Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0601053

Ranking Participants in Tournaments by means of Rating Functions

M. Brozos-Vazquez·M. A. Campo-Cabana·J. C. Diaz-Ramos·J. Gonzalez-Diaz
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we bring a novel approach to the theory of tournament rankings. We combine two different theories that are widely used to establish rankings of populations after a given tournament. First, we use the statistical approach of paired comparison analysis to define the performance of a player in a natural way. Then, we determine a ranking (and rating) of the players in the given tournament. Finally, we show, among other properties, that the new ranking method is the unique one satisfying a natural consistency requirement.

Explore related subjects

math.GMmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

M. Brozos-Vazquez, M. A. Campo-Cabana, J. C. Diaz-Ramos, J. Gonzalez-Diaz. 2006-12-02. Ranking Participants in Tournaments by means of Rating Functions. https://arxiv.org/abs/math/0601053

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$ via a Rank-Zero Elliptic Curve

In this paper we study the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ introduced by R. A. Sharipov in the first-cuboid specialization of his cuboid equations. For nonzero integers $a,u$ with $u^2\neq a^2$ we prove that $P_{a,u}(t)$ is irreducible in $\mathbb{Z}[t]$ (equivalently, in $\mathbb{Q}[t]$), thus confirming Sharipov's irreducibility conjecture in this two-parameter case. Over $K=\mathbb{Q}(\sqrt2)$ we have a factorization $P_{a,u}(t)=H_-(t)H_+(t)$ into two conjugate quartics. We show that reducibility of either $H_\pm$ would force the discriminant of a certain quadratic in $S=t^2$ to be a square in $K$, which in turn implies (via $τ=(au/Δ)^2$, where $Δ=u^2-a^2$) the existence of a rational point $(y,v)\in\mathcal{C}(\mathbb{Q})$ on the genus-one quartic $\mathcal{C}: v^2=16y^4+136y^2+1$ with $y^2=τ$. We give an explicit isomorphism $\overline{\mathcal{C}}\simeq E$ with the elliptic curve $E: Y^2=X(X-8)(X-9)$, which has conductor $48$ and Mordell-Weil rank $0$. Enumerating $E(\mathbb{Q})$ and tracing back to $\mathcal{C}(\mathbb{Q})$ rules out the only possible values $τ\in\{0,1/4\}$, and hence excludes any further factorization of $H_\pm$ in $K[t]$. A quadratic Galois descent then yields the irreducibility of $P_{a,u}(t)$ over $\mathbb{Q}$ and $\mathbb{Z}$.

math.GM↗
arXiv2026

Irreducibility of the Cuboid Polynomial $Q_{p,q}(t)$ via the Rank-One Chabauty-Coleman Method

Let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial, with coprime positive integers $p\ne q$. We prove that $Q_{p,q}(t)$ is irreducible over $\mathbb{Q}$, and hence over $\mathbb{Z}$. More generally, the normalized associated quintic $P_s(x)$ and its composition $P_s(t^2)$ are irreducible over $\mathbb{Q}$ for every $s\in\mathbb{Q}\setminus\{0,\pm1\}$. The proof reduces linear and quadratic factors to rational points and horizontal conjugate pairs on one fixed genus-$2$ curve. A certified rank upper bound for its Jacobian, together with an exact divisor calculation, gives a rank-one annihilating differential at the prime $3$. For the quadratic-factor case, the symmetric coordinates of a horizontal pair lie on an explicit double cover of an elliptic quartic. All rational affine points on this model are $3$-adically integral and reduce to six smooth residue classes. The pulled-back differential is a unit in each class, proving that six displayed points form the complete list. Their coordinates exclude every admissible parameter. A negative real root of the irreducible quintic then supplies the additional degree argument required for irreducibility of the degree-$10$ polynomial. The rank computation and all exact arithmetic certificates are specified explicitly for reproduction.

math.GM↗