The Fary-Milnor theorem in Hadamard manifolds

The Fary-Milnor theorem is generalized: Let be a simple closed curve in a complete simply connected Riemannian 3-manifold of nonpositive sectional curvature. If has total curvature less than or equal to , then is the boundary of an embedded disk. The example of a trefoil knot which moves back and forth abritrarily close to a geodesic segment shows that the bound is sharp in any such space. The original theorem was for closed curves in Euclidean 3-space and the proof by integral geometry did not apply to spaces of variable curvature. Now, instead, a combinatorial proof has been devised.

     

Recent BES results on hadron spectroscopy

Light hadron spectroscopy is a powerful tool in hadronic physics.Several recent BES results on hadron spectroscopy are reported in this proceeding,including the confirmation of the enhancement in pp invariant mass in radiative J/ψ decays,the observation of a charged κ ± decaying to K ± π 0 and the measurement of exclusive decays of the χ c0 and χ c2 to π ^0 π^ 0 and ηη.     

Scaled Gromov hyperbolic graphs

In this article, the δ‐hyperbolic concept, originally developed for infinite graphs, is adapted to very large but finite graphs. Such graphs can indeed exhibit properties typical of negatively curved spaces, yet the traditional δ‐hyperbolic concept, which requires existence of an upper bound on the fatness δ of the geodesic triangles, is unable to capture those properties, as any finite graph has finite δ. Here the idea is to scale δ relative to the diameter of the geodesic triangles and use the Cartan–Alexandrov–Toponogov (CAT) theory to derive the thresholding value of δdiam below which the geometry has negative curvature properties. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 157–180, 2008     

Arc-smooth generalized universal covering spaces

Let Y be a path-connected subset of a CAT(0) space Z, allowing for a map to a 1-dimensional separable metric space X, such that the nontrivial point preimages of f form a null sequence of convex subsets of Z. Such Y need not be homotopy equivalent to a 1-dimensional space.

We prove that Y admits a generalized universal covering space, which we equip with an arc-smooth structure by consistently and continuously selecting one tight representative from each path homotopy class of Y. It follows that all homotopy groups of Y vanish in dimensions greater than 1.     

一种箱体类零件装配变形的测试与分析方法

箱体类零件装配后,可能发生变形,改变其冷加工时的质量控制。本文以６１０５柴油机机体曲轴箱为例,介绍一种测试和描述箱体类零件装配变形的计算机辅助方法     

Warped Products and Conformal Boundaries of CAT(0)-Spaces

We discuss the conformal boundary of a warped product of two length spaces and provide a method to calculate this in terms of the individual conformal boundaries. This technique is then applied to produce CAT(0)-spaces with complicated conformal boundaries. Finally, we prove that the conformal boundary of an Hadamard n-manifold is always simply connected for n≥3, thus providing a bound for the level of complication of the boundary of such a manifold.      

The Tits boundary of a 2-complex

We investigate the Tits boundary of -complexes that have only a finite number of isometry types of cells. In particular, we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the -complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two -complexes are quasi-isometric, then the cores of their Tits boundaries are bi-Lipschitz.

     

A Geometric Proof of Stallings' Theorem on Groups with More than One End

Stallings showed that a finitely generated group which has more than one end splits as an amalgamated free product or an HNN extension over a finite subgroup. Dunwoody gave a new geometric proof of the theorem for the class of almost finitely presented groups, and separately, using somewhat different methods, generalised it to a larger class of splittings. Here we adapt the geometric method to the class of finitely generated groups using Sageev's generalisation of Bass Serre theory concerning group pairs with more than one end, and show that this new proof simultaneously establishes Dunwoody's generalisation.