Related papers: A M\"obius invariant discretization of O'Hara's M\…

We considered random discrete approximation of O'Hara energy. O'Hara energy is the energy defined for a knot, and O'Hara energy was introduced for defining the standard shape for each knot class (equivalence class by ambient isotopy) by…

The M\"{o}bius energy, defined by O'Hara, is one of the knot energies, and named after the M\"{o}bius invariant property which was shown by Freedman-He-Wang. The energy can be decomposed into three parts, each of which is M\"{o}bius…

O'Hara introduced several functionals as knot energies. One of them is the M\"{o}bius energy. We know its M\"{o}bius invariance from Doyle-Schramm's cosine formula. It is also known that the M\"{o}bius energy was decomposed into three…

In this short article, we extend the cosine formula for the M\"{o}bius energy to generalized O'Hara energies. The newly derived formula gives us a condition for which the right circle minimizes the energy under the length-constraint.…

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 investigate a discrete version of the M\"obius energy, that is of geometric interest in its own right and is defined on equilateral polygons with $n$ segments. We show that the $\Gamma$-limit regarding $L^{q}$ or $W^{1,q}$ convergence,…

In the present paper we investigate generalizations of O'Hara's M\"obius energy on curves \cite{ohara_1991a}, to M\"obius-invariant energies on non-smooth subsets of $\R^n$ of arbitrary dimension and co-dimension. In particular, we show…

The O'Hara energies, introduced by Jun O'Hara in 1991, were proposed to answer the question of what is a "good" figure in a given knot type. A property of the O'Hara energies is that the "better" the figure of a knot is, the less the energy…

The M\"{o}bius energy is one of the knot energies, and is named after its M\"{o}bius invariant property. It is known to have several different expressions. One is in terms of the cosine of conformal angle, and is called the cosine formula.…

This work is motivated by the classical discrete elastic rod model by Audoly et al. We derive a discrete version of the Kirchhoff elastic energy for rods undergoing bending and torsion and prove $\Gamma$-convergence to the continuous model.…

Using the Wilson formulation of lattice gauge theories, a gauge invariant grid discretization of a one-particle Hamiltonian in the presence of an external electromagnetic field is proposed. This Hamiltonian is compared both with that…

We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of $\Gamma$-convergence. Variants of this discrete energy have been…

The present chapter gives an overview on results for discrete knot energies. These discrete energies are designed to make swift numerical computations and thus open the field to computational methods. Additionally, they provide an…

O'Hara's energies, introduced by Jun O'Hara, were proposed to answer the question of what is the canonical shape in a given knot type, and were configured so that the less the energy value of a knot is, the "better" its shape is. The…

We present two semidiscretizations of the Camassa-Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line,…

We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…

Under suitable technical conditions we show that minimisers of the discrete interaction energy for attractive-repulsive potentials converge to minimisers of the corresponding continuum energy as the number of particles goes to infinity. We…

Let $M$ be a smooth manifold and $\Gamma$ a group acting on $M$ by diffeomorphisms; which means that there is a group morphism $\rho:\Gamma\rightarrow \mathrm{Diff}(M)$ from $\Gamma$ to the group of diffeomorphisms of $M$. For any such…

Using the method of the "exact discretization" of the Schr\"odinger equation, we propose a particular discretized version of the N=2 Supersymmetric Quantum Mechanics. After defining the corresponding shape invariance condition, we show that…

We derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal…