Acknowledgments In the path for pursuing my Ph.D, there are many people supporting me, guiding me and helping me. Ph.D, all the years offering the financial support to me along the path to achieving it and teaching me to prepare excellent talks. I feel lucky in his research group that I have the opportunities to exchange and cooperate with many other research groups from all over the world. Pro...

Let A be a compact d-rectifiable set embedded in Euclidean space R, d ≤ p. For a given continuous distribution σ(x) with respect to d-dimensional Hausdorff measure on A, our earlier results provided a method for generating N -point configurations on A that have asymptotic distribution σ(x) as N →∞; moreover such configurations are “quasi-uniform” in the sense that the ratio of the covering radi...

We consider a generalization of the point-to-point (and single-source) shortest path problem to instances where the shortest path must satisfy a formal language constraint. Given an alphabet Σ, a (directed) network G whose edges are weighted and Σ-labeled, and a regular grammar L ⊆ Σ, the Regular Language Constrained Shortest Path problem consists of finding a shortest path p in G complying wit...

The projected gravitational potential of galaxy clusters is reflected in both their X–ray emission and their imprint on the images of background sources due to their gravitational lensing effects. Since these projections of the potential are weighted differently along the line-of-sight, we propose a method to combine them and remove the degeneracy between two cases: (i) a cluster consisting of ...

Gimbel and Thomassen asked whether 3-colorability of a triangle-free graph drawn on a fixed surface can be tested in polynomial time. We settle the question by giving a linear-time algorithm for every surface which combined with previous results gives a lineartime algorithm to compute the chromatic number of such graphs. Our algorithm is based on a structure theorem that for a triangle-free gra...

Local network community detection is the task of finding a single community of nodes concentrated around few given seed nodes in a localized way. Conductance is a popular objective function used in many algorithms for local community detection. This paper studies a continuous relaxation of conductance. We show that continuous optimization of this objective still leads to discrete communities. W...

The relationship between the X-ray determined temperature T of the intracluster gas and the optical measured velocity dispersion σ of the cluster galaxies is often believed to be not only a straightforward but also robust test for the dynamical properties of galaxy clusters. Here, we present the σT relationship using the 94 clusters drawn from the largest sample of 149 clusters in literature, f...

An IFS (iterated function system) such that there exists a positive bounded function h : [0, 1] → R and a probability ν on [0, 1] satisfying P u (h) = ρh, P * u (ν) = ρν. There is no meaning to ask if the probabilities ν on [0, 1] arising in IFS are invariant for a dynamical system, but, we can ask if probabil-itiesˆν on [0, 1] × Σ are holonomic forˆσ. A probabilityˆν on [0, 1] × Σ is called ho...