On the mean width of log-concave functions

In this work we present a new, natural, definition for the mean width of log-concave functions. We show that the new definition coincide with a previous one by B. Klartag and V. Milman, and deduce some properties of the mean width, including an Urysohn type inequality. Finally, we prove a functional version of the finite volume ratio estimate and the low-M* estimate.Comment: 15 page

Almost Euclidean sections of the N-dimensional cross-polytope using O(N) random bits

It is well known that R^N has subspaces of dimension proportional to N on which the \ell_1 norm is equivalent to the \ell_2 norm; however, no explicit constructions are known. Extending earlier work by Artstein--Avidan and Milman, we prove that such a subspace can be generated using O(N) random bits.Comment: 16 pages; minor changes in the introduction to make it more accessible to both Math and CS reader

Trajectories of depressive symptoms after hip fracture

BACKGROUND: Hip fracture is often complicated by depressive symptoms in older adults. We sought to characterize trajectories of depressive symptoms arising after hip fracture and examine their relationship with functional outcomes and walking ability. We also investigated clinical and psychosocial predictors of these trajectories. METHOD: We enrolled 482 inpatients, aged ≥60 years, who were admitted for hip fracture repair at eight St Louis, MO area hospitals between 2008 and 2012. Participants with current depression diagnosis and/or notable cognitive impairment were excluded. Depressive symptoms and functional recovery were assessed with the Montgomery–Asberg Depression Rating Scale and Functional Recovery Score, respectively, for 52 weeks after fracture. Health, cognitive, and psychosocial variables were gathered at baseline. We modeled depressive symptoms using group-based trajectory analysis and subsequently identified correlates of trajectory group membership. RESULTS: Three trajectories emerged according to the course of depressive symptoms, which we termed ‘resilient’, ‘distressed’, and ‘depressed’. The depressed trajectory (10% of participants) experienced a persistently high level of depressive symptoms and a slower time to recover mobility than the other trajectory groups. Stressful life events prior to the fracture, current smoking, higher anxiety, less social support, antidepressant use, past depression, and type of implant predicted membership of the depressed trajectory. CONCLUSIONS: Depressive symptoms arising after hip fracture are associated with poorer functional status. Clinical and psychosocial variables predicted membership of the depression trajectory. Early identification and intervention of patients in a depressive trajectory may improve functional outcomes after hip fracture

A characterization of the support map

AbstractIn this short note we give a characterization of the support map from classical convexity. We show it is the unique additive transformation from the class of closed convex sets in Rn which include 0 to the class of positive 1-homogeneous functions on Rn. This will be a consequence of a theorem about transforms from the class of convex sets to itself which preserve Minkowski addition

Differential classification of states of consciousness using envelope- and phase-based functional connectivity

The development of sophisticated computational tools to quantify changes in the brain\u27s oscillatory dynamics across states of consciousness have included both envelope- and phase-based measures of functional connectivity (FC), but there are very few direct comparisons of these techniques using the same dataset. The goal of this study was to compare an envelope-based (i.e. Amplitude Envelope Correlation, AEC) and a phase-based (i.e. weighted Phase Lag Index, wPLI) measure of FC in their classification of states of consciousness. Nine healthy participants underwent a three-hour experimental anesthetic protocol with propofol induction and isoflurane maintenance, in which five minutes of 128-channel electroencephalography were recorded before, during, and after anesthetic-induced unconsciousness, at the following time points: Baseline; light sedation with propofol (Light Sedation); deep unconsciousness following three hours of surgical levels of anesthesia with isoflurane (Unconscious); five minutes prior to the recovery of consciousness (Pre-ROC); and three hours following the recovery of consciousness (Recovery). Support vector machine classification was applied to the source-localized EEG in the alpha (8-13 Hz) frequency band in order to investigate the ability of AEC and wPLI (separately and together) to discriminate i) the four states from Baseline; ii) Unconscious ( deep unconsciousness) vs. Pre-ROC ( light unconsciousness); and iii) responsiveness (Baseline, Light Sedation, Recovery) vs. unresponsiveness (Unconscious, Pre-ROC). AEC and wPLI yielded different patterns of global connectivity across states of consciousness, with AEC showing the strongest network connectivity during the Unconscious epoch, and wPLI showing the strongest connectivity during full consciousness (i.e., Baseline and Recovery). Both measures also demonstrated differential predictive contributions across participants and used different brain regions for classification. AEC showed higher classification accuracy overall, particularly for distinguishing anesthetic-induced unconsciousness from Baseline (83.7 ± 0.8%). AEC also showed stronger classification accuracy than wPLI when distinguishing Unconscious from Pre-ROC (i.e., deep from light unconsciousness) (AEC: 66.3 ± 1.2%; wPLI: 56.2 ± 1.3%), and when distinguishing between responsiveness and unresponsiveness (AEC: 76.0 ± 1.3%; wPLI: 63.6 ± 1.8%). Classification accuracy was not improved compared to AEC when both AEC and wPLI were combined. This analysis of source-localized EEG data demonstrates that envelope- and phase-based FC provide different information about states of consciousness but that, on a group level, AEC is better able to detect relative alterations in brain FC across levels of anesthetic-induced unconsciousness compared to wPLI