Daniel Savu (USC Columbia)
Title: On Sparse Approximation in Banach Spaces.
Abstract: The sparse approximation problems ask for complete recovery of functions in a given space that are supported by few of the elements of a system of generators for the space or for approximate recovery that involves a limited number of generators. This is made in regard with redundant systems which offer convenience of representation as well as better rates of approximation. The redundancy raises, in turn, very difficult theoretical problems. We give answers to some of these problems in the very general setting of Banach spaces. The theoretical results complete the previous findings in greedy approximation in this setting and show, for the algorithms considered, the same general recovery properties as the ones known in the particular case of Hilbert spaces. Moreover, we provide a novel idea of improvement of the geometry of the redundant systems by switching to a different setting than the standard Hilbert space. This improvement would translate in better recovery properties as we are able to prove the same efficiency of the greedy approach in the new setting.
Wei-Kai Lai (USC Salkehatchie)
Title: The Radon-Nikodym Property for Positive Tensor Products of Banach Lattices.
Abstract: In 1950’s, Grothendieck started the theory of projective and injective tensor products of Banach spaces. From the positivity perspective, Fremlin and Wittstock extended the theory to projective and injective tensor products of Banach lattices in 1972 and 1974 respectively. In this talk, we are going to discuss the Radon-Nikodym property for Fremlin and Wittstock’s versions of tensor product of Banach lattices.
Wei-Tian Li (USC Columbia)
Title: Lubell Function and Forbidden Subposets (part II).
Abstract: In 1928, Sperner proved that the size of a largest antichain in the Boolean lattice Bn is equal to n choose n/2. Since then, the results on largest sizes families not containing some specific posets were discovered gradually. The well-known inequality, the LYM-inequality, was individually used by Lubell, Yamamoto, and Meshalkin to reprove Sperner's Theorem. We will introduce the Lubell function, derived from LYM-inequality, and use it to estimate the maximal sizes of families do not contain some posets P. (This is a joint work with Jerrold R. Griggs and Lincoln Lu.)
Yiting Yang (USC Columbia)
Title: On the second order Randic index of trees.
Abstract: Let G be a simple graph. The second Randic index of G is defined as
2R(G)=∑xyz 1/(dxdydz)1/2
where the summation runs over all paths xyz of length two, contained in G. It was first considered by chemists Randic, Kier and Hall in the study of branching properties of alkanes. One interesting problem on it is to find the maximum and minimum 2R value and its corresponding graphs among classes of graphs. In this talk, we will talk about the maximum and minimum 2R value on trees with fixed size.
Pictures