Related papers: Recent topics on the O'Hara energies

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing…

We present sufficient criteria for the equivalence of tame knots at low regularity. To this end, we introduce a localized version of Gromov's distortion for any closed path-connected subset of $\R^n$. If two such sets have local Gromov…

Energy randomness is a notion of partial randomness introduced by Diamondstone and Kjos-Hanssen to characterize the sequences that can be elements of a Martin-L\"of random closed set (in the sense of Barmpalias, Brodhead, Cenzer, Dashti,…

This article represents the first installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on sharp cubic energy estimates. Precisely, we introduce and…

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,…

By introducing a new topology, a representation formula of the Gamma limit of the Kobayashi-Warren-Carter energy is given in a multi-dimensional domain. A key step is to study the Gamma limit of a single-well Modica-Mortola functional. The…

We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second…

We prove the existence of symmetric critical torus knots for O'Hara's knot energy family $E_\alpha$, $\alpha\in (2,3)$ using Palais' classic principle of symmetric criticality. It turns out that in every torus knot class there are at least…

The Moebius energy of a knot is an energy functional for smooth curves based on an idea of self-repelling. If a knot has a thick tubular neighborhood, we would intuitively expect the energy to be low. In this paper, we give explicit bounds…

The factorization method was introduced by Schroedinger in 1940. Its use in bound-state problems is widely known, including in supersymmetric quantum mechanics; one can create a factorization chain, which simultaneously solves a sequence of…

We consider variational energies of the form \[E_H(u)=\frac12\int_\Omega H^2(\nabla u)\,dx\] defined on the Sobolev space $H^1_0(\Omega)$, where $H$ is a general seminorm. Our primary objective is to investigate optimization problems…

The energy of gravitating systems has been an issue since Einstein proposed general relativity: considered to be ill defined, having no proper local density. Energy-momentum is now regarded as \emph{quasi-local} (associated with a closed…

Light received from a cosmological source is redshifted with an apparent loss of energy, a problem first pointed out by Edwin Hubble in 1936. A new type of energy called Hubble Energy is introduced to restore the principle of energy…

We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(\Omega)}+\int_\Omega W(u)\,dx, $$ where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$…

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 an article about some elementary geometric and combinatorial natures of various knot energies. A related "new" knot invariant -- the X-crossing number -- is introduced.

We establish long-time existence of Banach gradient flows for generalised integral Menger curvatures and tangent-point energies, and for O'Hara's self-repulsive potentials $E^{\alpha,p}$. In order to do so, we employ the theory of curves of…

The relation between the derivative of the energy with respect to occupation number and the orbital energy, $\partial E/\partial n_i = \epsilon_i$, was first introduced by Slater for approximate total energy expressions such as Hartree-Fock…

This essay is dedicated to Prof. K.K. Phua on the occasion of his 80th birthday. While the contents of this essay are based on our recent papers$^{1,2}$ published in the International Journal of Modern Physics A, I have added many personal…

Let $(X,\omega)$ be a compact normal K\"ahler space, with Hodge metric $\omega$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical K\"ahler metrics, we introduce a geodesic…