Related papers: Collatz convergence is a Hydra game

In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new…

Let $q$ be an odd prime, and let $T_{q}:\mathbb{Z}\rightarrow\mathbb{Z}$ be the Shortened $qx+1$ map, defined by $T_{q}\left(n\right)=n/2$ if $n$ is even and $T_{q}\left(n\right)=\left(qn+1\right)/2$ if $n$ is odd. The study of the dynamics…

Driven by recent successes in two-player, zero-sum game solving and playing, artificial intelligence work on games has increasingly focused on algorithms that produce equilibrium-based strategies. However, this approach has been less…

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then…

In decision-dependent games, multiple players optimize their decisions under a data distribution that shifts with their joint actions, creating complex dynamics in applications like market pricing. A practical consequence of these dynamics…

Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a "quantum-classical game"--that is, a non-local game of two…

An alternative computational approach to the Collatz (3n+1) conjecture is presented that may be theoretically capable of confirming the conjecture.

The Collatz map (or the $3n{+}1$-map) $f$ is defined on positive integers by setting $f(n)$ equal to $3n+1$ when $n$ is odd and $n/2$ when $n$ is even. The Collatz conjecture states that starting from any positive integer $n$, some iterate…

Identifying a full basis of operators to a given order is key to the generality of Effective Field Theory (EFT) and is by now a problem of known solution in terms of the Hilbert series. The present work is concerned with hidden symmetry in…

We propose a notion of concavity in two-sided many-to-one matching, which is an analogue to the balancedness condition in cooperative games. A stable matching exists when the market is concave. We provide a class of concave markets. In the…

Artificial intelligence algorithms are increasingly used by firms to set prices. Previous research shows that they can exhibit collusive behaviour, but how quickly they can do so has so far remained an open question. I show that a modern…

We introduce a geometric dynamical system where iteration is defined as a cycling composition of different maps acting on a space composed of three or more lines in $\mathbb{R}^2$. This system is motivated by the dynamics of iterated…

We introduce an infinite set of integer mappings that generalize the well-known Collatz-Ulam mapping and we conjecture that an infinite subset of these mappings feature the remarkable property of the Collatz conjecture, namely that they…

Significant efforts have been devoted in the last decade towards improving the predictivity of coarse-grained models in molecular dynamics simulations and providing a rigorous justification of their use, through a combination of theoretical…

We explore the behaviour emerging from learning agents repeatedly interacting strategically for a wide range of learning dynamics, including $Q$-learning, projected gradient, replicator and log-barrier dynamics. Going beyond the better…

Convergence of the policy iteration method for discrete and continuous optimal control problems holds under general assumptions. Moreover, in some circumstances, it is also possible to show a quadratic rate of convergence for the algorithm.…

Lloyd's algorithm is an iterative method that solves the quantization problem, i.e. the approximation of a target probability measure by a discrete one, and is particularly used in digital applications. This algorithm can be interpreted as…

Combinatorial Game Theory is a branch of mathematics and theoretical computer science that studies sequential 2-player games with perfect information. Normal play is the convention where a player who cannot move loses. Here, we generalize…

Physical systems can fail. For this reason the problem of identifying and reacting to faults has received a large attention in the control and computer science communities. In this paper we study the fault diagnosis problem for hybrid…

Evolutionary game theory is a powerful mathematical framework to study how intelligent individuals adjust their strategies in collective interactions. It has been widely believed that it is impossible to unilaterally control players'…