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In this book the details of many calculations are provided for access to work in quantum groups, algebraic differential calculus, noncommutative geometry, fuzzy physics, discrete geometry, gauge theory, quantum integrable systems, braiding, finite topological spaces, some aspects of geometry and quantum mechanics and gravity. TOC:Preface.- 1. Basics for quantum groups.- 2. Calculus and algebra.- 3. Differential Calculi.- 4. More on Q analysis.- 5. Remarks on quantum transmutation.- 6. Discretization and quantum mechanics.- 7. Discrete calculus.- 8. Fuzzy physics and matrix geometry.- 9. Further aspects of FQM.- 10. Integrable systems.- 11. Aspects of gauge theory.- 12. Pointless spaces and quantum gravity.- Bibliography.- Index.