Reducible means and reducible inequalities

It is well-known that if a real valued function acting on a convex set satisfies the $n$-variable Jensen inequality, for some natural number $n\geq 2$, then, for all $k\in\{1,\dots, n\}$, it fulfills the $k$-variable Jensen inequality as well. In other words, the arithmetic mean and the Jensen inequality (as a convexity property) are both reducible. Motivated by this phenomenon, we investigate this property concerning more general means and convexity notions. We introduce a wide class of means which generalize the well-known means for arbitrary linear spaces and enjoy a so-called reducibility property. Finally, we give a sufficient condition for the reducibility of the $(M,N)$-convexity property of functions and also for H\"older--Minkowski type inequalities

The Bregman chord divergence

Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of testing one by one the entries of an ever-expanding dictionary of {\em ad hoc} distances, one rather prefers to consider parametric classes of distances that are exhaustively characterized by axioms derived from first principles. Bregman divergences are such a class. However fine-tuning a Bregman divergence is delicate since it requires to smoothly adjust a functional generator. In this work, we propose an extension of Bregman divergences called the Bregman chord divergences. This new class of distances does not require gradient calculations, uses two scalar parameters that can be easily tailored in applications, and generalizes asymptotically Bregman divergences.Comment: 10 page

Entropic Uncertainty Relations in Quantum Physics

Uncertainty relations have become the trademark of quantum theory since they were formulated by Bohr and Heisenberg. This review covers various generalizations and extensions of the uncertainty relations in quantum theory that involve the R\'enyi and the Shannon entropies. The advantages of these entropic uncertainty relations are pointed out and their more direct connection to the observed phenomena is emphasized. Several remaining open problems are mentionedComment: 35 pages, review pape

What Should Vaccine Developers Ask? Simulation of the Effectiveness of Malaria Vaccines

A number of different malaria vaccine candidates are currently in pre-clinical or clinical development. Even though they vary greatly in their characteristics, it is unlikely that any of them will provide long-lasting sterilizing immunity against the malaria parasite. There is great uncertainty about what the minimal vaccine profile should be before registration is worthwhile; how to allocate resources between different candidates with different profiles; which candidates to consider combining; and what deployment strategies to consider.We use previously published stochastic simulation models, calibrated against extensive epidemiological data, to make quantitative predictions of the population effects of malaria vaccines on malaria transmission, morbidity and mortality. The models are fitted and simulations obtained via volunteer computing. We consider a range of endemic malaria settings with deployment of vaccines via the Expanded program on immunization (EPI), with and without additional booster doses, and also via 5-yearly mass campaigns for a range of coverages. The simulation scenarios account for the dynamic effects of natural and vaccine induced immunity, for treatment of clinical episodes, and for births, ageing and deaths in the cohort. Simulated pre-erythrocytic vaccines have greatest benefits in low endemic settings (<EIR of 10.5) where between 12% and 14% of all deaths are averted when initial efficacy is 50%. In some high transmission scenarios (>EIR of 84) PEV may lead to increased incidence of severe disease in the long term, if efficacy is moderate to low (<70%). Blood stage vaccines (BSV) are most useful in high transmission settings, and are comparable to PEV for low transmission settings. Combinations of PEV and BSV generally perform little better than the best of the contributing components. A minimum half-life of protection of 2–3 years appears to be a precondition for substantial epidemiological effects. Herd immunity effects can be achieved with even moderately effective (>20%) malaria vaccines (either PEV or BSV) when deployed through mass campaigns targeting all age-groups as well as EPI, and especially if combined with highly efficacious transmission-blocking components.We present for the first time a stochastic simulation approach to compare likely effects on morbidity, mortality and transmission of a range of malaria vaccines and vaccine combinations in realistic epidemiological and health systems settings. The results raise several issues for vaccine clinical development, in particular appropriateness of vaccine types for different transmission settings; the need to assess transmission to the vector and duration of protection; and the importance of deployment additional to the EPI, which again may make the issue of number of doses required more critical. To test the validity and robustness of our conclusions there is a need for further modeling (and, of course, field research) using alternative formulations for both natural and vaccine induced immunity. Evaluation of alternative deployment strategies outside EPI needs to consider the operational implications of different approaches to mass vaccination

Gauss’ theorem concerning the center of gravity and its application

Gauss’ theorem, Center of gravity, Complex plane,