Related papers: Combinatorial distance between braid words
Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability.…
We prove the sharp bound for the probability that two experts who have access to different information, represented by different $\sigma$-fields, will give radically different estimates of the probability of an event. This is relevant when…
We explore a natural extension of braid arrangements in the context of determinantal arrangements. We show that these determinantal arrangements are free divisors. Additionally, we prove that free determinantal arrangements defined by the…
A well-known construction of associahedra comes from truncations of simplices. Motivated by compactifications of point configurations, we show associahedra as truncations of certain products of simplices. This is then used to provide a…
In this paper, we study a classical construction of lattices from number fields and obtain a series of new results about their minimum distance and other characteristics by introducing a new measure of algebraic numbers. In particular, we…
We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.
We present an algorithm to generate positive braids of a given length as words in Artin generators with a uniform probability. The complexity of this algorithm is polynomial in the number of strands and in the length of the generated…
The set of Schr\"oder words (Schr\"oder language) is endowed with a natural partial order, which can be conveniently described by interpreting Schr\"oder words as lattice paths. The resulting poset is called the Schr\"oder pattern poset. We…
We obtain a new bound on certain double sums of multiplicative characters improving the range of several previous results. This improvement comes from new bounds on the number of collinear triples in finite fields, which is a classical…
Two words $p$ and $q$ are avoided by the same number of length-$n$ words, for all $n$, precisely when $p$ and $q$ have the same set of border lengths. Previous proofs of this theorem use generating functions but do not provide an explicit…
We consider the problem of bounding away from 0 the minimum value m taken by a polynomial P of Z[X_1,...,X_k] over the standard simplex, assuming that m>0. Recent algorithmic developments in real algebraic geometry enable us to obtain a…
We study bracket words, which are a far-reaching generalisation of Sturmian words, along Hardy field sequences, which are a far-reaching generalisation of Piatetski--Shapiro sequences $\lfloor n^c \rfloor$. We show that thus obtained…
Occam's Razor tells us to pick the simplest model that fits our observations. In order to make sense of his process mathematically, we interpret it in the context of posets of functions. Our approach leads to some unusual new combinatorial…
A simple multivariable version of the reduced Burau matrix is constructed for any braid. It is shown how the multivariable Alexander polynomial for the closure of the braid can be found directly from this matrix.
We first define the concept of Lie algebroid in the convenient setting. In reference to the finite dimensional context, we adapt the notion of prolongation of a Lie algebroid over a fibred manifold to a convenient Lie algebroid over a…
We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for this invariant. The bounds differ by a multiplicative constant. The main motivation comes from…
We give a lower bound for the degree of an irreducible factor of a given polynomial. This improves and generalizes the results obtained in [4, On the irreducible factors of a polynomial, Proc. Amer. Math. Soc., 148 (2020] 1429 -- 1437].
First we consider pair-wise distances for literal objects consisting of finite binary files. These files are taken to contain all of their meaning, like genomes or books. The distances are based on compression of the objects concerned,…
We classify closed curves on a once-punctured torus with a single self-intersection from a combinatorial perspective. We determine the number of closed curves with given word-length and with zero, one, and arbitrary self-intersections.
We construct Poisson brackets at boundaries of open strings and membranes with constant background fields which are compatible with their boundary conditions. The boundary conditions are treated as primary constraints which give infinitely…