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 packages for performing symbolic computations.

Description-Table Of Contents

1. Introduction. 1.1. Notational preliminaries -- 2. Combinatorial algebra. 2.1. Six group and semigroup algebras. 2.2. Clifford and Grassmann algebras. 2.3. The symmetric Clifford algebra Cl[symbol]. 2.4. The idempotent-generated algebra Cl[symbol]. 2.5. The n-particle zeon algebra Cl[symbol]. 2.6. Generalized zeon algebras -- 3. Norm inequalities on Clifford algebras. 3.1. Norms on Cl[symbol]. 3.2. Generating functions. 3.3. Clifford matrices and the Clifford-Frobenius norm. 3.4. Powers of Clifford matrices -- 4. Specialized adjacency matrices. 4.1. Essential graph theory. 4.2. Clifford adjacency matrices. 4.3. Nilpotent adjacency matrices -- 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. 6.2. From graphs to quantum random variables. 6.3. Connected components in graph processes -- 7. Geometric graph processes. 7.1. Preliminaries. 7.2. Dynamic graph processes. 7.3. Time-homogeneous walks on random geometric graphs -- 8. Time-homogeneous random walks. 8.1. Cl[symbol] and random walks on hypercubes. 8.2. Multiplicative walks on Cl[symbol]. 8.3. Induced additive walks on Cl[symbol] -- 9. Dynamic walks in Clifford algebras. 9.1. Preliminaries. 9.2. Expectation. 9.3. Limit theorems -- 10. Iterated stochastic integrals. 10.1. Preliminaries. 10.2. Stochastic integrals in Cl[symbol]. 10.3. Graph-theoretic iterated stochastic integrals -- 11. Partition-dependent stochastic measures. 11.1. Preliminaries. 11.2. Cycle covers, independent sets, and partitions. 11.3. Computations on lattices of partitions. 11.4. Free cumulants -- 12. Appell systems in Clifford algebras. 12.1. Essential background. 12.2. Operator calculus on Clifford algebras. 12.3. Generalized raising and lowering operators. 12.4. Clifford Appell systems. 12.5. Fermion algebras and the fermion field -- 13. Operator homology and cohomology. 13.1. Introduction. 13.2. Clifford homology and cohomology. 13.3. Homology and lowering operators. 13.4. Cohomology and raising operators. 13.5. Matrix representations of lowering and raising operators. 13.6. Graphs of raising and lowering operators. 13.7. Operators as quantum random variables -- 14. Multivector-level complexity. 14.1. Preliminaries. 14.2. Graph problems. 14.3. A matrix-free approach to representing graphs. 14.4. Other combinatorial applications -- 15. Blade-level complexity. 15.1. Blade operations. 15.2. Counting cycles. 15.3. Further remarks on complexity -- 16. Operator calculus approach to minimal path problems. 16.1. Path-identfying nilpotent adjacency matrices. 16.2. Operator calculus approach to multi-constrained paths. 16.3. Minimal path algorithms. 16.4. Application: precomputed routing in a store-and-forward satellite constellation -- 17. Symbolic computations with Mathematica. 17.1. CliffMath: Computations in Clifford Algebras of Arbitrary Signature. 17.2. CliffSymNil: A Companion Package. 17.3. CliffOC: Operator Calculus on Clifford Algebras. 17.4. """"Fast Zeon"""" implementation.