On the Crossing Number of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> □ <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Crossing numbers of graphs are in general very difficult to compute. There are several known exact results on the crossing number of the Cartesian products of paths, cycles or stars with small graphs. In this paper we study cr(K_m □ P_n), the crossing number of the Cartesian product K_m □ P_n. We prove that for m ≥ 3,n ≥ 1 and cr(K_m □ P_n)≥ (n − 1)cr(K_m+2 − e) + 2cr(K_m+1). For m≤ 5, according to Klešč, Jendrol and Ščerbová, the equality holds. In this paper, we also prove that the equality holds for m = 6, i.e., cr(K₆ □ P_n) = 15n + 3. Research supported by NFSC (60373096, 60573022).     

Connecting colored point sets

We study the following Ramsey-type problem. Let S=B∪R be a two-colored set of n points in the plane. We show how to construct, in time, a crossing-free spanning tree T(B) for B, and a crossing-free spanning tree T(R) for R, such that both the number of crossings between T(B) and T(R) and the diameters of T(B) and T(R) are kept small. The algorithm is conceptually simple and is implementable without using any non-trivial data structure. This improves over a previous method in Tokunaga [Intersection number of two connected geometric graphs, Inform. Process. Lett. 59 (1996) 331-333] that is less efficient in implementation and does not guarantee a diameter bound. Implicit to our approach is a new proof for the result in the reference above on the minimum number of crossings between T(B) and T(R).     

On-Surface Synthesis of NBN-Doped Zigzag-Edged Graphene Nanoribbons

We report the first bottom-up synthesis of NBN-doped zigzag-edged GNRs (NBN-ZGNR1 and NBN-ZGNR2) through surface-assisted polymerization and cyclodehydrogenation based on two U-shaped molecular precursors with an NBN unit preinstalled at the zigzag edge. The resultant zigzag-edge topologies of GNRs are elucidated by high-resolution scanning tunneling microscopy (STM) in combination with noncontact atomic force microscopy (nc-AFM). Scanning tunneling spectroscopy (STS) measurements and density functional theory (DFT) calculations reveal that the electronic structures of NBN-ZGNR1 and NBN-ZGNR2 are significantly different from those of their corresponding pristine fully-carbon-based ZGNRs. Additionally, DFT calculations predict that the electronic structures of NBN-ZGNRs can be further tailored to be gapless and metallic through one-electron oxidation of each NBN unit into the corresponding radical cations. This work reported herein provides a feasible strategy for the synthesis of GNRs with stable zigzag edges yet tunable electronic properties.     

隧道下穿既有结构物引起的地表沉降控制标准研究

解决好隧道下穿既有结构物引起的地表沉降问题,对城市地下交通和高速铁路的建设具有重要的意义。在调研国内大量隧道下穿开挖引起的地表沉降控制标准和方法的基础上,根据隧道下穿公路、铁路、隧道和建筑物时引起的地表沉降的不同特点,结合隧道的施工、开挖面积、埋深和工程地质条件等因素,对隧道下穿不同的结构物提出不同的控制沉降措施和建议沉降标准。作者认为,目前的隧道下穿引起的地表最大沉降控制标准是不合理的,沉降控制标准应综合考虑既有结构物的特点、地质条件和施工特点等因素。     

Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges

A set of planar graphs {G₁(V,E₁),…,Gk(V,Ek)} admits a simultaneous embedding if they can be drawn on the same pointset P of order n in the Euclidean plane such that each point in P corresponds one-to-one to a vertex in V and each edge in Ei does not cross any other edge in Ei (except at endpoints) for i∈{1,…,k}. A fixed edge is an edge (u,v) that is drawn using the same simple curve for each graph Gi whose edge set Ei contains the edge (u,v). We give a necessary and sufficient condition for two graphs whose union is homeomorphic to K₅ or K3,3 to admit a simultaneous embedding with fixed edges (SEFE). This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide O(n⁴)-time algorithms to compute a SEFE.     

Evaluation of fiber tracking from subsampled q-space data in diffusion spectrum imaging

Diffusion spectrum imaging (DSI) is capable of resolving crossing and touching fiber bundles in a given voxel. Acquisition of DSI data involves sampling large number of points in the q-space which significantly increases scan times. The scan times can be reduced by exploiting the symmetry of the q-space. In this study the fiber pathways for five (fornix, cingulum, superior longitudinal fasciculus, corticospinal tract, and crossing fibers in the centrum semiovale region) fiber bundles derived using three subsampled data sets of different sizes derived from the 257 samples in the q-space are compared. The coefficient of variation of the ratio of the number of fiber pathways for each subsample data set to the original data points, averaged over all the 10 subjects, was used for quantitatively investigating the effect of subsampling on the tractography. The effect of threshold angles on tractography is also investigated. The effect of subsampling on the orientation distribution function (ODF) was quantitatively evaluated using both scalar and vector measures derived from the ODF. A streamline tractography method that improves the curvature problem and reduces the local truncation error to further improve the mapping of fiber pathways is adapted. Analysis of the fiber pathways in ten normal subjects, based on qualitative and quantitative methods, shows that the 129 and 198 q-space points provide very similar result with angle of threshold between 41° and 45°. Based on the scan time advantage, 129 subsampled points appear to be adequate for tractography.     

Minimal Obstructions for 1‐Immersions and Hardness of 1‐Planarity Testing

A graph is 1‐planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge (and any pair of crossing edges cross only once). A non‐1‐planar graph G is minimal if the graph is 1‐planar for every edge e of G. We construct two infinite families of minimal non‐1‐planar graphs and show that for every integer , there are at least nonisomorphic minimal non‐1‐planar graphs of order n. It is also proved that testing 1‐planarity is NP‐complete.     

On sensitive elliptic singular perturbation problems and logarithmic oscillations: the case of shells with edges

This work deals with singular perturbation problems depending on small positive parameter ?. The limit problem as ? → 0 has no solution within the classical theory of PDEs, which uses distribution theory. A very particular and less‐known phenomenon appears: large oscillations. These problems exhibit some kind of instability; very small and smooth variations of the data imply large singular perturbations of the solution. That kind of problems appears in elasticity for highly compressible two‐dimensional bodies and thin shells with elliptic middle surface with a part of the boundary free. Here, we consider certain properties of that oscillations and extend the theory to shells with edges. Copyright © 2012 John Wiley & Sons, Ltd.