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18 pages, 572 KiB

Open AccessArticle

Comparative Analysis of the Existence and Uniqueness Conditions of Parameter Estimation in Paired Comparison Models

by László Gyarmati, Éva Orbán-Mihálykó and Csaba Mihálykó

Axioms 2023, 12(6), 575; https://doi.org/10.3390/axioms12060575 - 9 Jun 2023

Cited by 2 | Viewed by 987

In this paper, paired comparison models with stochastic background are investigated. We focus on the models that allow three options for choice and the parameters are estimated by maximum likelihood method. The existence and uniqueness of the estimator are key issues of the evaluation. In the case of two options, a necessary and sufficient condition is given by Ford in the Bradley–Terry model. We generalize this statement for the set of strictly log-concave distribution. Although in the case of three options the necessary and sufficient condition is not known, there are two different sufficient conditions that are formulated in the literature. In this paper, we generalize them; moreover, we compare these conditions. Their capacities to indicate the existence of the maximum were analyzed using a large number of computer simulations. These simulations support that the new condition indicates the existence of the maximum much more frequently than the previously known ones. Full article

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26 pages, 1071 KiB

Open AccessArticle

Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics

by Torsten Asselmeyer-Maluga

Symmetry 2019, 11(10), 1298; https://doi.org/10.3390/sym11101298 - 15 Oct 2019

Cited by 11 | Viewed by 4748

Abstract

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement [...] Read more.

C (K) = S^{3} \ (K \times D^{2})

of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of

S^{3}

branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk

D^{3}

branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld–Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson–Thompson model). Full article

(This article belongs to the Special Issue Number Theory and Symmetry)

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