Related papers: Landau's Theorem Revisited Again

Sumner's universal tournament conjecture states that every $(2n-2)$-vertex tournament should contain a copy of every $n$-vertex oriented tree. If we know the number of leaves of an oriented tree, or its maximum degree, can we guarantee a…

We study the following rearrangement problem: Given $n$ words, rearrange and concatenate them so that the obtained string is lexicographically smallest (or largest, respectively). We show that this problem reduces to sorting the given words…

Sumner's universal tournament conjecture states that any tournament on $2n-2$ vertices contains a copy of any directed tree on $n$ vertices. We prove an asymptotic version of this conjecture, namely that any tournament on $(2+o(1))n$…

The Wang-Landau algorithm aims at sampling from a probability distribution, while penalizing some regions of the state space and favoring others. It is widely used, but its convergence properties are still unknown. We show that for some…

A knockout tournament is one of the most simple and popular forms of competition. Here, we are given a binary tournament tree where all leaves are labeled with seed position names. The players participating in the tournament are assigned to…

The traveling tournament problem (TTP) is to minimize the total traveling distance of all teams in a double round-robin tournament. In this paper, we focus on TTP-2, in which each team plays at most two consecutive home games and at most…

For many types of graphs, criteria have been discovered that give necessary and sufficient conditions for an integer sequence to be the degree sequence of such a graph. These criteria tend to take the form of a set of inequalities, and in…

The odds theorem and the corresponding solution algorithm (odds algorithm) are tools to solve a wide range of optimal stopping problems. Its generality and tractability have caught much attention. (Google for instance "Bruss odds" to obtain…

We study statistics of the knockout tournament, where only the winner of a fixture progresses to the next. We assign a real number called competitiveness to each contestant and find that the resulting distribution of prize money follows a…

In this paper we present a new proof of a proposition presented in the article: Complexes of tournaments, directionality filtrations and persistent homology. This paper is part of Joaqu\'in Casta\~neda's undergraduate thesis.

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 prove a couple of related theorems including Legendre's and Andrica's conjecture. Key to the proofs is an algorithm that delivers the exact upper bound on the greatest gap that can occur in a combinatorial game with the set of P primes…

Negative dependence of sequences of random variables is often an interesting characteristic of their distribution, as well as a useful tool for studying various asymptotic results, including central limit theorems, Poisson approximations,…

Suppose $\mathbb{F}$ is a field and let $\mathbf{a} := (a_1, a_2, \dotsc)$ be a sequence of non-zero elements in $\mathbb{F}$. For $\mathbf{a}_n := (a_1, \dotsc, a_n)$, we consider the family $\mathcal{M}_n(\mathbf{a})$ of $n \times n$…

We study the termination problem for nondeterministic recursive probabilistic programs. First, we show that a ranking-supermartingales-based approach is both sound and complete for bounded terminiation (i.e., bounded expected termination…

We consider a random knockout tournament among players $1, \ldots, n$, in which each match involves two players. The match format is specified by the number of matches played in each round, where the constitution of the matches in a round…

The problem of electing a leader from among $n$ contenders is one of the fundamental questions in distributed computing. In its simplest formulation, the task is as follows: given $n$ processors, all participants must eventually return a…

We consider the following problem posed by Volkmann in 2007: How close to regular must a c-partite tournament be, to secure a strongly connected subtournament of order $c$? We give sufficient conditions on the regularity of balanced…

We propose a proof of the downward L\"owenheim-Skolem that relies on strategies deriving from evaluation games instead of the Skolem normal forms. This proof is simpler, and easily understood by the students, although it requires, when…

We continue the study of selection and sorting of $n$ numbers under the adversarial comparator model, where comparisons can be adversarially tampered with if the arguments are sufficiently close. We derive a randomized sorting algorithm…