表观复杂性的谱简单性，第一部分：预测的非对角化元代数学
Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
摘要：
几乎所有的问题，人们可以问一个随机过程的行为和结构的复杂性减少到一个线性代数框架的时间演化由一个适当的隐马尔可夫过程发生器。每一种类型的问题-相关性、可预测性、预测成本、观察者同步等-都会产生一个不同的生成器类。然后，答案是类适当的转换动态的函数。不幸的是，这些动力学一般是非正规的，nondiagonalizable，奇异的，等等，Tractably分析这些动力学依赖于适应最近推出的亚纯泛函微积分，它指定的谱分解的nondiagonalizable线性算子的功能，即使当功能极点和零点符合运营商的频谱。一路上，我们建立特殊的属性的投影算子，展示他们如何捕捉组织的子过程中的一个复杂系统。这就避免了替代演算的虚假无穷大，在第二部分的续集中，这将导致复杂性测度的第一个封闭形式的表达式，表达为Drazin逆（奇异算子的负一次方）或适当的过渡动力学的特征值和投影算子。
英文摘要：
Virtually all questions that one can ask about the behavioral and structural complexity of a stochastic process reduce to a linear algebraic framing of a time evolution governed by an appropriate hidden-Markov process generator. Each type of question---correlation, predictability, predictive cost, observer synchronization, and the like---induces a distinct generator class. Answers are then functions of the class-appropriate transition dynamic. Unfortunately, these dynamics are generically nonnormal, nondiagonalizable, singular, and so on. Tractably analyzing these dynamics relies on adapting the recently introduced meromorphic functional calculus, which specifies the spectral decomposition of functions of nondiagonalizable linear operators, even when the function poles and zeros coincide with the operator's spectrum. Along the way, we establish special properties of the projection operators that demonstrate how they capture the organization of subprocesses within a complex system. Circumventing the spurious infinities of alternative calculi, this leads in the sequel, Part II, to the first closed-form expressions for complexity measures, couched either in terms of the Drazin inverse (negative-one power of a singular operator) or the eigenvalues and projection operators of the appropriate transition dynamic.
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