Past Differential Equations and Applications Seminar

3 June 2010
16:30
Alexander Movchan
Abstract
Bloch Floquet waves are considered in structured media. Such waves are dispersive and the dispersion diagrams contain stop bands. For an example of a harmonic lattice, we discuss dynamic band gap Green’s functions characterised by exponential localisation. This is followed by simple models of exponentially localised defect modes. Asymptotic models involving uniform asymptotic approximations of physical fields in structured media are compared with homogenisation approximations.
  • Differential Equations and Applications Seminar
27 May 2010
16:30
Abstract
We report the numerical realization and demonstration of robustness of certain 2-component structures in Bose-Einstein Condensates in 2 and 3 spatial dimensions with non-trivial topological charge in one of the components. In particular, we identify a stable symbiotic state in which a higher-dimensional bright soliton exists even in a homogeneous setting with defocusing interactions, as a result of the effective potential created by a stable vortex in the other component. The resulting vortex-bright solitary waves, which naturally generalize the recently experimentally observed dark-bright solitons, are examined both in the homogeneous medium and in the presence of parabolic and periodic external confinement and are found to be very robust.
  • Differential Equations and Applications Seminar
20 May 2010
16:30
Gero Miesenboeck
Abstract
An emerging set of methods enables an experimental dialogue with biological systems composed of many interacting cell types---in particular, with neural circuits in the brain. These methods are sometimes called “optogenetic” because they employ light-responsive proteins (“opto-“) encoded in DNA (“-genetic”). Optogenetic devices can be introduced into tissues or whole organisms by genetic manipulation and be expressed in anatomically or functionally defined groups of cells. Two kinds of devices perform complementary functions: light-driven actuators control electrochemical signals; light-emitting sensors report them. Actuators pose questions by delivering targeted perturbations; sensors (and other measurements) signal answers. These catechisms are beginning to yield previously unattainable insight into the organization of neural circuits, the regulation of their collective dynamics, and the causal relationships between cellular activity patterns and behavior.
  • Differential Equations and Applications Seminar
13 May 2010
16:30
Abstract
In the first part of my presentation, I plan to review several applications modelled by delay differential equations (DDEs) starting from familiar examples such as traffic flow problems to physiology and industrial problems. Although delay differential equations have the reputation to be difficult mathematical problems, there is a renewed interest for both old and new problems modelled by DDEs. In the second part of my talk, I’ll emphasize the need of developing asymptotic tools for DDEs in order to guide our numerical simulations and help our physical understanding. I illustrate these ideas by considering the response of optical optoelectronic oscillators that have been studied both experimentally and numerically.
  • Differential Equations and Applications Seminar
6 May 2010
16:30
Abstract
How does form emerge from cellular processes? Using cell-based mechanical models of growth, we investigated the geometry of leaf vasculature and the cellular arrangements at the shoot apex. These models incorporate turgor pressure, wall mechanical properties and cell division. In connection with experimental data, they allowed us to, on the one hand, account for characteristic geometrical property of vein junctions, and, on the other hand, speculate that growth is locally regulated.
  • Differential Equations and Applications Seminar
11 March 2010
16:30
Richard Braun (University of Delaware)
Abstract
We study dynamics from models for the human tear film in one and two dimensional domains. The tear film is roughly a few microns thick over a domain on a centimeter scale; this separation of scales makes lubrication models desirable. Results on one-dimensional blinking domains are presented for multiple blink cycles. Results on two-dimensional stationary domains are presented for different boundary conditions. In all cases, the results are sensitive to the boundary conditions; this is intuitively satisfying since the tear film seems to be controlled primarily from the boundary and its motion. Quantitative comparison with in vivo measurement will be given in some cases. Some discussion of tear film properties will also be given, and results for non-Newtonian models will be given as available, as well as some future directions.
  • Differential Equations and Applications Seminar

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