2000 sasken all rights reserved mathematical strategies p.s.subramanian csrd group 21 jan 2001, iit/...

2000 SASKEN All Rights Reserved

Mathematical Strategies

P.S.Subramanian

CSRD group

21 Jan 2001, IIT/ Mumbai

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

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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