Bobo Hua

Towards Grothendieck Constants and LHV Models in Quantum Mechanics

We adopt a continuous model to estimate the Grothendieck constants. An analytical formula to compute the lower bounds of Grothendieck constants has been explicitly derived for arbitrary orders, which improves previous bounds. Moreover, our lower bound of the Grothendieck constant of order three gives a refined bound of the threshold value for the nonlocality of the two-qubit Werner states.

A notion of nonpositive curvature for general metric spaces

We introduce a new definition of nonpositive curvature in metric spaces and study its relationship to the existing notions of nonpositive curvature in comparison geometry. The main feature of our definition is that it applies to all metric spaces and does not rely on geodesics. Moreover, a scaled and a relaxed version of our definition are appropriate in discrete metric spaces, and are believed to be of interest in geometric data analysis.

Sharp Davies-Gaffney-Grigor'yan Lemma on Graphs

In this note, we prove the sharp Davies–Gaffney–Grigor’yan Lemma for minimal heat kernels on graphs.

Spectral distances on graphs

By assigning a probability measure via the spectrum of the normalized Laplacian to each graph and using L p Wasserstein distances between probability measures, we define the corresponding spectral distances d p on the set of all graphs. This approach can even be extended to measuring the distances between infinite graphs. We prove that the diameter of the set of graphs, as a pseudo-metric space equipped with d 1 , is one. We further study the behavior of d 1 when the size of graphs tends to infinity by interlacing inequalities aiming at exploring large real networks. A monotonic relation between d 1 and the evolutionary distance of biological networks is observed in simulations.

Liouville theorems for f-harmonic maps into Hadamard spaces

In this paper, we study harmonic functions on weighted manifolds and harmonic maps from weighted manifolds into Hadamard spaces introduced by Korevaar and Schoen. We prove various Liouville theorems for these har- monic maps.

Polynomial growth harmonic functions on finitely generated abelian groups

In the present paper, we develop geometric analysis techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We provide a geometric analysis proof of the classical Heilbronn theorem (Heilbronn in Proc Camb Philos Soc 45:194–206, 1949) and the recent Nayar theorem (Nayar in Bull Pol Acad Sci Math 57:231–242, 2009) on polynomial growth harmonic functions on lattices

On the symmetry of the Laplacian spectra of signed graphs

\mathbb Z ^n

Quantum Nonlocality of Arbitrary Dimensional Bipartite States.

that does not use a representation formula for harmonic functions. In the abelian group case, by Yau’s gradient estimate, we actually give a simplified proof of a general polynomial growth harmonic function theorem of (Alexopoulos in Ann Probab 30:723–801, 2002). We calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups by linear algebra, rather than by Floquet theory Kuchment and Pinchover (Trans Am Math Soc 359:5777–5815, 2007). While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself. Moreover, we also calculate the dimension of solutions to higher order Laplace operators.