A Surface of Heteroclinic Connections Between Two Saddle Slow Manifolds in the Olsen Model

The Olsen model for the biochemical peroxidase-oxidase reaction has a parameter regime where one of its four variables evolves much slower than the other three. It is characterized by the existence of periodic orbits along which a large oscillation is followed by many much smaller oscillations before the process repeats. We are concerned here with a crucial ingredient for such mixed-mode oscillations (MMOs) in the Olsen model: a surface of connecting orbits that is followed closely by the MMO periodic orbit during its global, large-amplitude transition back to another onset of small oscillations. Importantly, orbits on this surface connect two one-dimensional saddle slow manifolds, which exist near curves of equilibria of the limit where the slow variable is frozen and acts as a parameter of the so-called fast subsystem. We present a numerical method, based on formulating suitable boundary value problems, to compute such a surface of connecting orbits. It involves a number of steps to compute the slow manifolds, certain submanifolds of their stable and unstable manifolds and, finally, a first connecting orbit that is then used to sweep out the surface by continuation. If it exists, such a surface of connecting orbits between two one-dimensional saddle slow manifolds is robust under parameter variations. We compute and visualize it in the Olsen model and show how this surface organizes the global return mechanism of MMO periodic orbits from the end of small oscillations back to a region of phase space where they start again.

Robust Stability of Multicomponent Membranes: the Role of Glycolipids

We present the multicomponent functionalized free energies that characterize the low-energy packings of amphiphilic molecules within a membrane through a correspondence to connecting orbits within a reduced dynamical system. To each connecting orbits we associate a manifold of low energy membrane-type configurations parameterized by a large class of admissible interfaces. The normal coercivity of the manifolds is established through criteria depending solely on the structure of the associated connecting orbit. We present a class of examples that arise naturally from geometric singular perturbation techniques, focusing on a model that characterizes the stabilizing role of cholesterol-like glycolipids within phospholipid membranes.

Differentiation between maxillary and malar midface position within the facial profile

Aims To define midfacial position differentiating maxillary and zygomatic regions and to evaluate the corresponding cephalometric characteristics discerning midfacial flatness and fullness. Material and Methods A total of 183 pretreatment lateral cephalometric radiographs of non-growing orthodontic patients (age 25.98 ± 8.43 years) screened at our university orthodontic clinic. The lateral cephalographs of the orthodontic patients were stratified in four groups: flat, normal toward flat, normal toward full, full,according to distances from nasion and sella to points J and G (NJ, SJ, NG and SG). J is the midpoint of the distance connecting orbitale to point A, and G the center of the triangle connecting orbit, key ridge and pterygomaxillary fissure. Statistics included the Kendall tau-b test for best associations among measurements. Results All measurements were statistically significantly different between flat and full groups. The highest associations were between NJ and SJ (τb = 0.71; p < 0.001) and NG and SG (τb = 0.70; p < 0.001). Flat midfaces were characterized by canting of the cranial base and palatal plane, hyperdivergent pattern and maxillary retrognathism. The opposite was true for fuller midfaces. Conclusion Midface skeletal location was assessed differentially in the naso-maxillary and malo-zygomatic structures differentially. Craniofacial characteristics were identified according to this stratification, indicating the potential for application in facial diagnosis and need for testing on 3D cone-beam computed tomography images.

Heegaard splittings and Morse-Smale flows

We describe three theorems which summarize what survives in three dimensions of Smale's proof of the higher-dimensional Poincaré conjecture. The proofs require Smale's cancellation lemma and a lemma asserting the existence of a2-gon. Such2-gons are the analogues in dimension two of Whitney disks in higher dimensions. They are also embedded lunes; an (immersed) lune is an index-one connecting orbit in the Lagrangian Floer homology determined by two embedded loops in a2-manifold.