Post-Lie Algebras, Factorization Theorems and Isospectral-Flows

In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of a second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang-Baxter equation. In this context, we prove a factorization theorem for group-like elements. An explicit exponential solution of the corresponding Lie bracket flow is presented, which is based on the aforementioned post-Lie Magnus expansion.Comment: 49 pages, no-figures, review articl

Exotic Smooth Structures on Small 4-Manifolds

Let M be either CP^2#3CP^2bar or 3CP^2#5CP^2bar. We construct the first example of a simply-connected symplectic 4-manifold that is homeomorphic but not diffeomorphic to M.Comment: 11 page

Knots, Braids and BPS States in M-Theory

In previous work we considered M-theory five branes wrapped on elliptic Calabi-Yau threefold near the smooth part of the discriminant curve. In this paper, we extend that work to compute the light states on the worldvolume of five-branes wrapped on fibers near certain singular loci of the discriminant. We regulate the singular behavior near these loci by deforming the discriminant curve and expressing the singularity in terms of knots and their associated braids. There braids allow us to compute the appropriate string junction lattice for the singularity and,hence to determine the spectrum of light BPS states. We find that these techniques are valid near singular points with N=2 supersymmetry.Comment: 38 page

Predicting mental imagery based BCI performance from personality, cognitive profile and neurophysiological patterns

Mental-Imagery based Brain-Computer Interfaces (MI-BCIs) allow their users to send commands to a computer using their brain-activity alone (typically measured by ElectroEncephaloGraphy— EEG), which is processed while they perform specific mental tasks. While very promising, MI-BCIs remain barely used outside laboratories because of the difficulty encountered by users to control them. Indeed, although some users obtain good control performances after training, a substantial proportion remains unable to reliably control an MI-BCI. This huge variability in user-performance led the community to look for predictors of MI-BCI control ability. However, these predictors were only explored for motor-imagery based BCIs, and mostly for a single training session per subject. In this study, 18 participants were instructed to learn to control an EEG-based MI-BCI by performing 3 MI-tasks, 2 of which were non-motor tasks, across 6 training sessions, on 6 different days. Relationships between the participants’ BCI control performances and their personality, cognitive profile and neurophysiological markers were explored. While no relevant relationships with neurophysiological markers were found, strong correlations between MI-BCI performances and mental-rotation scores (reflecting spatial abilities) were revealed. Also, a predictive model of MI-BCI performance based on psychometric questionnaire scores was proposed. A leave-one-subject-out cross validation process revealed the stability and reliability of this model: it enabled to predict participants’ performance with a mean error of less than 3 points. This study determined how users’ profiles impact their MI-BCI control ability and thus clears the way for designing novel MI-BCI training protocols, adapted to the profile of each user

The contributions of local and remote atmospheric moisture fluxes to East Asian precipitation and its variability

We investigate the contribution of the local and remote atmospheric moisture fluxes to East Asia (EA) precipitation and its interannual variability during 1979-2012. We use and expand the \citet{Brubaker_etal_JC_1993} method, which connects the area-mean precipitation to area-mean evaporation and the horizontal moisture flux into the region. Due to its large landmass and hydrological heterogeneity, EA is divided into five sub-regions: Southeast (SE), Tibetan Plateau (TP), Central East (CE), Northwest (NW) and Northeast (NE). For each region, we first separate the contributions to precipitation of local evaporation from those of the horizontal moisture flux by calculating the precipitation recycling ratio: the fraction of precipitation over a region that originates as evaporation from the same region. Then, we separate the horizontal moisture flux across the region's boundaries by direction. We estimate the contributions of the horizontal moisture fluxes from each direction, as well as the local evaporation, to the mean precipitation and its interannual variability. We find that the major contributors to the mean precipitation are not necessarily those that contribute most to the precipitation interannual variability. Over SE, the moisture flux via the southern boundary dominates the mean precipitation and its interannual variability. Over TP, in winter and spring, the moisture flux via the western boundary dominates the mean precipitation; however, variations in local evaporation dominate the precipitation interannual variability. The western moisture flux is the dominant contributor to the mean precipitation over CE, NW and NE. However, the southern or northern moisture flux or the local evaporation dominates the precipitation interannual variability over these regions, depending on the season. Potential mechanisms associated with interannual variability in the moisture flux are identified for each region. The methods and results presented in this study can be readily applied to model simulations, to identify simulation biases in precipitation that relate to the simulated moisture supplies and transport

On diagrammatic bounds of knot volumes and spectral invariants

In recent years, several families of hyperbolic knots have been shown to have both volume and $\lambda_1$ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized twist number of a diagram can provide two-sided bounds on either the volume or $\lambda_1$. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on $\lambda_1$. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded.Comment: 16 pages, 7 figure

From Rota-Baxter Algebras to Pre-Lie Algebras

Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight 1 on complex associative algebras in dimension $\leq 3$ and their corresponding pre-Lie algebras.Comment: 23 pages, appear in Journal of Physics A; Mathematical and Theoretica

Optic Nerve Head Change in Non-Arteritic Anterior Ischemic Optic Neuropathy and Its Influence on Visual Outcome

To evaluate changes in cup/disc (C/D) diameter ratios and parapapillary atrophy in patients with non-arteritic anterior ischemic optic neuropathy (NA-AION), using morphometric methods.The clinical non-interventional study included 157 patients with unilateral or bilateral NA-AION. Optic disc photographs taken from both eyes at the end of follow-up were morphometrically examined.Follow-up was 86.3±70.3 months. Horizontal and vertical disc diameters (P = 0.30;P = 0.61, respectively), horizontal and vertical C/D ratios (P = 0.47;P = 0.19,resp.), and size of alpha zone and beta zone of parapapillary atrophy (P = 0.27;P = 0.32,resp.) did not differ significantly between affected eyes and contralateral normal eyes in patients with unilateral NA-AION. Similarly, horizontal and vertical disc diameters, horizontal and vertical C/D ratios, and size of alpha zone and beta zone did not vary significantly (all P>0.05) between the unaffected eyes of patients with unilateral NA-AION and the eyes of patients with bilateral NA-AION. Optic disc diameters, C/D ratios, size of alpha zone or beta zone of parapapillary atrophy were not significantly associated with final visual outcome in the eyes affected with NA-AION (all P>0.20) nor with the difference in final visual acuity between affected eyes and unaffected eyes in patients with unilateral NA-AION (all P>0.25).NA-AION did not affect C/D ratios nor alpha zone and beta zone of parapapillary atrophy. Optic disc size was not related to the final visual acuity outcome in NA-AION

Improving higher education standards through reengineering in West African universities–A case study of Nigeria

This article examines the context of higher education (HE), policies and challenges in the West African context. A multi-level framework and analysis of reengineering, leading change in complexity, activity-based view of the University Business Model and Pedagogical Content Knowledge enable the development of deep connections between the macro- and meso-level and -micro challenges of Higher Education System (HES). These include elements of effective leadership, structures and curriculum and learning pedagogies. Drawing on the analyses of interviews from 25 overseas trained senior academics from Nigerian universities, a preliminary refinement of the philosophy of reengineering, re-thinking and revaluing the higher education system (HES) is offered. These have traditionally been addressed in a piecemeal perspective in HE policy and the academic literature; such a traditional approach has not been the systematic rethinking advocated in the philosophy of reengineering