|2/8/11||Dror Speiser||Computing||Say you wanted to check if a specific transform is ergodic (or, better yet, you wanted to compute zeros of the Riemann zeta function!). The transform is complicated. You tried some sums, some integrals, yielding nothing. Maybe you're just not good enough. So you want to try and test this on a computer. Numerically apply the transform many times over many numbers, and see what happens. You multiply some numbers, apply cosine twice, and divide by pi. Wait, how did you multiply some numbers? Apply cosine?! Pi?!?! This lecture will be an introduction to the most advanced techniques for the most basic operations of computing. Pi?!?!|
|16/8/11||Naomi Feldheim||Discrete Complex Analysis||How would you define a harmonic function on the grid Z^2? what about an analytic
function? Do they still have nice relationships between them, such as: "real part" is
harmonic, or that the derivative and the integral of an analytic function is also
analytic? May we add and multiply such functions? What happens in the limit when we take a
finer and finer grid?
We will try to answer those questions, which are a basis to a beautiful subject which is very much applicable nowadays in statistical mechanics. As an application, we shall prove the Riemann mapping theorem.
Following a lecture by S. Smirnov.
|6/9/11||Ohad Noy Feldheim||The Firefighter Problem||The firefighter problem deals with monotone dynamic processes in
graphs. A fire spreads through a graph, from burning vertices to
their unprotected neighbors. In every round, a small amount of
unburnt vertices can be permanently protected by firefighters.
For infinite graphs, the problem is deciding what is the minimal number of vertices the firefighters need to protect every turn in order to eventually stop the fire from advancing?
We use potential functions to prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.
Joint work with Rani Hod.
|20/9/11||Liat Even-Dar Mandel||Direct Fully Discrete Energy Estimates for Solutions to Finite Difference Schemes for Quasilinear Symmetric Hyperbolic Systems||
Nonlinear hyperbolic symmetric partial differential equations occur in many fields of
applied mathematics. In particular the Euler equations, describing the flow of inviscid
fluids, are important in the aerospace industry. The existence and uniqueness of the
solution of these PDE's have been proven in the past. Approximation for their solution
can be obtained by numerically solving appropriate nonlinear finite difference schemes.
Existence and uniqueness of solutions for those schemes have been proven in the past but
always with the assumption that the continuous PDE's solutions are very smooth and the
finite difference schemes solutions are estimated in a space which has fewer derivatives.
Our aim is to prove a stronger theorem of existence and uniqueness of discrete solutions for the finite difference schemes without assuming higher smoothness of the continuous solution. As in the continuous case, the crucial part of the proof is obtaining the appropriate energy estimates for the schemes. We successfully prove the existence and uniqueness of solutions in a discrete Sobolev space of the same order as in the continuous case, for a semi implicit scheme. In addition, we proved convergence of the interpolation of the discrete solution to the continuous solution in this smooth space.