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On new characterization of inextensible flows of space-like curves in de Sitter space

100%
Yeneroğlu M.
Open Mathematics
|
2016
|
tom 14
|
nr 1
946-954
EN
Elastica and inextensible flows of curves play an important role in practical applications. In this paper, we construct a new characterization of inextensible flows by using elastica in space. The inextensible flow is completely determined for any space-like curve in de Sitter space [...] S 1 3 $\mathbb{S}_{1}^{3}$. Finally, we give some characterizations for curvatures of a space-like curve in de Sitter space [...] S 1 3 $\mathbb{S}_{1}^{3}$.
2
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Discrete thickness

64%
Scholtes S.
Molecular Based Mathematical Biology
|
2014
|
tom 2
|
nr 1
EN
We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are deffned on equilateral polygons with n vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the Γ-limit of the discrete ropelength for n → ∞, regarding the topology induced by the Sobolev norm ‖ · ‖ W1,∞(S1,ℝd). This result directly implies the convergence of almost minimizers of the discrete energies in a fixed knot class to minimizers of the smooth energy.Moreover,we show that the unique absolute minimizer of inverse discrete thickness is the regular n-gon.
3
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Some minimization problems for planar networks of elastic curves

64%
Dall’Acqua A., Pluda A.
Geometric Flows
|
2017
|
tom 2
|
nr 1
EN
In this note we announce some results that will appear in [6] on the minimization of the functional F(Γ) = ∫Γk2 + 1 ds, where Γ is a network of three curves with fixed equal angles at the two junctions. The informal description of the results is accompanied by a partial review of the theory of elasticae and a diffuse discussion about the onset of interesting variants of the original problem passing from curves to networks. The considered energy functional F is given by the elastic energy and a term that penalize the total length of the network.We will show that penalizing the length is tantamount to fix it. The paper is concluded with the explicit computation of the penalized elastic energy of the “Figure Eight”, namely the unique closed elastica with self-intersections (see Figure 1).
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