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Forthcoming events in this series
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A generalisation of Reifenberg's theorem in 3-space
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.
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Divergence-Measure Fields, Geometric Measures,
and Conservation Laws
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.
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Gradient flows as a selection criterion for equilibria of non-convex
material models.
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.
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Coupled Systems: Theory and Examples
Abstract
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?
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.
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On a conjectured estimate for solutions of the three-dimensional Stokes equations with a constant that is optimal and independen
Abstract
/notices/events/abstracts/applied-analysis/tt05/Heywood.pdf
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A 2D compressible membrane theory as a Gamma-limit of a nonlinear elasticity model for incompressible membranes in 3D
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.
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A mathematical description of the invasion of Bacteriophage T4
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, @email
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.
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On the one-dimensional Perona-Malek equation
Abstract
We use the partial differential inclusion method to establish existence of
infinitely many weak solutions to the one-dimensional version of the
Perona-Malek anisotropic diffusion model in the theory of image processing. We
consider the homogeneous Neumann problem as the model requires.
.
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On a class of quasilinear parabolic equations
Abstract
An important class of nonlinear parabolic equations is the class of quasi-linear equations, i.e., equations with a leading second-order (in space) linear part (e.g., the Laplacian) and a nonlinear part which depends on the first-order spatial derivatives of the unknown function. This class contains the Navier-Stokes system of fluid dynamics, as well as "viscous" versions (or "regularized") of the Hamilton-Jacobi equation, nonlinear hyperbolic conservation laws and more. The talk will present various recent results concerning existence/uniqueness (and nonexistence/nonuniqueness) of global solutions. In addition, a new class of "Bernstein-type" estimates of derivatives will be presented. These estimates are independent of the viscosity parameter and thus lead to results concerning the "zero-viscosity" limit.
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Rigidity estimates for two wells and applications to thin films
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Existence and regularity results for Landau-Lifschitz equations in R^3
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Neumann problem in a perforated layer - Sieving "ad infinitum"
Abstract
/notices/events/abstracts/applied-analysis/ht05/gudrun.pdf
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Nonlinear Capacity and Blow-up for Nonlinear PDE's
Abstract
/notices/events/abstracts/applied-analysis/ht05/pohozaev.shtml
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A variational model for dislocations in the line tension limit
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Nonlinear magnetoelasticity of magneto-sensitive solids
Abstract
/notices/events/abstracts/applied-analysis/ht05/ogden.shtml
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Dynamics of coherent structures in oscillatory media
Abstract
Coherent structures, or defects, are interfaces between wave trains with
possibly different wavenumbers: they are time-periodic in an appropriate
coordinate frame and connect two, possibly different, spatially-periodic
travelling waves. We propose a classification of defects into four
different classes which have all been observed experimentally. The
characteristic distinguishing these classes is the sign of the group
velocities of the wave trains to either side of the defect, measured
relative to the speed of the defect. Using a spatial-dynamics description
in which defects correspond to homoclinic and heteroclinic orbits, we then
relate robustness properties of defects to their spectral stability
properties. If time permits, we will also discuss how defects interact with
each other.
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Marstrand's Theorem for Polytope density
Abstract
Marstrand's Theorem is a one of the classic results of Geometric Measure Theory, amongst other things it says that fractal measures do not have density. All methods of proof have used symmetry properties of Euclidean space in an essential way. We will present an elementary history of the subject and state a version of Marstrand's theorem which holds for spaces whose unit ball is a polytope.
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Classifying crystal structures: geometric and arithmetic approach
Abstract
While the classification of crystals made up by just one atom per cell is well-known and understood (Bravais lattices), that for more complex structures is not. We present a geometric way classifying these crystals and an arithmetic one, the latter introduced in solid mechanics only recently. The two approaches are then compared. Our main result states that they are actually equivalent; this way a geometric interpretation of the arithmetic criterion in given. These results are useful for the kinematic description of solid-solid phase transitions. Finally we will reformulate the arithmetic point of view in terms of group cohomology, giving an intrinsic view and showing interesting features.