Toric symplectic ball packing

We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric-combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory. Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section.Comment: 17 pages, 6 figure

Symplectic bifurcation theory for integrable systems

This paper develops a symplectic bifurcation theory for integrable systems in dimension four. We prove that if an integrable system has no hyperbolic singularities and its bifurcation diagram has no vertical tangencies, then the fibers of the induced singular Lagrangian fibration are connected. The image of this singular Lagrangian fibration is, up to smooth deformations, a planar region bounded by the graphs of two continuous functions. The bifurcation diagram consists of the boundary points in this image plus a countable collection of rank zero singularities, which are contained in the interior of the image. Because it recently has become clear to the mathematics and mathematical physics communities that the bifurcation diagram of an integrable system provides the best framework to study symplectic invariants, this paper provides a setting for studying quantization questions, and spectral theory of quantum integrable systems.Comment: 42 pages, 19 figure

Risk Taking in Hospitalized Patients with Acute and Severe Traumatic Brain Injury

Rehabilitation can improve cognitive deficits observed in patients with traumatic brain injury (TBI). However, despite rehabilitation, the ability of making a choice often remains impaired. Risk taking is a daily activity involving numerous cognitive processes subserved by a complex neural network. In this work we investigated risk taking using the Balloon Analogue Risk Task (BART) in patients with acute TBI and healthy controls. We hypothesized that individuals with TBI will take less risk at the BART as compared to healthy individuals. We also predicted that within the TBI group factors such as the number of days since the injury, severity of the injury, and sites of the lesion will play a role in risk taking as assessed with the BART. Main findings revealed that participants with TBI displayed abnormally cautious risk taking at the BART as compared to healthy subjects. Moreover, healthy individuals showed increased risk taking throughout the task which is in line with previous work. However, individuals with TBI did not show this increased risk taking during the task. We also investigated the influence of three patients’ characteristics on their performance at the BART: Number of days post injury, Severity of the head injury, and Status of the frontal lobe. Results indicate that performance at the BART was influenced by the number of days post injury and the status of the frontal lobe, but not by the severity of the head injury. Reported findings are encouraging for risk taking seems to naturally improve with time postinjury. They support the need of conducting longitudinal prospective studies to ultimately identify impaired and intact cognitive skills that should be trained postinjury