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 $q=p$

Accelerometery as a measure of modifiable physical activity in high-risk elderly preoperative patients: a prospective observational pilot study.

OBJECTIVES: To use wrist-worn accelerometers (Axivity AX3) to establish normative physical activity (PA) and acceptability data for the high-risk elderly preoperative population, to assess whether PA could be modified by a prehabilitation intervention as part of routine care, to assess any correlation between accelerometer-measured PA and self-reported PA and to assess the acceptability of wearing wrist-worn accelerometers in this population. STUDY DESIGN: Prospective, observational, pilot study. SETTING: Single National Health Service Hospital. PARTICIPANTS: Frail patients≥65 years awaiting major surgery referred to a multidisciplinary preoperative clinic at which they received a routine intervention aimed at improving their PA. 35 patients were recruited. Average age 79.9 years (SD=5.6). PRIMARY OUTCOMES: Normative PA data measured as a mean daily Euclidean norm minus one (ENMO) in milli-gravitational units (mg). SECONDARY OUTCOMES: Measure PA levels (mg) following a routine preoperative intervention. Determine correlation between patient-reported PA (measured using the Physical Activity Scale for the Elderly) and accelerometer-measured PA (mg). Assess acceptability of wearing a wrist-worn accelerometer measured using Visual Analogue Scale (VAS) questionnaire and device wear time (hours). RESULTS: Median baseline daily PA was 14.3 mg (IQR 9.75-22.04) with an improvement in PA detected following the intervention (median ENMO post intervention 20.91 mg (IQR 14.83-27.53), p=0.022). There was no significant correlation between accelerometer-measured and self-reported PA (baseline ρ=0.162 (p=0.4), post intervention ρ=-0.144 (p=0.5)). We found high acceptability ratings (median score of 10/10 on VAS, IQR 8-10) and wear-time compliance (163.2 hours (IQR 150-167.5) preintervention and 166.1 hours (IQR 162.5-167) post intervention). CONCLUSIONS: Accelerometery is acceptable to this population and increases in PA levels measured following an unoptimised routine clinical intervention which indicates that health behavioural change interventions may be successful during the preoperative period. Accelerometers may therefore be a useful tool to design and validate interventions for improving PA in this setting. TRIAL REGISTRATION NUMBER: NCT03737903

Increase of efficiency of information support of the organization processes based on quality management and information technologies

In this article, the role and influence of the integrated information environment on the organization processes was considered. The interrelation of information technologies and quality management system is shown on the basis of system and process approaches. The sequence of building IMS in the enterprise was determined and the main stages of information support of the company's QMS were identified