Can we make a Finsler metric complete by a trivial projective change?

A trivial projective change of a Finsler metric $F$ is the Finsler metric $F + df$. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change. The problem actually came from lorentz geometry and mathematical relativity: it was observed that it is possible to understand the light-line geodesics of a (normalized, standard) stationary 4-dimensional space-time as geodesics of a certain Finsler Randers metric on a 3-dimensional manifold. The trivial projective change of the Finsler metric corresponds to the choice of another 3-dimensional slice, and the existence of a trivial projective change that is forward and backward complete is equivalent to the global hyperbolicity of the space-time.Comment: 11 pages, one figure, submitted to the proceedings of VI International Meeting on Lorentzian Geometry (Granada

Clinical phenotypes of acute heart failure based on signs and symptoms of perfusion and congestion at emergency department presentation and their relationship with patient management and outcomes

Objective To compare the clinical characteristics and outcomes of patients with acute heart failure (AHF) according to clinical profiles based on congestion and perfusion determined in the emergency department (ED). Methods and results Overall, 11 261 unselected AHF patients from 41 Spanish EDs were classified according to perfusion (normoperfusion = warm; hypoperfusion = cold) and congestion (not = dry; yes = wet). Baseline and decompensation characteristics were recorded as were the main wards to which patients were admitted. The primary outcome was 1-year all-cause mortality; secondary outcomes were need for hospitalisation during the index AHF event, in-hospital all-cause mortality, prolonged hospitalisation, 7-day post-discharge ED revisit for AHF and 30-day post-discharge rehospitalisation for AHF. A total of 8558 patients (76.0%) were warm+ wet, 1929 (17.1%) cold+ wet, 675 (6.0%) warm+ dry, and 99 (0.9%) cold+ dry; hypoperfused (cold) patients were more frequently admitted to intensive care units and geriatrics departments, and warm+ wet patients were discharged home without admission. The four phenotypes differed in most of the baseline and decompensation characteristics. The 1-year mortality was 30.8%, and compared to warm+ dry, the adjusted hazard ratios were significantly increased for cold+ wet (1.660; 95% confidence interval 1.400-1.968) and cold+ dry (1.672; 95% confidence interval 1.189-2.351). Hypoperfused (cold) phenotypes also showed higher rates of index episode hospitalisation and in-hospital mortality, while congestive (wet) phenotypes had a higher risk of prolonged hospitalisation but decreased risk of rehospitalisation. No differences were observed among phenotypes in ED revisit risk. Conclusions Bedside clinical evaluation of congestion and perfusion of AHF patients upon ED arrival and classification according to phenotypic profiles proposed by the latest European Society of Cardiology guidelines provide useful complementary information and help to rapidly predict patient outcomes shortly after ED patient arrival

Influence of the length of hospitalisation in post-discharge outcomes in patients with acute heart failure: Results of the LOHRCA study

Objective: To investigate the relationship between length of hospitalisation (LOH) and post-discharge outcomes in acute heart failure (AHF) patients and to ascertain whether there are different patterns according to department of initial hospitalisation. Methods: Consecutive AHF patients hospitalised in 41 Spanish centres were grouped based on the LOH (15 days). Outcomes were defined as 90-day post-discharge all-cause mortality, AHF readmissions, and the combination of both. Hazard ratios (HRs), adjusted by chronic conditions and severity of decompensation, were calculated for groups with LOH >6 days vs. LOH <6 days (reference), and stratified by hospitalisation in cardiology, internal medicine, geriatrics, or short-stay units. Results: We included 8563 patients (mean age: 80 (SD = 10) years, 55.5% women), with a median LOH of 7 days (IQR 4–11): 2934 (34.3%) had a LOH 15 days. The 90-day post-discharge mortality was 11.4%, readmission 32.2%, and combined endpoint 37.4%. Mortality was increased by 36.5% (95%CI = 13.0–64.9) when LOH was 11–15 days, and by 72.0% (95%CI = 42.6–107.5) when >15 days. Conversely, no differences were found in readmission risk, and the combined endpoint only increased 21.6% (95%CI = 8.4–36.4) for LOH >15 days. Stratified analysis by hospitalisation departments rendered similar post-discharge outcomes, with all exhibiting increased mortality for LOH >15 days and no significant increments in readmission risk. Conclusions: Short hospitalisations are not associated with worse outcomes. While post-discharge readmissions are not affected by LOH, mortality risk increases as the LOH lengthens. These findings were similar across hospitalisation departments