Related papers: The lonely runner conjecture holds for eight runne…
We show that the bunkbed conjecture remains true when gluing along a vertex. As immediate corollaries, we obtain that the bunkbed conjecture is true for forests and that a minimal counterexample to the bunkbed conjecture is 2-connected.
The work takes another look at the number of runs that a string might contain and provides an alternative proof for the bound. We also propose another stronger conjecture that states that, for a fixed order on the alphabet, within every…
We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of what was known as the "runs" conjecture (Kolpakov \& Kucherov (FOCS '99)), which states that the…
Predicting the future performance of young runners is an important research issue in experimental sports science and performance analysis. We analyse a data set with annual seasonal best performances of male middle distance runners for a…
Consider a transient near-critical (1,2) random walk on the positive half line. We give a criteria for the finiteness of the number of the skipped points (the points never visited) by the random walk. This result generalizes (partially) the…
We establish new upper bounds for the length of runs of consecutive positive integers each with exactly $k$ divisors, where $k$ is a given positive integer of some special forms. Also we have found exact values of the maximum possible runs…
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…
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…
Recently, a short and elegant proof was presented showing that a binary word of length $n$ contains at most $n-3$ runs. Here we show, using the same technique and a computer search, that the number of runs in a binary word of length $n$ is…
In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.
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…
A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.
We give a short proof of the strong law of large numbers based on duality for random walk
We survey recent developments on the Restriction conjecture.
A Condorcet voting scheme chooses a winning candidate as one who defeats all others in pairwise majority rule. We provide a review which includes the rigorous mathematical treatment for calculating the limiting probability of a Condorcet…
We extend Carpi's results by showing that Dejean's conjecture holds for n >= 30.
For a long time, Collatz Conjecture has been assumed to be true, although a formal proof has eluded all efforts to date. In this article, evidence is presented that suggests such an assumption is incorrect. By analysing the stopping times…
We show that Dejean's conjecture holds for n>=27. This brings the final resolution of the conjecture by the approach of Moulin Ollagnier within range of the computationally feasible.
The Wiegold conjecture holds for the small Ree groups for $k$-tuples where $k \geq 5$.
The recent lottery ticket hypothesis proposes that there is one sub-network that matches the accuracy of the original network when trained in isolation. We show that instead each network contains several winning tickets, even if the initial…