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Knot theory. --- Low-dimensional topology. --- Manifolds and cell complexes -- Low-dimensional topology -- Knots and links in $S 3$. --- Algebraic topology -- Classical topics -- Degree, winding number. --- Group theory and generalizations -- Other generalizations of groups -- Loops, quasigroups. --- Group theory and generalizations -- Permutation groups -- General theory for finite groups. --- Algebraic topology -- Homology and cohomology theories -- Other homology theories. --- Manifolds and cell complexes -- Low-dimensional topology -- Fundamental group, presentations, free differential calculus. --- Manifolds and cell complexes -- Low-dimensional topology -- Invariants of knots and 3-manifolds. --- Group theory and generalizations -- Other generalizations of groups -- Sets with a single binary operation (groupoids). --- Manifolds and cell complexes -- PL-topology -- Knots and links (in high dimensions).

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This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.

Geometry --- Mathematics --- Physical Sciences & Mathematics --- Mathematics. --- Differential geometry. --- Algebraic topology. --- Manifolds (Mathematics). --- Complex manifolds. --- Differential Geometry. --- Algebraic Topology. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Differential geometry --- Math --- Science --- Global differential geometry. --- Cell aggregation --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation

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This book contains papers presented at the Workshop on the Analysis of Large-scale, High-Dimensional, and Multi-Variate Data Using Topology and Statistics, held in Le Barp, France, June 2013. It features the work of some of the most prominent and recognized leaders in the field who examine challenges as well as detail solutions to the analysis of extreme scale data.   The book presents new methods that leverage the mutual strengths of both topological and statistical techniques to support the management, analysis, and visualization of complex data. It covers both theory and application and provides readers with an overview of important key concepts and the latest research trends.   Coverage in the book includes multi-variate and/or high-dimensional analysis techniques, feature-based statistical methods, combinatorial algorithms, scalable statistics algorithms, scalar and vector field topology, and multi-scale representations. In addition, the book details algorithms that are broadly applicable and can be used by application scientists to glean insight from a wide range of complex data sets.

Mathematics. --- Topology. --- Statistical Theory and Methods. --- Applications of Mathematics. --- Algorithms. --- Visualization. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Cell aggregation --- Mathematical statistics. --- Mathématiques --- Algorithmes --- Visualisation --- Topologie --- Statistique mathématique --- Cell aggregation_xMathematics. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Applied mathematics. --- Engineering mathematics. --- Manifolds (Mathematics). --- Complex manifolds. --- Statistics. --- Imagery (Psychology) --- Imagination --- Visual perception --- Polyhedra --- Set theory --- Algebras, Linear --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Algorism --- Algebra --- Arithmetic --- Math --- Science --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Foundations --- Statistical methods --- Statistics . --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Engineering --- Engineering analysis --- Statistical analysis --- Statistical data --- Statistical science --- Econometrics

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Motivated by a variational model concerning the depth of the objects in a picture and the problem of hidden and illusory contours, this book investigates one of the central problems of computer vision: the  topological and algorithmic reconstruction of a smooth three dimensional scene starting from the visible part of an apparent contour. The authors focus their attention on the manipulation of apparent contours using a finite set of elementary moves, which correspond to diffeomorphic deformations of three dimensional scenes. A large part of the book is devoted to the  algorithmic part, with implementations, experiments, and computed examples. The book is intended also as a user's guide to the software code appcontour, written for the manipulation of apparent contours and their invariants. This book is addressed to theoretical and applied scientists working in the field of mathematical models of image segmentation.

Mathematics. --- Mathematical Applications in Computer Science. --- Image Processing and Computer Vision. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Calculus of Variations and Optimal Control; Optimization. --- Mathematical Software. --- Computer vision. --- Computer software. --- Mathematical optimization. --- Cell aggregation --- Mathématiques --- Vision par ordinateur --- Logiciels --- Optimisation mathématique --- Geometry -- Data processing. --- Three-dimensional imaging. --- Engineering & Applied Sciences --- Computer Science --- Geometry --- Data processing. --- Machine vision --- Vision, Computer --- 3-D imaging --- 3D imaging --- Three-dimensional imaging systems --- Three-dimensional imaging techniques --- Three-dimensional visualization --- Visualization, Three-dimensional --- Image processing. --- Computer science --- Computer mathematics. --- Calculus of variations. --- Manifolds (Mathematics). --- Complex manifolds. --- Imaging systems --- Artificial intelligence --- Image processing --- Pattern recognition systems --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Software, Computer --- Computer systems --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Computer science—Mathematics. --- Optical data processing. --- Isoperimetrical problems --- Variations, Calculus of --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Optical computing --- Visual data processing --- Bionics --- Electronic data processing --- Integrated optics --- Photonics --- Computers --- Computer mathematics --- Mathematics --- Optical equipment