Connections between the stability of a Poincare map and boundedness of certain associate sequences

Let $m\ge 1$ and $N\ge 2$ be two natural numbers and let ${\mathcal{U}}=\{U(p, q)\}_{p\ge q\ge 0}$ be the $N$-periodic discrete evolution family of $m\times m$ matrices, having complex scalars as entries, generated by ${\mathcal{L}}(\mathbb{C}^m)$-valued, $N$-periodic sequence of $m\times m$ matrices $(A_n).$ We prove that the solution of the following discrete problem $$y_{n+1}=A_ny_n+e^{i\mu n}b,\quad n\in\mathbb{Z}_+,\quad y_0=0$$ is bounded for each $\mu\in\mathbb{R}$ and each $m$-vector $b$ if the Poincare map $U(N, 0)$ is stable. The converse statement is also true if we add a new assumption to the boundedness condition. This new assumption refers to the invertibility for each $\mu\in\mathbb{R}$ of the matrix $V_{\mu}:=\sum\nolimits_{\nu=1}^NU(N, \nu)e^{i\mu \nu}.$ By an example it is shown that the assumption on invertibility cannot be removed. Finally, a strong variant of Barbashin's type theorem is proved

Initial injectable therapy in type 2 diabetes: Key considerations when choosing between glucagon-like peptide 1 receptor agonists and insulin

Managing type 2 diabetes is complex and necessitates careful consideration of patient factors such as engagement in self-care, comorbidities and costs. Since type 2 diabetes is a progressive disease, many patients will require injectable agents, usually insulin. Recent ADA-EASD guidelines recommend glucagon-like peptide 1 receptor agonists (GLP-1 RAs) as first injectable therapy in most cases. The basis for this recommendation is the similar glycemic efficacy of GLP-1 RAs and insulin, but with GLP-1 RAs promoting weight loss instead of weight gain, at lower hypoglycemia risk, and with cardiovascular benefits in patients with pre-existing cardiovascular disease. GLP-1 RAs also reduce burden of glucose self-monitoring. However, tolerability and costs are important considerations, and notably, rates of drug discontinuation are often higher for GLP-1 RAs than basal insulin. To minimize risk of gastrointestinal symptoms patients should be started on lowest doses of GLP-1 RAs and up-titrated slowly. Overall healthcare costs may be lower with GLP-1 RAs compared to insulin. Though patient-level costs may still be prohibitive, GLP-1 RAs can replace 50–80 units of insulin daily and reduce costs associated with glucose self-monitoring. Decisions regarding initiating injectable therapy should be individualized. This review provides a framework to guide decision-making in the real-world setting

Sex-disaggregated data matters: tracking the impact of COVID-19 on the health of women and men

Sex and gender matter to health outcomes, but despite repeated commitments to sex-disaggregate data in health policies and programmes, a persistent and substantial absence of such data remains especially in lower-income countries. This represents a missed opportunity for monitoring and identifying gender-responsive, evidence-informed solutions to address a key driver of the pandemic. In this paper we review the availability of national sex-disaggregated surveillance data on COVID-19 and examine trends on the testing-to-outcome pathway. We further analyse the availability of data according to the economic status of the country and investigate the determinants of sex differences, including the national gender inequality status (according to a global index) in each country. Results are drawn from 18 months of global data collection from over 200 countries. We find differences in COVID-19 prevention behaviours and illness outcomes by sex, with lower uptake of vaccination and testing plus an elevated risk of severe disease and death among men. Supporting and maintaining the collection, collation, interpretation and presentation of sex-disaggregated data requires commitment and resources at subnational, national and global levels, but provides an opportunity for identifying and taking gender-responsive action on health inequities. As a first step the global health community should recognise, value and support the importance of sex-disaggregated data for identifying and tackling an inequitable pandemic

Duration of type 2 diabetes does not appear to moderate hypoglycaemia rate with insulin degludec versus insulin glargine U100

In the DEVOTE and SWITCH 2 trials, insulin degludec 100 units/mL (degludec) was superior to insulin glargine 100 units/mL (glargine U100) with respect to the rates of severe (DEVOTE; across trial) and overall symptomatic (SWITCH 2; during the maintenance period of the trial) hypoglycaemia in individuals with type 2 diabetes. In this post hoc analysis, data from 7635 individuals from DEVOTE and 720 individuals from SWITCH 2 were analysed by subgroups of diabetes duration at baseline (<10, ≥10–<15, ≥15–<20 and ≥20 years) using prespecified models from both trials. There was a trend towards lower rates of hypoglycaemia with degludec versus glargine U100 across all diabetes duration subgroups in both trials, with the difference being statistically significant in some subgroups in DEVOTE and SWITCH 2. Overall, however, no significant interaction was observed between diabetes duration and treatment (DEVOTE interaction, P =.496; SWITCH 2 interaction, P =.144). Therefore, in this post hoc analysis of DEVOTE and SWITCH 2, diabetes duration did not appear to affect the reduction in rates of hypoglycaemia observed with degludec compared with glargine U100

Evidence-informed health policy: are we beginning to get there at last

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Prochlo: Strong Privacy for Analytics in the Crowd

The large-scale monitoring of computer users' software activities has become commonplace, e.g., for application telemetry, error reporting, or demographic profiling. This paper describes a principled systems architecture---Encode, Shuffle, Analyze (ESA)---for performing such monitoring with high utility while also protecting user privacy. The ESA design, and its Prochlo implementation, are informed by our practical experiences with an existing, large deployment of privacy-preserving software monitoring. (cont.; see the paper