On imbedding differentiable manifolds in Euclidean space

Assume n,k,m,q are positive integers. Let M^n denote a smooth differentiable n-manifold and R^k Euclidean k-space. (a) If M^n is open it imbeds smoothly in R^k, k=2n-1 (b) If M^n is open and parallelizable it immerses in R^n (c) Assume M^n is closed and (m-1)-connected, 1< 2m-n < n+1. If a neighborhood of the (n-m)-skeleton immerses in R^q, a>2n-2m, then the complement of a point of M^n imbeds smoothly in R^q.

Immersions in the stable range

Immersions of a differentiable m-manifold M in a differentiable n-manifold N, 2n > 3m+1, are classified up to regular homotopy by the homotopy classes of fibre maps F: T(M) ----> T(N) such that F(-X)=-F(X) and F(X) is nonzero of X is nonzero.

ANOSOV MAPS, POLYCYCLIC GROUPS AND HOMOLOGY

Many manifolds that do not admit Anosov diffeomorphisms are constructed. For example: the Cartesian product of the Klein bottle and a torus.

A map of North America : from the latest surveys and maps /

Relief shown pictorially.

Status of important farmlands maps as of January 1, 1982 : Massachusetts /

Title in right upper margin: Important farmlands maps.

Grain coarsening maps: A new tool to predict microfabric evolution of polymineralic rocks

Polymineralic rocks undergo grain coarsening with increasing temperature in both static and deformational environments, as long as no mineral reactions occur. The grain coarsening in such rocks is complex because the different phases influence each other, and it is this interaction that controls the rate of grain coarsening of the entire aggregate. We present a mathematical approach to investigate coupled grain coarsening using a set of microstructural parameters, including grain size and volume fraction of both second phases and matrix mineral in combination with temperature information. Based on samples from polymineralic carbonate mylonites that were deformed at different temperatures, we demonstrate how the mathematical relation can be calibrated for this natural system. Using such data sets for other lithologies, grain coarsening maps can be generated, which allow the prediction of microstructural evolution in polymineralic rocks. Such predictions are crucial for all subdisciplines in the earth sciences that require fundamental knowledge about microstructural changes and rheology of an orogen at different depths, such as structural geology, geophysics, geodynamics, and metamorphic petrology.

Statue of Liberty National Monument, maps /

In text: In 1986 the Statue will be 100 years old.

A three-dimensional representation method for noisy point clouds based on growing self-organizing maps accelerated on GPUs

The research described in this thesis was motivated by the need of a robust model capable of representing 3D data obtained with 3D sensors, which are inherently noisy. In addition, time constraints have to be considered as these sensors are capable of providing a 3D data stream in real time. This thesis proposed the use of Self-Organizing Maps (SOMs) as a 3D representation model. In particular, we proposed the use of the Growing Neural Gas (GNG) network, which has been successfully used for clustering, pattern recognition and topology representation of multi-dimensional data. Until now, Self-Organizing Maps have been primarily computed offline and their application in 3D data has mainly focused on free noise models, without considering time constraints. It is proposed a hardware implementation leveraging the computing power of modern GPUs, which takes advantage of a new paradigm coined as General-Purpose Computing on Graphics Processing Units (GPGPU). The proposed methods were applied to different problem and applications in the area of computer vision such as the recognition and localization of objects, visual surveillance or 3D reconstruction.