Contents -- Preface -- Acknowledgments -- 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) algebras -- 2.2.2 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 circuits -- 4.3.2 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)-components -- 6.3.3 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 walks -- 8.3 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 -- 11.1 Preliminaries -- 11.2 Cycle Covers, Independent Sets, and Partitions -- 11.3 Computations on Lattices of Partitions -- 11.3.1 Computations on lattice segments -- 11.3.2 Computations on restricted lattice segments -- 11.4 Free Cumulants -- Operator Calculus -- 12. Appell Systems in Clifford Algebras -- 12.1 Essential Background -- 12.1.1 Appell systems -- 12.1.2 Clifford algebras -- 12.2 Operator Calculus on Clifford Algebras -- 12.3 Generalized Raising and Lowering Operators -- 12.4 Clifford Appell Systems -- 12.4.1 Heterogeneous Clifford Appell systems.

12.4.2 Role of blade factorization in the construction of 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 -- Symbolic Computations -- 14. Multivector-Level Complexity -- 14.1 Preliminaries -- 14.2 Graph Problems -- 14.2.1 Cycles and paths -- 14.2.2 Edge-disjoint cycle decompositions of graphs -- 14.3 A Matrix-Free Approach to Representing Graphs -- 14.4 Other Combinatorial Applications -- 14.4.1 Computing the permanent -- 14.4.2 The set packing and set covering problems -- 15. Blade-Level Complexity -- 15.1 Blade Operations -- 15.2 Counting Cycles -- 15.2.1 Cycles of fixed length -- 15.2.2 Remarks on space complexity -- 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 -- 16.2.1 Feasible and optimal paths in m-weighted graphs -- 16.2.2 The dynamic multi-constrained path problem -- 16.3 Minimal Path Algorithms -- 16.4 Application: Precomputed Routing in a Store-and- Forward Satellite Constellation -- 16.4.1 Operator calculus implementation -- 16.4.2 The results -- 17. Symbolic Computations with Mathematica -- 17.1 CliffMath' : Computations in Clifford Algebras of Arbitrary Signature -- 17.1.1 CliffMath' procedures -- 17.1.2 Examples -- 17.2 CliffSymNil': A Companion Package -- 17.2.1 CliffSymNil' procedures -- 17.2.2 Examples -- 17.3 CliffOC': Operator Calculus on Clifford Algebras -- 17.3.1 CliffOC' procedures -- 17.3.2 Examples.

17.4 "Fast Zeon" Implementation -- Bibliography -- Index.