On the existence threshold for positive solutions of p-laplacian equations with a concave-convex nonlinearity

We study the following boundary value problem with a concave-convex nonlinearity: \begin{equation*} \left\{ \begin{array}{r c l l} -\Delta_p u & = & \Lambda\,u^{q-1}+ u^{r-1} & \textrm{in }\Omega, \\ u & = & 0 & \textrm{on }\partial\Omega. \end{array}\right. \end{equation*} Here $\Omega \subset \mathbb{R}^n$ is a bounded domain and $1<q<p<r<p^*$. It is well known that there exists a number $\Lambda_{q,r}>0$ such that the problem admits at least two positive solutions for $0<\Lambda<\Lambda_{q,r}$, at least one positive solution for $\Lambda=\Lambda_{q,r}$, and no positive solution for $\Lambda > \Lambda_{q,r}$. We show that $$\lim_{q \to p} \Lambda_{q,r} = \lambda_1(p),$$ where $\lambda_1(p)$ is the first eigenvalue of the p-laplacian. It is worth noticing that $\lambda_1(p)$ is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case $q=p$

On Automated Lemma Generation for Separation Logic with Inductive Definitions

Separation Logic with inductive definitions is a well-known approach for deductive verification of programs that manipulate dynamic data structures. Deciding verification conditions in this context is usually based on user-provided lemmas relating the inductive definitions. We propose a novel approach for generating these lemmas automatically which is based on simple syntactic criteria and deterministic strategies for applying them. Our approach focuses on iterative programs, although it can be applied to recursive programs as well, and specifications that describe not only the shape of the data structures, but also their content or their size. Empirically, we find that our approach is powerful enough to deal with sophisticated benchmarks, e.g., iterative procedures for searching, inserting, or deleting elements in sorted lists, binary search tress, red-black trees, and AVL trees, in a very efficient way

Abnormalities of sodium handling and of cardiovascular adaptations during high salt diet in patients with mild heart failure.

BACKGROUND: Sodium retention and hormonal activation are fundamental hallmarks in congestive heart failure. The present study was designed to assess the ability of patients with asymptomatic to mildly symptomatic heart failure and no signs or symptoms of congestion to excrete ingested sodium and to identify possible early abnormalities of hormonal and hemodynamic mechanisms related to sodium handling. METHODS AND RESULTS: The effects of a high salt diet (250 mEq/day for 6 days) on hemodynamics, salt-regulating hormones, and renal excretory response were investigated in a balanced study in 12 untreated patients with idiopathic or ischemic dilated cardiomyopathy and mild heart failure (NYHA class I-II, ejection fraction < 50%) (HF) and in 12 normal subjects, who had been previously maintained a 100 mEq/day NaCl diet. In normal subjects, high salt diet was associated with significant increases of echocardiographically measured left ventricular end-diastolic volume, ejection fraction, and stroke volume (all P < .001) and with a reduction of total peripheral resistance (P < .001). In addition, plasma atrial natriuretic factor (ANF) levels increased (P < .05), and plasma renin activity and aldosterone concentrations fell (both P < .001) in normals in response to salt excess. In HF patients, both left ventricular end-diastolic and end-systolic volumes increased in response to high salt diet, whereas ejection fraction and stroke volume failed to increase, and total peripheral resistance did not change during high salt diet. In addition, plasma ANF levels did not rise in HF in response to salt loading, whereas plasma renin activity and aldosterone concentrations were as much suppressed as in normals. Although urinary sodium excretions were not significantly different in the two groups, there was a small but systematic reduction of daily sodium excretion in HF, which resulted in a significantly higher cumulative sodium balance in HF than in normals during the high salt diet period (P < .001). CONCLUSIONS: These results show a reduced ability to excrete a sodium load and early abnormalities of cardiac and hemodynamic adaptations to salt excess in patients with mild heart failure and no signs or symptoms of congestion

Abatement of an Azo Dye on Structured C-Nafion/Fe-Ion Surfaces by Photo-Fenton Reactions Leading to Carboxylate Intermediates with a Remarkable Biodegradability Increase of the Treated Solution

A novel C-Nafton/Fe-ion structured fabric capable of mediating Orange II decomposition in Fenton-immobilized photoassisted reactions is presented. The catalyst preparation requires the right balance between the amount of the Nafion necessary to protect the C-surface and the minimum encapsulation of the Fe-cluster catalytic sites inside the Nafion to allow the photocatalysis to proceed. The C-Nafion/Fe fabric can be used up to pH 10 under light to photocatalyze the disappearance of Orange II in the presence of H2O2. The photocatalysis mediated by the C-Nafion/Fe-ion fabric increased with the applied light intensity and reaction temperature in the reaction needing an activation energy of 9.8 kcal/mol. This indicates that ion- and radical-molecule reactions take place during Orange II disappearance. The build up and decomposition of intermediate iron complexes under light involves the recycling of Fe2+ and was detected by infrared spectroscopy (FTIR). This observation, along with other experimental results, allows us to suggest a surface mechanism for the dye degradation on the C-Nafion/Fe-ion fabrics. The C-Nafion/Fe-ion fabric in the presence of H2O2 under solar simulated light transforms the totally nonbiodegradable Orange II into a biocompatible material with a very high BOD5/COD value. X-ray photoelectron spectroscopy (XPS) and sputtering by Ar+-ions of the upper surface layer of the C-Nafion/Fe-ion fabric allow us to describe the intervention of the photocatalyst down to the molecular level. Most of the Fe clusters examined by transmission electron microscopy (TEM) showed particle sizes close to 4 nm due to their encapsulation into the Gierke cages of the Nafion thin film observed by scanning electron microscopy (SEM) and optical microscopy (OM)

Competing-risk analysis of coronavirus disease 2019 in-hospital mortality in a Northern Italian centre from SMAtteo COvid19 REgistry (SMACORE)

An accurate prediction of the clinical outcomes of European patients requiring hospitalisation for Coronavirus Disease 2019 (COVID-19) is lacking. The aim of the study is to identify predictors of in-hospital mortality and discharge in a cohort of Lombardy patients with COVID-19. All consecutive hospitalised patients from February 21st to March 30th, 2020, with confirmed COVID-19 from the IRCCS Policlinico San Matteo, Pavia, Lombardy, Italy, were included. In-hospital mortality and discharge were evaluated by competing risk analysis. The Fine and Gray model was fitted in order to estimate the effect of covariates on the cumulative incidence functions (CIFs) for in-hospital mortality and discharge. 426 adult patients [median age 68 (IQR 56 to 77&nbsp;years)] were admitted with confirmed COVID-19 over a 5-week period; 292 (69%) were male. By 21 April 2020, 141 (33%) of these patients had died, 239 (56%) patients had been discharged and 46 (11%) were still hospitalised. Among these 46 patients, updated as of 30 May, 2020, 5 (10.9%) had died, 8 (17.4%) were still in ICU, 12 (26.1%) were transferred to lower intensity care units and 21 (45.7%) were discharged. Regression on the CIFs for in-hospital mortality showed that older age, male sex, number of comorbidities and hospital admission after March 4th were independent risk factors associated with in-hospital mortality. Older age, male sex and number of comorbidities definitively predicted in-hospital mortality in hospitalised patients with COVID-19

The eigenvalue problem for the ∞-Bilaplacian

We consider the problem of finding and describing minimisers of the Rayleigh quotient Λ∞:=infu∈W2,∞(Ω)∖{0}∥Δu∥L∞(Ω)∥u∥L∞(Ω), Λ∞:=infu∈W2,∞(Ω)∖{0}‖Δu‖L∞(Ω)‖u‖L∞(Ω), where Ω⊆RnΩ⊆Rn is a bounded C1,1C1,1 domain and W2,∞(Ω)W2,∞(Ω) is a class of weakly twice differentiable functions satisfying either u=0u=0 on ∂Ω∂Ω or u=|Du|=0u=|Du|=0 on ∂Ω∂Ω . Our first main result, obtained through approximation by LpLp -problems as p→∞p→∞ , is the existence of a minimiser u∞∈W2,∞(Ω)u∞∈W2,∞(Ω) satisfying {Δu∞∈Λ∞Sgn(f∞)Δf∞=μ∞ a.e. in Ω, in D′(Ω), {Δu∞∈Λ∞Sgn(f∞) a.e. in Ω,Δf∞=μ∞ in D′(Ω), for some f∞∈L1(Ω)∩BVloc(Ω)f∞∈L1(Ω)∩BVloc(Ω) and a measure μ∞∈M(Ω)μ∞∈M(Ω) , for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue Λ∞Λ∞ on the domain, establishing the validity of a Faber–Krahn type inequality: among all C1,1C1,1 domains with fixed measure, the ball is a strict minimiser of Ω↦Λ∞(Ω)Ω↦Λ∞(Ω) . This result is shown to hold true for either choice of boundary conditions and in every dimension

The Precarity of Progress: Implications of a Shifting Gendered Division of Labor for Relationships and Well-Being as a Function of Country-Level Gender Equality

The onset of the COVID-19 pandemic saw a shift toward a more traditional division of labor–one where women took greater responsibility for household tasks and childcare than men. We tested whether this regressive shift was more acutely perceived and experienced by women in countries with greater gender equality. Cross-cultural longitudinal survey data for women and men (N = 10,238) was collected weekly during the first few months of the pandemic. Multilevel modelling analyses, based on seven waves of data collection, indicated that a regressive shift was broadly perceived but not uniformly felt. Women and men alike perceived a shift toward a more traditional division of household labor during the first few weeks of the pandemic. However, this perception only undermined women’s satisfaction with their personal relationships and subjective mental health if they lived in countries with higher levels of economic gender equality. Among women in countries with lower levels of economic gender equality, the perceived shift predicted higher relationship satisfaction and mental health. There were no such effects among men. Taken together, our results suggest that subjective perceptions of disempowerment, and the gender role norms that underpin them, should be considered when examining the gendered impact of global crisis

Medium Chain Acylcarnitines Dominate the Metabolite Pattern in Humans under Moderate Intensity Exercise and Support Lipid Oxidation

Background: Exercise is an extreme physiological challenge for skeletal muscle energy metabolism and has notable health benefits. We aimed to identify and characterize metabolites, which are components of the regulatory network mediating the beneficial metabolic adaptation to exercise