Related papers: Critical exponents of plane meanders

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. For most concentrations of the scatterers the trajectories close…

The critical behaviour of correlation functions near a boundary is modified from that in the bulk. When the boundary is smooth this is known to be characterised by the surface scaling dimension $\xt$. We consider the case when the boundary…

Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to…

We consider a special case of the n-component cubic model on the square lattice, for which an expansion exists in Ising-like graphs. We construct a transfer matrix and perform a finite-size-scaling analysis to determine the critical points…

Classical vector analysis is the predominant formalism used by engineers of computational electromagnetism, despite the fact that manifold as a theoretical concept has existed for a century. This paper discusses the benefits of manifolds…

A meander can be seen as a pair of transversally intersecting simple closed curves on a 2-sphere. We consider pairs of transversally intersecting simple closed curves on a closed oriented surface of arbitrary genus g. The number of such…

Closed meanders are planar configurations of one or several disjoint closed Jordan curves intersecting a given line or curve transversely. They arise as shooting curves of parabolic PDEs in one space dimension, as trajectories of Cartesian…

In this paper, we introduce a prime factorization of open meanders, articulated through the framework of 2-colored operads. We demonstrate that each open meander can be canonically constructed from building blocks of two types: iterated…

An algorithm for optimizing the MERA tensor network in an infinite system is presented. Using this technique we compute the critical exponents of Ising and XXZ model.

We show that for $m$ points and $n$ lines in the real plane, the number of distinct distances between the points and the lines is $\Omega(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove that for any $m$ points in the…

We introduced concept of meander knots, 2-component meander links and multi-component meander links and derived different families of meander knots and links from open meanders with at most 16 crossings. We also defined semi-meander knots…

The multiplier $\lambda_n$ of a periodic orbit of period $n$ can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials $p_c(z)=z^2+c$. We provide a numerical algorithm for computing critical…

We establish an exact formula for the average number of edges appearing on the boundary of the global convex hull of n independent Brownian paths in the plane. This requires the introduction of a counting criterion which amounts to "cutting…

In this paper we study some Erdos type problems in discrete geometry. Our main result is that we show that there is a planar point set of n points such that no four are collinear but no matter how we choose a subset of size $n^{5/6+o(1)} $…

A set $L$ of straight lines and a set $P$ of points in the Euclidean plane define an arrangement $\mathcal{A}$ = ($L$, $P$) of construction lines and registration marks, if and only if: (1) any point in $P$ is a point of intersection of at…

We consider representations of tensors as sums of decomposable tensors or, equivalently, decomposition of multilinear forms into one--forms. In this short note we show that there exists a particular finite strongly orthogonal decomposition…

The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition.…

We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical…

We consider the hermitian matrix model with an external field entering the quadratic term $\tr(\Lambda X\Lambda X)$ and Penner--like interaction term $\alpha N(\log(1+X)-X)$. An explicit solution in the leading order in $N$ is presented.…

Computing mountain passes is a standard way of finding critical points. We describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite…