Exact Results Regarding the Physics of Complex Systems via Linear Algebra, Hidden Markov Models, and Information Theory

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Complex Systems, Impoverished Observables, and HMMs 1 1.1 Probability, Stochastic Dynamics, and HMMs . . . . . . . . . . . . . . . 3 1.1.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Shannon Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.4 Markovity and Markov order . . . . . . . . . . . . . . . . . . . . . 8 1.1.5 HMMs in general . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.6 Welcome to the jungle: Zoology of HMMs . . . . . . . . . . . . . 10 1.1.7 Probabilities, conditional probabilities, and linear algebra . . . . . . 11 1.1.8 Beyond normal: Probabilities invoke non-normal linear operators 13 2 Beyond the Spectral Theorem 14 2.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Spectral Primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Functional Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.2 Holomorphic functional calculus . . . . . . . . . . . . . . . . . . . 24 2.4.3 Meromorphic functional calculus . . . . . . . . . . . . . . . . . . . 25 2.5 Meromorphic Spectral Decomposition . . . . . . . . . . . . . . . . . . . . 26 2.5.1 Partial fraction expansion of the resolvent . . . . . . . . . . . . . 26 2.5.2 Decomposing the identity . . . . . . . . . . . . . . . . . . . . . . 28 2.5.3 Dunford decomposition, decomposed . . . . . . . . . . . . . . . . 28

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