06493nam 2200541 450 99649986930331620240125144012.09783031066641(electronic bk.)9783031066634(MiAaPQ)EBC7144122(Au-PeEL)EBL7144122(CKB)25430602200041(PPN)26635341X(EXLCZ)992543060220004120230329d2022 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierAlgebraic foundations for applied topology and data analysis /Hal SchenckCham, Switzerland :Springer,[2022]©20221 online resource (231 pages)Mathematics of Data ;v.1Print version: Schenck, Hal Algebraic Foundations for Applied Topology and Data Analysis Cham : Springer International Publishing AG,c2022 9783031066634 Includes bibliographical references and index.Intro -- 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.4.3.1 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.8.3.2 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.Mathematics of DataAlgebraic topologyTopological algebrasTopologia algebraicathubÀlgebres topològiquesthubLlibres electrònicsthubAlgebraic topology.Topological algebras.Topologia algebraicaÀlgebres topològiques514.2Schenck Hal1264645MiAaPQMiAaPQMiAaPQ996499869303316Algebraic Foundations for Applied Topology and Data Analysis2965585UNISA03986nam 22007575 450 991030023930332120250609110933.03-319-15645-410.1007/978-3-319-15645-3(CKB)3710000000402898(EBL)2094420(SSID)ssj0001501391(PQKBManifestationID)11921006(PQKBTitleCode)TC0001501391(PQKBWorkID)11524651(PQKB)10963165(DE-He213)978-3-319-15645-3(MiAaPQ)EBC2094420(PPN)185485502(MiAaPQ)EBC3110033(EXLCZ)99371000000040289820150424d2015 u| 0engur|n|---|||||txtccrColor Atlas of Head and Neck Surgery A Step-by-Step Guide /by Siba P. Dubey, Charles P. Molumi1st ed. 2015.Cham :Springer International Publishing :Imprint: Springer,2015.1 online resource (190 p.)Description based upon print version of record.3-319-15644-6 Includes bibliographical references and index.Surgery of the Nose and Paranasal sinus -- Laryngeal Surgery -- Thyroid Surgery -- Salivary Gland surgery -- Neck Dissection -- Local flaps in reconstruction of oral defects -- Surgery of the Mandible -- Facial plastic surgery -- Temporal bone carcinoma surgery.   .This surgical atlas, featuring a wealth of color photographs, provides detailed step-by-step descriptions of a wide range of open head and neck procedures, including radical and conservative (organ preservation, functional) approaches, aesthetic and reconstructive surgeries with the use of axial and free flaps, and surgery within the narrow confines of the skull base. Individual chapters are dedicated to surgery of the nose and paranasal sinuses, laryngeal surgery, thyroid surgery, surgery of the salivary glands, surgery of the mandible, surgery for temporal bone malignancy, facial plastic surgery, neck dissection, and surgery of the lip and oral cavity. The atlas will be a comprehensive practical reference for clinicians in the various specialties involved in head and neck surgery, including otolaryngologists, head and neck surgeons, plastic surgeons, maxillofacial surgeons, and surgical oncologists. It will assist practitioners in achieving the high level of competence that is essential owing to the large number of vital structures in the head and neck region.  .OtolaryngologyOtolaryngology, OperativeSurgery, PlasticMouthSurgeryMaxillaSurgeryOtorhinolaryngologyhttps://scigraph.springernature.com/ontologies/product-market-codes/H46007Head and Neck Surgeryhttps://scigraph.springernature.com/ontologies/product-market-codes/H46015Plastic Surgeryhttps://scigraph.springernature.com/ontologies/product-market-codes/H59087Oral and Maxillofacial Surgeryhttps://scigraph.springernature.com/ontologies/product-market-codes/H59060Otolaryngology.Otolaryngology, Operative.Surgery, Plastic.MouthSurgery.MaxillaSurgery.Otorhinolaryngology.Head and Neck Surgery.Plastic Surgery.Oral and Maxillofacial Surgery.610617.5617.5/1059617.51617.522059617952Dubey Siba Pauthttp://id.loc.gov/vocabulary/relators/aut789393Molumi Charles Pauthttp://id.loc.gov/vocabulary/relators/autBOOK9910300239303321Color Atlas of Head and Neck Surgery2512373UNINA