scholarly journals Euler equations and trace properties of minimizers of a functional for motion compensated inpainting

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Riccardo March ◽  
Giuseppe Riey
Keyword(s):  
Motion Estimation ◽  
Euler Equations ◽  
Vector Fields ◽  
Video Inpainting ◽  
Singular Sets ◽  
Vectorial Functions ◽  
Functions Of Bounded Variations

<p style='text-indent:20px;'>We compute the Euler equations of a functional useful for simultaneous video inpainting and motion estimation, which was obtained in [<xref ref-type="bibr" rid="b17">17</xref>] as the relaxation of a modified version of the functional proposed in [<xref ref-type="bibr" rid="b16">16</xref>]. The functional is defined on vectorial functions of bounded variations, therefore we also get the Euler equations holding on the singular sets of minimizers, highlighting in particular the conditions on the jump sets. Such conditions are expressed by means of traces of geometrically meaningful vector fields and characterized as pointwise limits of averages on cylinders with axes parallel to the unit normals to the jump sets.</p>

scholarly journals C-Flows on a Lie Group for Euler Equations

1970 ◽  
Vol 40 ◽  
pp. 67-84
Author(s):  
Yoshihei Hasegawa
Keyword(s):  
Euler Equations ◽  
Lie Group ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Invariant Vector ◽  
Left Invariant Vector ◽  
Left Invariant Riemannian Metric ◽  
Invariant Vector Fields

The purpose of this paper is to determine left-invariant vector fields on a Lie group G with a left-invariant Riemannian metric which induces C- flows on G.

Structure and stability of solutions of the Euler equations: a lagrangian approach

1990 ◽  
Vol 333 (1631) ◽  
pp. 321-342 ◽  
Keyword(s):  
Euler Equations ◽  
Vector Fields ◽  
Saddle Points ◽  
Relaxation Processes ◽  
Two Dimensional ◽  
Scalar And Vector Fields ◽  
Euler Flows ◽  
Relaxation Procedure ◽  
Unbounded Fluid

This paper reviews methods that are essentially lagrangian in character for determination of solutions of the Euler equations having prescribed topological characteristics. These methods depend in the first instance on the existence of lagrangian invariants for convected scalar and vector fields. Among these, the helicity invariant for a convected or ‘frozen-in’ vector field has particular significance. These invariants, and the associated topological interpretation are discussed in §§1 and 2. In §3 the method of magnetic relaxation to magnetostatic equilibria of prescribed topology is briefly described. This provides a powerful method for determining steady Euler flows through the well-known exact analogy between Euler flows and magnetostatic equilibria. Stability considerations relating to magnetostatic equilibria obtained in this way and to the analogous Euler flows are reviewed in §4. In §5 the related relaxation procedure is discussed; for two-dimensional and axisymmetric situations this technique provides stable solutions of the Euler equations for which the vorticity field has prescribed topology. The concept of flow signature is described in §6: this is the relevant topological characteristic for two-dimensional or axisymmetric situations, which is conserved during frozen-field relaxation processes. In §§7 and 8, the formation of tangential discontinuities as a normal part of the relaxation process when saddle points of the frozen-field are present is discussed. Section 9 considers briefly the application of these ideas to the theory of vortons, i.e. rotational disturbances that propagate without change of structure in an unbounded fluid. The paper concludes with a brief discussion, with comment on the possible development of the results in the context of turbulence.

Sectional-hyperbolic systems

2008 ◽  
Vol 28 (5) ◽  
pp. 1587-1597 ◽  
Author(s):  
R. METZGER ◽  
C. MORALES
Keyword(s):  
Vector Fields ◽  
Hyperbolic Systems ◽  
Negative Answer ◽  
Lorenz Attractor ◽  
Closed Orbits ◽  
Higher Dimensional ◽  
Singular Sets ◽  
Uniform Hyperbolicity ◽  
Mathematical Sciences ◽  
Lorenz Attractors

AbstractWe introduce a class of vector fields onn-manifolds containing the hyperbolic systems, the singular-hyperbolic systems on 3-manifolds, the multidimensional Lorenz attractors and the robust transitive singular sets in Liet al[Robust transitive singular sets via approach of an extended linear Poincaré flow.Discrete Contin. Dyn. Syst.13(2) (2005), 239–269]. We prove that the closed orbits of a system in such a class are hyperbolic in a persistent way, a property which is false for higher-dimensional singular-hyperbolic systems. We also prove that the singularities in the robust transitive sets in Liet alare similar to those in the multidimensional Lorenz attractor. Our results will give a partial negative answer to Problem 9.26 in Bonattiet al[Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III). Springer, Berlin, 2005].

LIFTS, JETS AND REDUCED DYNAMICS

2011 ◽  
Vol 08 (02) ◽  
pp. 331-344 ◽  
Author(s):  
OĞUL ESEN ◽  
HASAN GÜMRAL
Keyword(s):  
Incompressible Fluid ◽  
Euler Equations ◽  
Continuum Mechanics ◽  
Vector Fields ◽  
Kinetic Equations ◽  
Ideal Incompressible Fluid ◽  
Hamiltonian Vector ◽  
Plasma Dynamics ◽  
Hamiltonian Vector Fields ◽  
Vlasov Equations

We show that complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics. We discuss Euler equations for ideal incompressible fluid and momentum-Vlasov equations of plasma dynamics in connection with the lifts of divergence-free and Hamiltonian vector fields, respectively. As a further application, we obtain kinetic equations of particles moving with the flow of contact vector fields both from Lie–Poisson reductions and with the techniques of present framework.

scholarly journals A characterization of 3D steady Euler flows using commuting zero-flux homologies

2020 ◽  
pp. 1-16
Author(s):  
DANIEL PERALTA-SALAS ◽  
ANA RECHTMAN ◽  
FRANCISCO TORRES DE LIZAUR
Keyword(s):  
Euler Equations ◽  
Vector Fields ◽  
Higher Dimensions ◽  
Riemannian Metric ◽  
Euler Flows ◽  
Homological Characterization ◽  
Steady Solutions ◽  
Volume Preserving

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved.

scholarly journals Temporal Adaptive Alignment Network for Deep Video Inpainting

2020 ◽  
Author(s):  
Ruixin Liu ◽  
Zhenyu Weng ◽  
Yuesheng Zhu ◽  
Bairong Li
Keyword(s):  
Deep Learning ◽  
Motion Estimation ◽  
Optical Flow ◽  
Reference Frame ◽  
Reference Frames ◽  
Qualitative Evaluation ◽  
Temporal Information ◽  
Temporal Alignment ◽  
Video Inpainting ◽  
Target Frame

Video inpainting aims to synthesize visually pleasant and temporally consistent content in missing regions of video. Due to a variety of motions across different frames, it is highly challenging to utilize effective temporal information to recover videos. Existing deep learning based methods usually estimate optical flow to align frames and thereby exploit useful information between frames. However, these methods tend to generate artifacts once the estimated optical flow is inaccurate. To alleviate above problem, we propose a novel end-to-end Temporal Adaptive Alignment Network(TAAN) for video inpainting. The TAAN aligns reference frames with target frame via implicit motion estimation at a feature level and then reconstruct target frame by taking the aggregated aligned reference frame features as input. In the proposed network, a Temporal Adaptive Alignment (TAA) module based on deformable convolutions is designed to perform temporal alignment in a local, dense and adaptive manner. Both quantitative and qualitative evaluation results show that our method significantly outperforms existing deep learning based methods.

scholarly journals Infinitesimal quantum group from helicity in fluid mechanics

2020 ◽  
Vol 35 (30) ◽  
pp. 2050245
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Quantum Group ◽  
Lie Algebra ◽  
Euler Equations ◽  
Ideal Fluid ◽  
Vector Fields ◽  
Classical Fluid ◽  
Manin Triple ◽  
Isotropic Subspaces

Arnold showed that the Euler equations of an ideal fluid describe geodesics in the Lie algebra of incompressible vector fields. We will show that helicity induces a splitting of the Lie algebra into two isotropic subspaces, forming a Manin triple. Viewed another way, this shows that there is an infinitesimal quantum group (a.k.a. Lie bi-algebra) underlying classical fluid mechanics.

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