Related papers: The lonely runner with seven runners

We show that the shifted Lonely Runner Conjecture (sLRC) holds for 5 runners. We also determine that there are exactly 3 primitive tight instances of the conjecture, only two of which are tight for the non-shifted conjecture (LRC). Our…

Lonely Runner Conjecture, proposed by J\"{o}rg M. Wills and so nomenclatured by Luis Goddyn, has been an object of interest since it was first conceived in 1967 : Given positive integers $k$ and $n_1,n_2,\ldots,n_k$ there exists a positive…

For each subtorus $T$ of $(\mathbb{R}/\mathbb{Z})^n$, let $D(T)$ denote the (infimal) $L^\infty$-distance from $T$ to the point $(1/2,\ldots, 1/2)$. The $n$-th Lonely Runner spectrum $\mathcal{S}(n)$ is defined to be the set of all values…

For $x$ real, let $ \{ x \}$ be the fractional part of $x$ (i.e. $\{x\} = x - \lfloor x \rfloor $). The lonely runner conjecture can be stated as follows: for any $n$ positive integers $ v_1 < v_2 < \dots < v_n $ there exists a real number…

We are concerned with the computational problem of determining the covering radius of a rational polytope. This parameter is defined as the minimal dilation factor that is needed for the lattice translates of the correspondingly dilated…

Empty buses are standing at a bus station. $n$ passengers arrive, and they each board a bus completely at random (meaning that they choose uniformly and independently). Then all buses depart. We show that the more buses there are, the more…

In some athletic races, such as cycling and types of speed skating races, athletes have to complete a relatively long distance at a high speed in the presence of direct opponents. To win such a race, athletes are motivated to hide behind…

Consider continuous-time random walks on Cayley graphs where the rate assigned to each edge depends only on the corresponding generator. We show that the limiting speed is monotone increasing in the rates for infinite Cayley graphs that…

Visualizing the trajectory of multiple runners with videos collected at different points in a race could be useful for sports performance analysis. The videos and the trajectories can also aid in athlete health monitoring. While the runners…

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the…

Consider $n$ runners running on a circular track of unit length with constant speeds such that $k$ of the speeds are distinct. We show that, at some time, there will exist a sector $S$ which contains at least $|S|n+ \Omega(\sqrt{k})$…

For a subtorus $T \subseteq (\mathbb{R}/\mathbb{Z})^n$, let $D(T)$ denote the $L^\infty$-distance from $T$ to the point $(1/2, \ldots, 1/2)$. For a subtorus $U \subseteq (\mathbb{R}/\mathbb{Z})^n$, define $\mathcal{S}_1(U)$, the Lonely…

Depending on the route, runners may experience frustration, freedom, or fulfilment. However, finding routes that are conducive to the psychological experience of running remains an unresolved task in the literature. In a mixed-method study,…

The aim of this paper is to bring a mathematical justification to the optimal way of organizing one's effort when running. It is well known from physiologists that all running exercises of duration less than 3mn are run with a strong…

Physical activity is known to help improve and maintain one's health. In particular, recreational running has become increasingly popular in recent years. Yet, lack of motivation often interferes with people's routines and thus may prohibit…

We study an infinite system of particles initially occupying a half-line $y\leq 0$ and undergoing random walks on the entire line. The right-most particle is called a leader. Surprisingly, every particle except the original leader may never…

Consider a walk in the plane made of $n$ unit steps, with directions chosen independently and uniformly at random at each step. Rayleigh's theorem asserts that the probability for such a walk to end at a distance less than 1 from its…

All experiments attempting to verify the invariance of speed of light directly are based on two-way speed measurement. The challenge in one-way speed measurement, the requirement of spatially separated synchronised clocks, can be possibly…

Answering a question by Angel, Holroyd, Martin, Wilson and Winkler, we show that the maximal number of non-colliding coupled simple random walks on the complete graph $K_N$, which take turns, moving one at a time, is monotone in $N$. We use…

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover…