LEADER 05429nam 2200673 a 450 001 9910451618603321 005 20200520144314.0 010 $a1-280-66906-3 010 $a9786613645999 010 $a1-84816-877-2 035 $a(CKB)2550000000101589 035 $a(EBL)919092 035 $a(OCoLC)794328387 035 $a(SSID)ssj0000657784 035 $a(PQKBManifestationID)12285316 035 $a(PQKBTitleCode)TC0000657784 035 $a(PQKBWorkID)10656911 035 $a(PQKB)11671444 035 $a(MiAaPQ)EBC919092 035 $a(WSP)00002630 035 $a(Au-PeEL)EBL919092 035 $a(CaPaEBR)ebr10563507 035 $a(CaONFJC)MIL364599 035 $a(EXLCZ)992550000000101589 100 $a20120611d2012 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aOperator calculus on graphs$b[electronic resource] $etheory and applications in computer science /$fRene? Schott, G. Stacey Staples 210 $aLondon $cImperial College Press$d2012 215 $a1 online resource (428 p.) 300 $aDescription based upon print version of record. 311 $a1-84816-876-4 320 $aIncludes bibliographical references and index. 327 $aPreface; 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 327 $a2.2.1 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 327 $a4.3.1 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 327 $a6.3.2.1 (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 327 $a8.2.3 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 327 $a11. Partition-Dependent Stochastic Measures 330 $aThis 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 606 $aCalculus 606 $aComputer science$xMathematics 608 $aElectronic books. 615 0$aCalculus. 615 0$aComputer science$xMathematics. 676 $a515 676 $a515.72 700 $aSchott$b Rene?$0352247 701 $aStaples$b G. Stacey$0998164 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910451618603321 996 $aOperator calculus on graphs$92289568 997 $aUNINA