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Algebraic foundations for applied topology and data analysis / / Hal Schenck
| Algebraic foundations for applied topology and data analysis / / Hal Schenck |
| Autore | Schenck Hal |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (231 pages) |
| Disciplina | 514.2 |
| Collana | Mathematics of Data |
| Soggetto topico |
Algebraic topology
Topological algebras Topologia algebraica Àlgebres topològiques |
| Soggetto genere / forma | Llibres electrònics |
| ISBN |
9783031066641
9783031066634 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
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. |
| Record Nr. | UNINA-9910631079903321 |
Schenck Hal
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| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Algebraic Geometry and Number Theory [[electronic resource] ] : Summer School, Galatasaray University, Istanbul, 2014 / / edited by Hussein Mourtada, Celal Cem Sarıoğlu, Christophe Soulé, Ayberk Zeytin
| Algebraic Geometry and Number Theory [[electronic resource] ] : Summer School, Galatasaray University, Istanbul, 2014 / / edited by Hussein Mourtada, Celal Cem Sarıoğlu, Christophe Soulé, Ayberk Zeytin |
| Edizione | [1st ed. 2017.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2017 |
| Descrizione fisica | 1 online resource (XI, 232 p.) |
| Disciplina | 516.35 |
| Collana | Progress in Mathematics |
| Soggetto topico |
Algebraic geometry
Number theory Commutative algebra Commutative rings Algebraic topology Algebraic Geometry Number Theory Commutative Rings and Algebras Algebraic Topology |
| ISBN | 3-319-47779-X |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Preface -- List of Participants -- p-adic Variation in Arithmetic Geometry: A Survey -- The Birational Geometry of Moduli Spaces -- On the Geometry of Hypersurfaces of Low Degrees in the Projective Space -- The Riemann–Roch Theorem in Arakelov Geometry -- Computing the Gysin Map Using Fixed Points -- On -adic Galois L-functions -- Class Number Problems and Lang Conjectures. |
| Record Nr. | UNINA-9910254282903321 |
| Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2017 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Algebraic graph theory [[electronic resource] ] : morphisms, monoids, and matrices / / by Ulrich Knauer
| Algebraic graph theory [[electronic resource] ] : morphisms, monoids, and matrices / / by Ulrich Knauer |
| Autore | Knauer U. <1942-> |
| Pubbl/distr/stampa | Berlin ; ; Boston, : De Gruyter, c2011 |
| Descrizione fisica | 1 online resource (324 p.) |
| Disciplina | 511/.5 |
| Collana | De Gruyter studies in mathematics |
| Soggetto topico |
Graph theory
Algebraic topology |
| Soggetto genere / forma | Electronic books. |
| ISBN |
1-283-40044-8
9786613400444 3-11-025509-X |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Frontmatter -- Preface -- Contents -- Chapter 1. Directed and undirected graphs -- Chapter 2. Graphs and matrices -- Chapter 3. Categories and functors -- Chapter 4. Binary graph operations -- Chapter 5. Line graph and other unary graph operations -- Chapter 6. Graphs and vector spaces -- Chapter 7. Graphs, groups and monoids -- Chapter 8. The characteristic polynomial of graphs -- Chapter 9. Graphs and monoids -- Chapter 10. Compositions, unretractivities and monoids -- Chapter 11. Cayley graphs of semigroups -- Chapter 12. Vertex transitive Cayley graphs -- Chapter 13. Embeddings of Cayley graphs - genus of semigroups -- Bibliography -- Index -- Index of symbols |
| Record Nr. | UNINA-9910457590903321 |
|
Knauer U. <1942->
|
||
| Berlin ; ; Boston, : De Gruyter, c2011 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Algebraic graph theory [[electronic resource] ] : morphisms, monoids, and matrices / / by Ulrich Knauer
| Algebraic graph theory [[electronic resource] ] : morphisms, monoids, and matrices / / by Ulrich Knauer |
| Autore | Knauer U. <1942-> |
| Pubbl/distr/stampa | Berlin ; ; Boston, : De Gruyter, c2011 |
| Descrizione fisica | 1 online resource (324 p.) |
| Disciplina | 511/.5 |
| Collana | De Gruyter studies in mathematics |
| Soggetto topico |
Graph theory
Algebraic topology |
| Soggetto non controllato |
Algebra
Graph Theory Matrices Monoids Morphisms |
| ISBN |
1-283-40044-8
9786613400444 3-11-025509-X |
| Classificazione | SK 890 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Frontmatter -- Preface -- Contents -- Chapter 1. Directed and undirected graphs -- Chapter 2. Graphs and matrices -- Chapter 3. Categories and functors -- Chapter 4. Binary graph operations -- Chapter 5. Line graph and other unary graph operations -- Chapter 6. Graphs and vector spaces -- Chapter 7. Graphs, groups and monoids -- Chapter 8. The characteristic polynomial of graphs -- Chapter 9. Graphs and monoids -- Chapter 10. Compositions, unretractivities and monoids -- Chapter 11. Cayley graphs of semigroups -- Chapter 12. Vertex transitive Cayley graphs -- Chapter 13. Embeddings of Cayley graphs - genus of semigroups -- Bibliography -- Index -- Index of symbols |
| Record Nr. | UNINA-9910781751803321 |
|
Knauer U. <1942->
|
||
| Berlin ; ; Boston, : De Gruyter, c2011 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Algebraic graph theory [[electronic resource] ] : morphisms, monoids, and matrices / / by Ulrich Knauer
| Algebraic graph theory [[electronic resource] ] : morphisms, monoids, and matrices / / by Ulrich Knauer |
| Autore | Knauer U. <1942-> |
| Pubbl/distr/stampa | Berlin ; ; Boston, : De Gruyter, c2011 |
| Descrizione fisica | 1 online resource (324 p.) |
| Disciplina | 511/.5 |
| Collana | De Gruyter studies in mathematics |
| Soggetto topico |
Graph theory
Algebraic topology |
| Soggetto non controllato |
Algebra
Graph Theory Matrices Monoids Morphisms |
| ISBN |
1-283-40044-8
9786613400444 3-11-025509-X |
| Classificazione | SK 890 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Frontmatter -- Preface -- Contents -- Chapter 1. Directed and undirected graphs -- Chapter 2. Graphs and matrices -- Chapter 3. Categories and functors -- Chapter 4. Binary graph operations -- Chapter 5. Line graph and other unary graph operations -- Chapter 6. Graphs and vector spaces -- Chapter 7. Graphs, groups and monoids -- Chapter 8. The characteristic polynomial of graphs -- Chapter 9. Graphs and monoids -- Chapter 10. Compositions, unretractivities and monoids -- Chapter 11. Cayley graphs of semigroups -- Chapter 12. Vertex transitive Cayley graphs -- Chapter 13. Embeddings of Cayley graphs - genus of semigroups -- Bibliography -- Index -- Index of symbols |
| Record Nr. | UNINA-9910818349703321 |
|
Knauer U. <1942->
|
||
| Berlin ; ; Boston, : De Gruyter, c2011 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Algebraic K-theory [e-book] : proceedings of a conference held at Oberwolfach, June 1980 : part I / edited by R. Keith Dennis
| Algebraic K-theory [e-book] : proceedings of a conference held at Oberwolfach, June 1980 : part I / edited by R. Keith Dennis |
| Pubbl/distr/stampa | Berlin : Springer, 1982 |
| Descrizione fisica | 1 online resource (x, 414 p.) |
| Disciplina | 514.2 |
| Altri autori (Persone) | Dennis, R. Keith |
| Collana | Lecture Notes in Mathematics, 0075-8434 ; 966 |
| Soggetto topico |
Mathematics
Algebraic topology |
| ISBN | 9783540395539 |
| Classificazione |
AMS 12A62
AMS 13D15 AMS 18F25 |
| Formato | Risorse elettroniche  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISALENTO-991002210859707536 |
| Berlin : Springer, 1982 | ||
| Risorse elettroniche | ||
| Lo trovi qui: Univ. del Salento | ||
| ||
Algebraic K-theory [e-book] : proceedings of a conference held at Oberwolfach, June 1980 : part II / edited by R. Keith Dennis
| Algebraic K-theory [e-book] : proceedings of a conference held at Oberwolfach, June 1980 : part II / edited by R. Keith Dennis |
| Pubbl/distr/stampa | Berlin : Springer, 1982 |
| Descrizione fisica | 1 online resource (viii, 412 p.) |
| Disciplina | 514.2 |
| Altri autori (Persone) | Dennis, R. Keith |
| Collana | Lecture Notes in Mathematics, 0075-8434 ; 967 |
| Soggetto topico |
Mathematics
Algebraic topology |
| ISBN | 9783540395560 |
| Classificazione |
AMS 12A62
AMS 13D15 AMS 18F25 |
| Formato | Risorse elettroniche |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISALENTO-991002210929707536 |
| Berlin : Springer, 1982 | ||
| Risorse elettroniche | ||
| Lo trovi qui: Univ. del Salento | ||
| ||
Algebraic k-theory Evanston 1980 [e-book] : proceedings of the conference held at Northwestern university Evanston, March 24–27, 1980 / edited by Eric M. Friedlander, Michael R. Stein
| Algebraic k-theory Evanston 1980 [e-book] : proceedings of the conference held at Northwestern university Evanston, March 24–27, 1980 / edited by Eric M. Friedlander, Michael R. Stein |
| Pubbl/distr/stampa | Berlin : Springer, 1981 |
| Descrizione fisica | 1 online resource (517 p.) |
| Disciplina | 514.2 |
| Altri autori (Persone) |
Friedlander, Eric M.
Stein, Michael R. |
| Collana | Lecture Notes in Mathematics, 0075-8434 ; 854 |
| Soggetto topico |
Mathematics
Algebraic topology |
| ISBN | 9783540386469 |
| Formato | Risorse elettroniche |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISALENTO-991002193589707536 |
| Berlin : Springer, 1981 | ||
| Risorse elettroniche | ||
| Lo trovi qui: Univ. del Salento | ||
| ||
Algebraic structure of knot modules [e-book] / by Jerome P. Levine
| Algebraic structure of knot modules [e-book] / by Jerome P. Levine |
| Autore | Levine, Jerome P. |
| Pubbl/distr/stampa | Berlin : Springer, 1980 |
| Descrizione fisica | 1 online resource (xiv, 110 p.) |
| Disciplina | 514.2 |
| Collana | Lecture Notes in Mathematics, 0075-8434 ; 772 |
| Soggetto topico |
Mathematics
Algebraic topology |
| ISBN | 9783540385554 |
| Formato | Risorse elettroniche |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISALENTO-991002191369707536 |
|
Levine, Jerome P.
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| Berlin : Springer, 1980 | ||
| Risorse elettroniche | ||
| Lo trovi qui: Univ. del Salento | ||
| ||
Algebraic topology : a student's guide / J. F. Adams
| Algebraic topology : a student's guide / J. F. Adams |
| Autore | Adams, John Frank |
| Pubbl/distr/stampa | Cambridge [Eng.] : Cambridge University Press, 1972 |
| Descrizione fisica | vi, 300 p. ; 24 cm |
| Disciplina | 514.2 |
| Collana | London Mathematical Society lecture note series, 0076-0552 ; 4 |
| Soggetto topico | Algebraic topology |
| ISBN | 0521080762 |
| Classificazione |
AMS 54-XX
LC QA612 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISALENTO-991000663549707536 |
|
Adams, John Frank
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| Cambridge [Eng.] : Cambridge University Press, 1972 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. del Salento | ||
| ||
Soggetto topico
-
Algebraic topology
(538)
- Algebraic Topology (150)
- Mathematics (83)
- Algebraic Geometry (37)
- Algebraic geometry (37)