scholarly journals On the convexity condition for the Semi-Geostrophic system

2021 ◽  
Author(s):  
Adrian Tudorascu
Keyword(s):  
Vector Field ◽  
Critical Points ◽  
Vector Fields ◽  
Energy Functional ◽  
Convexity Condition ◽  
Local Minimizers ◽  
Flow Maps ◽  
Divergence Free Vector ◽  
Absolute Density ◽  
The Absolute

We show that conservative distributional solutions to the Semi-Geostrophic systems in a rigid domain are in some well-defined sense critical points of a time-shifted energy functional involving measure-preserving rearrangements of the absolute density and momentum, which arise as one-parameter flow maps of continuously differentiable, compactly supported divergence free vector fields. We also show directly, with no recourse to Monge- Kantorovich theory, that the convexity requirement on the modified pressure potentials arises naturally if these critical points are local minimizers of said energy functional for any admis- sible vector field. The obligatory connection with the Monge-Kantorovich theory is addressed briefly.

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.

scholarly journals Least-squares gradient calculation from multi-point observations of scalar and vector fields: methodology and applications with Cluster in the plasmasphere

2007 ◽  
Vol 25 (4) ◽  
pp. 971-987 ◽  
Author(s):  
J. De Keyser ◽  
F. Darrouzet ◽  
M. W. Dunlop ◽  
P. M. E. Décréau
Keyword(s):  
Vector Field ◽  
Least Squares ◽  
Measurement Errors ◽  
Vector Fields ◽  
Total Error ◽  
Weighted Least Squares ◽  
General Purpose ◽  
Small Scale ◽  
Divergence Free Vector ◽  
Gradient Calculation

Abstract. This paper describes a general-purpose algorithm for computing the gradients in space and time of a scalar field, a vector field, or a divergence-free vector field, from in situ measurements by one or more spacecraft. The algorithm provides total error estimates on the computed gradient, including the effects of measurement errors, the errors due to a lack of spatio-temporal homogeneity, and errors due to small-scale fluctuations. It also has the ability to diagnose the conditioning of the problem. Optimal use is made of the data, in terms of exploiting the maximum amount of information relative to the uncertainty on the data, by solving the problem in a weighted least-squares sense. The method is illustrated using Cluster magnetic field and electron density data to compute various gradients during a traversal of the inner magnetosphere. In particular, Cluster is shown to cross azimuthal density structure, and the existence of field-aligned currents in the plasmasphere is demonstrated.

scholarly journals Vector fields satisfying the barycenter property

2018 ◽  
Vol 16 (1) ◽  
pp. 429-436 ◽  
Author(s):  
Manseob Lee
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Axiom A ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Volume Preserving

AbstractWe show that if a vector fieldXhas theC1robustly barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, if a genericC1-vector field has the barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, we apply the results to the divergence free vector fields. It is an extension of the results of the barycenter property for generic diffeomorphisms and volume preserving diffeomorphisms [1].

scholarly journals ENERGY OF GLOBAL FRAMES

2008 ◽  
Vol 84 (2) ◽  
pp. 155-162
Author(s):  
FABIANO G. B. BRITO ◽  
PABLO M. CHACÓN
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Lower Bound ◽  
Vector Fields ◽  
Singular Vector ◽  
Unit Vector ◽  
Energy Functional ◽  
Closed Riemannian Manifold ◽  
Total Scalar Curvature ◽  
Dimension 2

AbstractThe energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k+1 which is not attained by any non-singular vector field for k>1. For k=1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

A pasting lemma and some applications for conservative systems

2007 ◽  
Vol 27 (5) ◽  
pp. 1399-1417 ◽  
Author(s):  
ALEXANDER ARBIETO ◽  
CARLOS MATHEUS
Keyword(s):  
Vector Field ◽  
Compact Manifold ◽  
Vector Fields ◽  
Three Dimensional ◽  
Invariant Set ◽  
Dominated Splitting ◽  
Divergence Free Vector ◽  
Conservative Systems ◽  
Divergence Free ◽  
Volume Preserving

AbstractWe prove that in a compact manifold of dimension n≥2, C1+α volume-preserving diffeomorphisms that are robustly transitive in the C1-topology have a dominated splitting. Also we prove that for three-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field cannot have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in three-dimensional manifolds are Anosov. For this, we prove a ‘pasting’ lemma, which allows us to make perturbations in conservative systems.

Harmonic vector fields on a weighted Riemannian manifold arising from a Finsler structure

2018 ◽  
Vol 9 (2) ◽  
pp. 131-141
Author(s):  
Neda Shojaee ◽  
Morteza Mirmohammad Rezaii
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Solution Space ◽  
Energy Functional ◽  
Laplacian Operator ◽  
Harmonic Vector ◽  
Harmonic Vector Field ◽  
Measure Spaces ◽  
Harmonic Vector Fields ◽  
Unit Norm

AbstractIn the present work, the harmonic vector field is defined on closed Finsler measure spaces through different approaches. At first, the weighted harmonic vector field is obtained as the solution space of a PDE system. Then a suitable Dirichlet energy functional is introduced. A σ-harmonic vector field is considered as the critical point of related action. It is proved that a σ-harmonic vector field on a closed Finsler space with an extra unit norm condition is an eigenvector of the defined Laplacian operator on vector fields. Moreover, we prove that a unit weighted harmonic vector field on a closed generalized Einstein manifold is a σ-harmonic vector field.

scholarly journals Representation of vector fields

2015 ◽  
Vol 3 (2) ◽  
pp. 73
Author(s):  
Alexander G. Ramm
Keyword(s):  
Vector Field ◽  
Explicit Formula ◽  
Simple Proof ◽  
Vector Fields ◽  
Smooth Domain ◽  
Gradient Field ◽  
Smooth Vector ◽  
Smooth Vector Field ◽  
Divergence Free Vector ◽  
Divergence Free

<p>A simple proof is given for the explicit formula which allows one to recover a \(C^2\) – smooth vector field \(A=A(x)\) in \(\mathbb{R}^3\), decaying at infinity, from the knowledge of its \(\nabla \times A\) and \(\nabla \cdot A\). The representation of \(A\) as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Similar results are obtained for a vector field in a bounded \(C^2\) - smooth domain.</p>

scholarly journals C3VFC: A Method for Tracing and Quantification of Microglia in 3D Temporal Images

2021 ◽  
Vol 11 (13) ◽  
pp. 6078
Author(s):  
Tiffany T. Ly ◽  
Jie Wang ◽  
Kanchan Bisht ◽  
Ukpong Eyo ◽  
Scott T. Acton
Keyword(s):  
Vector Field ◽  
Critical Points ◽  
State Of The Art ◽  
Temporal Data ◽  
Entire Structure ◽  
Tracing Algorithm ◽  
Vector Field Convolution ◽  
Multiple Cells ◽  
Tracing Method ◽  
Over Time

Automatic glia reconstruction is essential for the dynamic analysis of microglia motility and morphology, notably so in research on neurodegenerative diseases. In this paper, we propose an automatic 3D tracing algorithm called C3VFC that uses vector field convolution to find the critical points along the centerline of an object and trace paths that traverse back to the soma of every cell in an image. The solution provides detection and labeling of multiple cells in an image over time, leading to multi-object reconstruction. The reconstruction results can be used to extract bioinformatics from temporal data in different settings. The C3VFC reconstruction results found up to a 53% improvement on the next best performing state-of-the-art tracing method. C3VFC achieved the highest accuracy scores, in relation to the baseline results, in four of the five different measures: Entire structure average, the average bi-directional entire structure average, the different structure average, and the percentage of different structures.

scholarly journals Pairings between bounded divergence-measure vector fields and BV functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Graziano Crasta ◽  
Virginia De Cicco ◽  
Annalisa Malusa
Keyword(s):  
Vector Field ◽  
Conservation Laws ◽  
Bounded Variation ◽  
Vector Fields ◽  
Hyperbolic Conservation Laws ◽  
Divergence Measure ◽  
Compensated Compactness ◽  
Hyperbolic Conservation ◽  
Bv Functions ◽  
Bounded Functions

AbstractWe introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], all the main properties and features (e.g. coarea, Leibniz, and Gauss–Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in \mathrm{BV}. We remark that the standard pairing in general does not share this property.

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