Related papers: Bounds on multiple self-avoiding polygons
We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or…
We investigate the motion of two overlapping polymers with self-avoidance confined in a narrow 2d box. A statistical model is constructed using blob free-energy arguments. We find spontaneous segregation under the condition: $L > R_{//}$,…
This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way…
Every equivelar abstract polytope of type $\{p_1, \ldots, p_{n-1}\}$ has at least $2p_1 \cdots p_{n-1}$ flags. Polytopes that attain this lower bound are called tight. Here we investigate the question of under what conditions there is a…
We study the Escape Problem for discrete-time linear dynamical systems over compact semialgebraic sets. We establish a uniform upper bound on the number of iterations it takes for every orbit of a rational matrix to escape a compact…
We present a computer simulation study of the compact self-avoiding loops as regards their length and topological state. We use a Hamiltonian closed path on the cubic-shaped segment of a 3D cubic lattice as a model of a compact polymer. The…
A surface embedded in space, in such a way that each point has a neighborhood within which the surface is a terrain, projects to an immersed surface in the plane, the boundary of which is a self-intersecting curve. Under what circumstances…
We consider a self-avoiding walk model of polymer adsorption where the adsorbed polymer can be desorbed by the application of a force. In this paper the force is applied normal to the surface at the last vertex of the walk. We prove that…
A rectilinear polygon is a polygon whose edges are axis-aligned. Walking counterclockwise on the boundary of such a polygon yields a sequence of left turns and right turns. The number of left turns always equals the number of right turns…
A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\ge 7$. Finding the largest small $n$-gon for a given number $n\ge 3$ can be formulated as a nonconvex quadratically…
We give exact relations for a number of amplitude combinations that occur in the study of self-avoiding walks, polygons and lattice trails. In particular, we elucidate the lattice-dependent factors which occur in those combinations which…
Exactly solvable two-dimensional polygon models, counted by perimeter and area, are described by $q$-algebraic functional equations. We provide techniques to extract the scaling behaviour of these models up to arbitrary order and apply them…
A polygon is \textit{small} if it has unit diameter. The maximal area of a small polygon with a fixed number of sides $n$ is not known when $n$ is even and $n\geq14$. We determine an improved lower bound for the maximal area of a small…
Let $G$ be an infinite, vertex-transitive lattice with degree $\lambda$ and fix a vertex on it. Consider all cycles of length exactly $l$ from this vertex to itself on $G$. Erasing loops chronologically from these cycles, what is the…
Over an algebraically closed base field $k$ of characteristic 2, the ring $R^G$ of invariants is studied, $G$ being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring $R$ of the…
We analyze new data for self-avoiding polygons, on the square and triangular lattices, enumerated by both perimeter and area, providing evidence that the scaling function is the logarithm of an Airy function. The results imply universal…
We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus…
We present a new and more efficient implementation of transfer-matrix methods for exact enumerations of lattice objects. The new method is illustrated by an application to the enumeration of self-avoiding polygons on the square lattice. A…
A closed plane meander of order $n$ is a closed self-avoiding curve intersecting an infinite line $2n$ times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm,…
E. Bayer-Fluckiger gave a necessary and sufficient condition for a polynomial to be realized as the characteristic polynomial of a semisimple isometry of an even unimodular lattice, by describing the local-global obstruction, and the author…