Related papers: Curves That Must Be Retraced

We present results and examples which show that the consideration of a certain tubular mutation is advantageous in the study of noncommutative curves which parametrize the simple regular representations of a tame bimodule. We classify all…

We introduce the semiring of values $\Gamma$ with respect to the tropical operations associated to an algebroid curve. As a set, $\Gamma$ determines and is determined by the well known semigroup of values $S$ and we prove that $\Gamma$ is…

We study a class of functional problems reducible to computing $f^{(n)}(x)$ for inputs $n$ and $x$, where $f$ is a polynomial-time bijection. As we prove, the definition is robust against variations in the type of reduction used in its…

Let $M$ be a graph manifold such that each piece of its JSJ decomposition has the $\Bbb H^2 \times \Bbb R$ geometry. Assume that the pieces are glued by isometries. Then, there exists a complete Riemannian metric on $\Bbb R \times M$ which…

Let $\Gamma$ be a rooted tree and let $t$ be a positive integer. We study algebraic invariants and properties of the path ideal generated by monomial corresponding to paths of length $(t-1)$ in $\Gamma$. In particular, we give a recursive…

The definition of $1$-planar graphs naturally extends graph planarity, namely a graph is $1$-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time,…

If $\Gamma$ is the range of a Jordan curve that bounds a convex set in $\mathbb{R}^2,$ then $\frac{1}{2}(\Gamma+\Gamma)=\mathsf{co}(\Gamma),$ where $+$ is the Minkowski sum and $\mathsf{co}$ is the convex hull. Answering a question of V.N.…

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…

A natural parametrization of smooth projective plane curves which tolerates the presence of sextactic points is the Forsyth-Laguerre parametrization. On a closed projective plane curve, which necessarily contains sextactic points, this…

Let $\Gamma$ denote a distance-regular graph with classical parameters $(D, b, \alpha, \beta)$ and $D\geq 3$. Assume the intersection numbers $a_1=0$ and $a_2\not=0$. We show $\Gamma$ is 3-bounded in the sense of the article [D-bounded…

We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that for any convex shape $K$, there exist four points on the boundary of $K$ such that the length of any curve…

Scarf's lemma is one of the fundamental results in combinatorics, originally introduced to study the core of an N-person game. Over the last four decades, the usefulness of Scarf's lemma has been demonstrated in several important…

A numerical scheme for computing arc-length parametrized curves of low bending energy that are confined to convex domains is devised. The convergence of the discrete formulations to a continuous model and the unconditional stability of an…

We construct a countable infinite graph G that does not contain cycles of length four having the property that the sequence of graphs $G_n$ induced by the first $n$ vertices has minimum degree $\delta(G_n)> n^{\sqrt{2}-1+o(1)}$.

We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool…

Let $\Gamma=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $\Gamma$ are integers, then we say that $\Gamma$ is an integral graph. A graph $\Gamma$ is determined by its spectrum if every graph cospectral to it…

It is known that if $n$ vertices can be removed from a connected graph $\Gamma$ so that no subgraphs isomorphic to the graph $K$ remain, then no more than $|V(K)|\cdot n$ vertices can be removed, forming a set invariant with respect to all…

We construct a (non-removable) Jordan curve $\Gamma$ and a non-M\"{o}bius homeomorphism of the Riemann sphere which is conformal on the complement of $\Gamma$ and maps the curve $\Gamma$ onto itself. The curve is flexible in the sense of…

We consider problems in which a simple path of fixed length, in an undirected graph, is to be shifted from a start position to a goal position by moves that add an edge to either end of the path and remove an edge from the other end. We…

A $\textit{polygonal curve}$ is a collection of $m$ connected line segments specified as the linear interpolation of a list of points $\{p_0, p_1, \ldots, p_m\}$. These curves may be obtained by sampling points from an oriented curve in…