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Lecture# Dynamical Approaches to Spectral Theory of Operators

Description

This lecture covers ergodic theory, spectral theory of operators, and almost periodic factors. It discusses the concepts of ergodicity, reducibility, and almost reducibility in the context of dynamical systems and spectral theory. The lecture also explores the global theory of operators and establishes deviation estimates for spectral measures.

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Related concepts (221)

Ergodicity

In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process.

Ergodic theory

Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc.

Election

An election is a formal group decision-making process by which a population chooses an individual or multiple individuals to hold public office. Elections have been the usual mechanism by which modern representative democracy has operated since the 17th century. Elections may fill offices in the legislature, sometimes in the executive and judiciary, and for regional and local government. This process is also used in many other private and business organisations, from clubs to voluntary associations and corporations.

Spectral sequence

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Motivated by problems in algebraic topology, Jean Leray introduced the notion of a sheaf and found himself faced with the problem of computing sheaf cohomology.

Globalization

Globalization, or globalisation (Commonwealth English; see spelling differences), is the process of interaction and integration among people, companies, and governments worldwide. The term globalization first appeared in the early 20th century (supplanting an earlier French term mondialization), developed its current meaning some time in the second half of the 20th century, and came into popular use in the 1990s to describe the unprecedented international connectivity of the post-Cold War world.

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