Related papers: Unique Tournaments and Radar Tracking
For a regular tournament $T$ of order $n,$ denote by $c_{8}(T)$ the number of cycles of length $8$ in $T.$ Let $DR_{n}$ be a doubly-regular tournament of order $n\equiv 3\mod4$ (so, the out-sets and in-sets of its vertices are also regular…
We consider a tournament $T=(V, A)$. For $X\subseteq V$, the subtournament of $T$ induced by $X$ is $T[X] = (X, A \cap (X \times X))$. An interval of $T$ is a subset $X$ of $V$ such that for $a, b\in X$ and $ x\in V\setminus X$, $(a,x)\in…
We propose and analyse a 2-parameter family of 2-player games on two heaps of tokens, and present a strategy based on a class of sequences. The strategy looks easy, but is actually hard. A class of exotic numeration systems is then used,…
The field of algorithmic randomness studies what it means for infinite binary sequences to be random for some given uncertainty model. Classically, martingale-theoretic notions of such randomness involve precise uncertainty models, and it…
We analyze different methods of sorting and selecting a set of objects by their intrinsic value, via pairwise comparisons whose outcome is uncertain. After discussing the limits of repeated Round Robins, two new methods are presented: The…
In this paper, the construction of finite-length binary sequences whose nonlinear complexity is not less than half of the length is investigated. By characterizing the structure of the sequences, an algorithm is proposed to generate all…
The frequency of planets in binaries is an important issue in the field of extrasolar planet studies because of its relevance in the estimation of the global planet population of our galaxy and the clues it can give to our understanding of…
In this paper we investigate a differential game in which countably many dynamical objects pursue a single one. All the players perform simple motions. The duration of the game is fixed. The controls of a group of pursuers are subject to…
Over the last decade, extensive research has been conducted on the algorithmic aspects of designing single-elimination (SE) tournaments. Addressing natural questions of algorithmic tractability, we identify key properties of input instances…
An extension to a recently introduced architecture of clique-based neural networks is presented. This extension makes it possible to store sequences with high efficiency. To obtain this property, network connections are provided with…
We describe a round robin scheduling problem for a competition played in two divisions, motivated by a scheduling problem brought to the second author by a local sports organisation. The first division has teams from 2n clubs, and is played…
We investigate uniformity properties of strategies. These properties involve sets of plays in order to express useful constraints on strategies that are not \mu-calculus definable. Typically, we can state that a strategy is…
Single-elimination (SE) tournaments are a popular format used in competitive environments and decision making. Algorithms for SE tournament manipulation have been an active topic of research in recent years. In this paper, we initiate the…
Suppose that $k$ runners having different constant speeds run laps on a circular track of unit length. The Lonely Runner Conjecture states that, sooner or later, any given runner will be at distance at least $1/k$ from all the other…
We investigate structure for pairs of randomizations that do not follow each other in a chain. These are unrandomized-inclusive, independent, coincident or double randomizations. This involves taking several structures that satisfy…
If $T$ is an $n$-vertex tournament with a given number of $3$-cycles, what can be said about the number of its $4$-cycles? The most interesting range of this problem is where $T$ is assumed to have $c\cdot n^3$ cyclic triples for some $c>0$…
A tournament is an orientation of a complete graph. We say that a vertex $x$ in a tournament $\vec T$ controls another vertex $y$ if there exists a directed path of length at most two from $x$ to $y$. A vertex is called a king if it…
Multiple teams participate in a random competition. In each round the winner receives one point. We study the times until ties occur among teams. We construct martingales and supermartingales that enable us to prove the results regarding…
We introduce a novel system of matching and scoring players in tournaments, called Multi-Tier Tournaments, illustrated by chess and based on the following rules: 1. Players are divided into skill-based tiers, based on their Elo ratings. 2.…
Continuing the program begun in the first paper of this series, we present a pair of non-reconstructable tournament on $p$ vertices for each $p=2^n \ge 4$.