On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in R3\mathbb{R}^3

In this paper we use the blow up method of Dumortier and Roussarie \cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of singularities of piecewise smooth dynamical systems \cite{filippov1988differential} in $\mathbb R^3$. Using the regularization method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the power of our approach by considering the case of a fold line. We quickly recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a simple manner. Then, for the two-fold singularity, we show that the regularized system only fully retains the features of the singular canards in the piecewise smooth system in the cases when the sliding region does not include a full sector of singular canards. In particular, we show that every locally unique primary singular canard persists the regularizing perturbation. For the case of a sector of primary singular canards, we show that the regularized system contains a canard, provided a certain non-resonance condition holds. Finally, we provide numerical evidence for the existence of secondary canards near resonance.Comment: To appear in SIAM Journal of Applied Dynamical System

Periodic orbits of period 3 in the disc

Let f be an orientation preserving homeomorphism of the disc D2 which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi /3 or 4 pi /3, or f has a periodic point of least period n for each n in N*.Comment: 7 page

Global dynamics in the Poincare ball of the Chen system having invariant algebraic surfaces

Agraïments: The second author is supported by CNPq-Brazil under the project 305204/2009-2. The third author is partially supported by CNPq and FAPESP. All the authors are supported by the Int. Coop. Proj. CAPES/MECD-TQED II and PHB-2009-0025

The informed road map to prevention of Alzheimer disease: A call to arms

Alzheimer disease (AD) prevention trials hold the promise to delay or prevent cognitive decline and dementia onset by intervening before significant neuronal damage occurs. In recent years, the first AD prevention trials have launched and are yielding important findings on the biology of targeting asymptomatic AD pathology. However, there are limitations that impact the design of these prevention trials, including the translation of animal models that recapitulate key stages and multiple pathological aspects of the human disease, missing target validation in asymptomatic disease, uncertain causality of the association of pathophysiologic changes with cognitive and clinical symptoms, and limited biomarker validation for novel targets. The field is accelerating advancements in key areas including the development of highly specific and quantitative biomarker measures for AD pathology, increasing our understanding of the course and relationship of amyloid and tau pathology in asymptomatic through symptomatic stages, and the development of powerful interventions that can slow or reverse AD amyloid pathology. We review the current status of prevention trials and propose key areas of needed research as a call to basic and translational scientists to accelerate AD prevention. Specifically, we review (1) sporadic and dominantly inherited primary and secondary AD prevention trials, (2) proposed targets, mechanisms, and drugs including the amyloid, tau, and inflammatory pathways and combination treatments, (3) the need for more appropriate prevention animal models and experiments, and (4) biomarkers and outcome measures needed to design human asymptomatic prevention trials. We conclude with actions needed to effectively move prevention targets and trials forward

Periods implying almost all periods, trees with snowflakes, and zero entropy maps

Let $X$ be a compact tree, $f$ be a continuous map from $X$ to itself, $End(X)$ be the number of endpoints and $Edg(X)$ be the number of edges of $X$. We show that if $n>1$ has no prime divisors less than $End(X)+1$ and $f$ has a cycle of period $n$, then $f$ has cycles of all periods greater than $2End(X)(n-1)$ and topological entropy $h(f)>0$; so if $p$ is the least prime number greater than $End(X)$ and $f$ has cycles of all periods from 1 to $2End(X)(p-1)$, then $f$ has cycles of all periods (this verifies a conjecture of Misiurewicz for tree maps). Together with the spectral decomposition theorem for graph maps it implies that $h(f)>0$ iff there exists $n$ such that $f$ has a cycle of period $mn$ for any $m$. We also define {\it snowflakes} for tree maps and show that $h(f)=0$ iff every cycle of $f$ is a snowflake or iff the period of every cycle of $f$ is of form $2^lm$ where $m\le Edg(X)$ is an odd integer with prime divisors less than $End(X)+1$

Darunavir/Cobicistat/Emtricitabine/Tenofovir Alafenamide Versus Dolutegravir/Abacavir/Lamivudine in Antiretroviral-Naive Adults (SYMTRI): A Multicenter Randomized Open-Label Study (PReEC/RIS-57)

Background. Darunavir/cobicistat/emtricitabine/tenofovir alafenamide (D/C/F/TAF) is the reference for combination therapy based on protease inhibitors due to its efficacy, tolerability, and convenience. Head-to-head randomized comparisons between D/C/F/TAF and combination therapy based on integrase inhibitors in antiretroviral-naive patients are lacking. Methods. Adult (>18 years old) human immunodeficiency virus-infected antiretroviral-naive patients (HLA-B∗5701 negative and hepatitis B virus negative), with viral load (VL) ≥500 c/mL, were centrally randomized to initiate D/C/F/TAF or dolutegravir/ abacavir/lamivudine (DTG/3TC/ABC) after stratifying by VL and CD4 count. Clinical and analytical assessments were performed at weeks 0, 4, 12, 24, and 48. The primary endpoint was VL <50 c/mL at week 48 in the intention-to-treat (ITT)-exposed population (US Food and Drug Administration snapshot analysis, 10% noninferiority margin). Results. Between September 2018 and 2019, 316 patients were randomized and 306 patients were included in the ITT-exposed analysis (151 D/C/F/TAF and 155 DTG/3TC/ABC). Almost all (94%) participants were male and their median age was 35 years. Forty percent had a baseline VL >100 000 copies/mL, and 13% had <200 CD4 cells/μL. Median weight was 73 kg and median body mass index was 24 kg/m2 . At 48 weeks, 79% (D/C/F/TAF) versus 82% (DTG/3TC/ABC) had VL <50 c/mL (difference, −2.4%; 95% confidence interval [CI], −11.3 to 6.6). Eight percent versus four percent experienced virologic failure but no resistance-associated mutations emerged. Four percent versus six percent had drug discontinuation due to adverse events. In the per-protocol analysis, 94% versus 96% of patients had VL <50 c/mL (difference, −2%; 95% CI, −8.1 to 3.5). There were no differences in CD4 cell count or weight changes. Conclusions. We could not demonstrate the noninferiority of D/C/F/TAF relative to DTG/ABC/3TC as initial antiretroviral therapy, although both regimens were similarly well tolerated

Darboux theory of integrability for a class of nonautonomous vector fields

The goal of this paper is to extend the classical Darboux theory of integrability from autonomous polynomial vector fields to a class of nonautonomous vector fields. We also provide sufficient conditions for applying this theory of integrability and we illustrate this theory in several examples.Postprint (published version