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001451502 020__ $$a9783031066641$$q(electronic bk.)
001451502 020__ $$a3031066642$$q(electronic bk.)
001451502 020__ $$z9783031066634
001451502 020__ $$z3031066634
001451502 0247_ $$a10.1007/978-3-031-06664-1$$2doi
001451502 035__ $$aSP(OCoLC)1351750760
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001451502 050_4 $$aQA612
001451502 08204 $$a514/.2$$223/eng/20221221
001451502 1001_ $$aSchenck, Hal,$$eauthor.
001451502 24510 $$aAlgebraic foundations for applied topology and data analysis /$$cHal Schenck.
001451502 264_1 $$aCham :$$bSpringer,$$c[2022]
001451502 264_4 $$c©2022
001451502 300__ $$a1 online resource (xii, 224 pages).
001451502 336__ $$atext$$btxt$$2rdacontent
001451502 337__ $$acomputer$$bc$$2rdamedia
001451502 338__ $$aonline resource$$bcr$$2rdacarrier
001451502 4901_ $$aMathematics of data ;$$vvolume 1
001451502 504__ $$aIncludes bibliographical references and index.
001451502 5050_ $$aIntro -- Preface -- Contents -- 1 Linear Algebra Tools for Data Analysis -- 1.1 Linear Equations, Gaussian Elimination, Matrix Algebra -- 1.2 Vector Spaces, Linear Transformations, Basis and Change of Basis -- 1.2.1 Basis of a Vector Space -- 1.2.2 Linear Transformations -- 1.2.3 Change of Basis -- 1.3 Diagonalization, Webpage Ranking, Data and Covariance -- 1.3.1 Eigenvalues and Eigenvectors -- 1.3.2 Diagonalization -- 1.3.3 Ranking Using Diagonalization -- 1.3.4 Data Application: Diagonalization of the Covariance Matrix -- 1.4 Orthogonality, Least Squares Fitting, Singular Value Decomposition -- 1.4.1 Least Squares -- 1.4.2 Subspaces and Orthogonality -- 1.4.3 Singular Value Decomposition -- 2 Basics of Algebra: Groups, Rings, Modules -- 2.1 Groups, Rings and Homomorphisms -- 2.1.1 Groups -- 2.1.2 Rings -- 2.2 Modules and Operations on Modules -- 2.2.1 Ideals -- 2.2.2 Tensor Product -- 2.2.3 Hom -- 2.3 Localization of Rings and Modules -- 2.4 Noetherian Rings, Hilbert Basis Theorem, Varieties -- 2.4.1 Noetherian Rings -- 2.4.2 Solutions to a Polynomial System: Varieties -- 3 Basics of Topology: Spaces and Sheaves -- 3.1 Topological Spaces -- 3.1.1 Set Theory and Equivalence Relations -- 3.1.2 Definition of a Topology -- 3.1.3 Discrete, Product, and Quotient Topologies -- 3.2 Vector Bundles -- 3.3 Sheaf Theory -- 3.3.1 Presheaves and Sheaves -- 3.3.2 Posets, Direct Limit, and Stalks -- 3.3.3 Morphisms of Sheaves and Exactness -- 3.4 From Graphs to Social Media to Sheaves -- 3.4.1 Spectral Graph Theory -- 3.4.2 Heat Diffusing on a Wire Graph -- 3.4.3 From Graphs to Cellular Sheaves -- 4 Homology I: Simplicial Complexes to Sensor Networks -- 4.1 Simplicial Complexes, Nerve of a Cover -- 4.1.1 The Nerve of a Cover -- 4.2 Simplicial and Singular Homology -- 4.2.1 Singular homology -- 4.3 Snake Lemma and Long Exact Sequence in Homology.
001451502 5058_ $$a4.3.1 Maps of complexes, Snake Lemma -- 4.3.2 Chain Homotopy -- 4.4 Mayer-Vietoris, Rips and Čech Complex, Sensor Networks -- 4.4.1 Mayer-Vietoris Sequence -- 4.4.2 Relative Homology -- 4.4.3 Čech Complex and Rips Complex -- 5 Homology II: Cohomology to Ranking Problems -- 5.1 Cohomology: Simplicial, Čech, de Rham Theories -- 5.1.1 Simplicial Cohomology -- 5.1.2 Čech Cohomology -- 5.1.3 de Rham Cohomology -- 5.2 Ranking, the Netflix Problem, and Hodge Theory -- 5.2.1 Hodge Decomposition -- 5.2.2 Application to Ranking -- 5.3 CW Complexes and Cellular Homology -- 5.4 Poincaré and Alexander Duality: Sensor Networks Revisited -- 5.4.1 Statement of Theorems and Examples -- 5.4.2 Alexander Duality: Proof -- 5.4.3 Sensor Networks Revisited -- 5.4.4 Poincaré Duality -- 6 Persistent Algebra: Modules Over a PID -- 6.1 Principal Ideal Domains and Euclidean Domains -- 6.2 Rational Canonical Form of a Matrix -- 6.3 Linear Transformations, K[t]-Modules, Jordan Form -- 6.4 Structure of Abelian Groups and Persistent Homology -- 6.4.1 Z-Graded Rings -- 7 Persistent Homology -- 7.1 Barcodes, Persistence Diagrams, Bottleneck Distance -- 7.1.1 History -- 7.1.2 Persistent Homology and the Barcode -- 7.1.3 Computation of Persistent Homology -- 7.1.4 Alpha and Witness Complexes -- 7.1.5 Persistence Diagrams -- 7.1.6 Metrics on Diagrams -- 7.2 Morse Theory -- 7.3 The Stability Theorem -- 7.4 Interleaving and Categories -- 7.4.1 Categories and Functors -- 7.4.2 Interleaving -- 7.4.3 Interleaving Vignette: Merge Trees -- 7.4.4 Zigzag Persistence and Quivers -- 8 Multiparameter Persistent Homology -- 8.1 Definition and Examples -- 8.1.1 Multiparameter Persistence -- 8.2 Graded Algebra, Hilbert Function, Series, Polynomial -- 8.2.1 The Hilbert Function -- 8.2.2 The Hilbert Series -- 8.3 Associated Primes and Zn-Graded Modules -- 8.3.1 Geometry of Sheaves.
001451502 5058_ $$a8.3.2 Associated Primes and Primary Decomposition -- 8.3.3 Additional Structure in the Zn-Graded Setting -- 8.4 Filtrations and Ext -- 9 Derived Functors and Spectral Sequences -- 9.1 Injective and Projective Objects, Resolutions -- 9.1.1 Projective and Injective Objects -- 9.1.2 Resolutions -- 9.2 Derived Functors -- 9.2.1 Categories and Functors -- 9.2.2 Constructing Derived Functors -- 9.2.3 Ext -- 9.2.4 The Global Sections Functor -- 9.2.5 Acyclic Objects -- 9.3 Spectral Sequences -- 9.3.1 Total Complex of Double Complex -- 9.3.2 The Vertical Filtration -- 9.3.3 Main Theorem -- 9.4 Pas de Deux: Spectral Sequences and Derived Functors -- 9.4.1 Resolution of a Complex -- 9.4.2 Grothendieck Spectral Sequence -- 9.4.3 Comparing Cohomology Theories -- 9.4.4 Cartan-Eilenberg Resolution -- A Examples of Software Packages -- A.1 Covariance and Spread of Data via R -- A.2 Persistent Homology via scikit-tda -- A.3 Computational Algebra via Macaulay2 -- A.4 Multiparameter Persistence via RIVET -- Bibliography -- Index.
001451502 506__ $$aAccess limited to authorized users.
001451502 520__ $$aThis book gives an intuitive and hands-on introduction to Topological Data Analysis (TDA). Covering a wide range of topics at levels of sophistication varying from elementary (matrix algebra) to esoteric (Grothendieck spectral sequence), it offers a mirror of data science aimed at a general mathematical audience. The required algebraic background is developed in detail. The first third of the book reviews several core areas of mathematics, beginning with basic linear algebra and applications to data fitting and web search algorithms, followed by quick primers on algebra and topology. The middle third introduces algebraic topology, along with applications to sensor networks and voter ranking. The last third covers key contemporary tools in TDA: persistent and multiparameter persistent homology. Also included is a users guide to derived functors and spectral sequences (useful but somewhat technical tools which have recently found applications in TDA), and an appendix illustrating a number of software packages used in the field. Based on a course given as part of a masters degree in statistics, the book is appropriate for graduate students. .
001451502 588__ $$aDescription based upon print version of record.
001451502 650_0 $$aAlgebraic topology.
001451502 650_0 $$aTopological algebras.
001451502 655_0 $$aElectronic books.
001451502 77608 $$iPrint version:$$aSchenck, Hal$$tAlgebraic Foundations for Applied Topology and Data Analysis$$dCham : Springer International Publishing AG,c2022$$z9783031066634
001451502 830_0 $$aMathematics of data ;$$vvolume 1
001451502 852__ $$bebk
001451502 85640 $$3Springer Nature$$uhttps://univsouthin.idm.oclc.org/login?url=https://link.springer.com/10.1007/978-3-031-06664-1$$zOnline Access$$91397441.1
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001451502 980__ $$aBIB
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