Full reconstruction of a vector field from restricted Doppler and first integral moment transforms in ℝn

2020 ◽  
Vol 28 (2) ◽  
pp. 173-184 ◽  
Author(s):  
Rohit Kumar Mishra
Keyword(s):  
Vector Field ◽  
Integral Geometry ◽  
Tensor Field ◽  
Vector Fields ◽  
First Integral ◽  
Tensor Fields ◽  
Line Complex ◽  
Full Reconstruction ◽  
Ill Posed ◽  
Fixed Curve

AbstractWe show that a vector field in {\mathbb{R}^{n}} can be reconstructed uniquely from the knowledge of restricted Doppler and first integral moment transforms. The line complex we consider consists of all lines passing through a fixed curve {\gamma\subset\mathbb{R}^{n}}. The question of reconstruction of a symmetric m-tensor field from the knowledge of the first {m+1} integral moments was posed by Sharafutdinov [Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser. 1, De Gruyter, Berlin, 1994, p. 78]. In this work, we provide an answer to Sharafutdinov’s question for the case of vector fields from restricted data comprising of the first two integral moment transforms.

Polarization tensor in de Sitter gauge gravity

2021 ◽  
pp. 2150035
Author(s):  
R. Raziani ◽  
M. V. Takook
Keyword(s):  
Tensor Field ◽  
Vector Fields ◽  
Polarization Tensor ◽  
De Sitter ◽  
Field Equations ◽  
Tensor Fields ◽  
Sitter Group ◽  
Generalized Polarization ◽  
Mixed Symmetry ◽  
Symmetry Rank

The gauge theory of the de Sitter group, [Formula: see text], in the ambient space formalism has been considered in this paper. This method is important to construction of the de Sitter super-conformal gravity and Quantum gravity. [Formula: see text] gauge vector fields are needed which correspond to [Formula: see text] generators of the de Sitter group. Using the gauge-invariant Lagrangian, the field equations of these vector fields have been obtained. The gauge vector field solutions are recalled. By using these solutions, the spin-[Formula: see text] gauge potentials has been constructed. There are two possibilities for presenting this tensor field: rank-[Formula: see text] symmetric and mixed symmetry rank-[Formula: see text] tensor fields. To preserve the conformal transformation, a spin-[Formula: see text] field must be represented by a mixed symmetry rank-[Formula: see text] tensor field, [Formula: see text]. This tensor field has been rewritten in terms of a generalized polarization tensor field and a de Sitter plane wave. This generalized polarization tensor field has been calculated as a combination of vector polarization, [Formula: see text], and tensor polarization of rank-2, [Formula: see text], which can be used in the gravitational wave consideration. There is a certain extent of arbitrariness in the choice of this tensor and we fix it in such a way that, in the limit, [Formula: see text], one obtains the polarization tensor in Minkowski spacetime. It has been shown that under some simple conditions, the spin-[Formula: see text] mixed symmetry rank-[Formula: see text] tensor field can be simultaneously transformed by unitary irreducible representation of de Sitter and conformal groups ([Formula: see text]).

scholarly journals Visualizing High-Order Symmetric Tensor Field Structure with Differential Operators

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
Tim McGraw ◽  
Takamitsu Kawai ◽  
Inas Yassine ◽  
Lierong Zhu
Keyword(s):  
Feature Extraction ◽  
Vector Field ◽  
Tensor Field ◽  
Differential Operators ◽  
Synthetic Data ◽  
Symmetric Tensor ◽  
Field Structure ◽  
Data Sets ◽  
Tensor Fields ◽  
New Techniques

The challenge of tensor field visualization is to provide simple and comprehensible representations of data which vary both directionallyandspatially. We explore the use of differential operators to extract features from tensor fields. These features can be used to generate skeleton representations of the data that accurately characterize the global field structure. Previously, vector field operators such as gradient, divergence, and curl have previously been used to visualize of flow fields. In this paper, we use generalizations of these operators to locate and classify tensor field degenerate points and to partition the field into regions of homogeneous behavior. We describe the implementation of our feature extraction and demonstrate our new techniques on synthetic data sets of order 2, 3 and 4.

scholarly journals RATIONAL FIRST INTEGRALS FOR POLYNOMIAL VECTOR FIELDS ON ALGEBRAIC HYPERSURFACES OF ℝn+1

2012 ◽  
Vol 22 (11) ◽  
pp. 1250270 ◽  
Author(s):  
JAUME LLIBRE ◽  
YUDY BOLAÑOS
Keyword(s):  
Algebraic Geometry ◽  
Vector Field ◽  
Linear Algebra ◽  
Vector Fields ◽  
First Integral ◽  
First Integrals ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Algebraic Hypersurfaces ◽  
Rational First Integral

Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in ℝn of degree m is at least [Formula: see text], then the vector field has a rational first integral. Llibre and Zhang used only Linear Algebra to provide a shorter and easier proof of the result given by Jouanolou. We use ideas of Llibre and Zhang to extend the Jouanolou result to polynomial vector fields defined on algebraic regular hypersurfaces of ℝn+1, this extended result completes the standard results of the Darboux theory of integrability for polynomial vector fields on regular algebraic hypersurfaces of ℝn+1.

Stueckelberg mechanism for antisymmetric tensor fields

2002 ◽  
Vol 80 (7) ◽  
pp. 767-779 ◽  
Author(s):  
S V Kuzmin ◽  
D.G.C. McKeon
Keyword(s):  
Gauge Invariance ◽  
Tensor Field ◽  
Vector Fields ◽  
Antisymmetric Tensor ◽  
Tensor Fields ◽  
Antisymmetric Tensor Field ◽  
Mass Terms

It is shown how vector Stueckelberg fields can be introduced to ensure gauge invariance for mass terms for an antisymmetric tensor field. Scalar Stueckelberg fields allow one to have gauge invariance for these vector fields. Both the Abelian and non-Abelian cases are considered. Fully antisymmetric rank-three tensor fields and symmetric rank-two tensor fields are also discussed. PACS No.: 11.15-1

A note on analytic integrability of planar vector fields

2012 ◽  
Vol 23 (5) ◽  
pp. 555-562 ◽  
Author(s):  
A. ALGABA ◽  
C. GARCÍA ◽  
M. REYES
Keyword(s):  
Vector Field ◽  
Integrable Systems ◽  
Analytic Functions ◽  
Vector Fields ◽  
First Integral ◽  
Analytic Integrability ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Analytic First Integral ◽  
Image Position

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systemswhereh,K, Ψ and ξ are analytic functions defined in the neighbourhood ofOwithK(O) ≠ 0 or Ψ(O) ≠ 0 andn≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.

scholarly journals VECTOR FIELDS ON QUANTUM GROUPS

1996 ◽  
Vol 11 (06) ◽  
pp. 1077-1100 ◽  
Author(s):  
PAOLO ASCHIERI ◽  
PETER SCHUPP
Keyword(s):  
Vector Field ◽  
Hopf Algebra ◽  
Quantum Group ◽  
Vector Fields ◽  
Lie Derivative ◽  
Tensor Fields ◽  
Invariant Vector ◽  
Invariant Vector Field ◽  
Left Invariant Vector ◽  
Invariant Vector Fields

We construct the space of vector fields on a generic quantum group. Its elements are products of elements of the quantum group itself with left-invariant vector fields. We study the duality between vector fields and one-forms and generalize the construction to tensor fields. A Lie derivative along any (also non-left-invariant) vector field is proposed and a puzzling ambiguity in its definition discussed. These results hold for a generic Hopf algebra.

scholarly journals Killing Tensor Fields of Third Rank on a Two-Dimensional Riemannian Torus

2021 ◽  
Vol 14 (1) ◽  
pp. 1
Author(s):  
Vladimir A. Sharafutdinov
Keyword(s):  
Tensor Field ◽  
Killing Vector ◽  
First Integral ◽  
Symmetric Tensor ◽  
Two Dimensional ◽  
Killing Tensor ◽  
Tensor Fields ◽  
Killing Field ◽  
Quadratic Equations ◽  
Isothermal Coordinates

A rank m symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative is equal to zero. Such a field determines the first integral of the geodesic flow which is a degree m homogeneous polynomial in velocities. There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form ds^2= λ(z)|dz|^2 in the coordinates. The torus admits a third rank Killing tensor field if and only if the function λ satisfies the equation R(∂/∂z（λ(c∆^-1λ_zz+a))= 0 with some complex constants a and c≠0. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function λ. If the functions λ and λ + λ_0 satisfy the equation for a real constant λ0, 0, then there exists a non-zero Killing vector field on the torus.

scholarly journals Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1890
Author(s):  
Lucian-Miti Ionescu ◽  
Cristina-Liliana Pripoae ◽  
Gabriel-Teodor Pripoae
Keyword(s):  
Differential Geometry ◽  
Vector Field ◽  
Vector Fields ◽  
Torsion Tensor ◽  
Holomorphic Functions ◽  
Tensor Fields ◽  
Pedagogical Tool ◽  
Complex Integral ◽  
Curvature And Torsion

We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.

Unique Normal Form for a Class of Three-Dimensional Nilpotent Vector Fields

2017 ◽  
Vol 27 (08) ◽  
pp. 1750131 ◽  
Author(s):  
Jing Li ◽  
Liying Kou ◽  
Duo Wang
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Linear Part ◽  
Three Dimensional ◽  
First Integral ◽  
Second Order ◽  
Lie Brackets ◽  
The Given

In this paper, the unique normal form for a class of three-dimensional nilpotent vector fields with symmetry is investigated. The key technique used is a combination of multiple Lie brackets, linear grading function, new notations of block matrices and first integral of the linear part of the given vector field which avoids complicated calculations. The first and the second order normal forms are computed successively, and the uniqueness of the second order normal form is proved under certain conditions.

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

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