INDUSTRIAL MATHEMATICS :- KEY TO KEY TECHNOLOGIES AND ITs IMPACT ON RESEARCH &

EDUCATION IN MATHEMATICS

AMIYA K. PANI,FNASc,FASc Department of Masthematics Industrial Mathematics Group

IIT BOMBAY

For UG Training Programme under NPDE-TCA, see URL site

www.math.iitb.ac.in/~npde-tca

“THE ADVANCE AND PERFECTINS OF MATHEMATICS ARE CLOSELY JOINED TO THE PROSPERITY OF NATIONS”

NAPOLEAN BONAPARTE

SKETCH OF THE TALK USEFULNESS OF MATHEMATICS AS CONCEIVED BY SOCIETY HOPE & CHALLENGES AHEAD SOME QUESTIONS

• WHY HAS MATHEMATICS BECOME SO IMPORTANT FOR INDUSTRY IN RECENT YEARS ? • WHICH KIND OF MATHEMATICS IS NEEDED FOR THIS PURPOSE ?

PROBLEMS SOLVING ACTIVITY

• WHAT ARE THE CONSEQUENCES FOR EDUCATION IN MATHEMATICS ? • IN ITS NEW AVTAR :

WILL IT LOOSE ITS BEAUTY THAT IS ITS RIGOUR SOME CASE STUDIES :-

• MODELLING OF CHEMICAL PLANT • ON DIALYSIS • ON POPULATION GROWTH

MATHEMATICS : QUEEN OF SCIENCES

Ask this to a common man or even a manager of an Industrial Organization or some one who is struggling to understand math's.

About Usefulness of Science : WHILE MATHS IS REGARDED AS USEFUL TO PRACTISE THE BRAIN OF PUPILS, - PHILOSOPHY - CONSIDER TO BE AN ABSTRACT ENTITY:

Is It so glamorous ?

ONE CAN NOT EXPECT THAT MATHEMATICS IS VERY OFTEN NAMED IN A TOP LIST

HARDLY CONCEIVED AS USEFUL TO THE DEVELOPMENT OF A SOCIETY

~ ECONOMIC IMPACT

IN WE MADE A SURVEY ON INDIAN INDUSTRIES ABOUT ECONOMIC BENEFITS OF MATHS.

Managers ~ NIL ~ABSTRACT ENTITY

( Based on their knowledge of mathematics in schools or colleges ) Metamorphosis Globalization Increase of awareness NRB Bearing Company ( Managing Director) How ??

~ 1999

1991-92

“We want to double our profit in a year or two, it is Mathematical effort which can help us to achieve this”

It is also slowly conceived by others :

“THE DEMAND OF A MAXIMUM OUTPUT OF INDUSTRIAL RESEARCH & DEVELOPMENT TODAY CAN ONLY BE FULFILLED BY AN INCREASING USE OF MATHEMATICAL METHODS”

WHY ? IS IT A DREAM OR REALITY ? “EXAMPLES are SIMULATION METHODS, WHICH ALLOW TO REDUCE THE EXPERIMENTAL & CONSTRUCTIVE effort for the development of complex products” ~ PILOT PLANT SIMULATION : ~ Key word ~ substitutes more & more real experiments But which of course must be evaluated by experiments

“MATHEMATICS IS ALWAYS IS THE CORE OF COMPUTER SIMULATION”

IN INDUSTRY PAST & PRESENT, PEOPLE USE SOFTWARE PACKAGES TO SIMULATE A PROCESS OR THE BEHAVIOUR OF A PRODUCT. ~ forget : Limitation of their tool (disadvantage for mathematicians) ROLE OF MATHS IN SIMULATION General task is formulated by J.L. Lion (1994): “ Mathematics helps to make things better, faster, safer, cheaper by the SIMULATION OF COMPLEX PHENOMENA, the REDUCTION OF FLOOD OF DATA, VISUALISATION” SO ~ THERE IS HOPE (!) & CHALLENGES FOR US

BASED ON OUR EXPERIENCE, WE ANALYSE THE FOLLOWING QUESTIONS :- Q.1. WHY HAS MATHEMATICS BECOME SO IMPORTANT FOR INDUSTRY & FINANCIAL INSTITUTES ? Q.2. WHICH KIND OF MATHEMATICS IS NEEDED ? Q.3. WHAT ARE THE CONSEQUENCES FOR RESEARCH & EDUCATION IN MATHEMATICS ? IN ITS NEW AVTAR Q.4. WILL IT LOOSE ITS RIGOUR i.e., the strength & beauty of the traditional culture ? Explain through examples from IMG Some Success ?

Key words for the Role of Mathematics in Industry :- SIMULATION, DATA REDUCATION, CONTROL, OPTIMIZATION AND VISUALISATION To provide an answer to ONLY REASON : “MODERN COMPUTES ALLOW THE EVALUATION OF REALISTIC MATHEMATICAL MODELS” Therefore, the behaviour of complex 3-D systems can be predicted, designed & optimize in a virtual reality ~ near to reality

‘VITRTUAL PROTOTYPING’ ~ Fashionable in Industry, but Its core is

Mathematics.

MAIN COMPONENTS V-P ARE :

• MATHEMATICAL MODELLING

• SCIENTIFIC COMPUTING

Q.No. 1

MODELLING :

“ THE TRANFORMATION OF REAL OBJECTS INTO MATHEMATICS

BY NEGLECTING DETAILS, WHICH ARE UNIMPORTANT WITH

RESPECT TO THE QUESTIONS WE POSE”

Should be correct & logically admissible.

~ not unique (scope for improvement)

Raw materials : Mathematics; Better raw materials : ~ better model ~

good mathematics entails good modelling ~

Modelling

Compromise

_-----_-____---------_--

--------------/-/-/-/--/-

/-/-/-/-/-Muddy/-/-/-

/-/-/-/-/-Pond/-/-/-/-

/-/-/-/-/-/-/-/-/-//--

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,Desert,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,

HIERARCHICAL MODELS :

Simple ~ Complicated ones

Simplification of geometry or Industry wants

taking asymptotic limits

* Easier to evaluate * Optimal design

* Provides qualitative prediction * Optimal parameters of a 3-D

seldom with reliable data systems

To evaluate the more complex model with the aid of

SCIENTIFIC COMPUTING :

• To design algorithms to solve equations representing the model

• To Implement in a proper systems.

• To organize data to produce so that we can extract information which we want

RUSSIAN MATHEMATICIAN A.A. SAMARSKII called the whole process

of NUMERICAL EXPERIEMENTS

~ a new Scientific method

A MAP : Real world Virtual mathematical world

Reln :between : Asymptotical & Numerical Methods Supplement each

other

J. Kellar (ICIAM) 1995 said:

“while the 1st half of 20th century was dominated by Asymptotic

methods, the second by Numerical methods, we are now entering a

century dominated by combination of and interaction by both of these

methods”

Model + Algorithm + Programme = MAP

“WE MISSED THE BUS WHEN OUR APPLIED FRIENDS REFUSED

TO ACCEPT THE CHANGE”

IN COMPUTATIONAL MATHS.:

BANACH SPACE THEORY (FA) IS AS IMPORTANT AS COMPUTERS

~ S.L. Sobolev (1950) “DIFFERENTIAL EQUATION IS THE LANGUAGE OF SCIENCE & TECHNOLOGY” ~ V.I. ARNOLD Modeling & Simulation Mathematics is its core PROBLEM SOLVING ACTIVITY Identify Formulate Solve Interpret (i) (ii) (iii) (iv) Golden Rule : Modeling Our strength Engineers talk to Comp. Scientists

Q. No. 2

I F S I F s

I. Identification of the problem in Scientific setting : ~ Recognize that there is a problem & what the problem is ?

~ To be done with the help of the Engineers & Scientists who posed

the problem

II. Formulation of a Mathematical Model

~ Crucial & difficult : Stages

• Identify : What is important & what is not ?

• Select a suitable mathematical structure

• Identification of between concept of real situation & those in

mathematical system

(i) ~ (ii)

Dive from the World of Reality into the World of Mathematics.

(ii) ~ (iii)

Swim in this World of Mathematics

HOWEVER IN FRANCE :

“PRESTIGE OF MATHS IS MUCH HIGHER” ~

Recall a ‘QUOTATION’ from a letter to an assembly of mathematicians in

1992 (by Former President Mitterrand) :

~ MAY BE A REALITY FRANCE

BUT WHAT ABOUT IN INDIA

“YOUR DISCUSSION REFER TO A VERY ESSENTIAL DOMAIN OF

RESEARCH, WHICH COMMANDS THE SCIENTIFIC & TECHNOLOGICAL

DEVELOPMENT OF A COUNTRY & THEY REFER TO A CENTRAL

DISCIPLINE OF OUR EDUCATIONAL SYSTEM”

DECISION MAKERS need not be skilled in the process of the

underlying Problem solving activity

“The importance of Mathematics is not self evident”

~ David (1984 )

~ disadvantage for mathematicians & industry fails to recognize the

important of the activity needed for decision making.

(iii) Solution Process (Our Strength)

• Mathematical & Statistical Techniques

• Computational Tools

~ No Computer Science Programme can replace this

All efforts are buried in this

This is the activity that is traditionally called “Applied Mathematics”

~ Much of it is indistinguishable on the surface from pure math.

~ Keep in mind that the mathematical problem has a connection with the

real world.

~ Key component “Computational Methods” hinges on

• deep understating of math theory

•skill to develop efficient & reliable algorithms.

(iv) Explanation & Interpretation of results in the context of the original

problems

• Communication skill

• Evaluation of results

Final Stage (IV):

is a climb back from the world of maths. Into the world

of reality and more importantly with a prediction in your teeth.

FIRST MODELLING PROBLEM

10

-02 Easy

08

Gopal had ten rupees and he bought a pencial for rupees two. So how

many rupees he has now ?

Identification.

Formulation : ( dissociate from this )

Gopal had ten 10

He paid two - 02

Solution Process

10

- 02

Interpretation : He had left with rupees eight

Validation : 8 + 2 = 10

Wha

Just out, but some more so than others.

Just about, but some more so than others.

What kind of mathematics is useful?

`` Every kind, but at Kodak partial differential

equations are useful more often than topology’’

– Peter Castro

Industry hired 50% of the 2001 PhDs in statistics,

43% in numerical analysis, and

10% of those in geometry/topology.

We never know what kind of

mathematics is the right

kinds, so an “algebraist for

life” is not the right kind of

mathematician.

An industrial mathematician must be a

generalist, learning whatever kind of

mathematics the problem calls for. She

should be interested in all kinds of

mathematics, and also in things other

than mathematics.

Depth in one area is certainly a plus,

especially if the area seems relevant to

the industry, but breadth is more

important.

logical thinking

the ability to abstract and recognize

underlying structure

knowing the right questions, recognizing the

wrong ones

familiarity with a wide variety of problem-

solving tools

Problems never come in formulated as

mathematical problems. A mathematician’s

biggest contribution to a team is often an

ability to state the right question.

Solve its problems.

There are countless problems in industry that

require deep mathematics, but almost none

that can be solved by mathematics alone.

The strength of the mathematical sciences is

that they are pervasive in many applications.

The challenge is that they are only a part of

each application. – Shmuel Winograd

a mathematician in industry must be part of

a team. communication skills and social skills

matter (while, according to popular opinion,

these are positively harmful for an academic

mathematician).

skills in modeling and problem formulation

flexibility to go where the problems leads

breadth of interest, interdisciplinarity

balance between breadth and depth

knowing when to stop

computational skills

written and oral communication skills

social skills, teamwork

SINGLE SPECIES POPULATION MODELS

Problem: Given statistics of USA population:

Table 1

Year USA Population ( x 106)

1790 3.9

1800 5.3

1810 7.2

How do we predict the population size for the next 100 years?

Assumptions:

i. The initial size of population is known

ii. There is no emigration and immigration, that is,

the population is closed to migration.

Validity : While taking census, the floating population

is not taken into consideration.

Note: The change in population is only due to birth or death.

iii.Each individual in the population has the same chance of

dying or of reproducing.

Validity: We attribute to each individual the average

reproductive and survival traits for all in the population. i.e.,

we have eliminated both age and sex from the model.

FORMULATION

Variables & Parameters

t : time

N (t) : the population density at t

Birth

Death

Relation Between Parameters & Variables

( Established by experiments & observations) : Physical laws (s)

Birth or Death α Population Size

Birth = b N(t) Δt

Death = d N(t) Δt

b: no. of birth per animal per unit time

d: no. of death per animal per unit time

From Assumption (iii)

Modeling Traits

N(t) : given size at time t ,

To compute population size at t + Δt : N (t + Δt) population

size at t + Δt

Use Assumption (ii):

Population size at t + Δt = size at t + no. of newborn

babies during Δt – no. of that died (Conservation

Principle).Thus:

N (t + Δt) = N (t) + b N (t) Δt – dN(t)Δt

OR

N (t + Δt) = N (t) + (b – d) N (t) Δt ( 1 )

INCONSISTENCY

While animals occurs in Integer units, there is no

reason to expect that N (t + Δt) will be an integer even if N(t) is

NOT THAT SERIOUS:

Larger the size, the small in the percentage difference

between two successive integers.

For large population, round it to the nearest integer.

Observe:

(b – d) ≥ 0 N ≥ 0

(b – d) ≤ 0, it means N ≥ 0 or N ≤ 0

N ≤ 0 means the extinction

From ( 1 )

r = b – d : intrinsic rate of population growth (easier to

compute from data than b & d)

From Assumption (i) at t = 0

N ( 0 ) = N0 is known.

Now the problem boils down to find solution to (2) for t ≥ 0.

Discrete Model

Given a different initial data N0, we need to calculate again to

reach upto a prescribed time. This process is tidious and

expensive as we start from the beginning.

UNIFICATION

Convert (2) to a DIFFERENTIAL EQUATION

So a transition

DISCRETE CONTINUOUS

Q. Why should we go for this, that is, from discrete to continuous ?

• DEs are well developed

• It brings under one umbrella simmingly different phenomena

( Growth or Decay Models:

• Radio active decay to find art forgeries (carbon dating)

• Spread of technological innovation

• Growth of Economy

• Pollution in Lake

• Cumulative effect of Compound Interest

• ……………….

•……………….

• Never ending list)

Assume : N is continuously differentiable.

Continuity : Roughly speaking, a small change in time gives

rise to small change in population

Similarly, Differentiability for large population size, these are

reasonable assumptions.

Now

N (0) = N0

( 3 )

Note: The time rate of change of population

is directly proportional to the size of population.

This is called Malthusian model.

It is named after Thomas Robert Malthus, a British

Economist.

Thomas Robert Malthus was a British economist and demographer, whose famous Theory of Population highlighted the potential dangers of overpopulation.

Malthus has become widely known for his theories about population and its increase or decrease in response to various factors.

The sixth editions of his An Essay on the Principle of Population, published from 1798 to 1826, observed that sooner or later population gets checked by famine and disease.

An Essay on the Principles of Population, Malthus stated that while 'the populations of the world would increase in geometric proportions the food resources available for them would increase only in arithmetic proportions'.

SOLUTION PROCESS

Separation of variables: N ≠ 0

dt rN

dN

Integrate

log | N (t) | = rt + C1

| N ( t ) | = ec1 ert = c ert C > 0 ; C ≠ 0

N ( t ) = C ert C ≠ 0

N ( t ) = 0 C = 0

Hence we obtain a family of solutions

N ( t ) = C ert , C Є R ------- ( 4 )

One parameter family of solutions ( General solution) use

Initial condition to fix a unique solution:

At t = 0

N (0) = N0

On substituting in (4)

N0 = C

Hence N ( t ) = N0 ert -------- (5)

Born in Leipzig. Self taught Mathematician

and Got his ph.d. at the age of 20.

was given symbol of derivative and integral.

Fundamental results in calculus and

published in 1684 before the publication of

Principia Mathematica in 1687.

Discovered method of separation of variables

in 1691 and procedure of solving first order

linear ODE etc….

Observations

r > 0 : exponential growth

r = 0 ; no change

N ( t ) = N0 ( b = d )

r < 0 : decay

Now from N ( t ) = N0 e

rt

Interpretation

Using or table, let us predict population size

t = 0 corresponds to 1790

N0 = 3.9 * 106

t = 1 corresponds to 10 years as working time interval is 10

years.

N ( 1 ) ---- 1800

N (1) = N0 er

= 3.9 * 106 er ( to find r)

5.3 * 106 = 3.9 * 106 er

r = log (5.3 / 3.9) ≈ 0.307

N ( 2 ) = 7.3 * 106 ≈ 7.2 * 106 (actual)

Continue to predict : see the table 2

Year Population (*106) Actual Predicted Population (*106)

1810 7.2 7.3

1820 9.6 10.0

1830 12.9 13.7 10% error

1840 17.1 18.7

1850 23.2 25.6

1860 31.4 35.0

1870 38.6 47.8 30% error

1880 50.2 65.5

1890 62.9 89.6

1900 76.0 122.5 model is of little use

1910 92.0 167.6

1920 106.5 229.3

1930 123.2 313.7

Table 2…

Observations

• Reasonable agreement upto 1850 ( 10% error )

• In 1870 30%

• > 1870 : the model is of little use

Modification

Look back to analyze

• Malthusian model predicts exponential growth as r > 0

• Factors like lack of food, overcrowding, insufficient energy and

other environmental factors are detrimental to growth.

CAN not expect unlimited growth

• Malthusian model is OK for short time

1837: Dutch Mathematical biologist Verhulst proposed a

modification taking into consideration of crowding factor.

Pierre François Verhulst was a Belgian mathematician and got his ph.d. in number theory from the University of Ghent in 1825.

introduced the Verhulst equation (also known as the logistic equation) to model human population growth in 1838. His model could not be verified until 1930 (demonstrated and verified with reasonably accuracy by R.Pearl) due to insufficient data.

As an undergraduate at the University of Ghent, he was awarded two academic prizes for his works on the Calculus of variations. Later, he published papers on number theory and physics.

His main work is Traité des fonctions elliptiques (1841), for which he was admitted with an unanimous vote to the Académie de Bruxelles. There he was elected president in 1848, shortly before his fragile health, which had troubled him for many years, tore him from life at the young age of 45.

Improved Model

population to limit upper N

N

N1

Rate of change of population is proportional to

(i) The current population size N (t)

(ii)The fraction of population resource still not used, i.e.

Model:

)N

NrN(1

dt

dN

N ( 0 ) = N0

---------- ( 6 )

Observation:

~ slows down the rapid growth

SOLUTION PROCESS

Separation of variables: N ≠ 0

dtr)N/NN(1

dN

dt r)N/NN(1

dN

Integrate

Write

SOLUTION PROCESS

Separation of variables: N ≠ 0

dtr)N/NN(1

dN

dt r)N/NN(1

dN

Integrate

Write

Then

1C rt dN N/NNN

1

111

1CrtN/N1

N

ln

))/(1(ln

0

01

NN

NC

On solving

At t = 0, N ( 0 ) = N0

On pluging in ( 6 )

--------- ( 6 )

Hence

1N

N1

NN(t)

0

e-rt ---------- ( 8 )

On simplification, we arrive at:

Observations

growth unlimited predict not does

one) (desired N ) N(t0 r t As

,

diminished is growth thelarger, becomes N as and

first, growth type lexponentia,NN If 0

0dt

dNNN

As long as

NN00

N ( t ) is monotonically increasing as dN/dt > 0

d2N/ dt2 = r2

N

N1 N

N

2N1

0

N

N1

N

2N1

increases dt

dN

2

NN0 For ;

dN / dt is increasing if d2N/dt2 > 0

i.e. if

decreases dt / dN , /2NN For

Graph of N: Sigmoid curve or logistic curve

i.e

growth daccelerate of Period:/2NN0

Table 3 – Prediction Using Verhulst Model

Year USA Population (106) Predicted (* 106)

1820 9.6 9.7

1830 12.9 13.0

1840 17.1 17.4

1850 23.2 23.0

1860 31.4 30.2

1870 38.6 38.1

1880 50.2 49.9

1890 62.9 62.4

1900 76.0 76.5

1910 92.0 91.6

1920 106.5 107.0

1930 123.2 122.0 (not so good)

r = 0.3134 Nα = 197 * 106

t = 1 (10 yr)

N0 = 3.9 * 106

Validation

N0 = 3.9 * 106

r = 0.3134

Nα = 197 * 106

See table 3 for prediction

Observe: Excellent correlation between actual and predicted

over 100 yrs.

After 1930, prediction goes astray

Reason: Nα : 197 * 106

Where the VSA population is beyond 200 * 106

Remarks

- Technological developments – Pollution and

sociological trends have impact on growth and hence,

change r and Nα periodically.

-For more accurate: age and sex must be taken into

consideration

- Suspectible to epidemics

-More useful for bacterial growth and fish population etc.

THANK YOU