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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
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