Operator Calculus on Graphs :: Theory and Applications in Computer Science.
This pioneering book presents a study of the interrelationships among operator calculus, graph theory, and quantum probability in a unified manner, with significant emphasis on symbolic computations and an eye toward applications in computer science. Presented in this book are new methods, built on...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Singapore :
World Scientific,
2012.
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| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | This pioneering book presents a study of the interrelationships among operator calculus, graph theory, and quantum probability in a unified manner, with significant emphasis on symbolic computations and an eye toward applications in computer science. Presented in this book are new methods, built on the algebraic framework of Clifford algebras, for tackling important real world problems related, but not limited to, wireless communications, neural networks, electrical circuits, transportation, and the world wide web. Examples are put forward in Mathematica throughout the book, together with pack. |
| Beschreibung: | 11. Partition-Dependent Stochastic Measures. |
| Beschreibung: | 1 online resource (428 pages) |
| Bibliographie: | Includes bibliographical references (pages 399-406) and index. |
| ISBN: | 9781848168770 1848168772 1280669063 9781280669064 |
Internformat
MARC
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| 245 | 1 | 0 | |a Operator Calculus on Graphs : |b Theory and Applications in Computer Science. |
| 260 | |a Singapore : |b World Scientific, |c 2012. | ||
| 300 | |a 1 online resource (428 pages) | ||
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| 505 | 0 | |a Preface; Acknowledgments; Contents; Combinatorial Algebras and Their Properties; 1. Introduction; 1.1 Notational Preliminaries; 2. Combinatorial Algebra; 2.1 Six Group and Semigroup Algebras; 2.1.1 The group of blades Bp, q; 2.1.1.1 Involutions; 2.1.1.2 The n-dimensional hypercube Qn; 2.1.2 The abelian blade group Bp, q sym; 2.1.3 The null blade semigroup; 2.1.4 The abelian null blade semigroup sym; 2.1.5 The semigroup of idempotent blades idem; 2.1.6 The path semigroup n; 2.1.7 Summary; 2.1.7.1 Algebras I-IV; 2.1.7.2 Algebra V; 2.1.7.3 Algebra VI; 2.2 Clifford and Grassmann Algebras. | |
| 505 | 8 | |a 2.2.1 Grassmann (exterior) algebras2.2.2 Clifford algebras; 2.2.3 Operator calculus on Clifford algebras; 2.3 The Symmetric Clifford Algebra sym; 2.4 The Idempotent-Generated Algebra idem; 2.5 The n-Particle Zeon Algebra nil; 2.6 Generalized Zeon Algebras; 3. Norm Inequalities on Clifford Algebras; 3.1 Norms on C p; q; 3.2 Generating Functions; 3.3 Clifford Matrices and the Clifford-Frobenius Norm; 3.4 Powers of Clifford Matrices; Combinatorics and Graph Theory; 4. Specialized Adjacency Matrices; 4.1 Essential Graph Theory; 4.2 Clifford Adjacency Matrices; 4.3 Nilpotent Adjacency Matrices. | |
| 505 | 8 | |a 4.3.1 Euler circuits4.3.2 Conditional branching; 4.3.3 Time-homogeneous random walks on finite graphs; 5. Random Graphs; 5.1 Preliminaries; 5.2 Cycles in Random Graphs; 5.3 Convergence of Moments; 6. Graph Theory and Quantum Probability; 6.1 Concepts; 6.1.1 Operators as random variables; 6.1.2 Operators as adjacency matrices; 6.2 From Graphs to Quantum Random Variables; 6.2.1 Nilpotent adjacency operators in infinite spaces; 6.2.2 Decomposition of nilpotent adjacency operators; 6.3 Connected Components in Graph Processes; 6.3.1 Algebraic preliminaries; 6.3.2 Connected components. | |
| 505 | 8 | |a 6.3.2.1 (k, d)-components6.3.3 Second quantization of graph processes; 7. Geometric Graph Processes; 7.1 Preliminaries; 7.2 Dynamic Graph Processes; 7.2.1 Vertex degrees in Gn; 7.2.2 Energy and Laplacian energy of geometric graphs; 7.2.3 Convergence conditions and a limit theorem; 7.3 Time-Homogeneous Walks on Random Geometric Graphs; Probability on Algebraic Structures; 8. Time-Homogeneous Random Walks; 8.1 sym and Random Walks on Hypercubes; 8.2 Multiplicative Walks on C p, q; 8.2.1 Walks on directed hypercubes; 8.2.2 Random walks on directed hypercubes with loops. | |
| 505 | 8 | |a 8.2.3 Properties of multiplicative walks8.3 Induced Additive Walks on C p, q; 8.3.1 Variance of N; 8.3.2 Variance of; 8.3.3 Central limit theorems; 9. Dynamic Walks in Clifford Algebras; 9.1 Preliminaries; 9.2 Expectation; 9.3 Limit Theorems; 9.3.1 Conditions for convergence; 9.3.2 Induced additive walks; 9.3.3 Central limit theorem; 10. Iterated Stochastic Integrals; 10.1 Preliminaries; 10.2 Stochastic Integrals in; 10.3 Graph-Theoretic Iterated Stochastic Integrals; 10.3.1 Functions on partitions; 10.3.2 The Clifford evolution matrix; 10.3.3 Orthogonal polynomials. | |
| 500 | |a 11. Partition-Dependent Stochastic Measures. | ||
| 520 | |a This pioneering book presents a study of the interrelationships among operator calculus, graph theory, and quantum probability in a unified manner, with significant emphasis on symbolic computations and an eye toward applications in computer science. Presented in this book are new methods, built on the algebraic framework of Clifford algebras, for tackling important real world problems related, but not limited to, wireless communications, neural networks, electrical circuits, transportation, and the world wide web. Examples are put forward in Mathematica throughout the book, together with pack. | ||
| 588 | 0 | |a Print version record. | |
| 504 | |a Includes bibliographical references (pages 399-406) and index. | ||
| 650 | 0 | |a Calculus, Operational. |0 http://id.loc.gov/authorities/subjects/sh85018805 | |
| 650 | 0 | |a Graph theory. |0 http://id.loc.gov/authorities/subjects/sh85056471 | |
| 650 | 0 | |a Quantum statistics. |0 http://id.loc.gov/authorities/subjects/sh85109468 | |
| 650 | 0 | |a Computer science. |0 http://id.loc.gov/authorities/subjects/sh89003285 | |
| 650 | 6 | |a Calcul symbolique. | |
| 650 | 6 | |a Statistique quantique. | |
| 650 | 6 | |a Informatique. | |
| 650 | 7 | |a MATHEMATICS |x Calculus. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Mathematical Analysis. |2 bisacsh | |
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| 650 | 7 | |a Computer science |2 fast | |
| 650 | 7 | |a Graph theory |2 fast | |
| 650 | 7 | |a Quantum statistics |2 fast | |
| 700 | 1 | |a Staples, G. Stacey. | |
| 758 | |i has work: |a Operator calculus on graphs (Text) |1 https://id.oclc.org/worldcat/entity/E39PCGhJmjYQmbPJkqVyYRXgXd |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Schott, Rene. |t Operator Calculus on Graphs : Theory and Applications in Computer Science. |d Singapore : World Scientific, ©2012 |z 9781848168763 |
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Datensatz im Suchindex
| DE-BY-FWS_katkey | ZDB-4-EBA-ocn794328387 |
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| author_facet | Schott, Rene Staples, G. Stacey |
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| contents | Preface; Acknowledgments; Contents; Combinatorial Algebras and Their Properties; 1. Introduction; 1.1 Notational Preliminaries; 2. Combinatorial Algebra; 2.1 Six Group and Semigroup Algebras; 2.1.1 The group of blades Bp, q; 2.1.1.1 Involutions; 2.1.1.2 The n-dimensional hypercube Qn; 2.1.2 The abelian blade group Bp, q sym; 2.1.3 The null blade semigroup; 2.1.4 The abelian null blade semigroup sym; 2.1.5 The semigroup of idempotent blades idem; 2.1.6 The path semigroup n; 2.1.7 Summary; 2.1.7.1 Algebras I-IV; 2.1.7.2 Algebra V; 2.1.7.3 Algebra VI; 2.2 Clifford and Grassmann Algebras. 2.2.1 Grassmann (exterior) algebras2.2.2 Clifford algebras; 2.2.3 Operator calculus on Clifford algebras; 2.3 The Symmetric Clifford Algebra sym; 2.4 The Idempotent-Generated Algebra idem; 2.5 The n-Particle Zeon Algebra nil; 2.6 Generalized Zeon Algebras; 3. Norm Inequalities on Clifford Algebras; 3.1 Norms on C p; q; 3.2 Generating Functions; 3.3 Clifford Matrices and the Clifford-Frobenius Norm; 3.4 Powers of Clifford Matrices; Combinatorics and Graph Theory; 4. Specialized Adjacency Matrices; 4.1 Essential Graph Theory; 4.2 Clifford Adjacency Matrices; 4.3 Nilpotent Adjacency Matrices. 4.3.1 Euler circuits4.3.2 Conditional branching; 4.3.3 Time-homogeneous random walks on finite graphs; 5. Random Graphs; 5.1 Preliminaries; 5.2 Cycles in Random Graphs; 5.3 Convergence of Moments; 6. Graph Theory and Quantum Probability; 6.1 Concepts; 6.1.1 Operators as random variables; 6.1.2 Operators as adjacency matrices; 6.2 From Graphs to Quantum Random Variables; 6.2.1 Nilpotent adjacency operators in infinite spaces; 6.2.2 Decomposition of nilpotent adjacency operators; 6.3 Connected Components in Graph Processes; 6.3.1 Algebraic preliminaries; 6.3.2 Connected components. 6.3.2.1 (k, d)-components6.3.3 Second quantization of graph processes; 7. Geometric Graph Processes; 7.1 Preliminaries; 7.2 Dynamic Graph Processes; 7.2.1 Vertex degrees in Gn; 7.2.2 Energy and Laplacian energy of geometric graphs; 7.2.3 Convergence conditions and a limit theorem; 7.3 Time-Homogeneous Walks on Random Geometric Graphs; Probability on Algebraic Structures; 8. Time-Homogeneous Random Walks; 8.1 sym and Random Walks on Hypercubes; 8.2 Multiplicative Walks on C p, q; 8.2.1 Walks on directed hypercubes; 8.2.2 Random walks on directed hypercubes with loops. 8.2.3 Properties of multiplicative walks8.3 Induced Additive Walks on C p, q; 8.3.1 Variance of N; 8.3.2 Variance of; 8.3.3 Central limit theorems; 9. Dynamic Walks in Clifford Algebras; 9.1 Preliminaries; 9.2 Expectation; 9.3 Limit Theorems; 9.3.1 Conditions for convergence; 9.3.2 Induced additive walks; 9.3.3 Central limit theorem; 10. Iterated Stochastic Integrals; 10.1 Preliminaries; 10.2 Stochastic Integrals in; 10.3 Graph-Theoretic Iterated Stochastic Integrals; 10.3.1 Functions on partitions; 10.3.2 The Clifford evolution matrix; 10.3.3 Orthogonal polynomials. |
| ctrlnum | (OCoLC)794328387 |
| dewey-full | 515 515.72 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 515.72 |
| dewey-search | 515 515.72 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| id | ZDB-4-EBA-ocn794328387 |
| illustrated | Not Illustrated |
| indexdate | 2024-11-27T13:18:24Z |
| institution | BVB |
| isbn | 9781848168770 1848168772 1280669063 9781280669064 |
| language | English |
| oclc_num | 794328387 |
| open_access_boolean | |
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| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (428 pages) |
| psigel | ZDB-4-EBA |
| publishDate | 2012 |
| publishDateSearch | 2012 |
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| publisher | World Scientific, |
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| spelling | Schott, Rene. Operator Calculus on Graphs : Theory and Applications in Computer Science. Singapore : World Scientific, 2012. 1 online resource (428 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Preface; Acknowledgments; Contents; Combinatorial Algebras and Their Properties; 1. Introduction; 1.1 Notational Preliminaries; 2. Combinatorial Algebra; 2.1 Six Group and Semigroup Algebras; 2.1.1 The group of blades Bp, q; 2.1.1.1 Involutions; 2.1.1.2 The n-dimensional hypercube Qn; 2.1.2 The abelian blade group Bp, q sym; 2.1.3 The null blade semigroup; 2.1.4 The abelian null blade semigroup sym; 2.1.5 The semigroup of idempotent blades idem; 2.1.6 The path semigroup n; 2.1.7 Summary; 2.1.7.1 Algebras I-IV; 2.1.7.2 Algebra V; 2.1.7.3 Algebra VI; 2.2 Clifford and Grassmann Algebras. 2.2.1 Grassmann (exterior) algebras2.2.2 Clifford algebras; 2.2.3 Operator calculus on Clifford algebras; 2.3 The Symmetric Clifford Algebra sym; 2.4 The Idempotent-Generated Algebra idem; 2.5 The n-Particle Zeon Algebra nil; 2.6 Generalized Zeon Algebras; 3. Norm Inequalities on Clifford Algebras; 3.1 Norms on C p; q; 3.2 Generating Functions; 3.3 Clifford Matrices and the Clifford-Frobenius Norm; 3.4 Powers of Clifford Matrices; Combinatorics and Graph Theory; 4. Specialized Adjacency Matrices; 4.1 Essential Graph Theory; 4.2 Clifford Adjacency Matrices; 4.3 Nilpotent Adjacency Matrices. 4.3.1 Euler circuits4.3.2 Conditional branching; 4.3.3 Time-homogeneous random walks on finite graphs; 5. Random Graphs; 5.1 Preliminaries; 5.2 Cycles in Random Graphs; 5.3 Convergence of Moments; 6. Graph Theory and Quantum Probability; 6.1 Concepts; 6.1.1 Operators as random variables; 6.1.2 Operators as adjacency matrices; 6.2 From Graphs to Quantum Random Variables; 6.2.1 Nilpotent adjacency operators in infinite spaces; 6.2.2 Decomposition of nilpotent adjacency operators; 6.3 Connected Components in Graph Processes; 6.3.1 Algebraic preliminaries; 6.3.2 Connected components. 6.3.2.1 (k, d)-components6.3.3 Second quantization of graph processes; 7. Geometric Graph Processes; 7.1 Preliminaries; 7.2 Dynamic Graph Processes; 7.2.1 Vertex degrees in Gn; 7.2.2 Energy and Laplacian energy of geometric graphs; 7.2.3 Convergence conditions and a limit theorem; 7.3 Time-Homogeneous Walks on Random Geometric Graphs; Probability on Algebraic Structures; 8. Time-Homogeneous Random Walks; 8.1 sym and Random Walks on Hypercubes; 8.2 Multiplicative Walks on C p, q; 8.2.1 Walks on directed hypercubes; 8.2.2 Random walks on directed hypercubes with loops. 8.2.3 Properties of multiplicative walks8.3 Induced Additive Walks on C p, q; 8.3.1 Variance of N; 8.3.2 Variance of; 8.3.3 Central limit theorems; 9. Dynamic Walks in Clifford Algebras; 9.1 Preliminaries; 9.2 Expectation; 9.3 Limit Theorems; 9.3.1 Conditions for convergence; 9.3.2 Induced additive walks; 9.3.3 Central limit theorem; 10. Iterated Stochastic Integrals; 10.1 Preliminaries; 10.2 Stochastic Integrals in; 10.3 Graph-Theoretic Iterated Stochastic Integrals; 10.3.1 Functions on partitions; 10.3.2 The Clifford evolution matrix; 10.3.3 Orthogonal polynomials. 11. Partition-Dependent Stochastic Measures. This pioneering book presents a study of the interrelationships among operator calculus, graph theory, and quantum probability in a unified manner, with significant emphasis on symbolic computations and an eye toward applications in computer science. Presented in this book are new methods, built on the algebraic framework of Clifford algebras, for tackling important real world problems related, but not limited to, wireless communications, neural networks, electrical circuits, transportation, and the world wide web. Examples are put forward in Mathematica throughout the book, together with pack. Print version record. Includes bibliographical references (pages 399-406) and index. Calculus, Operational. http://id.loc.gov/authorities/subjects/sh85018805 Graph theory. http://id.loc.gov/authorities/subjects/sh85056471 Quantum statistics. http://id.loc.gov/authorities/subjects/sh85109468 Computer science. http://id.loc.gov/authorities/subjects/sh89003285 Calcul symbolique. Statistique quantique. Informatique. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Calculus, Operational fast Computer science fast Graph theory fast Quantum statistics fast Staples, G. Stacey. has work: Operator calculus on graphs (Text) https://id.oclc.org/worldcat/entity/E39PCGhJmjYQmbPJkqVyYRXgXd https://id.oclc.org/worldcat/ontology/hasWork Print version: Schott, Rene. Operator Calculus on Graphs : Theory and Applications in Computer Science. Singapore : World Scientific, ©2012 9781848168763 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=457191 Volltext |
| spellingShingle | Schott, Rene Operator Calculus on Graphs : Theory and Applications in Computer Science. Preface; Acknowledgments; Contents; Combinatorial Algebras and Their Properties; 1. Introduction; 1.1 Notational Preliminaries; 2. Combinatorial Algebra; 2.1 Six Group and Semigroup Algebras; 2.1.1 The group of blades Bp, q; 2.1.1.1 Involutions; 2.1.1.2 The n-dimensional hypercube Qn; 2.1.2 The abelian blade group Bp, q sym; 2.1.3 The null blade semigroup; 2.1.4 The abelian null blade semigroup sym; 2.1.5 The semigroup of idempotent blades idem; 2.1.6 The path semigroup n; 2.1.7 Summary; 2.1.7.1 Algebras I-IV; 2.1.7.2 Algebra V; 2.1.7.3 Algebra VI; 2.2 Clifford and Grassmann Algebras. 2.2.1 Grassmann (exterior) algebras2.2.2 Clifford algebras; 2.2.3 Operator calculus on Clifford algebras; 2.3 The Symmetric Clifford Algebra sym; 2.4 The Idempotent-Generated Algebra idem; 2.5 The n-Particle Zeon Algebra nil; 2.6 Generalized Zeon Algebras; 3. Norm Inequalities on Clifford Algebras; 3.1 Norms on C p; q; 3.2 Generating Functions; 3.3 Clifford Matrices and the Clifford-Frobenius Norm; 3.4 Powers of Clifford Matrices; Combinatorics and Graph Theory; 4. Specialized Adjacency Matrices; 4.1 Essential Graph Theory; 4.2 Clifford Adjacency Matrices; 4.3 Nilpotent Adjacency Matrices. 4.3.1 Euler circuits4.3.2 Conditional branching; 4.3.3 Time-homogeneous random walks on finite graphs; 5. Random Graphs; 5.1 Preliminaries; 5.2 Cycles in Random Graphs; 5.3 Convergence of Moments; 6. Graph Theory and Quantum Probability; 6.1 Concepts; 6.1.1 Operators as random variables; 6.1.2 Operators as adjacency matrices; 6.2 From Graphs to Quantum Random Variables; 6.2.1 Nilpotent adjacency operators in infinite spaces; 6.2.2 Decomposition of nilpotent adjacency operators; 6.3 Connected Components in Graph Processes; 6.3.1 Algebraic preliminaries; 6.3.2 Connected components. 6.3.2.1 (k, d)-components6.3.3 Second quantization of graph processes; 7. Geometric Graph Processes; 7.1 Preliminaries; 7.2 Dynamic Graph Processes; 7.2.1 Vertex degrees in Gn; 7.2.2 Energy and Laplacian energy of geometric graphs; 7.2.3 Convergence conditions and a limit theorem; 7.3 Time-Homogeneous Walks on Random Geometric Graphs; Probability on Algebraic Structures; 8. Time-Homogeneous Random Walks; 8.1 sym and Random Walks on Hypercubes; 8.2 Multiplicative Walks on C p, q; 8.2.1 Walks on directed hypercubes; 8.2.2 Random walks on directed hypercubes with loops. 8.2.3 Properties of multiplicative walks8.3 Induced Additive Walks on C p, q; 8.3.1 Variance of N; 8.3.2 Variance of; 8.3.3 Central limit theorems; 9. Dynamic Walks in Clifford Algebras; 9.1 Preliminaries; 9.2 Expectation; 9.3 Limit Theorems; 9.3.1 Conditions for convergence; 9.3.2 Induced additive walks; 9.3.3 Central limit theorem; 10. Iterated Stochastic Integrals; 10.1 Preliminaries; 10.2 Stochastic Integrals in; 10.3 Graph-Theoretic Iterated Stochastic Integrals; 10.3.1 Functions on partitions; 10.3.2 The Clifford evolution matrix; 10.3.3 Orthogonal polynomials. Calculus, Operational. http://id.loc.gov/authorities/subjects/sh85018805 Graph theory. http://id.loc.gov/authorities/subjects/sh85056471 Quantum statistics. http://id.loc.gov/authorities/subjects/sh85109468 Computer science. http://id.loc.gov/authorities/subjects/sh89003285 Calcul symbolique. Statistique quantique. Informatique. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Calculus, Operational fast Computer science fast Graph theory fast Quantum statistics fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85018805 http://id.loc.gov/authorities/subjects/sh85056471 http://id.loc.gov/authorities/subjects/sh85109468 http://id.loc.gov/authorities/subjects/sh89003285 |
| title | Operator Calculus on Graphs : Theory and Applications in Computer Science. |
| title_auth | Operator Calculus on Graphs : Theory and Applications in Computer Science. |
| title_exact_search | Operator Calculus on Graphs : Theory and Applications in Computer Science. |
| title_full | Operator Calculus on Graphs : Theory and Applications in Computer Science. |
| title_fullStr | Operator Calculus on Graphs : Theory and Applications in Computer Science. |
| title_full_unstemmed | Operator Calculus on Graphs : Theory and Applications in Computer Science. |
| title_short | Operator Calculus on Graphs : |
| title_sort | operator calculus on graphs theory and applications in computer science |
| title_sub | Theory and Applications in Computer Science. |
| topic | Calculus, Operational. http://id.loc.gov/authorities/subjects/sh85018805 Graph theory. http://id.loc.gov/authorities/subjects/sh85056471 Quantum statistics. http://id.loc.gov/authorities/subjects/sh85109468 Computer science. http://id.loc.gov/authorities/subjects/sh89003285 Calcul symbolique. Statistique quantique. Informatique. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh Calculus, Operational fast Computer science fast Graph theory fast Quantum statistics fast |
| topic_facet | Calculus, Operational. Graph theory. Quantum statistics. Computer science. Calcul symbolique. Statistique quantique. Informatique. MATHEMATICS Calculus. MATHEMATICS Mathematical Analysis. Calculus, Operational Computer science Graph theory Quantum statistics |
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| work_keys_str_mv | AT schottrene operatorcalculusongraphstheoryandapplicationsincomputerscience AT staplesgstacey operatorcalculusongraphstheoryandapplicationsincomputerscience |