? 2. If X has some non-algebraic finitely curved orbit L what is the classification of X? Problem 1 is related to the following question: Let C <FONT FACE=Symbol>Ì</FONT> C² be a holomorphic curve which has finite total Gaussian curvature. IsC contained in an algebraic curve?

scholarly journals Classification of Ricci solitons on Euclidean hypersurfaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450104 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Potential Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Position Vector ◽  
Ricci Solitons ◽  
Euclidean Submanifolds

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

scholarly journals A simple characterization of doubly twisted spacetimes

2021 ◽  
pp. 2150079
Author(s):  
Carlo Alberto Mantica ◽  
Luca Guido Molinari
Keyword(s):  
Vector Fields ◽  
Simple Classification ◽  
Simple Characterization

In this note, we characterize [Formula: see text] doubly twisted spacetimes in terms of “doubly torqued” vector fields. They extend Bang–Yen Chen’s characterization of twisted and generalized Robertson–Walker spacetimes with torqued and concircular vector fields. The result is a simple classification of [Formula: see text] doubly-twisted, doubly-warped, twisted and generalized Robertson–Walker spacetimes.

The geometry of the trajectories

2020 ◽  
Vol 17 (04) ◽  
pp. 2050051
Author(s):  
Mohammadreza Molaei
Keyword(s):  
Black Hole ◽  
Vector Field ◽  
Vector Fields ◽  
Second Step ◽  
Curvature Scalar ◽  
Schwarzschild Black Hole ◽  
Integral Curves ◽  
Trajectory Manifold

In this paper, we use of the geometry of a class of the nature flows to define trajectory manifolds. Trajectory connections as a generalization of the Levi-Civita connections are considered. A method for determining the geometry of the flows created by the integral curves of a vector field is presented. The method contains two steps, the first step is finding the connection by the trajectories of a vector field, and the second step is finding a trajectory metric corresponding to the deduced connection. We show that doing the first step is possible, but for some of the vector fields, the second step may not be possible. In the case of existence of a trajectory manifold a new kind of curvature which we called it “trajectory curvature scalar” appears. We calculate trajectory connections for some vector fields and by an example we show that the trajectory curvature scalar for a trajectory manifold may not be equal to the curvature scalar of it. We find trajectory connection for a vector field close to the Schwarzschild black hole.

N-DIMENSIONAL CANONICAL CHUA'S CIRCUIT

1993 ◽  
Vol 03 (01) ◽  
pp. 239-258 ◽  
Author(s):  
LJ. KOCAREV ◽  
LJ. KARADZINOV ◽  
L. O. CHUA
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Measure Zero ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Electrical Circuit ◽  
Continuous Vector ◽  
Linear Vector ◽  
Linear Dynamic ◽  
Canonical Chua’S Circuit

In this paper we present an n-dimensional canonical piecewise-linear electrical circuit. It contains 2n two-terminal elements: n linear dynamic elements (capacitors and inductors), n - 1 linear resistors and one nonlinear (piecewise-linear) resistor. This circuit can realize any prescribed eigenvalue pattern, except for a set of measure zero, associated with (i) any n-dimensional two-region continuous piecewise-linear vector fields and (ii) any n-dimensional three-region symmetric (with respect to the origin) piecewise-linear continuous vector fields. We also proved a theorem that specifies the conditions for a vector field, realized with our canonical circuit, to have two or three equilibrium points.

scholarly journals ∞-jets of diffeomorphisms preserving orbits of vector fields

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Sergiy Maksymenko
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Homogeneous Polynomials ◽  
Local Flow ◽  
Linear Vector ◽  
Identity Component ◽  
Hamiltonian Vector Fields

AbstractLet F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = .In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O.We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.

scholarly journals Complete lift of vector fields and sprays to T∞M

2015 ◽  
Vol 12 (10) ◽  
pp. 1550113 ◽  
Author(s):  
Ali Suri ◽  
Somaye Rastegarzadeh
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Dimensional Manifold ◽  
Closed Curves ◽  
Infinite Dimensional ◽  
Integral Curves ◽  
Complete Lift ◽  
Infinite Dimensional Manifold

In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on the geometry of its iterated tangent bundle TrM, r ∈ ℕ ∪ {∞}. First we endow TrM with a canonical atlas using that of M. Then the concepts of vertical and complete lifts for functions and vector fields on TrM are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply T∞M with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on M, can be lifted to a vector field or (semi)spray on T∞M. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on T∞M including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite-dimensional case of manifold of closed curves.

scholarly journals The enhanced principal rank characteristic sequence over a field of characteristic 2

2017 ◽  
Vol 32 ◽  
pp. 273-290 ◽  
Author(s):  
Xavier Martínez-Rivera
Keyword(s):  
Symmetric Matrix ◽  
Complete Characterization ◽  
Symmetric Matrices ◽  
Characteristic Sequence ◽  
The Matrix ◽  
Characteristic 2

The enhanced principal rank characteristic sequence (epr-sequence) of an $n \times n$ symmetric matrix over a field $\F$ was recently defined as $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_k$ is either $\tt A$, $\tt S$, or $\tt N$ based on whether all, some (but not all), or none of the order-$k$ principal minors of the matrix are nonzero. Here, a complete characterization of the epr-sequences that are attainable by symmetric matrices over the field $\Z_2$, the integers modulo $2$, is established. Contrary to the attainable epr-sequences over a field of characteristic $0$, this characterization reveals that the attainable epr-sequences over $\Z_2$ possess very special structures. For more general fields of characteristic $2$, some restrictions on attainable epr-sequences are obtained.

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