05429nam 2200673 a 450 991045161860332120200520144314.01-280-66906-397866136459991-84816-877-2(CKB)2550000000101589(EBL)919092(OCoLC)794328387(SSID)ssj0000657784(PQKBManifestationID)12285316(PQKBTitleCode)TC0000657784(PQKBWorkID)10656911(PQKB)11671444(MiAaPQ)EBC919092(WSP)00002630 (Au-PeEL)EBL919092(CaPaEBR)ebr10563507(CaONFJC)MIL364599(EXLCZ)99255000000010158920120611d2012 uy 0engur|n|---|||||txtccrOperator calculus on graphs[electronic resource] theory and applications in computer science /René Schott, G. Stacey StaplesLondon Imperial College Press20121 online resource (428 p.)Description based upon print version of record.1-84816-876-4 Includes bibliographical references and index.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 Algebras2.2.1 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 Matrices4.3.1 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 components6.3.2.1 (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 loops8.2.3 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 polynomials11. Partition-Dependent Stochastic MeasuresThis 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 packCalculusComputer scienceMathematicsElectronic books.Calculus.Computer scienceMathematics.515515.72Schott René352247Staples G. Stacey998164MiAaPQMiAaPQMiAaPQBOOK9910451618603321Operator calculus on graphs2289568UNINA