國立清華大學 數學系

鄒文明Wenming Zou (Tsinghua University, Beijing) Title: Some Progress on Schrodinger Systems Abstract: Some recent results on Schrodinger Systems related to the Bose-Einstein condensates will be reported. 章梅榮Meirong Zhang (Tsinghua University, Beijing) Title: Strong Continuity of Eigenvalues of Sturm-Liouville Problems in Potentials and Optimal Estimates of Eigenvalues Abstract: In this talk, we will present some strong continuity results of eigenvalues of Sturm- Liouville problems in potentials, i.e., the continuity of eigenvalues in potentials when the weak topologies are considered for potentials. Based on these, we will use the variational method and the limiting approach to obtain optimal lower and upper bounds for eigenvalues by using the norms of potentials. Some interesting systems of degree- two-of-freedom which are resulted from the optimal eigenvalue gaps will be introduced as an open problem. 蘇寧Ning Su (Tsinghua University, Beijing) Title: Entropy Solutions of Degenerate Parabolic-Hyperbolic Equations Abstract: We are interested in a class of nonlinear parabolic-hyperbolic equations arising from practical and engineering fields. Due to the strong degeneration of the equations, the investigation should be set into the framework of entropy solutions. We investigate existence and uniqueness for initial-boundary value problems, and show that under quite general conditions, the entropy solutions possess L1 contraction property, and hence obey comparison principles. Moreover, in some circumstance there exist maximal entropy solutions and minimal entropy solutions. 陸玫Mei Lu (Tsinghua University, Beijing) Title: The Guessing Number of (directed) Graphs Abstract: In 2007, Riis introduced a guessing game for graphs which is equivalent to finding protocols for network coding. Riis also proposed three open problems. In this talk, we present lower and upper bounds of guessing number for (directed) graphs. Based on this result, we give counter examples for two of Riis' open problems and show that one of his Problem is valid for Cayley graphs. 陳大廣Daguang Chen (Tsinghua University, Beijing) Title: The Sharp Estimates of Eigenvalues of Polyharmonic Operator Abstract: In this talk, we will talk aobut some lower bounds for the sums of eigenvalues of the polyharmonic operator, which are sharper than the recent results. At the same time, we will give some certain bounds for the sums of positive and negative powers of eigenvalues of the polyharmonic operator