Georgia Institute of Technology College of Sciences

We present algorithms for performing sparse univariate polynomial interpolation with errors in the evaluations of the polynomial. Our interpolation algorithms use as a substep an algorithm that originally is by R. Prony from the French Revolution (Year III, 1795) for interpolating exponential sums and which is rediscovered to decode digital error correcting BCH codes over finite fields (1960). Since Prony's algorithm is quite simple, we will give a complete description, as an alternative for Lagrange/Newton interpolation for sparse polynomials. When very few errors in the evaluations are permitted, multiple sparse interpolants are possible over finite fields or the complex numbers, but not over the real numbers. The problem is then a simple example of list-decoding in the sense of Guruswami-Sudan. Finally, we present a connection to the Erdoes-Turan Conjecture (Szemeredi's Theorem). This is joint work with Clement Pernet, Univ. Grenoble.

This is a part of GT MAP activities. There will be light refreshments through out the event. This seminar will be held at Skiles 006.

3:00 PM - 3:45PM  Prof. Patricio Vela will give a talk  on

    Revisiting Averaging Theory for Control of Biologically Inspired Robots

3:45PM -- 4:00PM Break with Discussions

4:00PM - 4:45PM  The second talk will be given by Alexander H. Chang on

           Modeling and Control of Robotic Snakes

4:45PM - 5PM Discussion of open problems stemming from the presentations.

First Talk Title: Revisiting Averaging Theory for Control of Biologically Inspired Robots

Abstract: Robotic locomotive mechanisms designed to mimic those of their biological counterparts differ from traditionally engineered systems. Though both require overcoming non-holonomic properties of the interaction dynamics, the nature of their non-holonomy differs.  Traditionally engineered systems have more direct actuation, in the sense that control signals directly lead to generated forces or torques, as in the case of rotors, wheels, motors, jets/ducted fans, etc. In contrast, the body/environment interactions that animals exploit induce forces  or torque that may not always align with their intended direction vector.Through periodic shape change animals are able to effect an overall force or torque in the desired direction. Deriving control equations for this class of robotic systems requires modelling the periodic interaction forces, then  applying averaging theory to arrive at autonomous nonlinear control models whose form and structure resembles that of traditionally engineered systems. Once obtained, classical nonlinear control methods may be applied, though some  attention is required since the control can no longer apply at arbitrary  time scales.The talk will cover the fundamentals of averaging theory and efforts to identify a generalized averaging strategy capable of recovering the desired control equations. Importantly, the strategy reverses the typical approach to averaged expansions, which significantly simplifies the procedure. Doing so  provides insights into feedback control strategies available for systems controlled through time-periodic signals.

Second Talk Title: Modeling and Control of Robotic Snakes

Speaker: Alexander H. Chang (GT ECE)

Abstract: Robotic snakes have the potential to navigate areas or environments that would be more challenging for traditionally engineered robots. To realize their potential requires deriving feedback control and path planning algorithms applicable to the diverse gait modalities possible. In turn, this requires equations of motion for snake movement that generalize across the gait types and their interaction dynamics. This talk will discuss efforts towards both obtaining general control equations for snake robots, and controlling them along planned trajectories. We model three-dimensional time- and spatially-varying locomotion gaits, utilized by snake-like robots, as planar continuous body curves. In so doing, quantities relevant to computing system dynamics are expressed conveniently and geometrically with respect to the planar body, thereby facilitating derivation of governing equations of motion. Simulations using the derived dynamics characterize the averaged, steady-behavior as a function of the gait parameters. These then inform an optimal trajectory planner tasked to generate viable paths through obstacle-strewn terrain. Discrete-time feedback control successfully guides the snake-like robot along the planned paths.

BIO:  Prof. Patricio Vela was born in Mexico City, Mexico and grew up in California. He earned his bachelor of science degree in 1998 and his doctorate in 2003 at the California Institute of Technology, where he did his graduate research on geometric nonlinear control and robotics. Dr. Vela came to Georgia Tech as a post-doctoral researcher in computer vision and joined the ECE faculty in 2005.

This is a part of GT MAP activities. There will be light refreshments through out the event. This seminar will be held at Skiles 006.

Members of Prof. Erturk's group will present their research

3:00 PM - 3:45PM  Prof. Alper Erturk will give a talk on

          Dynamics of Next-Generation Smart- and Meta-Structures

3:45PM -- 4:00PM Break with Discussions

4:00PM - 4:45PM  The second talk will be given  by Chris Sugino (PhD Student).

4:45PM - 5PM Discussion of open problems stemming from the presentations.

Abstract] The first part of this talk will review our recent efforts on the electroelastodynamics of smart structures for various applications ranging from nonlinear energy harvesting, bio-inspired actuation, and acoustic power transfer to elastic wave guiding and vibration attenuation via metamaterials. We will discuss how to exploit nonlinear dynamic phenomena for frequency bandwidth enhancement to outperform narrowband linear-resonant devices in applications such as vibration energy harvesting for wireless electronic components. We will also cover inherent nonlinearities (material and internal/external dissipative), and their interactions with intentionally designed nonlinearities, as well as electrical circuit nonlinearities. Electromechanical modeling efforts will be presented, and approximate analysis results using the method of harmonic balance will be compared with experimental measurements. Our recent efforts on phononic crystal-enhanced elastic wave guiding and harvesting, wideband vibration attenuation via locally resonant metamaterials, contactless acoustic power transfer, bifurcation suppression using nonlinear circuits, and exploiting size effects via strain-gradient induced polarization (flexoelectricity) in centrosymmetric elastic dielectrics will be summarized.

The second part of the talk will be centered on low-frequency vibration attenuation in finite structures by means of locally resonant elastic and electroelastic metamaterials. Locally resonant metamaterials are characterized by bandgaps at wavelengths that are much larger than the lattice size, enabling low-frequency vibration/sound attenuation. Typically, bandgap analyses and predictions rely on the assumption of waves traveling in an infinite medium, and do not take advantage of modal representations commonly used for the analysis of the dynamic behavior of finite structures. We will present a novel argument for estimating the locally resonant bandgap in metamaterial-based finite structures (i.e. meta-structures with prescribed boundary conditions) using modal analysis, yielding a simple closed-form expression for the bandgap frequency and size. A method for understanding the importance of the resonator locations and mass distribution will be discussed in the context of a Riemann sum approximation of an integral. Numerical and experimental results will be presented regarding the effects of mass ratio, non-uniform spacing of resonators, and parameter variations among the resonators. Electromechanical counterpart of the problem will also be summarized for piezoelectric structures.

Biography: Dr. Alper Erturk is an Associate Professor of Mechanical Engineering at Georgia Institute of Technology where he leads the Smart Structures and Dynamical Systems Laboratory. His theoretical and experimental research program is centered on the intersection of smart structures and dynamical systems for various interdisciplinary applications ranging from energy harvesting and bio-inspired actuation to electroelastic metamaterials and vibration attenuation. Prior to his current (tenured) appointment, Dr. Erturk was an Assistant Professor (2011-2016) of Mechanical Engineering at Georgia Tech and a Research Scientist (2009-2011) in the Center for Intelligent Material Systems & Structures at Virginia Tech.

Each spring semester the Georgia Tech School of Mathematics hosts the Institute's annual High School Mathematics Competition. The competition covers a wide range of high school math topics, including algebra, Euclidean geometry, combinatorics, number theory, and basic calculus. Schools from any state are welcome to compete. A school may register up to five teams with up to five contestants per team.

Begin your registration process by first CREATING YOUR ACCOUNT. You must have an account before you can add your team. After the account is created, you will receive an email explaining how to log in and add your students.

The workshop will launch the thematic semester on Materials for GT-MAP activities. 

View PHOTOS from the event.

This is a three day workshop: The first two days (Wed, Thurs) focusing on the theme of Materials, and third day including broad research topics, open to introducing your research. There will be light refreshments through out the event.

This event is free, however, please register HERE to help us make appropriate arrangements.   

Click here for tentative schedule with title and abstract.

Confirmed Speakers>>

Plenary Speakers: Prof. John Lowengrub (UCI, Math) and Prof. Zhonglin Wang (GT, MSE)

Invited Speakers from GT on topic of Materials:

  Chemistry and Biochemistry Prof. Angus Wilkinson

  Earth and Atmospheric Sciences Prof. Zhigang Peng

  Physics Prof. Paul Goldbart

  Physics Prof. Markus Kindermann

  Aerospace Engineering Prof. Julian Rimoli

  Civil and Environmental Engineering Prof. Phanish Suryanarayana

  Civil and Environmental Engineering Prof. Kimberly Kurtis (Cancellation)

  Civil and Environmental Engineering Prof. Arash Yavari

  Electrical and Computer Engineering Prof. Farrokh Ayazi  

  Materials Science and Engineering Prof. Hamid Garmestani

  Mechanical Engineering Prof. Surya Kalidindi

Invited Speakers from GT to highlight their research:

  Biology Prof. Annalise Paaby

  Biology Prof. Will Ratcliff

  Earth and Atmospheric Sciences Prof. Annalisa Bracco

  Earth and Atmospheric Sciences Prof. Athanasios Nenes

  Physics Prof. Predrag Cvitanovic

  Physics Prof. Harold Kim

  Physics Prof. Peter Yunker

  Computational Science and Engineering Prof. Edmond Chow

  Civil and Environmental Engineering  Prof. Francesco Fedele

  Electrical and Computer Engineering Prof. Erik Verriest

  Mechanical Engineering Prof. Alper Erturk

The School of Mathematics will host the inaugural International Conference on Dynamics of Differential Equations during the Georgia Tech 2013 spring break, March 16-20. The event will be held in the Klaus Advanced Computing Building, 266 Ferst Drive.

The conference will be preceded by four 2-hour tutorial sessions on Friday, March 15. Everyone, especially graduate students, advanced undergraduate students and young researchers, is invited to attend the tutorials.

The format of the conference consists of plenary lectures, invited lectures, contributed lectures and a poster session. There will also be two Hale memorial lectures delivered by Professor Geneviève Raugel

The organizers are:

• Luca Dieci, Georgia Tech
• Rafael De la Llave, Georgia Tech
• Kening Lu, Brigham Young University
• John Mallet-Paret, Brown University
• Konstantin Mischaikow, Rutgers University
• Erik Van Vleck, University of Kansas
• Yingfei Yi, Georgia Tech