Mathematical Strategies
P.S.Subramanian
CSRD group
21 Jan 2001, IIT/ Mumbai
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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
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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.
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Some Common Strategies
Encapsulation for representation
independence
Step-wise refinement
Coordinatisation (Cartesian, Positional and
Mixed)
Reuse
Linearisation
Localisation
Crowding
Dualisation
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Encapsulation
Need to study properties independent
of the `representation’.
In Computer Science the essence of
OOP
Representation = Implementation
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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?
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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.
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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
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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.
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Stepwise Refinement
In particular
if the tail of the sequence has problems only
from the set P
then we can solve Q.
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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.
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Stepwise Refinement
 Component based Software (and Hardware)
 Engineering
 is an important and evolving area.
 Sample reference see http://www.kestrel.edu
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Co-ordinatisation
Cartesian
Positional
Mixed
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Cartesian
 Synthetic Projective Geometry
 Underlying `Mathematics’ is
Wedderburn’s Representation Theorem of
Semi-simple rings in terms of Matrix rings over
division algebras.
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Cartesian
 The idea of coordinatising
 the Space of Functions
 enables us to transport
 many ideas from the usual coordinate geometry
 to these spaces.
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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.
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Mixed
 Krohn- Rhodes Theorem in Automata Theory
and it’s generalisations
 Underlying Mathematics is the theory of
Semigroup Decompositions
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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.
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Reuse
 Example (NOT historically accurate!)
 Galois Theory (again)
 Original Domain - Groups
 Problem- Stepwise Refinement
 New Domain - Fields
 Suitable Connection - Galois Connection
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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.
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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
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Mathematical Strategies
 Localisation - Sheaf Theory
Representation Theorem of Rings
Minkowski-Hasse on Quadratic Forms
Many Computer Science uses of Sheaf Theory
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Mathematical Strategies
 Crowding - Contraction Maps, Ramsey Theory
Fixed point Theorems and their uses.
 Duality- Fourier Transforms, Spectral Methods, Chu
Spaces, Ramsey = Discontinuous Duality,
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Mathematical Strategies
Conclusion
 One gets more insight into Mathematics and it’s
applications by reflecting on the strategies.
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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
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