Past Applied Analysis and Mechanics Seminar

7 November 2005
17:00
Abstract
Two dimensional minimal cones were fully classified by Jean Taylor in the mid 70's.  In joint work with G. David and T. De Pauw we prove that a closed set which is close to a minimal cone at all scales and at all locations is locally a bi-Hoelder image of a minimal cone.  This result is analogous to Reifenberg's disk theorem.  A couple of applications will be discussed.
  • Applied Analysis and Mechanics Seminar
31 October 2005
17:00
Gui-Qiang Chen
Abstract
In this talk we will discuss a theory of divergence-measure fields and related geometric measures, developed recently, and its applications to some fundamental issues in mathematical continuum physics and nonlinear conservation laws whose solutions have very weak regularity, including hyperbolic conservation laws, degenerate parabolic equations, degenerate elliptic equations, among others.
  • Applied Analysis and Mechanics Seminar
24 October 2005
17:00
Christoph Ortner
Abstract
For atomistic (and related) material models, global minimization gives the wrong qualitative behaviour; a theory of equilibrium solutions needs to be defined in different terms. In this talk, a process based on gradient flow evolutions is presented, to describe local minimization for simple atomistic models based on the Lennard- Jones potential. As an application, it is shown that an atomistic gradient flow evolution converges to a gradient flow of a continuum energy, as the spacing between the atoms tends to zero. In addition, the convergence of the resulting equilibria is investigated, in the case of both elastic deformation and fracture.
  • Applied Analysis and Mechanics Seminar
10 October 2005
17:00
Martin Golubitsky
Abstract
A coupled cell system is a collection of interacting dynamical systems.<br> Coupled cell models assume that the output from each cell is important and that signals from two or more cells can be compared so that patterns of synchrony can emerge. We ask: How much of the qualitative dynamics observed in coupled cells is the product of network architecture and how much depends on the specific equations?<br> <br> The ideas will be illustrated through a series of examples and theorems. One theorem classifies spatio-temporal symmetries of periodic solutions and a second gives necessary and sufficient conditions for synchrony in terms of network architecture.
  • Applied Analysis and Mechanics Seminar
6 June 2005
17:00
Georg Dolzmann
Abstract
We derive a two-dimensional compressible elasticity model for thin elastic sheets as a Gamma-limit of a fully three-dimensional incompressible theory. The energy density of the reduced problem is obtained in two steps: first one optimizes locally over out-of-plane deformations, then one passes to the quasiconvex envelope of the resulting energy density. This work extends the results by LeDret and Raoult on smooth and finite-valued energies to the case incompressible materials. The main difficulty in this extension is the construction of a recovery sequence which satisfies the nonlinear constraint of incompressibility pointwise everywhere. This is joint work with Sergio Conti.
  • Applied Analysis and Mechanics Seminar
30 May 2005
17:00
Richard D James
Abstract
Bacteriophage T4 is a virus that attacks bacteria by a unique mechanism. It lands on the surface of the bacterium and attaches its baseplate to the cell wall. Aided by Brownian motion and chemical bonding, its tail fibres stick to the cell wall, producing a large moment on the baseplate. This triggers an amazing phase transformation in the tail sheath, of martensitic type, that causes it to shorten and fatten. The transformation strain is about 50%. With a thrusting and twisting motion, this transformation drives the stiff inner tail core through the cell wall of the bacterium. The DNA of the virus then enters the cell through the hollow tail core, leading to the invasion of the host. This is a natural machine. As we ponder the possibility of making man-made machines that can have intimate interactions with natural ones, on the scale of biochemical processes, it is an interesting prototype. We present a mathematical theory of the martensitic transformation that occurs in T4 tail sheath. Following a suggestion of Pauling, we propose a theory of an active protein sheet with certain local interactions between molecules. The free energy is found to have a double-well structure. Using the explicit geometry of T4 tail sheath we introduce constraints to simplify the theory. Configurations corresponding to the two phases are found and an approximate formula for the force generated by contraction is given. The predicted behaviour of the sheet is completely unlike macroscopic sheets. To understand the position of this bioactuator relative to nonbiological actuators, the forces and energies are compared with those generated by inorganic actuators, including nonbiological martensitic transformations. Joint work with Wayne Falk, WF@ddt.biochem.umn.edu Wayne Falk and R. D. James, An elasticity theory for self-assembled protein lattices with application to the martensitic transformation in Bacteriophage T4 tail sheath, preprint. K. Bhattacharya and R. D. James, The material is the machine, Science 307 (2005), pp. 53-54.
  • Applied Analysis and Mechanics Seminar

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