Past Forthcoming Seminars

21 September 2004
Gian Luca Delzanno
The role of electron emission (either thermionic, secondary or photoelectric) in charging an object immersed in a plasma is investigated, both theoretically and numerically. In fact, recent work [1] has shown how electron emission can fundamentally affect the shielding potential around the object. In particular, depending on the physical parameters of the system (that were chosen such to correspond to common experimental conditions), the shielding potential can develop an attractive potential well. The conditions for the formation of the well will be reviewed, based on a theoretical model of electron emission from the grain. Furthermore, simulations will be presented regarding specific laboratory, space and astrophysical applications. [1] G.L. Delzanno, G. Lapenta, M. Rosenberg, Phys. Rev. Lett., 92, 035002 (2004).
16 September 2004
Prof Donald L Turcotte
Time delays are associated with rock fracture and earthquakes. The delay associated with the initiation of a single fracture can be attributed to stress corrosion and a critical stress intensity factor [1]. Usually, however, the fracture of a brittle material, such as rock, results from the coalescence and growth of micro cracks. Another example of time delays in rock is the systematic delay before the occurrence of earthquake aftershocks. There is also a systematic time delay associated with rate-and-state friction. One important question is whether these time delays are related. Another important question is whether the time delays are thermally activated. In many cases systematic scaling laws apply to the time delays. An example is Omori92s law for the temporal decay of after shock activity. Experiments on the fracture of fiber board panels, subjected instantaneously to a load show a systematic power-law decrease in the delay time to failure as a function of the difference between the applied stress and a yield stress [2,3]. These experiments also show a power-law increase in energy associated with acoustic emissions prior to rupture. The frequency-strength statistics of the acoustic emissions also satisfy the power-law Gutenberg-Richter scaling. Damage mechanics and dynamic fibre-bundle models provide an empirical basis for understanding the systematic time delays in rock fracture and seismicity [4-7]. We show that these approachesgive identical results when applied to fracture, and explain the scaling obtained in the fibre board experiments. These approaches also give Omori92s type law. The question of precursory activation prior to rock bursts and earthquakes is also discussed. [1] Freund, L. B. 1990. Dynamic Fracture Mechanics, Cambridge University Press, Cambridge.20 <br> [2] Guarino, A., Garcimartin, A., and Ciliberto, S. 1998. An experimental test of the critical behaviour of fracture precursors. Eur. Phys. J.; B6:13-24.20 <br> [3] Guarino, A., Ciliberto, S., and Garcimartin, A. 1999. Failure time and micro crack nucleation. Europhys. Lett.; 47: 456.20 <br> [4] Kachanov, L. M. 1986. Introduction to Continuum Damage Mechanics, Martinus Nijhoff, Dordrecht, Netherlands.20 <br> [5] Krajcinovic, D. 1996. Damage Mechanics, Elsevier, Amsterdam.20 <br> [6] Turcotte, D. L., Newman, W. I., and Shcherbakov, R. 2002. Micro- and macroscopic models of rock fracture, Geophys. J. Int.; 152: 718-728. <br> [7] Shcherbakov, R. and Turcotte, D. L. 2003. Damage and self-similarity in fracture. Theor. and Appl. Fracture Mech.; 39: 245-258.
  • Confronting Models with Data
18 June 2004
In this talk we discuss the analytic approximation to the loss distribution of large conditionally independent heterogeneous portfolios. The loss distribution is approximated by the expectation of some normal distributions, which provides good overall approximation as well as tail approximation. The computation is simple and fast as only numerical integration is needed. The analytic approximation provides an excellent alternative to some well-known approximation methods. We illustrate these points with examples, including a bond portfolio with correlated default risk and interest rate risk. We give an analytic expression for the expected shortfall and show that VaR and CVaR can be easily computed by solving a linear programming problem where VaR is the optimal solution and CVaR is the optimal value.
  • Mathematical Finance Seminar
17 June 2004
Prof Gilbert Strang
An essential first step in many problems of numerical analysis and computer graphics is to cover a region with a reasonably regular mesh. We describe a short MATLAB code that begins with a "distance function" to describe the region: $d(x)$ is the distance to the boundary (with d < 0 inside the region). The algorithm decides the location of nodes and the choice of edges. At each iteration the mesh becomes a truss (with forces in the edges that move the nodes). Then the Delaunay algorithm resets the topology (the choice of edges). The input includes a size function $h(x)$ to vary the desired meshlengths over the region. \\ \\ The code is adaptable and public ( It extends directly to $n$ dimensions. The 2D meshes are remarkably regular. In higher dimensions it is difficult to avoid "slivers". The theory is not complete. \\ \\ We look for the "right proof" of a known result about the inverse of a tridiagonal matrix $T$: Every 2 by 2 submatrix on and above the diagonal of inv($T$), or on and below the diagonal, has determinant = zero. This is the case $p = 1, k = 1$ of a general result for (band + low rank) matrices: All submatrices $B$ above the $p$th superdiagonal of $T$ have rank($B$) < $k$ if and only if all submatrices $C$ above the $p$th subdiagonal of inv($T$) have rank($C$) < $p+k$. \\ \\ A quick proof uses the Nullity Theorem. The complementary subspaces $Ba$ and $C$ have the same nullity. These full matrices inv($T$) appear in integral equations and in the boundary element method. The large low rank submatrices allow fast multiplication. The theory of decay away from the main diagonal could extend to include the property of low rank.
  • Computational Mathematics and Applications Seminar