Vortex filament equation for a regular polygon

In this paper, we study the evolution of the vortex filament equation,$$X_t = X_s \wedge X_{ss},$$with $X(s, 0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that $X(s, t)$ is also a polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gau$\beta$ sum. We also study the fractal behaviour of $X(0, t)$, relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal

On the Relationship between the One-Corner Problem and the M−M-Corner Problem for the Vortex Filament Equation

In this paper, we give evidence that the evolution of the vortex filament equation (VFE) for a regular M-corner polygon as initial datum can be explained at infinitesimal times as the superposition of M one-corner initial data. This fact is mainly sustained with the calculation of the speed of the center of mass; in particular, we show that several conjectures made at the numerical level are in agreement with the theoretical expectations. Moreover, due to the spatial periodicity, the evolution of VFE at later times can be understood as the nonlinear interaction of infinitely many filaments, one for each corner; and this interaction turns out to be some kind of nonlinear Talbot effect. We also give very strong numerical evidence of the transfer of energy and linear momentum for the M-corner case; and the numerical experiments carried out provide new arguments that support the multifractal character of the trajectory defined by one of the corners of the initial polygon

The Vortex Filament Equation as a Pseudorandom Generator

In this paper, we consider the evolution of the so-called vortex filament equation (VFE), $$X_t = X_s \wedge X_{ss},$$ taking a planar regular polygon of M sides as initial datum. We study VFE from a completely novel point of view: that of an evolution equation which yields a very good generator of pseudorandom numbers in a completely natural way. This essential randomness of VFE is in agreement with the randomness of the physical phenomena upon which it is based

Vortex Filament Equation for a regular polygon in the hyperbolic plane

The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case

On the Evolution of the Vortex Filament Equation for regular M-polygons with nonzero torsion

In this paper, we consider the evolution of the Vortex Filament equa- tion (VFE): Xt = Xs ∧ Xss, taking M-sided regular polygons with nonzero torsion as initial data. Us- ing algebraic techniques, backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; more- over, the multifractal trajectory of the point X(0,t) is not planar, and appears to be a helix for large times. These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as the one already established for circles. This is accomplished by showing the existence of variants of the so-called Rie- mann’s non-differentiable function that are as close to smooth curves as desired, when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the selfsimilar solutions of VFE have finite renormalized energy

Ultrathin carbon nanotube with single, double, and triple bonds

A metastable carbon nanotube with single, double, and triple bonds has been predicted from abinitio simulation. It results from the relaxation of an ideal carbon nanotube with chirality (2,1), without any potential barrier between the ideal nanotube and the new structure. Ten-membered carbon rings are formed by breaking carbon bonds between adjacent hexagons; eight-membered rings, already present in the ideal structure, become the smallest rings. This structure is stable in molecular dynamics simulations at temperatures up to 1000K. Raman, infrared, and optical absorption spectra are simulated to allow its identification in the laboratory. The structure can be described as a double helical chain with alternating single, double, and triple bonds, where the chains are bridged by single bondsThis work was supported by Grants No. SB2010-0119 (MEC), No. CTQ2010-19232 (MICIN), and No. A1/035856/11 (AECID

Complementarity between dark matter direct searches and CEν\nuNS experiments in U(1)′U(1)' models

We explore the possibility of having a fermionic dark matter candidate within $U(1)'$ models for CE$\nu$NS experiments in light of the latest COHERENT data and the current and future dark matter direct detection experiments. A vector-like fermionic dark matter has been introduced which is charged under $U(1)'$ symmetry, naturally stable after spontaneous symmetry breaking. We perform a complementary investigation using CE$\nu$NS experiments and dark matter direct detection searches to explore dark matter as well as $Z^{\prime}$ boson parameter space. Depending on numerous other constraints arising from the beam dump, LHCb, BABAR, and the forthcoming reactor experiment proposed by the SBC collaboration, we explore the allowed region of $Z^{\prime}$ portal dark matter.Comment: Matches published version. 21 pages, 5 figures, 2 table

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

In this paper we develop general LP and ILP techniques to find an approximate solution with improved objective value close to an existing solution. The task of improving an approximate solution is closely related to a classical theorem of Cook et al. in the sensitivity analysis for LPs and ILPs. This result is often applied in designing robust algorithms for online problems. We apply our new techniques to the online bin packing problem, where it is allowed to reassign a certain number of items, measured by the migration factor. The migration factor is defined by the total size of reassigned items divided by the size of the arriving item. We obtain a robust asymptotic fully polynomial time approximation scheme (AFPTAS) for the online bin packing problem with migration factor bounded by a polynomial in $\frac{1}{\epsilon}$. This answers an open question stated by Epstein and Levin in the affirmative. As a byproduct we prove an approximate variant of the sensitivity theorem by Cook at el. for linear programs

Gomphrena colosacana Hunz. & Subils var. colosacana

Tama, La Ventana, Lomas de ColosacánpublishedVersio