Mathematical Strategies
description
Transcript of Mathematical Strategies
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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 refinementCoordinatisation (Cartesian, Positional and
Mixed)ReuseLinearisationLocalisationCrowdingDualisation
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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 toSasken
Separating the strands in Signal Processing.
Generalising Shannon’s Information Theory
New Coding Techniques
Mathematics of Image processing
Mathematical aspects of Componentisation