1.  2000 SASKEN All Rights Reserved Mathematical Strategies P.S.Subramanian CSRD group 21 Jan 2001, IIT/ Mumbai

2.  2000 SASKEN All Rights Reserved 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.  2000 SASKEN All Rights Reserved 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.  2000 SASKEN All Rights Reserved Some Common Strategies  Encapsulation for representation independence  Step-wise refinement  Coordinatisation (Cartesian, Positional and Mixed)  Reuse  Linearisation  Localisation  Crowding  Dualisation

5.  2000 SASKEN All Rights Reserved Encapsulation  Need to study properties independent of the `representation’.  In Computer Science the essence of OOP Representation = Implementation

6.  2000 SASKEN All Rights Reserved Encapsulation - Example  Injectivity of function  f : A —› B, where A, B are Sets  un-encapsulated definition is  a, b in A, f(a) = f(b) => a = b  Can we give a definition without in?

7.  2000 SASKEN All Rights Reserved 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.  2000 SASKEN All Rights Reserved 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.  2000 SASKEN All Rights Reserved 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.  2000 SASKEN All Rights Reserved Stepwise Refinement  In particular  if the tail of the sequence has problems only from the set P  then we can solve Q.

11.  2000 SASKEN All Rights Reserved 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.  2000 SASKEN All Rights Reserved Stepwise Refinement  Component based Software (and Hardware)  Engineering  is an important and evolving area.  Sample reference-  see http://www.kestrel.edu

13.  2000 SASKEN All Rights Reserved Co-ordinatisation  Cartesian  Positional  Mixed

14.  2000 SASKEN All Rights Reserved Cartesian  Synthetic Projective Geometry  Underlying `Mathematics’ is Wedderburn’s Representation Theorem of Semi-simple rings in terms of Matrix rings over division algebras.

15.  2000 SASKEN All Rights Reserved Cartesian  The idea of coordinatising  the Space of Functions  enables us to transport  many ideas from the usual coordinate geometry  to these spaces.

16.  2000 SASKEN All Rights Reserved 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.  2000 SASKEN All Rights Reserved Mixed  Krohn- Rhodes Theorem in Automata Theory and it’s generalisations  Underlying Mathematics is the theory of Semigroup Decompositions

18.  2000 SASKEN All Rights Reserved 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.  2000 SASKEN All Rights Reserved Reuse  Example (NOT historically accurate!)  Galois Theory (again)  Original Domain - Groups  Problem- Stepwise Refinement  New Domain - Fields  Suitable Connection - Galois Connection

20.  2000 SASKEN All Rights Reserved 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.  2000 SASKEN All Rights Reserved 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.  2000 SASKEN All Rights Reserved Mathematical Strategies  Localisation - Sheaf Theory Representation Theorem of Rings Minkowski-Hasse on Quadratic Forms Many Computer Science uses of Sheaf Theory

23.  2000 SASKEN All Rights Reserved Mathematical Strategies  Crowding - Contraction Maps, Ramsey Theory Fixed point Theorems and their uses.  Duality- Fourier Transforms, Spectral Methods, Chu Spaces, Ramsey = Discontinuous Duality,

24.  2000 SASKEN All Rights Reserved Mathematical Strategies  Conclusion  One gets more insight into Mathematics and it’s applications by reflecting on the strategies.

25.  2000 SASKEN All Rights Reserved Some Mathematical Topics relevant to Sasken  Separating the strands in Signal Processing.  Generalising Shannon’s Information Theory  New Coding Techniques  Mathematics of Image processing  Mathematical aspects of Componentisation