Georgia Institute of Technology College of Sciences

In a joint work with D.Tamarkin we study analytic continuability of solutions of theLaplace-transformed Schroedinger equation by methods of Kashiwara-Schapira style microlocal theoryof sheaves.

The transformation of vibrations into low-power electricity has received growing attention over the last decade. The goal in this research field is to enable self-powered electronic components by harvesting the vibrational energy available in their environment. This talk will be focused on linear and nonlinear vibration-based energy harvesting using piezoelectric materials, including the modeling and experimental validation efforts. Electromechanical modeling discussions will involve both distributed-parameter and lumped-parameter approaches for quantitative prediction and qualitative representation. An important issue in energy harvesters employing linear resonance is that the best performance of the device is limited to a narrow bandwidth around the fundamental resonance frequency. If the excitation frequency slightly deviates from the resonance condition, the power output is drastically reduced. Energy harvesters based on nonlinear configurations (e.g., monostable and bistable Duffing oscillators with electromechanical coupling) offer rich nonlinear dynamic phenomena and outperform resonant energy harvesters under harmonic excitation over a range of frequencies. High-energy limit-cycle oscillations and chaotic vibrations in strongly nonlinear bistable beam and plate configurations are of particular interest. Inherent material nonlinearities and dissipative nonlinearities will also be discussed. Broadband random excitation of energy harvesters will be summarized with an emphasis on stochastic resonance in bistable configurations. Recent efforts on aeroelastic energy harvesting as well as underwater thrust and electricity generation using fiber-based flexible piezoelectric composites will be addressed briefly.

In this talk we will present a numerical algorithm for the computation of (hyperbolic) periodic orbits of the 1-D K-S equation u_t+v*u_xxxx+u_xx+u*u_x = 0, with v>0. This numerical algorithm consists on apply a suitable Newton scheme for a given approximate solution. In order to do this, we need to rewrite the invariance equation that must satisfy a periodic orbit in a form that its linearization around an approximate solution is a bounded operator. We will show also how this methodology can be used to compute rigorous estimates of the errors of the solutions computed.

I'll show how on metric measure spaces with Ricci curvature bounded from below in the sense of Lott-Sturm-Villani there is a well defined notion of Heat flow, and how the study of the properties of this flow leads to interesting geometric and analytic properties of the spaces themselves. A particular attention will be given to the class of spaces where the Heat flow is linear. (From a collaboration with Ambrosio and Savare')