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Mathematical Strategies

Mathematical Strategies. P.S.Subramanian CSRD group 21 Jan 2001, IIT/ Mumbai. Mathematical Strategies -. Strategy vs Tactics - in Chess Tactics is situation specific and concrete Strategy is generic and abstract Pros and Cons of Strategy and Tactics. Mathematical Strategies -.

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Mathematical Strategies

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  1. Mathematical Strategies P.S.Subramanian CSRD group 21 Jan 2001, IIT/ Mumbai

  2. Mathematical Strategies- • Strategy vs Tactics - in Chess • Tactics is situation specific and concrete • Strategy is generic and abstract • Pros and Cons of Strategy and Tactics

  3. Mathematical Strategies - • Why study the Strategies of Mathematics? • Helps us to `see the forest for the trees’. • Makes the learning of `new’ topics easier. • Makes the study of `History of Mathematics’ more meaningful.

  4. Some Common Strategies • Encapsulation for representation independence • Step-wise refinement • Coordinatisation (Cartesian, Positional and Mixed) • Reuse • Linearisation • Localisation • Crowding • Dualisation

  5. Encapsulation • Need to study properties independent of the `representation’. • In Computer Science the essence of OOP Representation = Implementation

  6. Encapsulation - Example • Injectivity of function • f : A —› B, where A, B are Sets • un-encapsulated definition is • a, binA, f(a) = f(b) => a = b • Can we give a definition without in?

  7. Encapsulation - example • Encapsulated Definition • let C be another set and • g , h : C —› A, be two maps • f is injective iff,f ° g= f ° h => g=h • Elements have vanished.

  8. Encapsulation • This line of thinking leads to `Category Theory’ • For a gentle introduction see `Conceptual Mathematics’ by William Lawvere - Prentice Hall. • Strongly Recommended for CS Students

  9. Step-wise Refinement • Given a collection of problems P which we know • how to solve, and a new problem Q • Find a sequence of subproblems with the • property that we have a method of transforming • the solution of problems occurring later in the • sequence to those of the earlier.

  10. Stepwise Refinement • In particular • if the tail of the sequence has problems only from the set P • then we can solve Q.

  11. Stepwise Refinement • Gaussian Elimination - What is P and Q? • Galois Theory - What is P and Q? • Let P be a set of Software specifications for which we have already written programs and Q is new specification for which we want to develop a program.

  12. Stepwise Refinement • Component based Software (and Hardware) • Engineering • is an important and evolving area. • Sample reference- • see http://www.kestrel.edu

  13. Co-ordinatisation • Cartesian • Positional • Mixed

  14. Cartesian • Synthetic Projective Geometry • Underlying `Mathematics’ is Wedderburn’s Representation Theorem of Semi-simple rings in terms of Matrix rings over division algebras.

  15. Cartesian • The idea of coordinatising • the Space of Functions • enables us to transport • many ideas from the usual coordinate geometry • to these spaces.

  16. Positional • Decimal Number System • Wavelets • Underlying Mathematics is that of Wreath Products • Krasner-Kaloujnine Theorem of Embedding a group in the wreath product of the factors of it’s composition series.

  17. Mixed • Krohn- Rhodes Theorem in Automata Theory and it’s generalisations • Underlying Mathematics is the theory of Semigroup Decompositions

  18. Reuse • If we have already solved a problem in some domain and if can establish a suitable connection between domains • then we can `reuse’ the solutions of problems of the former domain.

  19. Reuse • Example (NOT historically accurate!) • Galois Theory (again) • Original Domain - Groups • Problem- Stepwise Refinement • New Domain - Fields • Suitable Connection - Galois Connection

  20. Reuse • The Specware software from the Kestrel Institute • provides mechanisms for reuse of • ideas in the domain of Algorithm Design. • But, contrary to Galois theory which is fully automatic • one has to provide the connection manually.

  21. Linearisation • Newton-Raphson Temporarily pretend that the situation is linear Generalisation - Kantorovich to Fn Spaces • Structural Linearisation - Algebraic Topology Linear to Module to Abelian Categories

  22. Mathematical Strategies • Localisation - Sheaf Theory Representation Theorem of Rings Minkowski-Hasse on Quadratic Forms Many Computer Science uses of Sheaf Theory

  23. Mathematical Strategies • Crowding - Contraction Maps, Ramsey Theory Fixed point Theorems and their uses. • Duality- Fourier Transforms, Spectral Methods, Chu Spaces, Ramsey = Discontinuous Duality,

  24. Mathematical Strategies • Conclusion • One gets more insightinto Mathematics and it’s applications by reflecting on the strategies.

  25. Some Mathematical Topics relevant toSasken • Separating the strands in Signal Processing. • Generalising Shannon’s Information Theory • New Coding Techniques • Mathematics of Image processing • Mathematical aspects of Componentisation

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