Georgia Institute of Technology College of Sciences

I will discuss how one can solve certain concrete problems in number theory, for example the Diophantine equation 2x^2 + 1 = 3^m, using p-adic analysis. No previous knowledge of p-adic numbers will be assumed. If time permits, I will discuss how similar p-adic analytic methods can be used to prove the famous Skolem-Mahler-Lech theorem: If a_n is a sequence of complex numbers satisfying some finite-order linear recurrence, then for any complex number b there are only finitely many n for which a_n = b.

We prove via explicitly constructed initial data that solutionsto the gravity-capillary wave system in R^3 representing a 2d air-waterinterface immediately fail to be C^3 with respect to the initial data ifthe initial (h_0, \psi_0) \in H^{s + 1/2} \times H^s for s<3, where h isthe free surface and \psi is the velocity potential.

Given F, finite set of square matrices of dimension n, it is possible to define the Joint Spectral Radius or simply JSR as a generalization of the well known spectral radius of a matrix. The JSR evaluation proves to be useful for instance in the analysis of the asymptotic behavior of solutions of linear difference equations with variable coefficients, in the construction of compactly supported wavelets of and many others contexts. This quantity proves, however, to be hard to compute in general. Gripenberg in 1996 proposed an algorithm for the computation of lower and upper bounds to the JSR based on a four member inequality and a branch and bound technique. In this talk we describe a generalization of Gripenberg's method based on ellipsoidal norms that achieve a tighter upper bound, speeding up the approximation of the JSR. We show the performance of this new algorithm compared with Gripenberg's one. This talk is based on joint work with V.Y.Protasov.

The workshop begins on November 12 with three 1-hour tutorial lectures and continues with morning and afternoon sessions until November 14. The aim of this workshop is to bring together these different communities from research (Internet Topology Measurement, Economics, Theoretical Computer Science, Network Science) and related industry (ISPs, Content Providers, CDNs etc.) to help narrow the gap between research and operational practice. See the complete program, list of speakers and register to attend.