Set-theoretic solutions to the Yang-Baxter equation and generalized semi-braces

This paper aims to introduce a construction technique of set-theoretic solutions of the Yang-Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions

Deformed solutions of the Yang-Baxter equation coming from dual weak braces and unital near-trusses

We show that a family of set-theoretic solutions comes from dual weak braces, including the recent ones provided in the context of skew braces. In particular, the solutions we give are obtained by ``deforming" the classical ones by certain parameters $z$. In any dual weak brace $(S, +, \circ)$, the parameters giving rise to deformed solutions are only those belonging to the right distributor of $S$, i.e., $\mathcal{D}_r(S)=\{z \in S \, \mid \, \forall \, a,b \in S \quad (a+b) \circ z=a\circ z-z+b \circ z\}$, that is a full inverse subsemigroup of $\left(S, \circ\right)$. Particular attention is posed to the class of skew braces and, in a natural way, of two-sided skew braces. In addition, we construct new arbitrary solutions on the algebraic structure of the unital near-truss determined by extending deformed solutions coming from skew braces

Set-theoretical solutions of the Yang-Baxter and pentagon equations on semigroups

The Yang-Baxter and pentagon equations are two well-known equations of Mathematical Physic. If $S$ is a set, a map $s:S\times S\to S\times S$ is said to be a set theoretical solution of the Yang-Baxter equation if $$s_{23}\, s_{13}\, s_{12} = s_{12}\, s_{13}\, s_{23},$$ where $s_{12}=s\times id_S$, $s_{23}=id_S\times s$, and $s_{13}=(id_S\times \tau)\,s_{12}\,(id_S\times \tau)$ and $\tau$ is the flip map, i.e., the map on $S\times S$ given by $\tau(x,y)=(y,x)$. Instead, $s$ is called a set-theoretical solution of the pentagon equation if $$s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}.$$ The main aim of this work is to display how solutions of the pentagon equation turn out to be a useful tool to obtain new solutions of the Yang-Baxter equation. Specifically, we present a new construction of solutions of the Yang-Baxter equation involving two specific solutions of the pentagon equation. To this end, we provide a method to obtain solutions of the pentagon equation on the matched product of two semigroups, that is a semigroup including the classical Zappa product

Solutions of the Yang-Baxter equation and strong semilattices of skew braces

We prove that any set-theoretic solution of the Yang-Baxter equation associated to a dual weak brace is a strong semilattice of non-degenerate bijective solutions. This fact makes use of the description of any dual weak brace $S$ we provide in terms of strong semilattice $Y$ of skew braces $B_\alpha$, with $\alpha \in Y$. Additionally, we describe the ideals of $S$ and study its nilpotency by correlating it to that of each skew brace $B_\alpha$

Inverse semi-braces and the Yang-Baxter equation

The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace, that is a triple $(S,+, \cdot)$ with $(S,+)$ a semigroup and $(S, \cdot)$ an inverse semigroup satisfying the relation $a \left(b + c\right) = a b + a\left(a^{-1} + c\right)$, for all $a,b,c \in S$, where $a^{-1}$ is the inverse of $a$ in $(S, \cdot)$. In particular, we give several constructions of inverse semi-braces which allow for obtaining solutions that are different from those until known.Comment: 43 page

Vaccine coverage and determinants of incomplete vaccination in children aged 12-23 months in dschang, west region, cameroon: a cross-sectional survey during a polio outbreak

Inadequate immunization coverage with increased risk of vaccine preventable diseases outbreaksremains a problem in Africa. Moreover, different factors contribute to incomplete vaccination status. This study wasperformed in Dschang (West Region, Cameroon), during the polio outbreak occurred in October 2013, in order toestimate the immunization coverage among children aged 12–23 months, to identify determinants for incompletevaccination status and to assess the risk of poliovirus spread in the study population.Methods:A cross-sectional household survey was conducted in November-December 2013, using the WHOtwo-stage sampling design. An interviewer-administered questionnaire was used to obtain information fromconsenting parents of children aged 12–23 months. Vaccination coverage was assessed by vaccination card andparents’recall. Chi-square test and multilevel logistic regression model were used to identify the determinants ofincomplete immunization status. Statistical significance was set atp90 %, and 73.4 % children completedthe recommended vaccinations before 1-year of age. In the final multilevel logistic regression model, factorssignificantly associated with incomplete immunization status were: retention of immunization card (AOR: 7.89;95 % CI: 1.08–57.37), lower mothers’utilization of antenatal care (ANC) services (AOR:1.25; 95 % CI: 1.07–63.75),being the≥3rdborn child in the family (AOR: 425.4; 95 % CI: 9.6–18,808), younger mothers’age (AOR: 49.55;95 % CI: 1.59–1544), parents’negative attitude towards immunization (AOR: 20.2; 95 % CI: 1.46–278.9), and poorerparents’exposure to information on vaccination (AOR: 28.07; 95 % CI: 2.26–348.1). Longer distance from the vaccinationcenters was marginally significant (p=0.05).Conclusion:Vaccination coverage was high; however, 1 out of 7 children was partially vaccinated, and 1 out of 4 didnot complete timely the recommended vaccinations. In order to improve the immunization coverage, it is necessary tostrengthen ANC services, and to improve parents’information and attitude towards immunization, targeting youngerparents and families living far away from vaccination centers, using appropriate communication strategies. Finally, theestimated OPV-3 coverage is reassuring in relation to the ongoing polio outbrea

meningococcal c conjugate vaccine effectiveness before and during an outbreak of invasive meningococcal disease due to neisseria meningitidis serogroup c cc11 tuscany italy

Abstract Introduction In Tuscany, Italy, where a universal immunization program with monovalent meningococcal C conjugate vaccine (MCC) was introduced in 2005, an outbreak of invasive meningococcal disease (IMD) due to the hypervirulent strain of Neisseria meningitidis C/cc11 occurred in 2015–2016, leading to an immunization reactive campaign using either the tetravalent (ACWY) meningococcal conjugate or the MCC vaccine. During the outbreak, IMD serogroup C (MenC) cases were also reported among vaccinated individuals. This study aimed to characterize meningococcal C conjugate vaccines (MenC-vaccines) failures and to estimate their effectiveness since the introduction (2005–2016) and during the outbreak (2015–2016). Methods MenC cases and related vaccine-failures were drawn from the National Surveillance System of Invasive Bacterial Disease (IBD) for the period 2006–2016. A retrospective cohort-study, including the Tuscany' population of the birth-cohorts 1994–2014, was carried out. Based on annual reports of vaccination, person-years of MenC-vaccines exposed and unexposed individuals were calculated by calendar-year, birth-cohort, and local health unit. Adjusted (by birth-cohort, local health unit, and calendar-year) risk-ratios (ARR) of MenC invasive disease for vaccinated vs unvaccinated were estimated by the Poisson model. Vaccine-effectiveness (VE) was estimated as: VE = 1-ARR. Results In the period 2006–2016, 85 MenC-invasive disease cases were reported; 61 (71.8%) from 2015 to 2016. Twelve vaccine failures occurred, all of them during the outbreak. The time-interval from immunization to IMD onset was 20 days in one case, from 9 months to 3 years in six cases, and ≥7 years in five cases. VE was, 100% (95%CI not estimable, p = 0.03) before the outbreak (2006–2014) and 77% (95%CI 36–92, p  Conclusions In Tuscany, MenC-vaccine failures occurred exclusively during the 2015–2016 outbreak. Most of them occurred several years after vaccination. VE during the outbreak-period was rather high supporting an effective protection induced by MenC-vaccines

Cluster of invasive Neisseria meningitidis infections on a cruise ship, Italy, October 2012.

We describe a cluster of four cases of invasive meningococcal disease that occurred on a cruise ship sailing along the Italian coast in October 2012. All four cases were hospitalised with severe illness and one of them died. This report illustrates the importance of rapid implementation of emergency control measures such as administration of prophylaxis to all crew members and passengers to prevent the spread of the disease in such a close environment