RESUMEN
Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. Seminal line-gap and point-gap classifications of non-Hermitian systems using K-theory have deepened the understanding of many physical phenomena. However, ample systems remain beyond this description; reference points and lines do not in general distinguish whether multiple non-Hermitian bands exhibit intriguing exceptional points, spectral braids and crossings. To address this we consider two different notions: non-Hermitian band gaps and separation gaps that crucially encompass a broad class of multi-band scenarios, enabling the description of generic band structures with symmetries. With these concepts, we provide a unified and comprehensive classification of both gapped and nodal systems in the presence of physically relevant parity-time (PT) and pseudo-Hermitian symmetries using homotopy theory. This uncovers new stable topology stemming from both eigenvalues and wave functions, and remarkably also implies distinct fragile topological phases. In particular, we reveal different Abelian and non-Abelian phases inPT-symmetric systems, described by frame and braid topology. The corresponding invariants are robust to symmetry-preserving perturbations that do not induce (exceptional) degeneracy, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneousPTsymmetry breaking is captured by Chern-Euler and Chern-Stiefel-Whitney descriptions, a fingerprint of unprecedented non-Hermitian topology previously overlooked. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platforms.
RESUMEN
BACKGROUND: The disease management program (DMP) for type 2 Diabetes Mellitus (T2DM) is the largest DMP in Germany. Our goal was to analyze regional differences in unenrollment rates, suggest areas for intervention and provide background information, which population groups in which locations are currently not enrolled in the DMP for T2DM. METHODS: In this study, we used data of the 1.7 mil. insurants of the AOK Nordost health insurance. For the visualization of enrollment potential, we used the Besag-York-Mollie model (BYM). The spatial scan statistic (SaTScan) was used to detect areas of unusually high rates of unenrolled diabetics to prioritize areas for intervention. To explore sociodemographic associations, we used Bayesian spatial global regression models. A Spatially varying coefficient model (SVC) revealed in how far the detected associations vary over space. RESULTS: The proportion of diabetics currently not enrolled in the DMP T2DM was 36.8% in 2019 and varied within northeastern Germany. Local clusters were detected mainly in Mecklenburg-West-Pomerania and Berlin. The main sociodemographic variables associated with unenrollment were female sex, younger age, being unemployed, foreign citizenship, small household size and the proportion of persons commuting to work outside their residential municipality. The SVC model revealed important spatially varying effects for some but not all associations. CONCLUSION: Lower socioeconomic status and foreign citizenship had an ubiquitous effect on not being enrolled. The DMP T2DM therefore does currently not reach those population groups, which have a higher risk for secondary diseases and possible avoidable hospitalizations. Logically, future interventions should focus on these groups. Our methodology clearly suggests areas for intervention and points out, which population group in which locations should be specifically approached.