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This book provides a broad understanding of the fundamental tools and methods from information theory and mathematical programming, as well as specific applications in 6G and beyond system designs. The contents focus on not only both theories but also their intersection in 6G. Motivations are from the multitude of new developments which will arise once 6G systems integrate new communication networks with AIoT (Artificial Intelligence plus Internet of Things). Design issues such as the intermittent connectivity, low latency, federated learning, IoT security, etc., are covered. This monograph provides a thorough picture of new results from information and optimization theories, as well as how their dialogues work to solve aforementioned 6G design issues