Relative Rota-Baxter operators of nonzero weights on 33-Lie algebras and 3-post-Lie algebras

In this paper, first we introduce the notion of a relative Rota-Baxter operator of nonzero weights on a $3$-Lie algebra with respect to an action on another 3-Lie algebra, which can be characterized by graphs of the semidirect product 3-Lie algebra constructed from the action. %We show that a relative Rota-Baxter operator of weight $\lambda$ induces a new $3$-Lie algebra structure, which is called the descendent 3-Lie algebra. Then we introduce a new algebraic structure, which is called a $3$-post-Lie algebra. A 3-post-Lie algebra consists of a 3-Lie algebra structure and a ternary operation such that some compatibility conditions are satisfied. We show that a relative Rota-Baxter operator of nonzero weights induces a $3$-post-Lie algebra naturally. Thus $3$-post-Lie algebras can be viewed as the underlying algebraic structures of relative Rota-Baxter operators of nonzero weights on $3$-Lie algebras. Moreover, a $3$-post-Lie algebra also gives rise to a new 3-Lie algebra, which is called the subadjacent 3-Lie algebra, and an action on the original 3-Lie algebra. Next we construct an $L_\infty$-algebra from an action of 3-Lie algebras whose Maurer-Cartan elements are relative Rota-Baxter operators of nonzero weights. Consequently, we obtain the twisted $L_\infty$-algebra that controls deformations of a given relative Rota-Baxter operator of nonzero weights on 3-Lie algebras. Finally, we construct a cohomology theory for a relative Rota-Baxter operator of nonzero weights on $3$-Lie algebras and use the second cohomology group to classify infinitesimal deformations.Comment: 22 pages. arXiv admin note: text overlap with arXiv:2107.13950, arXiv:2108.0262

"All This is For My Child, For My Family"

Social discrimination and limited health care resources drive many people living with HIV/AIDS in China to seek help from their own families. This paper, which is based on a qualitative study of the experiences of this population, examines how the notion of family obligation has influenced how these individuals organise various aspects of their daily lives. Viewing family obligation as a matter of reciprocal relationships, a closer look is taken at the ways in which these individuals exercise their autonomy to fulfil those obligations and pursue their moral self despite their health status. The study suggests the significance as well as the limitations of incorporating family support into AIDS interventions in the Chinese context

Cohomology and the controlling algebra of crossed homomorphisms on 3-Lie algebras

In this paper, first we give the notion of a crossed homomorphism on a 3-Lie algebra with respect to an action on another 3-Lie algebra, and characterize it using a homomorphism from a Lie algebra to the semidirect product Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on 3-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an $L_\infty$-algebra whose Maurer-Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted $L_\infty$-algebra that controls deformations of a given crossed homomorphism on 3-Lie algebras.Comment: arxiv admin note: substantial text overlap with arXiv:2204.0219

« Je fais ça pour mon enfant, pour ma famille »

La discrimination sociale et une protection médicale limitée poussent la majorité des personnes vivant avec le VIH/SIDA à chercher de l’aide auprès de leur famille. Cet article, à partir d’une enquête qualitative sur l’expérience de la maladie, traite de la notion d’obligation familiale et de son influence sur l’organisation de la vie quotidienne de ces individus. En envisageant la notion comme une réciprocité des relations entre les membres d’une famille, nous nous penchons plus particulièrement sur la façon dont les personnes vivant avec le VIH préservent leur autonomie afin de remplir leurs devoirs et de se construire comme des sujets éthiques malgré leur état de santé. Cette étude souligne l’importance et les limites de l'intégration du soutien familial dans l’action contre le SIDA en Chine

Model averaging based on weighted generalized method of moments with missing responses

Model averaging based on the least squares estimator or the maximum likelihood estimator has been widely followed, while model averaging based on the generalized method of moments is almost rarely addressed. This paper is concerned with a model averaging method based on the weighted generalized method of moments for missing responses problem. The weight vector for model averaging is obtained via minimizing the leave-one-out cross validation criterion. With some mild conditions, the asymptotic optimality of the proposed method in the sense that it can achieve the lowest squared error asymptotically is proved. Some numerical experiments are conducted to evaluate the proposed method with the existing related ones, and the results suggest that the proposed method performs relatively well