Related papers: The lonely runner conjecture holds for nine runner…
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…
The article presents the proof of Casas-Alvero conjecture.
In this note, we provide a short proof of Feige's conjecture for identically distributed random variables.
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…
This paper presents a sharp approximation of the density of long runs of a random walk conditioned on its end value or by an average of a functions of its summands as their number tends to infinity. The conditioning event is of moderate or…
We formulate and discuss a conjecture which would extend a classical inequality of Bernstein.
The Jones unknot conjecture states that the Jones polynomial distinguishes the unknot from nontrivial knots. We prove it for knots up to 23 crossings.
We prove an analogue of the classical ballot theorem that holds for any random walk in the range of attraction of the normal distribution. Our result is best possible: we exhibit examples demonstrating that if any of our hypotheses are…
A proof of the continuous martingale convergence theorem is provided. It relies on a classical martingale inequality and the almost sure convergence of a uniformly bounded non-negative super-martingale, after a truncation argument.
Several results about the union-closed sets conjecture are presented.
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…
Associated to a random walk on $\mathbb{Z}$ and a positive integer $n$, there is a return probability of the random walk returning to the origin after $n$ steps. An interesting question is when the set of return probabilities uniquely…
We prove an interesting fact about Lottery: the winning 6 numbers (out of 49) in the game of the Lottery contain two consecutive numbers with a surprisingly high probability (almost 50%).
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 show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [8] holds while the classical Borel Conjecture fails.
The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…
In this paper, we conjecture a monotonicity property that we call monotonicity under coarse-graining for a class of multi-partite entanglement measures. We check these properties by computing the measures for various types of states using…
In this paper, we prove that the conductor formula of Bloch implies the conjecture of Deligne on Milnor numbers of isolated singularities. In particular, thanks to the work of Bloch on his conjecture, our result implies that this so-called…
We develop techniques to deal with monotonicity of sequences z_{n+1}/z_n and \sqrt[n]{z_n}. A series of conjectures of Zhi-Wei Sun and of Amdeberhan et al. are verified in certain unified approaches.
We survey the state of the union-closed sets conjecture.