Related papers: Moebius energy of thick knots
We present a numerical analysis of an incompressible decaying magnetohydrodynamic turbulence run on a grid of 1536^3 points. The Taylor Reynolds number at the maximum of dissipation is ~1100, and the initial condition is a superposition of…
In this work the thermodynamic properties of short polymer knots (up to 120 segments) defined on a simple cubic lattice are studied with the help of the Wang-Landau Monte Carlo algorithm. The sampling process is performed using pivot…
Simulations of decaying magnetohydrodynamic (MHD) turbulence are performed with a fluid and a kinetic code. The initial condition is an ensemble of long-wavelength, counter-propagating, shear-Alfv\'{e}n waves, which interact and rapidly…
We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two…
We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary. In particular, we obtain potentially sharp bounds on the complexity of torus knot complements.
We have advanced the energy and flux budget (EFB) turbulence closure theory that takes into account a two-way coupling between internal gravity waves (IGW) and the shear-free stably stratified turbulence. This theory is based on the budget…
We explore the energetics of microwaves interacting with a double quantum dot photodiode and show wave-particle aspects in photon-assisted tunneling. The experiments show that the single photon energy sets the relevant absorption energy in…
The M\"{o}bius strip, a long sheet of paper whose ends are glued together after a $180^{\circ}$ twist, has remarkable geometric and topological properties. Here, we consider dielectric M\"{o}bius strips of finite width and investigate the…
In this brief note, we consider a wave equation that has both trapping and a complex potential. For this problem, we prove a uniform bound on the energy and a Morawetz (or integrated local energy decay) estimate. The equation is a model…
The FitzHugh-Nagumo equation provides a simple mathematical model of cardiac tissue as an excitable medium hosting spiral wave vortices. Here we present extensive numerical simulations studying long-term dynamics of knotted vortex string…
The present work studies the isotropic and homogeneous turbulence for incompressible fluids through a specific Lyapunov analysis, assuming that the turbulence is due to the bifurcations associated to the velocity field. The analysis…
In this paper, we propose a discrete version of O'Hara's knot energy defined on polygons embedded in the Euclid space. It is shown that values of the discrete energy of polygons inscribing the curve which has bounded O'Hara's energy…
We prove that various families of energies relevant in geometric knot theory satisfy the Palais-Smale condition (PS) on submanifolds of arclength para\-metrized knots. These energies include linear combinations of the Euler-Bernoulli…
We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the M\"obius energy. For the M\"obius energy, due to the celebrated work of Freedman, He,…
We consider the waveguiding by thin patterned slabs embedded in a homogeneous medium. In the longwave limit, the wave spectra of slabs are found to be well described by a single frequency-independent parameter, which we call the "guiding…
For a general dark-energy equation of state, we estimate the maximum possible radius of massive structures that are not destabilized by the acceleration of the cosmological expansion. A comparison with known stable structures constrains the…
We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of "tight" knotted tubes with minimal length and fixed circular cross-section. Our curves approximately…
A ribbon is, intuitively, a smooth mapping of an annulus $S^1 \times I$ in 3-space having constant width $\varepsilon$. This can be formalized as a triple $(x,\varepsilon, \mathbf{u})$ where $x$ is smooth curve in 3-space and $\mathbf{u}$…
The vacuum polarization energy is the leading quantum correction to the classical energy of a soliton. We study this energy for two-component solitons in one space dimension as a function of the soliton's topological charge. We find that…
The search for solutions to the theory of weakly non-linear internal gravity wave turbulence is an active research topic. It is notably stimulated by the fact that this regime could drive fine-scale ocean dynamics for which the…