We consider geometric decompositions of aspherical 4manifolds which fibre over 2-orbifolds. We show first that no such manifold admits infinitely many fibrations over hyperbolic base orbifolds. If E is Seifert fibred over a hyperbolic surface B and either B has at most one cone point of order 2 or the monodromy has image in SL(2,Z) then E it has a decomposition induced from a decomposition of B...

For simplicity, we adopt the following rules: x denotes a set, r, s denote real numbers, n denotes a natural number, V denotes a real linear space, v, u, w, p denote vectors of V , A, B denote subsets of V , A1 denotes a finite subset of V , I denotes an affinely-independent subset of V , I1 denotes a finite affinelyindependent subset of V , F denotes a family of subsets of V , and L1, L2 denot...

The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or “geometric” knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In both of these cases, each knot space consists of five components, but contains...

In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on “convenient” vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,ω), we construct a weak symplectic structure on each leaf Iw of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree fo...

traditional building technologies have much to teach us about how to design regionally appropriate structures. the orosi is one of these useful technologies, which has been used for many centuries in order to control the harsh sunlight in iran. this architectural element was rather important because the intensive solar radiation could easily damage valuable carpets used in most spaces in persia...

We study tent spaces on general measure spaces (Ω, μ). We assume that there exists a semigroup of positive operators on Lp(Ω, μ) satisfying a monotone property but do not assume any geometric/metric structure on Ω. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman’s H1-BMO du...

This extended abstract should be considered as an overview of some results, and the more precise statements and proofs are found in the references. Hence the approach here is less rigorous than in the references. The new results presented at Dagstuhl are in [1] and [3]. The subject is directed topology in spaces which have been subdivided into cubes. Such spaces are used as the geometric model ...

In machine learning, high dimensional data generally should have a high degree of freedom. However, recent experiments in machine learning show that real world data in high dimensions is usually governed by a surprisingly low dimensions. We believe that in high dimensions, geometry information, for example, the “shape” of data distribution, can help learning algorithms to perform better. A geom...

When Einstein was thinking about the theory of general relativity based on the elimination of especial relativity constraints (especially the geometric relationship of space and time), he understood the first limitation of especial relativity is ignoring changes over time. Because in especial relativity, only the curvature of the space was considered. Therefore, tensor calculations should be to...

The aim of this paper is to present some existence results for nearest-point and farthestpoint problems, in connection with some geometric properties of Banach spaces. The idea goes back to Efimov and Stečkin who, in a series of papers (see [28, 29, 30, 31]), realized for the first time that some geometric properties of Banach spaces, such as strict convexity, uniform convexity, reflexivity, an...