scholarly journals Quantization of a particle on a two-dimensional manifold of constant curvature

2014 ◽  
Vol 55 (10) ◽  
pp. 102102 ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Constant Curvature ◽  
Dimensional Manifold ◽  
Two Dimensional

scholarly journals Random Assignment Problems on 2d Manifolds

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
D. Benedetto ◽  
E. Caglioti ◽  
S. Caracciolo ◽  
M. D’Achille ◽  
G. Sicuro ◽  
...  
Keyword(s):  
Assignment Problem ◽  
Unit Area ◽  
Constant Term ◽  
Beltrami Operator ◽  
Explicit Calculation ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Random Assignment ◽  
Assignment Problems ◽  
Theoretical Formulation

AbstractWe consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold $$\Omega $$ Ω of unit area. It is known that the average cost scales as $$E_{\Omega }(N)\sim {1}/{2\pi }\ln N$$ E Ω ( N ) ∼ 1 / 2 π ln N with a correction that is at most of order $$\sqrt{\ln N\ln \ln N}$$ ln N ln ln N . In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $$\Omega $$ Ω -dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on $$\Omega $$ Ω . We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.

TOPONOGOV-TYPE AREA COMPARISON THEOREM OF 2-DIMENSIONAL MANIFOLDS

2013 ◽  
Vol 15 (03) ◽  
pp. 1350007
Author(s):  
XIAOLE SU ◽  
HONGWEI SUN ◽  
YUSHENG WANG
Keyword(s):  
Riemannian Manifold ◽  
Comparison Theorem ◽  
Constant Curvature ◽  
Dimensional Manifold ◽  
Geodesic Triangle ◽  
Type Area ◽  
Simply Connected

Let △p1p2p3 be a geodesic triangle on M, a complete 2-dimensional Riemannian manifold of curvature ≥ k, and let [Formula: see text] be its comparison triangle on [Formula: see text] (a complete and simply connected 2-dimensional manifold of constant curvature k). Our main result is that if △p1p2p3 is areable, then its area is not less than that of [Formula: see text].

scholarly journals Paracontact Metric κ , μ -Manifold Satisfying the Miao-Tam Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Dehe Li ◽  
Jiabin Yin
Keyword(s):  
Constant Curvature ◽  
Dimensional Manifold ◽  
Critical Equation ◽  
Negative Constant

In this paper, we classified the paracontact metric κ , μ -manifold satisfying the Miao-Tam critical equation with κ > − 1 . We proved that it is locally isometric to the product of a flat n + 1 -dimensional manifold and an n -dimensional manifold of negative constant curvature − 4 .

scholarly journals Two-dimensional Manifold with Point-like Defects

2015 ◽  
Vol 74 ◽  
pp. 32-35 ◽  
Author(s):  
V.A. Gani ◽  
A.E. Dmitriev ◽  
S.G. Rubin
Keyword(s):  
Dimensional Manifold ◽  
Two Dimensional

scholarly journals GENERALIZATION OF HASIMOTO'S TRANSFORMATION

2009 ◽  
Vol 06 (04) ◽  
pp. 625-630 ◽  
Author(s):  
MATHIEU MOLITOR
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Constant Curvature ◽  
Schrodinger Equation ◽  
Three Dimensional ◽  
Nonlinear Schrodinger Equation ◽  
Vortex Filament ◽  
Dimensional Manifold ◽  
Moving Frames ◽  
Natural Complex

In this paper, we generalize the famous Hasimoto's transformation by showing that the dynamics of a closed unidimensional vortex filament embedded in a three-dimensional manifold M of constant curvature, gives rise under Hasimoto's transformation to the nonlinear Schrödinger equation. We also give a natural interpretation of the function ψ introduced by Hasimoto in terms of moving frames associated to a natural complex bundle over the filament.

Energy transfer in rotating turbulence

1997 ◽  
Vol 337 ◽  
pp. 303-332 ◽  
Author(s):  
CLAUDE CAMBON ◽  
N. N. MANSOUR ◽  
F. S. GODEFERD
Keyword(s):  
Energy Transfer ◽  
Rotation Axis ◽  
Physical Space ◽  
Nonlinear Effects ◽  
Spectral Energy ◽  
Rossby Number ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Plane Normal

The influence of rotation on the spectral energy transfer of homogeneous turbulence is investigated in this paper. Given the fact that linear dynamics, e.g. the inertial waves regime found in an RDT (rapid distortion theory) analysis, cannot affect a homogeneous isotropic turbulent flow, the study of nonlinear dynamics is of prime importance in the case of rotating flows. Previous theoretical (including both weakly nonlinear and EDQNM theories), experimental and DNS (direct numerical simulation) results are collected here and compared in order to give a self-consistent picture of the nonlinear effects of rotation on turbulence.The inhibition of the energy cascade, which is linked to a reduction of the dissipation rate, is shown to be related to a damping of the energy transfer due to rotation. A model for this effect is quantified by a model equation for the derivative-skewness factor, which only involves a micro-Rossby number Roω=ω′/(2Ω) – ratio of r.m.s. vorticity and background vorticity – as the relevant rotation parameter, in accordance with DNS and EDQNM results.In addition, anisotropy is shown also to develop through nonlinear interactions modified by rotation, in an intermediate range of Rossby numbers (RoL<1 and Roω>1), which is characterized by a macro-Rossby number RoL based on an integral lengthscale L and the micro-Rossby number previously defined. This anisotropy is mainly an angular drain of spectral energy which tends to concentrate energy in the wave-plane normal to the rotation axis, which is exactly both the slow and the two-dimensional manifold. In addition, a polarization of the energy distribution in this slow two-dimensional manifold enhances horizontal (normal to the rotation axis) velocity components, and underlies the anisotropic structure of the integral length-scales. Finally a generalized EDQNM (eddy damped quasi-normal Markovian) model is used to predict the underlying spectral transfer structure and all the subsequent developments of classic anisotropy indicators in physical space. The results from the model are compared to recent LES results and are shown to agree well. While the EDQNM2 model was developed to simulate ‘strong’ turbulence, it is shown that it has a strong formal analogy with recent weakly nonlinear approaches to wave turbulence.

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