Mathematics and Its Applications

1. Automata Networks.- 1.1. Introduction.- 1.2. Definitions Regarding Automata Networks.- 1.3. Cellular Automata.- 1.4. Complexity Results for Automata Networks.- 1.5. Neural Networks.- 1.6. Examples of Automata Networks.- 1.6.1. XOR Networks.- 1.6.2. Next Majority Rule.- 1.6.3. Multithreshold Automaton.- 1.6.4. The Ising Automaton.- 1.6.5. Bounded Neural Network (BNN).- 1.6.6. Bounded Majority Network.- 2. Algebraic Invariants on Neural Networks.- 2.1. Introduction.- 2.2. K-Chains in 0-1 Periodic Sequences.- 2.3. Covariance in Time.- 2.4. Algebraic Invariants of Synchronous Iteration on Neural Networks.- 2.5. Algebraic Invariants of Sequential Iteration on Neural Networks.- 2.6. Block Sequential Iteration on Neural Networks.- 2.7. Iteration with Memory.- 2.8. Synchronous Iteration on Majority Networks.- 3. Lyapunov Functionals Associated to Neural Networks.- 3.1. Introduction.- 3.2. Synchronous Iteration.- 3.3. Sequential Iteration.- 3.4. Tie Rules for Neural Networks.- 3.5. Antisymmetrical Neural Networks.- 3.6. A Class of Symmetric Networks with Exponential Transient Length for Synchronous Iteration.- 3.7. Exponential Transient Classes for Sequential Iteration.- 4. Uniform One and Two Dimensional Neural Networks.- 4.1. Introduction.- 4.2. One-Dimensional Majority Automata.- 4.3. Two-Dimensional Majority Cellular Automata.- 4.3.1. 3-Threshold Case.- 4.3.2. 2-Threshold Case.- 4.4. Non-Symmetric One-Dimensional Bounded Neural Networks.- 4.5. Two-Dimensional Bounded Neural Networks.- 5. Continuous and Cyclically Monotone Networks.- 5.1. Introduction.- 5.2. Positive Networks.- 5.3. Multithreshold Networks.- 5.4. Approximation of Continuous Networks by Multithreshold Networks.- 5.5. Cyclically Monotone Networks.- 5.6. Positive Definite Interactions. The Maximization Problem.- 5.7. Sequential Iteration for Decreasing Real Functions and Optimization Problems.- 5.8. A Generalized Dynamics.- 5.9. Chain-Symmetric Matrices.- 6. Applications on Thermodynamic Limits on the Bethe Lattice.- 6.1. Introduction.- 6.2. The Bethe Lattice.- 6.3. The Hamiltonian.- 6.4. Thermodynamic Limits of Gibbs Ensembles.- 6.5. Evolution Equations.- 6.6. The One-Site Distribution of the Thermodynamic Limits.- 6.7. Distribution of the Thermodynamic Limits.- 6.8.Period ? 2 Limit Orbits of Some Non Linear Dynamics on $$ mathbb{R}_{ + }^{s} $$.- 7. Potts Automata.- 7.1. The Potts Model.- 7.2. Generalized Potts Hamiltonians and Compatible Rules.- 7.2.1. Majority Networks.- 7.2.2. Next Majority Rule.- 7.2.3. Median Rule.- 7.2.4. Threshold Functions.- 7.3. The Complexity of Synchronous Iteration on Compatible Rules.- 7.3.1. Logic Calculator.- 7.3.2. Potts Universal Automaton.- 7.4. Solvable Classes for the Synchronous Update.- 7.4.1. Maximal Rules.- 7.4.1.1. Majority Networks.- 7.4.1.2. Local Coloring Rules.- 7.4.2. Smoothing Rules.- 7.4.3. The Phase Unwrapping Algorithm.- References.- Author and Subject Index.