Gagliardo-Nirenberg Inequalities for Differential Forms in Heisenberg Groups

The L 1-Sobolev inequality states that the L n/(n--1)-norm of a compactly supported function on Euclidean n-space is controlled by the L 1-norm of its gradient. The generalization to differential forms (due to Lanzani & Stein and Bourgain & Brezis) is recent, and states that a the L n/(n--1)-norm of a compactly supported differential h-form is controlled by the L 1-norm of its exterior differential du and its exterior codifferential $\delta$u (in special cases the L 1-norm must be replaced by the H 1-Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms

L1L^1-Poincar\'e and Sobolev inequalities for differential forms in Euclidean spaces

In this paper, we prove Poincar\'e and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to Bourgain-Brezis.Comment: Accepted for publication in Science China Mathematics. arXiv admin note: text overlap with arXiv:1902.0481

L1-Poincar\ue9 and Sobolev inequalities for differential forms in Euclidean spaces

In this paper, we prove Poincar\ue9 and Sobolev inequalities for differential forms in L1(\u211dn). The singular integral estimates that it is possible to use for Lp, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis (2007)

L1L^1-Poincar\'e inequalities for differential forms on Euclidean spaces and Heisenberg groups

In this paper, we prove interior Poincar{\'e} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis in Euclidean spaces, and to Chanillo-van Schaftingen in Heisenberg groups

Orlicz spaces and endpoint Sobolev-Poincaré inequalities for differential forms in Heisenberg groups

In this paper we prove Poincar´e and Sobolev inequalities for differential forms in the Rumin’s contact complex on Heisenberg groups. In particular, we deal with endpoint values of the exponents, obtaining finally estimates akin to exponential Trudinger inequalities for scalar function. These results complete previous results obtained by the authors away from the exponential case. From the geometric point of view, Poincaré and Sobolev inequalities for differential forms provide a quantitative formulation of the vanishing of the cohomology. They have also applications to regularity issues for partial differential equations

J Clin Sleep Med

Non-24-hour sleep-wake disorder is 1 of several chronic circadian rhythm sleep-wake disorders. It is defined as progressive daily shifts in sleep onset and wake times. It mainly affects patients who are sight-impaired, is relatively rare in sighted patients, and is difficult to treat, with no guidelines. This case report discusses non-24-hour sleep-wake disorder in a sighted young man who complained of alternating severe insomnia and excessive sleepiness, with a sleep agenda and actigraphic data showing a daily delay of approximately 2 hours. A novel therapy by total sleep deprivation followed by a combination of morning light therapy and nocturnal melatonin administration was efficient in stopping his free-running sleep-wake pattern both immediately and in the long term. The treatment combination for 6 months resulted in stable circadian entrainment to a 24-hour cycle. Compliance with chronotherapy was maintained over the course of follow-up

Design and Preliminary Study of a Neurofeedback Protocol to reduce Drowsiness

International audienceNeuroFeedback (NF) consists in using electroencephalographic (EEG) measurements to guide users to perform a cognitive learning using information coming from their own brain activity, by means of a real-time sensory feedback (e.g., visual or auditory)[4]. Many NF approaches have been studied to improve attentional abilities, notably for Attention Deficit Hy-peractivity Disorder [1, 2]. However, to our knowledge, no NF solution has been proposed to specifically reduce drowsiness. Thus, we propose a complete EEG-NF solution to train users to self-regulate an EEG marker of drowsiness. This marker is based on a ratio of beta over theta/alpha power in Cz electrode. In addition to this EEG marker of drowsiness, we also carefully selected and designed the duration, the sequencing, the objective evaluation metrics and the visual and audio feedback to use in for each NF session. Preliminary study with five healthy subjects showed that three of them could learn to self-regulate this EEG marker with a relatively short number of NF sessions (up to 8 sessions of 40 min). Clinical trials with sleep-deprived subjects are expected to begin in 2019 to study possible cognitive and clinical benefits of this self-regulation. The implementation of this NF solution is available for free 1 , with the OpenViBE platform [3], under the AGPL-3.0 license

Non-REM Sleep Characteristics Predict Early Cognitive Impairment in an Aging Population

Objective: Recent research suggests that sleep disorders or changes in sleep stages or EEG waveform precede over time the onset of the clinical signs of pathological cognitive impairment (e.g., Alzheimer's disease). The aim of this study was to identify biomarkers based on EEG power values and spindle characteristics during sleep that occur in the early stages of mild cognitive impairment (MCI) in older adults.Methods: This study was a case-control cross-sectional study with 1-year follow-up of cases. Patients with isolated subjective cognitive complaints (SCC) or MCI were recruited in the Bordeaux Memory Clinic (MEMENTO cohort). Cognitively normal controls were recruited. All participants were recorded with two successive polysomnography 1 year apart. Delta, theta, and sigma absolute spectral power and spindle characteristics (frequency, density, and amplitude) were analyzed from purified EEG during NREM and REM sleep periods during the entire second night.Results: Twenty-nine patients (8 males, age = 71 ± 7 years) and 29 controls were recruited at T0. Logistic regression analyses demonstrated that age-related cognitive impairment were associated with a reduced delta power (odds ratio (OR) 0.072, P < 0.05), theta power (OR 0.018, P < 0.01), sigma power (OR 0.033, P < 0.05), and spindle maximal amplitude (OR 0.002, P < 0.05) during NREM sleep. Variables were adjusted on age, gender, body mass index, educational level, and medication use. Seventeen patients were evaluated at 1-year follow-up. Correlations showed that changes in self-reported sleep complaints, sleep consolidation, and spindle characteristics (spectral power, maximal amplitude, duration, and frequency) were associated with cognitive impairment (P < 0.05).Conclusion: A reduction in slow-wave, theta and sigma activities, and a modification in spindle characteristics during NREM sleep are associated very early with a greater risk of the occurrence of cognitive impairment. Poor sleep consolidation, lower amplitude, and faster frequency of spindles may be early sleep biomarkers of worsening cognitive decline in older adults