Related papers: Partially Directed Snake Polyominoes
The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous…
We present general algorithms (fully implemented in Maple) for calculations of various quantities related to constrained directed walks for a general set of steps on the square lattice in two dimensions. As a special case, we rederive…
I introduce a new type of continuous-time quantum walk on graphs called the quantum snake walk, the basis states of which are fixed-length paths (snakes) in the underlying graph. First I analyze the quantum snake walk on the line, and I…
Trying to enumerate all of the walks in a 2D lattice is a fun combinatorial problem and there are numerous applications, from polymers to sports. Computers provide a wonderful tool for analyzing these walks; we provide a Maple package for…
Solutions to the Markov equation appear in many mathematical contexts. We aim to build on the understanding of them by proving a recent conjecture about Markov polynomials; solutions to a generalised version of the Markov equation. The…
Snake robots are inspired by the ability of biological snakes to move over rock, grass, leaves, soil, up trees, along pavement and more. Their ability to move in multiple distinct environments is due to their legless locomotion strategy,…
For $d \geq 2$ and $n \in \mathbb{N}$, let $\mathsf{W}_n$ denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice $\mathbb{Z}^d$, and write $\Gamma$ for a $\mathsf{W}_n$-distributed walk. We show that…
We optimize three-dimensional snake kinematics for locomotor efficiency. We assume a general space-curve representation of the snake backbone with small-to-moderate lifting off the ground and negligible body inertia. The cost of locomotion…
In this paper, we propose a method for generating undulatory gaits for snake robots. Instead of starting from a pre-defined movement pattern such as a serpenoid curve, we use a Model Predictive Control approach to automatically generate…
We investigate the snaking of localised patterns, seen in numerous physical applications, using a variational approximation. This method naturally introduces the exponentially small terms responsible for the snaking structure, that are not…
Let there be given a probability measure $\mu$ on the unit circle $\TT$ of the complex plane and consider the inner product induced by $\mu$. In this paper we consider the problem of orthogonalizing a sequence of monomials $\{z^{r_k}\}_k$,…
The aim of this work is the study of the class of periodic parallelogram polyominoes, and two of its variantes. These objets are related to 321-avoiding affine permutations. We first provide a bijection with the set of triangles under Dyck…
We define what we call an on/off Brownian snake. We use this to construct on/off super Brownian motion recently introduced to the literature by Blath and Jacobi and which is a measure-valued branching process with a dormant state and an…
Nature suggests that exploiting the elasticities and natural dynamics of robotic systems could increase their locomotion efficiency. Prior work on elastic snake robots supports this hypothesis, but has not fully exploited the nonlinear…
In this paper we study the widely considered endomorphisms and weak endomorphisms of a finite undirected path from monoid generators perspective. Our main aim is to determine the ranks of the monoids $wEnd P_n$ and $End P_n$ of all weak…
Various subsets of self-avoiding walks naturally appear when investigating existing methods designed to predict the 3D conformation of a protein of interest. Two such subsets, namely the folded and the unfoldable self-avoiding walks, are…
We study the parameterized complexity of a variant of the classic video game Snake that models real-world problems of motion planning. Given a snake-like robot with an initial position and a final position in an environment (modeled by a…
It is known when we call a poset P, a $\mathcal{P}$-chain permutational poset, given a subset of permutations $\mathcal{P}$ of the symmetric group $S_{n}$. In this work, we use the same idea to study subsets of words of length $n$, that are…
We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey \emph{two-step rules,} which allow (or forbid) steps in a given direction to be followed by steps in another direction. We classify these rules according to a…
We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a…