Topological Persistence In Geometry And Analysis
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Topological Persistence In Geometry And Analysis
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Author : Leonid Polterovich
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-05-11
Topological Persistence In Geometry And Analysis written by Leonid Polterovich and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-11 with Education categories.
The theory of persistence modules originated in topological data analysis and became an active area of research in algebraic topology. This book provides a concise and self-contained introduction to persistence modules and focuses on their interactions with pure mathematics, bringing the reader to the cutting edge of current research. In particular, the authors present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, they discuss topological function theory, which provides new insight into oscillation of functions. The book is accessible to readers with a basic background in algebraic and differential topology.
Mathematical Principles Of Topological And Geometric Data Analysis
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Author : Parvaneh Joharinad
language : en
Publisher: Springer Nature
Release Date : 2023-07-29
Mathematical Principles Of Topological And Geometric Data Analysis written by Parvaneh Joharinad and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-29 with Mathematics categories.
This book explores and demonstrates how geometric tools can be used in data analysis. Beginning with a systematic exposition of the mathematical prerequisites, covering topics ranging from category theory to algebraic topology, Riemannian geometry, operator theory and network analysis, it goes on to describe and analyze some of the most important machine learning techniques for dimension reduction, including the different types of manifold learning and kernel methods. It also develops a new notion of curvature of generalized metric spaces, based on the notion of hyperconvexity, which can be used for the topological representation of geometric information. In recent years there has been a fascinating development: concepts and methods originally created in the context of research in pure mathematics, and in particular in geometry, have become powerful tools in machine learning for the analysis of data. The underlying reason for this is that data are typically equipped with some kind of notion of distance, quantifying the differences between data points. Of course, to be successfully applied, the geometric tools usually need to be redefined, generalized, or extended appropriately. Primarily aimed at mathematicians seeking an overview of the geometric concepts and methods that are useful for data analysis, the book will also be of interest to researchers in machine learning and data analysis who want to see a systematic mathematical foundation of the methods that they use.
Symplectic Geometry
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Author : Helmut Hofer
language : en
Publisher: Springer Nature
Release Date : 2022-12-05
Symplectic Geometry written by Helmut Hofer and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-12-05 with Mathematics categories.
Over the course of his distinguished career, Claude Viterbo has made a number of groundbreaking contributions in the development of symplectic geometry/topology and Hamiltonian dynamics. The chapters in this volume – compiled on the occasion of his 60th birthday – are written by distinguished mathematicians and pay tribute to his many significant and lasting achievements.
Handbook Of Discrete And Computational Geometry
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Author : Csaba D. Toth
language : en
Publisher: CRC Press
Release Date : 2017-11-22
Handbook Of Discrete And Computational Geometry written by Csaba D. Toth and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-22 with Computers categories.
The Handbook of Discrete and Computational Geometry is intended as a reference book fully accessible to nonspecialists as well as specialists, covering all major aspects of both fields. The book offers the most important results and methods in discrete and computational geometry to those who use them in their work, both in the academic world—as researchers in mathematics and computer science—and in the professional world—as practitioners in fields as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed significantly to the growth of discrete mathematics in recent years. This has been fueled partly by the advent of powerful computers and by the recent explosion of activity in the relatively young field of computational geometry. This synthesis between discrete and computational geometry lies at the heart of this Handbook. A growing list of application fields includes combinatorial optimization, computer-aided design, computer graphics, crystallography, data analysis, error-correcting codes, geographic information systems, motion planning, operations research, pattern recognition, robotics, solid modeling, and tomography.
Geometric And Topological Mesh Feature Extraction For 3d Shape Analysis
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Author : Jean-Luc Mari
language : en
Publisher: John Wiley & Sons
Release Date : 2019-12-05
Geometric And Topological Mesh Feature Extraction For 3d Shape Analysis written by Jean-Luc Mari and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-12-05 with Mathematics categories.
Three-dimensional surface meshes are the most common discrete representation of the exterior of a virtual shape. Extracting relevant geometric or topological features from them can simplify the way objects are looked at, help with their recognition, and facilitate description and categorization according to specific criteria. This book adopts the point of view of discrete mathematics, the aim of which is to propose discrete counterparts to concepts mathematically defined in continuous terms. It explains how standard geometric and topological notions of surfaces can be calculated and computed on a 3D surface mesh, as well as their use for shape analysis. Several applications are also detailed, demonstrating that each of them requires specific adjustments to fit with generic approaches. The book is intended not only for students, researchers and engineers in computer science and shape analysis, but also numerical geologists, anthropologists, biologists and other scientists looking for practical solutions to their shape analysis, understanding or recognition problems.
Geometry And Topology In Music
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Author : Moreno Andreatta
language : en
Publisher: CRC Press
Release Date : 2024-11-01
Geometry And Topology In Music written by Moreno Andreatta and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-11-01 with Mathematics categories.
This book introduces path-breaking applications of concepts from mathematical topology to music-theory topics including harmony, chord progressions, rhythm, and music classification. Contributions address topics of voice leading, Tonnetze (maps of notes and chords), and automatic music classification. Focusing on some geometrical and topological aspects of the representation and formalisation of musical structures and processes, the book covers topological features of voice-leading geometries in the most recent advances in this mathematical approach to representing how chords are connected through the motion of voices, leading to analytically useful simplified models of high-dimensional spaces; It generalizes the idea of a Tonnetz, a geometrical map of tones or chords, and shows how topological aspects of these maps can correspond to many concepts from music theory. The resulting framework embeds the chord maps of neo-Riemannian theory in continuous spaces that relate chords of different sizes and includes extensions of this approach to rhythm theory. It further introduces an application of topology to automatic music classification, drawing upon both static topological representations and time-series evolution, showing how static and dynamic features of music interact as features of musical style. This volume will be a key resource for academics, researchers, and advanced students of music, music analyses, music composition, mathematical music theory, computational musicology, and music informatics. It was originally published as a special issue of the Journal of Mathematics and Music.
Discrete Geometry And Mathematical Morphology
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Author : Joakim Lindblad
language : en
Publisher: Springer Nature
Release Date : 2021-05-15
Discrete Geometry And Mathematical Morphology written by Joakim Lindblad and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-05-15 with Computers categories.
This book constitutes the proceedings of the First IAPR International Conference on Discrete Geometry and Mathematical Morphology, DGMM 2021, which was held during May 24-27, 2021, in Uppsala, Sweden. The conference was created by joining the International Conference on Discrete Geometry for computer Imagery, DGCI, with the International Symposium on Mathematical Morphology, ISMM. The 36 papers included in this volume were carefully reviewed and selected from 59 submissions. They were organized in topical sections as follows: applications in image processing, computer vision, and pattern recognition; discrete and combinatorial topology; discrete geometry - models, transforms, visualization; discrete tomography and inverse problems; hierarchical and graph-based models, analysis and segmentation; learning-based approaches to mathematical morphology; multivariate and PDE-based mathematical morphology, morphological filtering. The book also contains 3 invited keynote papers.
Topological Methods In Data Analysis And Visualization Ii
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Author : Ronald Peikert
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-01-10
Topological Methods In Data Analysis And Visualization Ii written by Ronald Peikert and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-10 with Mathematics categories.
When scientists analyze datasets in a search for underlying phenomena, patterns or causal factors, their first step is often an automatic or semi-automatic search for structures in the data. Of these feature-extraction methods, topological ones stand out due to their solid mathematical foundation. Topologically defined structures—as found in scalar, vector and tensor fields—have proven their merit in a wide range of scientific domains, and scientists have found them to be revealing in subjects such as physics, engineering, and medicine. Full of state-of-the-art research and contemporary hot topics in the subject, this volume is a selection of peer-reviewed papers originally presented at the fourth Workshop on Topology-Based Methods in Data Analysis and Visualization, TopoInVis 2011, held in Zurich, Switzerland. The workshop brought together many of the leading lights in the field for a mixture of formal presentations and discussion. One topic currently generating a great deal of interest, and explored in several chapters here, is the search for topological structures in time-dependent flows, and their relationship with Lagrangian coherent structures. Contributors also focus on discrete topologies of scalar and vector fields, and on persistence-based simplification, among other issues of note. The new research results included in this volume relate to all three key areas in data analysis—theory, algorithms and applications.
Computational Topology For Data Analysis
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Author : Tamal Krishna Dey
language : en
Publisher: Cambridge University Press
Release Date : 2022-03-10
Computational Topology For Data Analysis written by Tamal Krishna Dey and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-03-10 with Computers categories.
This book provides a computational and algorithmic foundation for techniques in topological data analysis, with examples and exercises.
Function Theory And P Spaces
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Author : Raymond Cheng
language : en
Publisher: American Mathematical Soc.
Release Date : 2020-05-28
Function Theory And P Spaces written by Raymond Cheng and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-28 with Education categories.
The classical ℓp sequence spaces have been a mainstay in Banach spaces. This book reviews some of the foundational results in this area (the basic inequalities, duality, convexity, geometry) as well as connects them to the function theory (boundary growth conditions, zero sets, extremal functions, multipliers, operator theory) of the associated spaces ℓpA of analytic functions whose Taylor coefficients belong to ℓp. Relations between the Banach space ℓp and its associated function space are uncovered using tools from Banach space geometry, including Birkhoff-James orthogonality and the resulting Pythagorean inequalities. The authors survey the literature on all of this material, including a discussion of the multipliers of ℓpA and a discussion of the Wiener algebra ℓ1A. Except for some basic measure theory, functional analysis, and complex analysis, which the reader is expected to know, the material in this book is self-contained and detailed proofs of nearly all the results are given. Each chapter concludes with some end notes that give proper references, historical background, and avenues for further exploration.