相关论文: Approximating Ropelength by Energy Functions
We describe several configurations of clasped ropes which are balanced and thus critical for the Gehring ropelength problem of arXiv:math.DG/0402212.
We consider the rope climber fall problem in two different settings. The simplest formulation of the problem is when the climber falls from a given altitude and is attached to one end of the rope while the other end of the rope is attached…
We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. Specifically, we relate w(K) to w(J), where w(K) is the width of K in the sense of Gabai. We use this information to demonstrate that thin…
Thermopower of molecular junctions is sensitive to details in the junction and may increase, decrease, or saturate with increasing chain length, depending on the system. Using McConnell's theory for exponentially suppressed transport…
Power grids exhibit patterns of reaction to outages similar to complex networks. Blackout sequences follow power laws, as complex systems operating near a critical point. Here, the tolerance of electric power grids to both accidental and…
We study the incompressible surfaces in the exterior of a cable knot and use this to compute the representativity and waist of most cable knots.
The slicing degree of a knot $K$ is defined as the smallest integer $k$ such that $K$ is $k$-slice in $\#^n \overline{\mathbb{CP}^2}$ for some $n$. In this paper, we establish bounds for the slicing degrees of knots using Rasmussen's…
We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof…
Hopf solitons in the Skyrme-Faddeev model are string-like topological solitons classified by the integer-valued Hopf charge. In this paper we introduce an approximate description of Hopf solitons in terms of elastic rods. The general form…
This paper deals with the approximation of discrete real-valued functions by first-degree splines (broken lines) with free knots for arbitrary $L_p$-norms ($1 \leq p \leq \infty)$. We prove the existence of best approximations und derive…
A measure called Physical Complexity is established and calculated for a population of sequences, based on statistical physics, automata theory, and information theory. It is a measure of the quantity of information in an organism's genome.…
Entropy and information provide natural measures of correlation among elements in a network. We construct here the information theoretic analog of connected correlation functions: irreducible $N$--point correlation is measured by a decrease…
A usual classification tool to study a fractal interface is the computation of its fractal dimension. But a recent method developed by Y. Heurteaux and S. Jaffard proposes to compute either weak and strong accessibility exponents or local…
In the present paper we prove a family of tight upper and lower bounds for the Shannon entropy and von Neumann entropy based on the p-norms. This allows us to have an entropy estimate, a criterion for the finiteness of it and a bound on the…
We develop a reinforcement learning pipeline for simplifying knot diagrams. A trained agent learns move proposals and a value heuristic for navigating Reidemeister moves. The pipeline applies to arbitrary knots and links; we test it on…
The problem of finding robust and effective methods for locating entanglement in embedded curves is relevant to both applications and theoretical investigations. Rather than focusing on an exact determination, we introduce the knot…
In this report, I will start by first giving a brief introduction on knots to build some intuition before beginning the more rigorous review in the Literature Review section. There, I will define knot equivalence, the Jones polynomial…
Ribbons are long narrow strips possessing three distinct material length scales (thickness, width, and length) which allow them to produce unique shapes unobtainable by wires or filaments. For example when a ribbon has half a twist and is…
We develop an instanton technique for calculations of correlation functions characterizing statistical behavior of the elastic string in disordered media and apply the proposed approach to correlations of string free energies corresponding…
We present a simple method to efficiently compute a lower limit of the topological entropy and its spatial distribution for two-dimensional mappings. These mappings could represent either two-dimensional time-periodic fluid flows or…