scholarly journals On generalization of Zermelo navigation problem on Riemannian manifolds

2019 ◽  
Vol 16 (04) ◽  
pp. 1950058 ◽  
Author(s):  
Piotr Kopacz
Keyword(s):  
Vector Field ◽  
Riemannian Manifolds ◽  
Velocity Vector ◽  
Inner Product ◽  
Finsler Metric ◽  
Velocity Vector Field ◽  
Navigation Problem ◽  
Navigation Data ◽  
Dimension 2

With the extended navigation data, we consider the generalized Zermelo navigation on Riemannian manifolds, admitting a space-dependent ship’s speed in the presence of perturbation determined by a weak velocity vector field, with application of Finsler metric of Randers type. The approach is shown via indicatrix and inner product. We also compare our findings in the context of conformality for the cases of weak and critical winds. The study is illustrated with the example in dimension 2.

scholarly journals A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation

2017 ◽  
Vol 25 (3) ◽  
pp. 107-123 ◽  
Author(s):  
Piotr Kopacz
Keyword(s):  
Vector Field ◽  
Critical Velocity ◽  
Riemannian Manifolds ◽  
Velocity Vector ◽  
Finsler Metric ◽  
Velocity Vector Field ◽  
Navigation Problem ◽  
Space Dependence

Abstract We generalize the Zermelo navigation on Riemannian manifolds (M; h), admitting a space dependence of a ship's speed 0 < |u(x)|h ≤ 1 in the presence of a perturbation W̃ determined by a strong (critical) velocity vector field satisfying |W̃ (x)|h = |u(x)|h, with application of Finsler metric of Kropina type.

EVOLUTION OF CENTRAL MOMENTS FOR A GENERAL-RELATIVISTIC BOLTZMANN EQUATION: THE CLOSURE BY ENTROPY MAXIMIZATION

2002 ◽  
Vol 14 (05) ◽  
pp. 469-510 ◽  
Author(s):  
ZBIGNIEW BANACH ◽  
WIESLAW LARECKI
Keyword(s):  
Vector Field ◽  
Boltzmann Equation ◽  
Velocity Vector ◽  
Hyperbolic Systems ◽  
Velocity Vector Field ◽  
Field Equations ◽  
Entropy Maximization ◽  
Central Moments ◽  
Relativistic Boltzmann Equation ◽  
Lorentzian Metric

Beginning from the relativistic Boltzmann equation in a curved space-time, and assuming that there exists a fiducial congruence of timelike world lines with four-velocity vector field u, it is the aim of this paper to present a systematic derivation of a hierarchy of closed systems of moment equations. These systems are found by using the closure by entropy maximization. Our concepts are primarily applied to the formalism of central moments because if an alternative and more familiar theory of covariant moments is taken into account, then the method of maximum entropy is ill-defined in a neighborhood of equilibrium states. The central moments are not covariant in the following sense: two observers looking at the same relativistic gas will, in general, extract two different sets of central moments, not related to each other by a tensorial linear transformation. After a brief review of the formalism of trace-free symmetric spacelike tensors, the differential equations for irreducible central moments are obtained and compared with those of Ellis et al. [Ann. Phys. (NY)150 (1983) 455]. We derive some auxiliary algebraic identities which involve the set of central moments and the corresponding set of Lagrange multipliers; these identities enable us to show that there is an additional balance law interpreted as the equation of balance of entropy. The above results are valid for an arbitrary choice of the Lorentzian metric g and the four-velocity vector field u. Later, the definition of u as in the well-known theory of Arnowitt, Deser, and Misner is proposed in order to construct a hierarchy of symmetric hyperbolic systems of field equations. Also, the Eckart and Landau–Lifshitz definitions of u are discussed. Specifically, it is demonstrated that they lead, in general, to the systems of nonconservative equations.

scholarly journals Blood flow velocity vector field reconstruction from dual-beam bidirectional Doppler OCT measurements in retinal veins

2015 ◽  
Vol 6 (5) ◽  
pp. 1599 ◽  
Author(s):  
Gerold C. Aschinger ◽  
Leopold Schmetterer ◽  
Veronika Doblhoff-Dier ◽  
Rainer A. Leitgeb ◽  
Gerhard Garhöfer ◽  
...  
Keyword(s):  
Blood Flow ◽  
Vector Field ◽  
Flow Velocity ◽  
Blood Flow Velocity ◽  
Velocity Vector ◽  
Velocity Vector Field ◽  
Field Reconstruction ◽  
Dual Beam ◽  
Doppler Oct

A new characterization for classical Bernoulli–Euler elastic curves in Rn

2019 ◽  
Vol 16 (07) ◽  
pp. 1950103
Author(s):  
Ayşe Altin
Keyword(s):  
Vector Field ◽  
Velocity Vector ◽  
Velocity Vector Field ◽  
Elastic Curve ◽  
Elastic Curves

In this paper, we show that the velocity vector field of classical Bernoulli–Euler elastic curve is harmonic in [Formula: see text] space. We propose a new characterization for classical Bernoulli–Euler elastic curves and plot graphs of examples that satisfy this characterization.

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