We develop a statistical model for the testing of disease prevalence in a population. The model assumes a binary test result, positive or negative, but allows for biases in sample selection and both ...type I (false positive) and type II (false negative) testing errors. Our model also incorporates multiple test types and is able to distinguish between retesting and exclusion after testing. Our quantitative framework allows us to directly interpret testing results as a function of errors and biases. By applying our testing model to COVID-19 testing data and actual case data from specific jurisdictions, we are able to estimate and provide uncertainty quantification of indices that are crucial in a pandemic, such as disease prevalence and fatality ratios.
This article is part of the theme issue ‘Data science approach to infectious disease surveillance’.
A new class of premature, partial latin squares Euler, Reinhardt
Bulletin mathématiques de la Société des sciences mathématiques de Roumanie,
04/2024, Letnik:
67, Številka:
115
Journal Article
Combining two well-known types, we present a new class of partial latin squares whichare not completable and minimal with respect to this property.
A tight map is a map with some of its vertices marked, such that every vertex of degree 1 is marked. We give an explicit formula for the number $N_{0,n}(d_1,…,d_n)$ of planar tight maps with $n$ ...labeled faces of prescribed degrees $d_1,…,d_n$, where a marked vertex is seen as a face of degree 0. It is a quasi-polynomial in $(d_1,…,d_n)$, as shown previously by Norbury. Our derivation is bijective and based on the slice decomposition of planar maps. In the non-bipartite case, we also rely on enumeration results for two-type forests. We discuss the connection with the enumeration of non necessarily tight maps. In particular, we provide a generalization of Tutte's classical slicings formula to all non-bipartite maps.
The dual braid monoid was introduced by Bessis in his work on complex reflection arrangements. The goal of this work is to show that Koszul duality provides a nice interplay between the dual braid ...monoid and the cluster complex introduced by Fomin and Zelevinsky. Firstly, we prove koszulity of the dual braid monoid algebra, by building explicitly the minimal free resolution of the ground field. This is done explicitly using some chains complexes defined in terms of the positive part of the cluster complex. Secondly, we derive various properties of the quadratic dual algebra. We show that it is naturally graded by the noncrossing partition lattice. We get an explicit basis, naturally indexed by positive faces of the cluster complex. Moreover, we find the structure constants via a geometric rule in terms of the cluster fan. Eventually, we realize this dual algebra as a quotient of a Nichols algebra. This latter fact makes a connection with results of Zhang, who used the same algebra to compute the homology of Milnor fibers of reflection arrangements.
Le monoïde dual des tresses a été introduit par Bessis dans le contexte des arrangements d’hyperplanscomplexes. Le but de ce travail est de montrer que la dualité de Koszul fournit une interaction remarquableavec le complexe d’amas introduit par Fomin et Zelevinsky. Premièrement, nous démontrons la koszulitéde l’algèbre du monoïde dual des tresses, en donnant explicitement la résolution libre minimale ducorps de base. Cette construction utilise des complexes de chaînes définis grâce à la partie positive ducomplexe d’amas. Deuxièmement, nous examinons diverses propriétés de l’algèbre quadratique duale.Nous démontrons qu’elle est naturellement graduée par le treillis des partitions non-croisées. Nousobtenons une base explicite, indicée par les faces positives du complexe d’amas. Les constantes destructure peuvent être décrites explicitement en termes de l’éventail des amas. Enfin, nous réalisons cettealgèbre duale comme un quotient d’une algèbre de Nichols. Ce dernier point se relie aux travaux deZhang, qui a utilisé cette algèbre pour un calcul d’homologie des fibres de Milnor d’un arrangement deCoxeter.
Parabolic Tamari Lattices in Linear Type B Fang, Wenjie; Mühle, Henri; Novelli, Jean-Christophe
The Electronic journal of combinatorics,
03/2024, Letnik:
31, Številka:
1
Journal Article
Recenzirano
We study parabolic aligned elements associated with the type-$B$ Coxeter group and the so-called linear Coxeter element. These elements were introduced algebraically in (Mühle and Williams, 2019) for ...parabolic quotients of finite Coxeter groups and were characterized by a certain forcing condition on inversions. We focus on the type-$B$ case and give a combinatorial model for these elements in terms of pattern avoidance. Moreover, we describe an equivalence relation on parabolic quotients of the type-$B$ Coxeter group whose equivalence classes are indexed by the aligned elements. We prove that this equivalence relation extends to a congruence relation for the weak order. The resulting quotient lattice is the type-$B$ analogue of the parabolic Tamari lattice introduced for type $A$ in (Mühle and Williams, 2019).
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such ...as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combinatorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session.
Given a mixed hypergraph $\mathcal{F}=(V,\mathcal{A}\cup \mathcal{E})$, a non-negative integer $k$ and functions $f,g:V\rightarrow \mathbb{Z}_{\geq 0}$, a packing of $k$ spanning mixed ...hyperarborescences of $\mathcal{F}$ is called $(k,f,g)$-flexible if every $v \in V$ is the root of at least $f(v)$ and at most $g(v)$ of the mixed hyperarborescences. We give a characterization of the mixed hypergraphs admitting such packings. This generalizes results of Frank and, more recently, Gao and Yang. Our approach is based on matroid intersection, generalizing a construction of Edmonds. We also obtain an algorithm for finding a minimum weight solution to the problem mentioned above.
In this paper, we first prove that a Rota-Baxter family algebra indexed by a semigroup induces an ordinary Rota-Baxter algebra structure on the tensor product with the semigroup algebra. We show that ...the same phenomenon arises for dendriform and tridendriform family algebras. Then we construct free dendriform family algebras in terms of typed decorated planar binary trees. Finally, we generalize typed decorated rooted trees to typed valently decorated Schröder trees and use them to construct free tridendriform family algebras.
PDF
