Connectivity by geodesics on a class of globally hyperbolic spacetimes

During the past years there has been a considerable amount of research related to the problem of geodesic connectedness of Lorentzian manifolds. This topic has wide applications in Physics, but for mathematicians its interest is essentially due to the peculiar di culty of this natural problem, which makes it challenging from both an analytical and a geometrical point of view. In this talk I discuss the geodesic connectedness problem on globally hyper- bolic spacetimes endowed with a complete, timelike or lightlike, Killing vector eld and a complete Cauchy hypersurface. Then I introduce the notion of open subset with convex boundary and present some applications of previous results.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A pseudo–index approach to fractional equations

We study the multiplicity of weak solutions of a quasilinear elliptic problem on an open bounded domain of RN with smooth boundary ∂ Ω. Both the non– resonant and the resonant case are analyzed. Furthermore, we consider an analogous problem for the fractional p–Laplacian and on RN .Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Recent results on (p,q)-Laplacian type equations

Start ing from a new sum decomposit ion of W1,p(RN ) ∩W1,q(RN ) and using a variat ional approach, we invest igate the existence of mult ipleweak solut ions of a (p, q)–Laplacian equat ion on RN , for 1 < q < p < N , with a sign–changing potent ial and a Carath´eodory react ion term sat isfying the celebrated Ambroset t i– Rabinowitz condit ion. Our assumpt ions are mild and different from those used in related papers and moreover our results improve or complement previous ones for the single p–Laplacian. 1Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Infinitely many solutions for non-local problems with broken symmetry

The aim of this paper is to investigate the existence of solutions of the non-local elliptic problem (Formula Presented) where s Ó (0, 1), n > 2s, is an open bounded domain of Rn with Lipschitz boundary ∂Ω, (-Δ)s is the nonlocal Laplacian operator, 2 < p < 2s and h Ó L2 (Ω). This problem requires the study of the eigenvalue problem related to the fractional Laplace operator, with or without potential.Fil: Bartolo, Rossella. Politecnico Di Bari; ItaliaFil: de Napoli, Pablo Luis. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales.; ArgentinaFil: Salvatore, Addolorata. Università degli Studi di Bari; Itali

A multi-state analysis of disease trajectories and mental health transitions in patients with type 2 diabetes: A population-based retrospective cohort study utilizing health administrative data

Aims: To investigate the risk of major depression and dementia in patients with type 2 diabetes, including dementia resulting from depression, and their impact on diabetes-related complications and mortality. Methods: We conducted a population-based retrospective cohort study including 11,441 incident cases of diabetes in 2015-2017, with follow-up until 2022. A multi-state survival analysis was performed on a seven-state model with 15 transitions to capture disease progression and onset of mental disorders. Results: Eight-year probabilities of depression, dementia, diabetes-related complications, and death were 9.7% (95% CI 8.7-10.7), 0.9% (95% CI 0.5-1.3), 10.4% (95% CI 9.5-11.4), and 14.8% (95% CI 13.9-15.7), respectively. Depression increased the risk of dementia up to 3.7% (95% CI 2.0-5.4), and up to 10.3% (95% CI 0.3-20.4) if coupled with diabetes complications. Eight-year mortality was 37.5% (95% CI 33.1-42.0) after depression, 74.1% (95% CI 63.7-84.5) after depression plus complications, 76.4% (95% CI 68.8-83.9) after dementia, and 98.6% (95% CI 96.1-100.0) after dementia plus complications. Conclusions: The interconnections observed across depression, dementia, complications, and mortality underscore the necessity for comprehensive and integrated approaches in managing diabetes. Early screening for depression, followed by timely and targeted interventions, may mitigate the risk of dementia and improve diabetes prognosis