Convergence between Categorical Representations of Reeb Space and Mapper

The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it compresses the components of the level sets of a multivariate mapping and obtains a summary representation of their relationships. A related construction called mapper, and a special case of the mapper construction called the Joint Contour Net have been shown to be effective in visual analytics. Mapper and JCN are intuitively regarded as discrete approximations of the Reeb space, however without formal proofs or approximation guarantees. An open question has been proposed by Dey et al. as to whether the mapper construction converges to the Reeb space in the limit. In this paper, we are interested in developing the theoretical understanding of the relationship between the Reeb space and its discrete approximations to support its use in practical data analysis. Using tools from category theory, we formally prove the convergence between the Reeb space and mapper in terms of an interleaving distance between their categorical representations. Given a sequence of refined discretizations, we prove that these approximations converge to the Reeb space in the interleaving distance; this also helps to quantify the approximation quality of the discretization at a fixed resolution

Agricultural and Small Business Retirement Plans and Needs for Information: Evidence from a Vermont Survey

A survey of Vermont agricultural and small business owners and managers was conducted to examine their retirement plans and needs for information. Results from 828 respondents indicate that 42% of them do not have any retirement plan and 45% of the surveyed businesses do not offer any retirement plan for their employees. Results also suggest that most small business owners and managers who do not have a retirement plan are not very interested in more information on retirement planning.Consumer/Household Economics,

Penalties for Emergency Medical Treatment and Labor Act Violations Involving Obstetrical Emergencies

Introduction: The Emergency Medical Treatment and Labor Act (EMTALA) was intended to prevent inadequate, delayed, or denied treatment of emergent conditions by emergency departments (ED). While controversies exist regarding the scope of the law, there is no question that EMTALA applies to active labor, a key tenet of the statute and the only medical condition – labor – specifically included in the title of the law. In light of rising maternal mortality rates in the United States, further exploration into the state of emergency obstetrical (OB) care is warranted. Understanding civil monetary penalty settlements levied by the Office of the Inspector General (OIG) related to EMTALA violations involving labor and other OB emergencies will help to inform the current state of access to and quality of OB emergency care.Methods: We reviewed descriptions of all EMTALA-related OIG civil monetary penalty settlements from 2002-2018. OB-related cases were identified using keywords in settlement descriptions. We described characteristics of settlements including the nature of the allegation and compared them with non-OB settlements.Results: Of 232 EMTALA-related OIG settlements during the study period, 39 (17%) involved active labor and other OB emergencies. Between 2002 and 2018 the proportion of settlements involving OB emergencies increased from 17% to 40%. Seven (18%) of these settlements involved a pregnant minor. Most OB cases involved failure to provide screening exam (82%) and/or stabilizing treatment (51%). Failure to arrange appropriate transfer was more common for OB  (36%) compared with non-OB settlements (21%) (p = 0.041). Fifteen (38%) involved a provider specifically directing a pregnant woman to proceed to another hospital, typically by private vehicle.Conclusion: Despite inclusion of the term “labor” in the law’s title, one in six settlements related to EMTALA violations involved OB emergencies. One in five settlements involved a pregnant minor, indicating that providers may benefit from education regarding obligations to evaluate and stabilize minors absent parental consent. Failure to arrange appropriate transfer was more common among OB settlements. Findings suggesting need for providers to understand EMTALA-specific requirements for appropriate transfer and for EDs at hospitals without dedicated OB services to implement policies for evaluation of active labor and protocols for transfer when indicated