Related papers: Polygonal approximation and energy of smooth knots
We consider random walk model of a semi-flexible polymer chain on a square and a cubic lattice to enumerate conformations of the polymer chain in two and three dimensions, respectively. The bending energy of the chain is assumed as the key…
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…
Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions, or they do not correctly account for intrinsic curvature, which leads to unnatural-looking…
Two spherical bubbles with changing radii are considered to be moving in ideal fluid along their center-line. The exact expression for the fluid kinetic energy is obtained. The Stokes stream function is expanded in Gegenbauer polynomials in…
We consider a geodesic $\gamma$ of length $2L$ in an oriented Riemannian manifold $(\mathcal M, g)$ and a thin tube $\Omega^*_h$ around $\gamma$ of radius $h$. We study an 'elastic' energy per unit volume $E_h(u)$ of maps $u$ from…
This is the first of three papers on the short-distance properties of the energy-momentum tensor in field theory. We study the energy-momentum tensor for renormalized field theory in curved space. We postulate an exact Ward identity of the…
In geometry processing, smoothness energies are commonly used to model scattered data interpolation, dense data denoising, and regularization during shape optimization. The squared Laplacian energy is a popular choice of energy and has a…
The M\"obius energy is a well-studied knot energy with nice regularity and self-repulsive properties. Stationary curves under the M\"obius energy gradient are of significant theoretical interest as they they can indicate equilibrium states…
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.
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
We study the notion of heterotopic energy defined as the limit of Sobolev energies of Sobolev mappings in a given homotopy class approximating almost everywhere a given Sobolev mapping. We show that the heterotopic energy is finite if and…
The Willmore energy plays a central role in the conformal geometry of surfaces in the conformal 3-sphere \(S^3\). It also arises as the leading term in variational problems ranging from black holes, to elasticity, and cell biology. In the…
The development of computational resources has made it possible to determine upper bounds for atomic and molecular energies with high precision. Yet, error bounds to the computed energies have been available only as estimates. In this…
A regular $n$-gon inscribing a knot is a sequence of $n$ points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular $n$-gon for any $n$.
We report on the geometry and mechanics of knotted stiff strings. We discuss both closed and open knots. Our two main results are: (i) Their equilibrium energy as well as the equilibrium tension for open knots depend on the type of knot as…
In this paper, we get the sharpest known to date lower bounds for the minimal Green energy of the compact harmonic manifolds of any dimension.
In this article we study point configurations minimizing the discrete energy on a compact Riemannian manifold, where the energy kernel is taken to be the Green's function for the Laplacian. We show that every point in a minimizing…
On a smooth compact connected $d$-dimensional Riemannian manifold $M$, if $0 < s < d$ then an asymptotically equidistributed sequence of finite subsets of $M$ that is also well-separated yields a sequence of Riesz $s$-energies that…
Minimizing the elastic free energy of a thin sheet of nematic polymer network among smooth isometric immersions is the strategy purported by the mainstream theory. In this paper, we broaden the class of admissible spontaneous deformations:…
We prove a coisotropic intersection result and deduce the following: 1. Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. 2. A stable non-squeezing…