Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact K\"ahler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called {\em real bisectional curvature} for Hermitian manifolds. When the metric is K\"ahler, this is just the holomorphic sectional curvature $H$, and when the metric is non-K\"ahler, it is slightly stronger than $H$. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.

Persistent homology is constrained to purely topological persistence, while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low-dimensional multiscale paradigm for revealing topological persistence and extracting geometric shapes from high-dimensional datasets. For a point-cloud dataset, a filtration procedure is used to generate a sequence of chain complexes and associated families of simplicial complexes and chains, from which we construct persistent combinatorial Laplacian matrices. We show that a full set of topological persistence can be completely recovered from the harmonic persistent spectra, that is, the spectra that have zero eigenvalues, of the persistent combinatorial Laplacian matrices. However, non-harmonic spectra of the Laplacian matrices induced by the filtration offer another powerful tool for data analysis, modeling, and prediction. In this work, fullerene stability is predicted by using both harmonic spectra and non-harmonic persistent spectra, while the latter spectra are successfully devised to analyze the structure of fullerenes and model protein flexibility, which cannot be straightforwardly extracted from the current persistent homology. The proposed method is found to provide excellent predictions of the protein B-factors for which current popular biophysical models break down.

We prove a finiteness property of the values of the skein polynomial of homogeneous knots that allows us to establish large classes of such knots to have arbitrarily unsharp Bennequin inequality (for the Thurston-Bennequin invariant of any of their Legendrian embeddings in the standard contact structure of R3). We also give a short proof that there are only finitely many such knots that have given genus and given braid index.

Since the traffic congestion become more and more serious in modern society due to the sharp increasing of private cars, how to improve the transportation efficiency and utilize the current road network more effectively has become a crucial issue. In this paper, a new dynamic route guidance algorithm was proposed in order to provide travelers humanized “optimal route” and to alleviate the loss caused by traffic jams. The study built a graph theory model for Beijing’s ring road transportation system, and proposed a evaluation function &σ=V_f/[k×(t+m^(ρ-ρ0))]& to describe the real time complex traffic flow, and realized the route searching by timed recomputation of classic Dijkstra algorithm. Meanwhile, due to the investigation of special features of ring roads, the study improved the priority of ring road nodes during the searching process. Comparing with Dijkstra algorithm, the time-complexity of this new algorithm decreases to 1/(16k^2) (k is the number of ring road in the road network), and extra mileage is less than 5%, which is more effective applying in large scale ring-road networks. The algorithm was realized by C++ language and connected to Google Earth’s map database with easy operation interface. (The operation of the algorithm program was manifested in the appended video)

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