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By studying the properties of $q$-series $\widehat Z$-invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to 3-manifolds with toral boundaries, and to BPS partition functions with line operators. This provides a new physical realization of logarithmic vertex algebras in the framework of the 3d-3d correspondence and opens new avenues for their future study. For example, we illustrate how invoking a knot-quiver correspondence for $\widehat{Z}$-invariants leads to many infinite families of new fermionic formulae for VOA characters., 85 pages, 3 figures, 6 tables

Supplemental material, TMJ964565_Supplemental_Material_CLN for Five-year relative survival by stage of breast and colon cancers in Italy by Lucia Mangone, Pamela Mancuso, Isabella Bisceglia, Barbara Braghiroli, Francesca Ferrari, Massimo Vicentini and Paolo Giorgi Rossi in Tumori Journal

Supplemental material, TMJ964565_Supplemental_Material_CLN for Five-year relative survival by stage of breast and colon cancers in Italy by Lucia Mangone, Pamela Mancuso, Isabella Bisceglia, Barbara Braghiroli, Francesca Ferrari, Massimo Vicentini and Paolo Giorgi Rossi in Tumori Journal

We introduce supergroup analogues of 3-manifold invariants $\hat{Z}$, also known as homological blocks, which were previously considered for ordinary compact semisimple Lie groups. We focus on superunitary groups, and work out the case of SU(2|1) in details. Physically these invariants are realized as the index of BPS states of a system of intersecting fivebranes wrapping a 3-manifold in M-theory. As in the original case, the homological blocks are q-series with integer coefficients. We provide an explicit algorithm to calculate these q-series for a class of plumbed 3-manifolds and study quantum modularity and resurgence properties for some particular 3-manifolds. Finally, we conjecture a formula relating the $\hat{Z}$ invariants and the quantum invariants constructed from a non-semisimple category of representation of the unrolled version of a quantum supergroup., Comment: 56 pages, 7 figures. v2: minor corrections in the text and formulas, references added

Background Childhood obesity prevention is a public health priority in industrialized countries. The Reggio Emilia Local Health Authority has implemented a program involving primary and secondary prevention as well as the care of obese children. There are many health-promoting mobile apps, but few are targeted to children and very few are sponsored by public health agencies. Objective The goal of the research was to describe the process and tools adopted to cocreate a mobile app sponsored by the Reggio Emilia Local Health Authority to be installed in parents’ phones aimed at promoting child health and preventing obesity. Methods After stakeholder mapping, a consulting committee including relevant actors, stakeholders, and users was formed. Key persons for childhood obesity prevention were interviewed, focus groups with parents and pediatricians were conducted, and community reporting storytelling was collected. The results of these activities were presented to the consulting committee in order to define the functionalities and contents of the mobile app. Results Three key trends emerged from community reporting: being active, playing, and being outdoors; time for oneself, family, and friends; and the pressures of life and work and not having time to be active and socialize. In focus groups, interviews, and labs, mothers showed a positive attitude toward using an app to manage their children's weight, while pediatricians expressed concerns that the app could increase their workload. When these findings were explored by the consulting committee, four key themes were extracted: strong relationships with peers, family members, and the community; access to safe outdoor spaces; children’s need for age-appropriate independence; and professional support should be nonjudgmental and stigma-free. It should be a dialogue that promotes family autonomy. The app functions related to these needs include the following: (1) newsletter with anticipatory guidance, recipes, and vaccination and well-child visit reminders; (2) regional map indicating where physical activity can be done; (3) information on how to manage emergencies (eg, falls, burns, fever); (4) module for reinforcing the counseling intervention conducted by pediatricians for overweight children; and (5) a function to build a balanced daily diet. Conclusions The pilot study we conducted showed that cocreation in health promotion is feasible, with the consulting committee being the key co-governance and cocreation tool. The involvement of stakeholders in this committee made it possible to expand the number of persons and institutions actively contributing to the project.

Figure S1. Confocal analysis of eGFP-tagged yâ +â LAT1 mutants in CHO cells. (PDF 484 kb)

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs., 119 pages, 10 figures and 20 tables