SET THEORY AND ITS APPLICATIONS IN PHYSICS AND COMPUTING

Why learn set theory? This book provides the answer — it is interesting, and also useful! Taking a new approach and looking from a fresh perspective, the discussion flows in a friendly and transparent way, supplemented with a lot of examples and figures. This makes the theory easily comprehensible: the proofs get vivid and visual, enveloped with interesting applications for students in (applied) math, physics, and engineering.

Given the theory and the applications, the book could serve as a textbook in four (undergraduate) math courses: Introduction to set theory and its application; Chaos theory and stability — a geometrical point of view; Functional analysis — Han-Banach theory; and Cryptography with quantum computing. It teaches set theory from the basics, including the axiom of choice, the well ordering theorem, and Zorn's lemma. Furthermore, it uses Cantor's set to introduce chaos theory from a geometrical point of view. Moreover, it introduces the binomial formula (and other related formulas), and uses them in quantum statistical mechanics. And finally, it uses Zorn's lemma in functional analysis, general relativity, and quantum mechanics. There are also practical applications in cryptography, error correction, quantum computing and programming.

Contents:

• Introduction to Set Theory:
  • Sets and Their Cardinality
  • Ordinals and Zorn's Lemma
• Applications in Functional Analysis:
  • Zorn's Lemma in Han-Banach Theory
• Cantor Set and Stability:
  • The Pigeonhole Principle and Stability and Its Applications in Calculus and Classical Mechanics
  • Cantor Set and Its Applications
  • Is the Universe Infinite?
  • Binary Trees and Chaos Theory
  • Entropy and Information
• The Binomial Formula and Quantum Statistical Mechanics:
  • Newton's Binomial and Trinomial Formulas
  • Applications in Quantum Statistical Mechanics
• Towards General Relativity and Quantum Mechanics:
  • Spacetime and Local Coordinates
  • Zorn's Lemma in Quantum Mechanics
• Applications in Cryptography and Error Correction:
  • Coding–Decoding: the RSA Key Exchange
  • Fast Fourier Transform: A Virtual Binary Tree
  • Error Correction: The Reed-Solomon Code
  • Application in Computational Biology
• Towards Quantum Computing:
  • Quantum FFT
  • Shor's Factoring Algorithm
  • Towards Feynman Diagrams
• Appendix: Applications in C++:
  • Dynamic Trees and Tensors in C++
  • Nonlinear Maxwell Solver in C++
• References
• Index

Readership: Advanced undergraduate and graduate students of set theory and its applications in chaos, functional analysis, cryptography, and related courses.

Key Features:

• Takes a new approach, looking at things from a fresh angle: not only theoretical, but also practical
• The discussion flows in a friendly and transparent way, supplemented with a lot of examples and figures
• Serve as a textbook in a few (undergraduate) math courses, thanks to the theory and the applications
• Teaches set theory from scratch, including the axiom of choice, the well ordering theorem, and Zorn's lemma
• Also included are practical applications in cryptography, error correction, quantum computing and programming
• Self-contained and requires no prerequisite at all