Active Interest Mentorship for Soon-to-Retire People: A Self-Sustaining Retirement Preparation Program

Retirement is a major life transition that often leads to maladjustments and mental health hazards. In this study, we developed an innovative retirement preparation program, the Active Interest Mentorship Scheme (AIMS), which utilized active interest development as a positive entry point through which to engage soon-to-retire people. Each retiree received a 1-year mentorship 6 months before retirement. Adopting a quasi-experimental design, the study aimed to evaluate the efficacy of the AIMS in protecting retirees’ well-being. The well-being status of 161 retirees was assessed at 4-month intervals. Measures included self-esteem, life satisfaction, positive affect, depression, anxiety, and somatic symptoms. Serial trend analysis revealed a general improvement in well-being at 4 months after mentorship, followed by a mark reversion in some variables at 2 months after retirement. Upon completion of the program, participants generally returned to a level of well-being that was comparable with or better than preretirement levels. The first 2 months after retirement appeared to be the most distressing. The findings support the efficacy, as well as feasibility of the innovative retirement preparation program.

A FAMILY OF GENERALIZED KAC–MOODY ALGEBRAS AND DEFORMATION OF MODULAR FORMS

We develop an analogue of Gindikin–Karpelevich formula for a family of generalized Kac–Moody algebras, attached to Borcherds–Cartan matrices consisting of only one positive entry in the diagonal. As an application, we obtain a deformation of Fourier coefficients of modular forms such as the modular j-function and Ramanujan τ-function.

Counting self-dual interval orders

International audience In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2+2)-free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part of the paper, we derive a generating function for the number of self-dual unlabeled interval orders, with respect to the same statistics. Our method is based on a bijective correspondence between interval orders and upper-triangular matrices in which each row and column has a positive entry. Dans cet article, on obtient une expression explicite pour la fonction génératrice du nombre des ensembles partiellement ordonnés (posets) qui évitent le motif (2+2). La fonction compte ces ensembles par rapport à plusieurs statistiques naturelles, incluant le nombre d'éléments, le nombre de niveaux, et le nombre d'éléments minimaux et maximaux. Dans la deuxième partie, on obtient une expression similaire pour la fonction génératrice des posets autoduaux évitant le motif (2+2). On obtient ces résultats à l'aide d'une bijection entre les posets évitant (2+2) et les matrices triangulaires supérieures dont chaque ligne et chaque colonne contient un élément positif.

Regular stochastic matrices and digraphs

This paper presents an algorithm to determine whether a stochastic matrix is regular. The main theorem is the following. Hypothesis: An n-by-n stochastic matrix has at least one positive entry off the main diagonal in every row and column. There is at most one row with n — 1 zeros and at most one column with n — 1 zeros. There are no j-by-k submatrices consisting entirely of zeros, where j and k are integers greater than 1, with j + k = n. Conclusion: The matrix is regular. Similar results hold for strongly connected digraphs.

Regular stochastic matrices and digraphs

This paper presents an algorithm to determine whether a stochastic matrix is regular. The main theorem is the following. Hypothesis: An n-by-n stochastic matrix has at least one positive entry off the main diagonal in every row and column. There is at most one row with n — 1 zeros and at most one column with n — 1 zeros. There are no j-by-k submatrices consisting entirely of zeros, where j and k are integers greater than 1, with j + k = n. Conclusion: The matrix is regular. Similar results hold for strongly connected digraphs.