Thresholds for hanger slackening and cable shortening in the Melan equation for suspension bridges

The Melan equation for suspension bridges is derived by assuming small displacements of the deck and inextensible hangers. We determine the thresholds for the validity of the Melan equation when the hangers slacken, thereby violating the inextensibility assumption. To this end, we preliminarily study the possible shortening of the cables: it turns out that there is a striking difference between even and odd vibrating modes since the former never shorten the cables. These problems are studied both on beams and plates

Solenoidal extensions in domains with obstacles: explicit bounds and applications to Navier-Stokes equations

We introduce a new method for constructing solenoidal extensions of fairly general boundary data in (2d or 3d) cubes that contain an obstacle. This method allows us to provide explicit bounds for the Dirichlet norm of the extensions. It runs as follows: by inverting the trace operator, we first determine suitable extensions, not necessarily solenoidal, of the data; then we analyze the Bogovskii problem with the resulting divergence to obtain a solenoidal extension; finally, by solving a variational problem involving the infinity-Laplacian and using ad hoc cutoff functions, we find explicit bounds in terms of the geometric parameters of the obstacle. The natural applications of our results lie in the analysis of inflow-outflow problems, in which an explicit bound on the inflow velocity is needed to estimate the threshold for uniqueness in the stationary Navier-Stokes equations and, in case of symmetry, the stability of the obstacle immersed in the fluid flow

Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations

Half a century after the appearance of the celebrated paper by Serrin about overdetermined boundary value problems in potential theory and related symmetry properties, we reconsider semilinear polyharmonic equations under Dirichlet boundary conditions in the unit ball of $\mathbb{R}^{n}$. We discuss radial properties (symmetry and monotonicity) of positive solutions of such equations and we show that, in conformal dimensions, the associated Green function satisfies elegant reflection and symmetry properties related to a suitable Kelvin transform (inversion about a sphere). This yields an alternative formula for computing the partial derivatives of solutions of polyharmonic problems. Moreover, it gives some hints on how to modify a counterexample by Sweers where radial monotonicity fails: we numerically recover strict radial monotonicity for the biharmonic equation in the unit ball of $\mathbb{R}^{4}$

Eight(y) mathematical questions on fluids and structures

Turbulence is a long-standing mystery. We survey some of the existing (and some-times contradictory) results and suggest eight natural questions whose answers would increase the mathematical understanding of this phenomenon; each of these questions, yet, gives rise to ten subquestions

The Impact of COVID-19 Quarantine on Patients With Dementia and Family Caregivers: A Nation-Wide Survey

Introduction: Previous studies showed that quarantine for pandemic diseases is associated with several psychological and medical effects. The consequences of quarantine for COVID-19 pandemic in patients with dementia are unknown. We investigated the clinical changes in patients with Alzheimer’s disease and other dementias, and evaluated caregivers’ distress during COVID-19 quarantine. Methods: The study involved 87 Italian Dementia Centers. Patients with Alzheimer’s Disease (AD), Dementia with Lewy Bodies (DLB), Frontotemporal Dementia (FTD), and Vascular Dementia (VD) were eligible for the study. Family caregivers of patients with dementia were interviewed by phone in April 2020, 45 days after quarantine declaration. Main outcomes were patients’ changes in cognitive, behavioral, and motor symptoms. Secondary outcomes were effects on caregivers’ psychological features. Results: 4913 patients (2934 females, 1979 males) fulfilled the inclusion criteria. Caregivers reported a worsening in cognitive functions in 55.1% of patients, mainly in subjects with DLB and AD. Aggravation of behavioral symptoms was observed in 51.9% of patients. In logistic regression analysis, previous physical independence was associated with both cognitive and behavioral worsening (odds ratio 1.85 [95% CI 1.42–2.39], 1.84 [95% CI 1.43–2.38], respectively). On the contrary, pandemic awareness was a protective factor for the worsening of cognitive and behavioral symptoms (odds ratio 0.74 [95% CI 0.65–0.85]; and 0.72 [95% CI 0.63–0.82], respectively). Approximately 25.9% of patients showed the onset of new behavioral symptoms. A worsening in motor function was reported by 36.7% of patients. Finally, caregivers reported a high increase in anxiety, depression, and distress. Conclusion: Our study shows that quarantine for COVID-19 is associated with an acute worsening of clinical symptoms in patients with dementia as well as increase of caregivers’ burden. Our findings emphasize the importance to implement new strategies to mitigate the effects of quarantine in patients with dementia