Math Magicians
Numbers that Matter
Inspired by Nature and Mastered by Mathematics
Games People Play
Hybrid Automotive Applications
Network Operating System (NOS)
Modeling of Human Arm Impedance
Magical Mathematics from Vedas
Life Without Mathematics
VOL. 5, ISSUE 2 APR - JUNE 2012
a quarterly journal of KPIT Cummins Infosystems Limited
Math Matters
Colophon
TechTalk@KPITCummins is a quarterly journal of
Science and Technology published by
KPIT Cummins Infosystems Limited, Pune, India.
Guest Editorial
Prof. Manindra Agrawal
Department of Computer Science and Engineering
Dean of Resource, Planning and Generation (DRPG)
Indian Institute of Technology,
Kanpur, India
Chief Editor
Dr. Vinay G. Vaidya
CTO-Engineering, VP
KPIT Cummins Infosystems Limited,
Pune, India
[email protected]
Editorial and Review Committee
Sanjyot Gindi
Aditi Athavale
Priti Ranadive
Chaitanya Rajguru
Nikhil Jotwani
Krishnan Kutty
Sudhakar Sah
Arun Nair
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The individual authors are solely responsible
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For Internal Circulation Only
TechTalk@KPIT Cummins
Contents
Editorial
Guest Editorial - Dr. Manindra Agrawal
2
Editorial - Dr. Vinay Vaidya
3
Profile of a Scientist
Srinivasa Ramanujan
Priti Ranadive
21
Book Review
Fermat’s Last Theorem
Sanjyot Gindi
39
Articles
4
Math Magicians
Nikhil Jotwani and Reenakumari Behera
Numbers that Matter
Aditi Athavale
10
Inspired by Nature and Mastered by Mathematics: Automobile Evolution
Ravi Rajan
16
Games People Play
Chaitanya S. Rajguru
22
Mathematics in Hybrid Automotive Applications
Tarun Kancharla
30
Mathematics in Network Operating System (NOS)
Arun S. Nair
34
Mathematical Modeling of Human Arm Impedance
Daniel Ruschen
40
Magical Mathematics from Vedas
Anuradha Dhumal
44
Life Without Math
Krishnan Kutty
46
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
1
Guest Editorial
Mathematics is the purest and most abstract of sciences. Mathematical formulas and
derivations may appear, to many, mere symbol juggling far from anything real. And they would
be right too, as mathematics concerns itself with deriving conclusions from certain
assumptions, irrespective of whether the assumptions are correct in real life or not.
Dr. Manindra Agarwal
Department of Computer
Science and Engineering
Dean of Resource, Planning
and Generation (DRPG)
Indian Institute of Technology,
Kanpur, India
However, surprisingly, the real world phenomenon can often be modeled mathematically,
investing tremendous power in mathematical calculations to analyze and predict these
phenomena. One of the most prominent examples of this is the Navier-Stokes equation
which models the flow of fluids, and is used to predict weather, design cars and aircrafts,
understanding tsunamis etc. One can safely say that without developing the mathematics of
this equation, none of the above would have been possible.
The impact of mathematics in our lives does not end here though. There are instances when
mathematical models, that do not seem to represent any phenomenon in real world, are
designed and studied due just to curiosity. However, several years (or decades) later, new
understanding of these phenomenon develops and it is realized that that the abstract models
developed earlier are the right one to capture them! Perhaps the most famous example of this
is the Riemannian Geometry developed by mathematician Bernhard Riemann in 19th century
which was found to capture our space-time reality after Einstein's general theory of relativity.
Yet another important way mathematics influences our life is by giving us tools to create new
technology. An example is linear codes -- these are linear functions that map any two input
“words” to two output “words” that the far apart from each other. These have enabled the
use of modern communication technology like phones and internet, as well as storage of
information in media like CDs and DVDs. Another example is modular arithmetic -arithmetic modulo a fixed number. The security of online transactions, be it stock market
trades, or credit card payments, relies on a method based on modular arithmetic.
One can enumerate several more examples where mathematics has made a profound impact
in our lives. This issue is devoted to highlighting some of these. Hopefully, these will provide a
good overview to the reader of the areas influenced and developed by mathematics.
Dr. Manindra Agrawal is a professor at the department of computer science and engineering and
the Dean of Resource, Planning and Generation (DRPG) at the Indian Institute of Technology,
Kanpur. He is a recipient of Clay Research Award (2002), Shanti Swarup Bhatnagar Award (2003),
ICTP Prize (2003), Fulkerson Prize (2006), Gödel Prize (2006) ,Infosys Prize for Mathematics
(2008), G D Birla Award (2009), H K Firodia Award (2011)
2
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Editorial
Dr. Vinay G. Vaidya
CTO - Engineering, VP
KPIT Cummins Infosystems
Limited, Pune, India
Commonly used yardstick of someone's research is the number of citations of the papers
published. Refinement of that is the H index. There is something beyond both these indices and
that is what I would like to propose as the 'Inspiration Index'. Inspiration index, if possible to
calculate, would measure the number of people who were influenced by the individual. This
inspiration can come from either direct communication, through seminars, through research
papers, or through books. Another word for this is 'childhood heroes'. The same term is
modified into 'role models' for adults. At any given time, one can always tell names of over a
dozen people that have influenced them, inspired them, and have made significant difference in
the way their lives have been shaped. During the bi-centennial of the United States in 1976,
Smithsonian had come up with a list of 100 people who had influenced the US most. One can
take a step forward and ask a large sample of population about a dozen people who have shaped
their lives. This data could be a beginning to form the 'Inspiration Index'.
For me mathematicians have always heavily influenced me. As a child, they were the magicians,
the gifted ones, and someone I would envy. Ramanujan's stories were a great influence.Gauss and
his probability distribution was something I did not trust even after passing engineering. Until
one day in a science center, I saw a demonstration of random balls actually forming a normal
distribution. Maxwell's ability to link and modify equations to create a universal
electromagnetism theory was unfathomable. They will always be a part of list of a dozen who
have influenced many many individuals.
Mathematics is precise and so are the reactions of many about mathematics. For some it is a
tough subject while for others it is a subject with logical steps that lead to compact meaningful
results. For some it is filled with frustration while for others it is full of fun. For some it would be a
great relief if mathematics were to be taken out of the curriculum while for others it is hard to
imagine life without mathematics.
Although, Mathematics is very important it is taken lightly since we often do not realize the
power of mathematics and how much time one can save in understanding many disciplines.
Writing any problem statement in mathematical form often gives different insight into the
problem as well as the possible solution set.
We have been fortunate to have a renowned mathematician, Prof. Agrawal, as our guest editor.
Prof. Agrawal is one of the most gifted mathematicians of our times. Prof. Agrawal, Kayal and
Saxena have solved one of the most interesting problems in mathematics and have come up with
a primality test for numbers. He has truly inspired many and he will continue to do so for many
more years.
Our goal in bringing this issue of TechTalk on 'Math Matters' is to share the joy of mathematics,
bring out the omnipresence of the subject, and above all inspire our readers. Hope you enjoy it.
Please send your feedback to :
[email protected]
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
3
Euclid
Narendra Karmarkar
Manindra Agrawal
Fibonacci
Ramanujan
Bhaskara II
4
Hypatia
Pythagoras
Math Magicians
About the Authors
Nikhil Jotwani
High Performance Computing,
CREST,
KPIT Cummins Infosystems Ltd.,
Pune, India.
Areas of Interest
Image processing,
Embedded systems,
Telecommunication
Reenakumari Behera
Vision Systems,
CREST,
KPIT Cummins Infosystems Ltd.,
Pune, India.
Areas of Interest
Resource Engineering, Computer vision
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
5
1. Introduction
Whenever we think of mathematics, we always
tend to remember the hard times we had to face in
understanding the complex equations, the struggle
to get the right answers. Not getting the derivative
right used to derive the juice out of our mind and
the symbol of integration looks like a snake behind
our life. However, we forget to understand that
these formulae, equations or analysis is readily
available to us these days. Just a search with a
couple of key words on the internet and we have
all the history on our screens. What about the time
when no one knew what an addition or a
subtraction meant. What went through the minds
of some people that they actually came up with
theories that have now become the backbone of
various fields? You will be astonished to know that
some of the work in mathematics dates back to
500 BC. Since then many people have worked in
this field and tremendous development has been
done in math. S.Gudder, a professor of
Mathematics once quoted“The essence of
mathematics is not to make simple things
complicated, but to make complicated things
simple”. More than 25000 mathematicians have
worked through eras to develop the field of
mathematics that we know today. Here you would
find an insight on some of the influential
mathematicians with whose work we would
struggle to do our daily chores.
Pythagoras(569 BC - 475 BC)
Geometry books invariably have a chapter on 'The
Pythagoras theorem'. Do you know that the Greek
mathematician Pythagoras was the first to give a
formal proof to this theorem even though it was
known to the Babylonians about a 1000 years
earlier. That is why this theorem is named after
him. The theorem holds such importance in our
lives. It forms one of the basic concepts of
mathematics. Pythagoras founded a school named
‘Semicircle' in the city of Samos, his birthplace. The
school is known by that name even today. Outside
the city, he had personal spot in a cave where he
spent most of his time doing research on the use of
mathematics. One of his contributions was that he
was able to establish a correlation between the
6
TechTalk@KPITCummins, Volume 5, Issue2, 2012
harmonious intervals in music to simple rational
numbers. He also discovered the simple parametric
form of Pythagorean triplets (xx-yy, 2xy, xx+yy).
The concepts of perfect and amicable numbers,
polygonal numbers, golden ratio, the five regular
solids, and irrational numbers, all have come from
Pythagorean School.
Euclid (325 BC - 265 BC), the Greek
mathematician, also famously known as 'Euclid of
Alexandria' is considered as the 'Father of
Geometry'. His work was very influential and
contributed a lot to the field of mathematics. Euclid
wanted to make a collection, which would serve as
a single reference for mathematical proof and other
works in the field of mathematics. One of his most
influential works called 'The Element', which served
as the main reference for teaching mathematics
until the late 19thcentury that served the purpose.
The discourse gives more emphasis on the concepts
of geometry. It also includes work done on number
theory, prime numbers etc.
Hypatia (370 – 415 AD)
There is a misconception that the field of
mathematics is purely male dominated. However,
many women have made significant contribution to
this analytical field. Hypatia was one of the first
women to make significant contribution in this field.
She was the daughter of the mathematician and
philosopher Theon. One of the remarkable facts
about Hypatia is that she became the head of the
Platonist school at Alexandria in 400 AD. Her major
contributions are the collaborative work, which she
did with her father. She also contributed to the book
'Diophantus's Arithmetica', which is a collection of
algebraic problems giving solutions to determine
equations. She also contributed to 'Ptolemy's
Almagest', which is a discourse on motion of stars,
and planetary path, written in the 2nd century.
Brahmagupta (598 - 670) was the author
of many important works in Mathematics and
Astronomy. The most famous writings by
Brahmagupta include Brahmasphutasiddhanta and
Khandakhadyaka. We know that when we subtract
two similar quantities the answer is zero. This is
because Brahmagupta defined it back then. His
approach of multiplication is similar to present day
multiplication. He also invented an algorithm
similar to the Newton-Raphson iterative formula
for computing square roots. He presented
methods to solve quadratic equations and
as equations having two solutions. He used a
binomial expansion based method for finding nth
roots of an equation. He also proved properties of
figures in non-Euclidean geometries. He extended
his work on ratios to include the multiplication of
ratios. He also raised the question of whether a
ratio can be regarded as a number.
indeterminate equations of the form ax + c = by.
He gave the formulae (n*(n+1)*(2n+1)/6) and
2
((n*(n+1)/2) ) as the sum of the squares of the first
n natural numbers and the sum of the cubes of the
first n natural numbers respectively.
Aryabhata II (920 -1000)
Space geometry may sound quite different.
Aryabhata II was an Indian mathematician who
contributed to this field. Mahasiddhanta is a
treatise on mathematical astronomy covering the
usual topics that Indian mathematicians worked on
during this period. It also includes topics such as
Geometry, Geography and Algebra with
applications to the longitudes of the planets. In
Mahasiddhanta, rules to solve the indeterminate
equation: ‘by = ax + c’ are also discussed. The
rules apply in a number of different cases such as
when ‘c’ is positive, when ‘c’ is negative, when the
number of the quotients of the mutual divisions is
even, when the number of quotients is odd, etc.
Omar Khayyam (1048-1122)
He was a philosopher, mathematician, astronomer
and a poet. Yes, these are the qualities describing
the next influential mathematician. He came from
Persia, a person who wrote a discourse on various
subjects like Mechanics, Geography, Mineralogy,
Music, Climatology and Theology. He is Omar
Bhaskara (1114-1185) is also known as
Bhaskara II or as Bhaskaracharya. He is one of the
most important mathematicians of the medieval
times. He named his book on arithmetic after his
daughter 'Lilavati'. Not everyone would believe this,
but there is a fascinating story about Lilavati.
Bhaskara was warned that if Lilavati's marriage did
not take place at the particular mentioned time, she
would be widowed. Therefore, to avoid this,
Bhaskara kept a bowl with a small hole at the
bottom in a vessel full of water. When the cup would
drown completely, that would be the particular
moment when the marriage should have taken
place. However, due to Lilavati's curiosity, a pearl
from her nose ring fell into the bowl disturbing the
setup and the cup sank before it was actually
supposed to and the marriage took place at the
wrong time. Believe it or not, she was widowed
after a few years of her marriage. Later she studied
mathematics from her father and became a wellknown mathematician. His works include
Bijaganita, which is on algebra; The
Siddhantasiromani, which is on mathematical
astronomy; The Vasanabhasya of Mitaksara, which is
Bhaskaracharya's own commentary on the
Siddhantasiromani; the Karanakutuhala or
Brahmatulya, which is a simplified version of the
Siddhantasiromani and the Vivarana, which is a
commentary on the Shishyadhividdhidatantra of
Lalla. One of the interesting results given by
Bhaskaracharya is:
Khayyam. He is the author of one of the most
important treatise named 'Treatise on
Demonstration of Problems of Algebra', which
includes geometrical solutions for cubic equations.
Khayyam predicted the number of days in a year to
be 365.24219858156 days. Intersecting conic
sections are used for classifying the cubic equations
Figure 1: A page from Bhaskara'sLilavati Source: [1]
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
7
1,1,2,3,5,8,13,21,34,55………Remember this
all these involvements, he was called 'Marvelous
sequence? Yes, it is the Fibonacci series.
Merchinston'.
Leonard of Pisa or Fibonacci (11701250 AD) was the one who gave us this series.
He produced 'Liber Abaci (Book of calculation)',
which consists of the Hindu-Arabic place-valued
decimal system. It also contains a large collection of
problems aimed at merchants. The Fibonacci
numbers are also introduced in this book. 'Practica
Geometriae' written in 1220 AD contains a large
collection of geometry problems, practical
information for surveyors and details of how to
Figure 3(a)
Figure 3(b)
Figure 3(a) Method of writing fractions in
1550's 3(b) Long division example
Source: [1]
calculate the height of tall objects using similar
Blaise Pascal (1623-1662 AD) was a
triangles. Recent study shows that Fibonacci series
French Mathematician and philosopher. He
is used for market analysis. We would become
worked on conic sections and projective geometry
millionaires if we know exactly how to use this
and in correspondence with Fermat; he laid the
series in determining the market study.
foundations for the theory of probability. When
other youngsters were playing football on the
ground, Pascal at the age of 16 wrote a concise
essay on conics, containing contributions to
projective geometry rivaling those of his teacher,
Desargues.
Figure 2(a)
Figure 2(b)
Figure 2(a): Triangular and square multiplication
tables used in 1530's 2(b) RIESE's Arithmetic book,
1538 Source: [1]
If
Source: [1]
John Napier were alive today, the headlines
would read like “Napier Log(s) out of school but
invents Logarithms”. Though he dint attend school,
he traveled a lot in Europe for his education. He has
great interest in Astronomy and Astrology. His
contributions include two formulas known as
Napier's analogies used in solving spherical triangles
and an invention called "Napier's bones" used for
mechanically multiplying, dividing and taking square
roots and cube roots. He also invented exponential
expressions for trigonometric functions, and
introduced the decimal notation for fractions.
Napier expected that the use of logarithms would
help in reducing errors of calculations. Along with
his research on Mathematics, he was well known
for finding mechanisms, which would help in
improving the quality of his crops and cattle. Due to
8
Figure 4: The Pascal's triangle
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Leonhard Euler (1707-1783 AD) was a
Swiss mathematician who made enormous
contributions to a wide range of mathematics and
physics including analytic geometry, trigonometry,
geometry, and calculus and number theory. He
wrote books on the calculus of variations; on the
calculation of planetary orbits; on artillery and
ballistics; on analysis; on shipbuilding and navigation;
on the motion of the moon and lectures on the
differential calculus. He was the first to consider
sine, cos etc. as functions rather than as chords. He
made decisive and formative contributions to
geometry, calculus and number theory. He
integrated Leibniz's differential calculus and
Newton's method of fluxions (method of
differential calculus) into mathematical analysis and
introduced beta and gamma functions, and
integrating factors for differential equations. We
owe to Euler the notation f (x) for a function (1734
AD), ‘e’ for the base of natural logs (1727 AD), ‘i’
for the square root of -1 (1777 AD), ‘π’ for pi, ‘∑’
for summation (1755 AD), the notation for finite
differences Dy and D2y and many others.
algorithm is called the 'Karmarkar's algorithm' and it
solves the linear programming problems in
polynomial time. He received the 'Paris Kanellakis
Award' for his algorithm. Having earned his PhD in
computer science, he is presently working on the
architecture of the new super computer.
Electronics students would have found digital
electronics as the toughest subject if George Boole
had not defined the Boolean algebra. They
Manindra Agrawal was born on 20 May 1966
definitely owe their high scores to him. George
algorithmic number theory and algebra. His best-
in Allahabad, India. Prof. Agarwal works in theory
of computation; specifically, in complexity theory,
Boole (1815-1864 AD) approached logic in
known work is in algorithmic number theory: along
a new way reducing it to a simple algebra,
incorporating logic into Mathematics. He also
worked on differential equations, the calculus of
finite differences and general methods in
probability. He pointed out the analogy between
algebraic symbols and those that represent logical
forms. It began the algebra of logic called Boolean
algebra that now finds application in computer
construction, switching circuits and many others.
with two of his students (Neeraj Kayal and Nitin
Saxena), he designed the first deterministic
polynomial time algorithm (AKS algorithm)for
testing the primality of a number. This discovery
resolved a long-standing problem of a fast test of
primality, which had been the subject of intense
study in mathematics and computer science
research [2].
Conclusion:
Ramanujan (1887-1920) made substantial
contributions to the analytical theory of numbers
and worked on elliptic functions, continued
fractions and infinite series. His most famous work
was on the number p(n) of partitions of an integer
n into summands. In a joint paper with Hardy,
Ramanujan gave an asymptotic formula for p(n).
Ramanujan left a number of unpublished
notebooks filled with theorems that
mathematicians have continued to study. G N
Watson, Mason Professor of Pure Mathematics at
Birmingham from 1918 to 1951 published 14
papers under the general title Theorems stated by
Ramanujan and in all he published nearly 30 papers
that were inspired by Ramanujan's work.
There is a great learning that everyone can achieve
from the way various mathematicians have
contributed to the growth of the science
Mathematics has its roots in as early as 500 BC. The
number of mathematicians listed in this article
is still a drop in the ocean. There are a plenty of
mathematicians who have dedicated their entire
lives in shaping Mathematics in the form that we see
it today.
References:
[1]
Charles Eames, Ray Redheffer, Men of
modern mathematics: a history chart of
mathematicians from 1000 to 1900,
International Business Machines
Corporation, 1966.
We all are aware that most of the scheduling
algorithms or optimization techniques use linear
programming methods as one of the intermediate
steps. However, it is highly time consuming and
using it for real time applications would be
doubtful. To counter this,
Narendra
Karmarkar , an Indian Mathematician
developed his own efficient algorithm. His
[2]
http://www-groups.dcs.stand.ac.uk/~history/Indexes/_500_AD.html
[3]
[4]
http://www.xtimeline.com/evt/view.aspx
Biographies:
'Baudhanya','Pythagoras','Khayyam','hypatia',
bhaskara 2', an internet article www.
history.mcs.st-and.ac.uk
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
9
=
1
2
-3
10
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
- 2
e
2
3
2
Numbers that Matter
About the Author
Aditi Athavale
High Performance Computing,
CREST,
KPIT Cummins Infosystems Ltd.,
Pune, India.
Areas of Interest
Cryptography,
Multicore programming
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
11
28
I. Introduction
“Number is the ruler of forms and ideas, and
the cause of gods and demons” ~Pythagoras
II. Important Numeric Constants
A. Euler's number 'e’
Though named after Leon Hard Euler, the credit of
discovering the value of 'e' goes to Jacob Bernoulli
(1654-1705). It is said that a banker friend of
Bernoulli had discussed with him on the ways of
luring more customers into banking business. After
a discussion on compound interest and interest
rates etc, Bernoulli analyzed a particular trend of
interest calculation.
'If one starts with a principle of $1 and the rate of
interest is 100%, then at the end of one year, one
would get $2. What will happen if the person is
allowed to withdraw the amount and reinvest it in the
same scheme?'
If one withdraws the amount after six months, the
principle would go up to 2.25 which is (1 + ½)2.
Similarly, if calculated for 'n' times, the principle
would be (1+1/n)n.
Bernoulli observed that the value approaches a
limit of 2.7182818 for large values of 'n', as shown
in figure 1. And that's how this constant came into
existence. Few years later, Euler named the
constant as 'e' and since then, it is also known as
'Euler's number'.
12
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
27
26
25
y
‘All numbers are equal (in importance, not in
value!), but some are more equal than others'!
Some numbers have repeatedly proven
themselves in all walks all life; may it be
architecture, engineering, music or finance. They
are here for thousands of years, and retain their
importance even in today's world. In this article,
we will see different important numbers and
constants, which find many applications in various
branches of engineering sciences. We will also
revisit few important classes of numbers including
Prime numbers, Random numbers and Complex
numbers. In order to emphasize their relevance
and importance, practical applications of those
numbers are also explained.
24
23
22
21
2
0
10
20
30
40
50
n
60
70
80
90
100
Figure 1: Reaching the limit of 2.718..
The beauty of e lies in its properties. 'e' can be
expanded into a Tailor series and it has a nice form
of 1+ ----+(x**n)/n!. Take a derivative of 'e' to the
power x, it is 'e' to the power x. Take an integral of
'e' to the power x, it is still the same. Multiply a
function by ‘e’ to the power (-sx) the result of which
takes one into different domain and it also has its
unique inverse. This is referred to as Fourier
transform or Laplace transform depending on the
limits of integration and the value of 's' as being
complex or real plus complex. 'e' has literally
opened doors to the new world for
mathematicians. No wonder it is called the king of
all functions.
B. Pi 'p'
The value of Pi is known since ancient times. The
values closer to approximated value of Pi appear
throughout the history of Babylonian, Egyptian,
Indian and Chinese mathematical literatures and
architectures. The great Indian Mathematician,
Srinivasa Ramanujan, computed several series for
rapidly calculating the value of Pi. These series are
used in many of the current fastest techniques to
compute Pi.
The angular measure 'radian' is closely related to Pi.
According to the Wolfram definition, radian is a 'unit
of angular measure which is equal to the angle
subtended by an arc on a unit circle to the center of
the circle' [6]. As the ratio of the circumference to
the radius of a circle is Pi, full angle (i.e. 3600) is
nothing but 2p radians. In simpler words, radian is
'radius being laid along the circumference' [5].
Hence, it provides a natural representation of
angles.
Apart from geometry, Pi is used in expansion of
various series and some important formulae in
Mathematics, Physics and Chemistry. Some of the
known examples are Euler's identity:
Heisenberg's uncertainty principle, Coulomb's law
for describing the electric force etc.
C. Golden Ratio 'f’
Golden ratio is believed to be aesthetically pleasing
and thus occurs at so many places in nature.
Leonardo da Vinci, one of the most eminent artists
has illustrated the golden ratio in 'De Divina
Proportione' as shown in figure 2 [7]. Human body
proportions drawn in a frame of pentagon also
illustrate the golden ratio [7]. The limit of ratio of
two successive Fibonacci numbers tending to
infinity also comes out to be same. Many
researchers relate the golden ratio to the human
genome DNA as well.
Fig.2. (a) Golden ratio as illustrated in 'De Divina
Proportione' by Leonardo da Vinci [7] (b)
Human body proportions
Let us assume that 'a' and 'b' are two numbers
where 'a' is greater than 'b'. Numbers 'a' and 'b' are
said to be in Golden Ratio if the ratio of their sum to
'a' is equal to the ratio of 'a' to 'b'. The golden ratio is
denoted by f.
D. Pythagoras's Constant
Pythagoras's constant is nothing but square root of
2. In order to visualize, it is the length of
hypotenuse of a triangle with the non-hypotenuse
sides with unit length. The discovery of the
number dates back to 1800-1600 BCE. One
peculiarity of Pythagoras's constant is that the
number is irrational. In other words, the number
cannot be expressed in the form of division of 2
integers. There are multiple methods which prove
the irrationality of square root of 2.It is proven that
Pythagoreans did have a proof for the same.
The value of the number is 1.414 (taken up to 3
decimal places). One of the interesting
representations of the number is
The number is very often used in Trigonometry as
As per ISO 216 standard, aspect ratio of A0, A1,..,
A10 papers follow Pythagoras's Constant.
I1I. Prime Numbers
Prime numbers are nothing but the basic building
blocks in the number theory. In simple words, they
are the numbers which are divisible only by two
numbers, one and that number itself. All other
natural numbers (except 1) which are not 'Prime'
are called as 'Composite numbers' and they can be
expressed as a combination of one or more prime
factors. There are infinite prime numbers in nature.
A proof of this fact was given by Euclid, the eminent
Greek Mathematician.
Primality tests are those which determine if a given
number is Prime or not. There are many of them,
but no single method can be efficiently applied to all
the numbers. Apart from the traditional approach,
'Sieve of Eratosthenes' is one of the oldest and
relatively practical approaches of finding prime
numbers in a given range of numbers. With an
iterative approach, the method eliminates all the
composite numbers which are multiple of
successive prime numbers. It starts with the first
prime number i.e. 2. In the given range, it eliminates
all the numbers which are multiple of 2. As 3 is not
eliminated, it is marked as a prime number and so
on. In this way, all the numbers which are not
eliminated are marked as prime numbers in the
given range.
Figure 3: Demonstration of 'Sieve of Eratosthenes' [1].
One more important test for checking primality is
developed by Dr. Manindra Agarwal, the Guest
Editor of this issue of TechTalk. The test is known as
AKS test (Agrawal-Kayal-Saxena test) of primality,
and has won many International accolades. Using
this test, it can be determined in polynomial time if a
number is prime or not.
In today's world, prime numbers are almost
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
13
irreplaceable when security of software and
embedded applications is to be ensured. 'Trapdoor
functions' are the ones which are easy to compute
only in one direction. Computing these functions in
inverse direction is close to impossible with limited
computational resources. An example of trapdoor
function is prime factorization. If we take two very
large prime numbers, it is very easy to multiply
them. On the other hand, if we try to get back
these two prime numbers from the result, it is very
difficult to do so with ordinary resources. One can
still proceed using the Brute Force method (i.e.
trial and error method). However; there is no
guarantee of conversion of this method in given
limited time. Public cryptography system, based
on RSA algorithm, is one of the most important
types of technique which is used to enforce
security in online transactions, is based on 'prime
factorization'. Many of the algorithms that are used
in Internet security are based on variants of prime
factorization problem.
IV. Complex Numbers
As we all know, complex number is a number
consisting of two parts, real and imaginary. If 'a' is
real component and 'b' is imaginary component of
a complex number 'c', it is represented by
Here 'i' is imaginary unit which is equal to -1.
"The imaginary number is a fine and
wonderful resource of the human
spirit, almost an amphibian between
being and not being." ~ Gottfried Wilhelm
Leibniz (1646-1716)
To have better understanding of complex numbers,
let us look at an example [2].
Figure 4: Projection of circular motion (a) side
view (b) with respect to time
Consider an object attached to a circular moving
plate. Figure 4 shows 2 instances of the assembly. If
we look at it from the side view, the object seems
to be moving only in vertical direction, in a to and
fro manner. If we plot a graph of position of the
14
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
plate with respect to time, it represents a sinusoidal
wave. Thus, the continuous rotating motion is
mapped to oscillations along vertical axis. However
if we have a look at only the oscillatory motion, it
requires some imagination to conclude that it is
actually a rotational motion. The imaginary number
is used to depict this unseen component of such
phenomena.
Figure 5 shows the phase associated with a complex
number, with respect to the current example.
Figure 5: (a) Phase angle as shown in the assembly
(b) Phase associated with the complex number
Similarly, the oscillatory voltages and currents in
electronic circuits are result of rotational
movement. The voltage is represented by
Here 'j' is the notation used for representing
imaginary number. The real part of this voltage is
given by V0 cos(2pft) and imaginary part is
represented by V0 sin(2pft).
V. Random Numbers
According the definition from Wolfram Mathworld
[4], a random number is a number selected by
chance from a large set of numbers which adhere to
a certain distribution pattern. Various examples of
such distributions are Gaussian distribution,
Uniform distribution, Poisson distribution and so
on..
Random numbers are observed across different real
life situations.
l
For example, normal or Gaussian distribution is
observed when we toss a set of coins 'n' number
of times, where 'n' is a large number. The
histogram of number of heads appeared during
the tossing follows a bell shaped curve, which is
Gaussian distribution.
References
l
Poisson distribution is observed with people
arriving at a bus stop during a specific interval of
[1] ‘Prime Number', From Wikipedia,
time with a known average rate.
http://en.wikipedia.org/wiki/Prime_number
l
Uniform distribution occurs when all events
[2] ‘More Complex Makes Simple', Jim Lesurf,
happen with an equal chance in a given interval
University of St. Andrews, Scotland,
of time.
http://www.standrews.ac.uk/~jcgl/Scots_Guide/info/signals
Random numbers and their distributions are
/complex/cmplx.html
widely used in various fields such as simulation,
data modeling, sampling, cryptography, security,
gaming theory and so on.
VI. Conclusion
In this article, we saw few specific numeric
constants like Euler's number, Pi, Golden ratio and
Pythagoras's number etc. We saw some classes of
numbers like prime numbers, complex numbers
and also their applications. There is no doubt about
the usage and the importance of these numbers in
every walk of our life. In today's modern world that
is influenced thoroughly by information and
technology, these numbers create a valuable link
between humans and machines.
[3] ‘Mathematical Quotes',
http://math.sfsu.edu/beck/quotes.html
[4] ‘Random Number', Wolfram MathWorld,
http://mathworld.wolfram.com/RandomNum
ber.html
[5] ‘Radian', Math is Fun,
http://www.mathsisfun.com/geometry/radian
s.html
[6] ‘Radian', Wolfram MathWorld,
http://mathworld.wolfram.com/Radian.html
[7] ‘Golden Ratio', From Wikipedia,
http://en.wikipedia.org/wiki/Golden_ratio
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
15
16
Inspired by Nature and
Mastered by Mathematics
Automobile Evolution:
About the Author
Ravi Ranjan
MEDS, Bangalore
Areas of Interest
Mechatronics, Mechanical Design,
Renewable energy and
Propulsion systems
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
17
From the Flintstone's car to the most advanced
input and output parameters. These models are
contraptions known to humankind, mobility has
used to investigate the ride comfort, handling
been redefined continuously, driven by
performance and other analyses. Subsystems have
mathematics like a catalyst in a chemical reaction.
evolved due to mathematical analysis from a simple
The greatest invention of mobility was the wheel,
passive system to complex adaptive ones.
which may have derived inspiration from biological
creatures that becomes circular either to save
themselves from becoming a prey or to run fast to
catch a prey as shown in figure 1.
Figure 3: Analytical representation of vehicle
Figure 1: Escher's wheel-like animal(L) and
Bronze Age disk wheel (2500 BC) ®
A simple dynamic equation for the vehicle motion
can be given by:
The ever-growing quest of better mobility led to
V0= [NwFt - Fv]/ Mv
development of car from carts and chariots.
Where Fv = wind drag force (function of vehicle
Mathematics had always been the driving force
velocity), Mv = vehicle mass,
behind this development starting from Nicolas
N w =Number of driving wheels (during
Joseph Cugnot's self-propelled road vehicle (1771)
acceleration) or the total number of wheels (during
to Gottlieb Daimler's first modern automobile
braking), and Ft= tire tractive force, which is the
(1887)
average friction force of the driving wheels for
acceleration and the average friction force of all
wheels for deceleration. Thus, the motion of
vehicle can be accurately predicted with given range
of input parameters.
Few of the subsystems that evolved to impress with
the performance are
Figure 2: 1771, Nicolas Joseph Cugnot's
steam-powered car
I. Evolution of Mathematical
Modeling for Automobiles
l
Anti lock braking system
l
Electric power steering
Engine management system
l
Traction control system
l
l
Active suspension system
Although automobiles were developed in the late
l
Adaptive headlamps etc.
18th century, the importance of mathematics was
felt only in the 20th century. This led to a non-linear
II. Anti lock braking system
evolution of automobiles from a cart with set of
Mathematical analysis of braking dynamics during
wheels to a sophisticated machine with ideal
panic braking resulted in Anti lock braking system.
combination of comfort, safety and power.
The mathematical model of subsystems (tire, road
Today, a numerical simulation of system dynamics
etc)explains how the locking of wheels results in slip
based on mathematical model is a basic building
and the effect of slip on other parameters like the
block in the design of an automobile. Each
braking distance and steering control.
subsystem is mathematically modeled on a set of
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TechTalk@KPITCummins, Volume 5, Issue 2, 2012
The mathematical relationship between these
Speed
Sensor
parameters provides a means to quantify the effects
of a cause.
Pump/Value
Thus, a mathematical relationship of the different
subsystem allows the ECU to decide on the amount
of assist torque required at a given speed.
IV. Nature Replication
Speed
Sensors
Figure 4:Anti lock braking system
Thus equations negotiating slip curve, friction
coefficient, breaking force, wheel lock etc.,
provoked a problem statement viz., to sense the
wheel lock in advance and release/rebuild pressure
to maintain optimum traction between tire and
ground. The result of this is a smart evolved
braking system that many automobiles now use.
Anti-lock braking system not only provides
efficient braking but also superior steerability
during braking.
No matter how fast we evolve, one can never
supersede nature's intelligence. The beauty is in
learning the nature and understanding its processes.
A simple thought inspired by a bird's flight took us to
complete new transportation means of air
transport, thereby shrinking the world into tiny
piece of land. Imagine infinite such natural processes
occurring in our vicinity that are yet unobserved and
unexplored as well. If we are able to observe and
successfully replicate more and more of these, the
development will be phenomenal. Let us take an
example of a cyclone. It is a naturally occurring flow
of winds but in a definite pattern, that generates a
huge amount of energy, as against a gentle breeze if
the flow were in some other pattern. It is important
III. Electric power steering
to study the geometry of a cyclone and understand
Electric Power Steering (EPS) has important
significance in improving the vehicle's dynamic and
static performances. Based on a mathematical
model and the characteristic curves of EPS System;
characteristics of EPS system are analyzed,
the process to tap energy from the same.
As shown in figure 6, Inverted Archimedean Spiral
can be the closest approximation of the geometry
of a cyclone. The equation of this curve can be used
to model a replica.
including the typical power curves which deal with
the relationship of the assist torque and the
steering wheel torque.
Steering
torque
Torque signal
Motor control number
Torque
sensor
Assist torque
Road reaction torque
Figure 6: Inverted Spiral
Figure 5: Electric power steering
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
19
V. Conclusion:
During the last century, automotive industry has
References
seen a great drift towards embracing scientific
approach. Quantifiable parameters are more
1."Effectiveness of ABS and Vehicle Stability
valued than intuition based design. Mathematics
Control Systems" (PDF). Royal Automobile
gave an insight into controllable systems and it has
Club of Victoria. April 2004, Retrieved 2010-
evolved smarter variants by providing set of
12-07.
equations to quantify the effect of cause and vice
versa.
2."Non-Skid Braking". FLIGHT International, 30
October 1953. pp. 587–588.
3.M.C. Escher 'Curl-Up' 2008 the M.C. Escher
Company - Holland.
4.http://www.ansoft.com
Figure 7: Ergonomic analysis
Not only in the performance of vehicles but it also
gave the parameters to elevate the passengers
5. http://www.mitsubishielectric.com
6.http://en.wikipedia.org/wiki/Spiral
comfort. Ergonomic analysis based on the
biological consideration of the occupants gave a
complete new dimension to the relevance of
passenger comfort.
Yet there is a lot more to be done. The underlying
idea is plain and simple:
Learn from nature and let Mathematics bridge the
gap!
A mathematician is flying non-stop from Edmonton to Frankfurt with AirTransat. The scheduled
flying time is nine hours.
Some time after taking off, the pilot announces
that one engine had to be turned off due to
mechanical failure: "Don't worry - we're safe.
The only noticeable effect this will have for us is
that our total flying time will be ten hours instead
of nine."
A few hours into the flight, the pilot informs the passengers that another engine had to be
turned off due to mechanical failure: "But don't worry - we're still safe. Only our flying time will
go up to twelve hours." Some time later, a third engine fails and has to be turned off. But the
pilot reassures the passengers: "Don't worry - even with one engine, we're still perfectly safe. It
just means that it will take sixteen hours total for this plane to arrive in Frankfurt." The
mathematician remarks to his fellow passengers: "If the last engine breaks down, too, then we'll
be in the air for twenty-four hours altogether!”
20
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Profile of a Scientist
S. Ramanujan's
achievements, in a very
short life span, are quite
exceptional. He had a
wealth of ideas that have
been inspirational to the
scientific community and
are topics of research even
now in the 21st century.
Born on 22nd December
Srinivasa Ramanujan 1887 in Tamil Nadu, he had
no formal education in
mathematics. However, he emerged as a great
mathematician from his poverty stricken
background only because of his perseverance,
determination and endless passion for
mathematics.
Ramanujan mastered trigonometry at the age of 12
and invented his own theorems when in school.
When in school, he was known to be an introvert
and kept to himself. One very interesting story that
is often mentioned about him during his school
days is his prodigious memory that would entertain
his friends. He could repeat the value of the
constant p to any number of decimal places; and
this was just the beginning of what had to come
from him. Ramanujan's work is mainly in the area of
number theory and summation formulae involving
constants like p prime numbers and partition
functions. The fastest algorithms currently being
used are based on Ramanujan’s series
of p convergence. This series for p converges
exponentially and truncating only the sum of the
first term also gives the approximation for
correct up to 6 decimal places.
Most of his formulae that were claimed to be
without any formal proofs were later proven to
hold true. One of his most important papers was
about connecting the computation of the value of
p to modular forms. This theory is very important
in modern mathematics even today. He had
published several papers in Indian mathematical
journals and then tried to interest European
mathematicians. In 1913 he wrote a letter to the
famous mathematician G. H. Hardy from
Cambridge, UK, that contained several
mathematical formulae he had invented. Hardy
viewed them skeptically at first, but soon realized
that it was exceptional work. Some of those
formulae were already proven by some other
mathematicians and others he did not understand
at all. Hardy wrote back to Ramanujan to provide
proofs for his inventions and also invited him to
Cambridge to carry on further research. Hardy
tried to fill in the formal education gap that
Ramanujan had. Their collaborative work was
published in more than 10 research papers.
Ramanujan published more than 30 research
papers individually, based on his 3 years work at
Cambridge. The most interesting work was on
partition function that counts the number of ways a
natural number can be decomposed into smaller
parts. Hardy and Ramanujan had developed a
method called the circle method to derive an
asymptotic formula for this function. This method is
today the most important tool of analytic number
theory and was used for major advances in the 20th
century to solve difficult problems such as
Goldbach's conjecture, Waring's conjecture, etc.
The circle method continues to inspire research
even today.
Another important method invented by the duo is
called the “normal order method”. This method
analyzes the behavior of additive arithmetical
functions. They also showed that a random natural
number usually has about log (log n) prime factors.
This led to an entirely new field of mathematics
called 'probabilistic number theory'. This theory
was largely developed in the 20th century by other
mathematicians.
There is another interesting anecdote about
Ramanujan and Hardy related to the number 1729.
Once, Hardy visited Ramanujan while he was ill and
mentioned that he had arrived in a taxi number
1729; which seemed to be an ordinary and
uninteresting number. Ramanujan immediately
replied that this number was actually quite
remarkable: it is the smallest integer that can be
represented in two ways by the sum of two cubes:
1729 = 13+ 123 = 93 + 103
Hardy wrote about Ramanujan's theories: “it is his
insight into algebraic formulae, transformations of
infinite series and so forth, that was most amazing.
On this side, most certainly I have never met his
equal, and I can compare him only with Euler or
Jacobi.”
Unfortunately, Ramanujan lived a short life due to
several health problems while in India and England.
This is said to be due to the lack of vegetarian food
during the first-world war, when he returned to
India in 1919 and died soon after in Kumbakonam.
While on the death bed he wrote his last letter to
Hardy, outlining a new theory of 'mock theta
functions'. This theory was ignored in the 20th
century and was mentioned only in a few papers
because Ramanujan's definitions of mock theta
functions were vague. Instead of definitions he put
forward 17 examples of these new functions and
formulated general conjectures around them. Many
mathematicians tried to prove this and were
successful to prove them individually. At a
framework that blended them together was missing
until 2002. The doctoral thesis of S. Zwegers in
2002 puts forward the groundwork for a new
theory of mock modular forms.
A legend like him will always inspire generations to
come and it is only apt, that the year 2012 has been
declared as the 'National Mathematical Year' by the
Prime Minister of India, commemorating the 125th
birth anniversary of the great mathematician
S. Ramanujan.
Priti Ranadive
High Performance Computing, CREST
21
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TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Games People Play
About the Author
Chaitanya S. Rajguru
High Performance Computing,
CREST,
KPIT Cummins Infosystems Limited,
Pune, India.
Areas of Interest
Computing Hardware and Algorithms,
VLSI Technology,
Energy & Power in Systems,
and Environmental Aspects of Technology
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
23
I. Introduction
The theme of this TechTalk issue is “Mathematics.”
What do games have to do with mathematics?
Well, mathematics is said to be a powerful language
that helps us describe various phenomena and
understand them better. The number system helps
us count and compare objects. Statistics give us a
handle on large-scale phenomena such as
populations. Similarly, a branch of mathematics
called “Game Theory” can describe and model
human interactions. It has many practical
applications in understanding individual and group
behavior in business negotiations, auctions,
international competition, and much more. While
the mathematics involved can be deep and
abstract, the insights that game theory provides
into human behavior are absolutely compelling and
well worth the mathematical effort.
Let us then begin with a game. OK, you and I both
contribute $100 and put the money in an envelope
on the table. You shall make the first move. You
must make me an offer of some amount up to
$200. Perhaps you offer me $50 as an example.
Next, it is my move, and I can either accept or
reject your offer. If I accept, then I get that amount
($50 in this example) and you get the rest ($150).
However, if I decline, neither of us gets anything,
and the money is given away to the next person
who walks by, never to be seen again. We shall play
the game only once and part as friends. Let us
assume that both of us are sane, logical, and try our
best to maximize our personal gains.
So now that you understand the rules, how much
will you offer me?
Think carefully … you could choose to take one of
several directions. As a first choice, you may offer
me a big amount, say $120. I am more likely to
accept your offer, but then you shall get only $80,
and end up with less cash than you put in. As a
second choice, if you offer me the small amount of
$40, you could get $160 … but only if I accept! If I
decline, you shall be left with nothing. The third
possibility is that you try to be “fair” by offering me
$100 and even out your chances, but then you are
24
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
back where you started. Moreover, I could decline
even the $100 offer, because I have the veto power;
I might only accept if I made a good profit.
So, what should your strategy be? To cut your losses
and make me a high offer? To break even at $100?
Or to milk the first-mover advantage and make me
a low offer? The answer is not obvious.
Now consider my perspective. What should my
strategy be? At first glance, I may be tempted to
accept any offer I get … after all, I would always get
some money by accepting whatever offer you
make, rather than getting zero by rejecting it.
However, that is a slippery slope: if I follow that
logic, I should accept even $50; … no, even $20; no,
even $10; and so on, all the way down to $1. Yet
accepting $1 out of $200 would leave me feeling
cheated! On the other hand, should I become
greedy and only accept an offer that makes me a
profit? That logic quickly spirals upwards, to
accepting a $199 offer and nothing below that. This
greedy strategy may work well if I can somehow
signal my aggressive intentions to you before we
play the game. But then, you too could declare your
intentions of squeezing me towards accepting $1.
So we are back at the beginning, trying to find yet
other ways to out do each other. This game is
quickly turning from a pastime into a real battle of
the wits!
II. What Is Game Theory?
Is your head spinning after all these “what-if”
questions and trying to second-guess your
opponent? However, you may actually be good at
negotiating the price of apples with a street vendor,
with whom you try to drive down the price as low as
possible without breaking the deal. Guess what …
both these interactions can be analyzed
mathematically. The branch of mathematics that
makes that possible is game theory. That real-life
feel is what is so endearing about this branch of
mathematics: it is all about how to make the right
decisions in interactions called “games”.
A “game” is a situation in which you try to win by
making smart decisions, but the outcome also
depends upon decisions made by others. Think
Rock-Paper-Scissors, or Tic-Tac-Toe. However,
games are not always trivial pastimes. They can
model a business negotiation. They can generate
an assessment of the likely outcome of a
democratic election. Games can aid a pricing
decision for the latest detergent to launch in the
competitive consumer goods market. Or they can
help you choose a winning strategy in a patent
infringement lawsuit. Now we begin to appreciate
the practical uses of game theory.
Game theory is a formal (mathematical) study of
conflict and cooperation [1]. The games take the
form of rules – chess rules or stock market rules,
for example. Players make choices during the
game in line with these rules, and try to achieve the
best outcome for themselves – a win or at least a
draw, more money, more votes, better chances of
survival, or anything else that they value. Players
may be individuals, groups, firms, countries,
animals, software programs, and so on. They can
choose to compete with other players, or
to cooperate by forming coalitions. The
popular American television serial, “Survivor”,
demonstrated some interesting competitioncooperation strategies, for example. Political and
industrial coalitions could be similarly motivated.
Conflict and cooperation are as old as life itself. In
nature, all organisms compete for food, space,
mates, and much else. Many cooperate for their
mutual benefit, such as members of bat colonies.
Human activities in all spheres are full of conflict
and cooperation decisions. Their formal study as
the field of game theory was launched in 1944 with
a book, “Games and Economic Behavior”, written
by John von Neumann and Oscar Morgenstern [2].
(John von Neumann is also revered as the father of
the digital computer, software, and algorithms)
Game theory has generated great interest and
activity since then, due to its multiple applications
and benefits.
III. Reality is a Game
Let us take a real-life situation and represent it as a
game. We enter the D'Light Photo Studio run by
Sunny and Shady, who work in morning and
evening shifts respectively. As each uses the photo
lab, it becomes increasingly cluttered and slows
them down. Hence, Sunny and Shady plan to clean
up the lab for an hour every day during their
respective shifts. If one of them forgets to clean the
lab, it takes the other three hours to clean it by
himself. If neither cleans the lab one day, each loses
two hours of productive time. So we have a
situation where each one's decision (whether to
clean the lab or not) affects the other. What is the
most likely outcome? Maybe game theory can tell
us, so let us model this situation as a game.
A handy way to represent this game is to use the
two-by-two matrix shown in figure 1. Sunny and
Shady each have two choices – clean the lab or skip and they choose independently. This results in
four possible outcomes. Sunny's choices are
represented on the top and Shady's on the left, and
the four outcomes by each of the boxes. The
numbers in the boxes represent the cost to each of
them in hours – maroon for Sunny, green for Shady.
These numbers match the costs described above.
For example, if Sunny chooses to skip and Shady
chooses to clean, then Sunny spends 0 hours but
Shady spends 3 hours.
Sunny
Clean
Clean
Skip
1
1
0
3
Shady
Skip
3
0
2
2
Figure 1: Game Representation
What would be the best decision for an individual in
this setup? We will assume that each worker would
prefer to minimize his work. When one of them, say
Shady, is deciding for himself, he will prefer to skip
cleaning. To understand why, let us assume Shady
initially chooses to clean. Then his cost is either 1 (if
Sunny cleans too) or 3 (if Sunny skips). Now Shady
considers changing his choice to “skip”. To his
pleasant surprise, he finds that his cost actually
reduces from 1 to 0, or from 3 to 2, depending upon
the starting point. Thus, he will always be inclined to
skip. Our game is symmetric, so Sunny would also
choose to skip by the same logic.
This means that the final outcome of this game is
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
25
that nobody cleans and everyone pays the price.
(Such a stable state is called a Nash equilibrium of
the game, in honor of the mathematician John
Nash, who invented the concept). It is stable
because no player will change their decision once
in the equilibrium state: if any were to change their
decisions, their costs would increase. Here we
have the very realistic result that independent,
selfish decisions by interdependent people actually
result in a sub-optimal system. This model can help
us understand how people may think about sharing
household chores, or what positions countries
may take regarding pollution limits.
IV. Applications of Game Theory
Negotiations
Business and economics were the original
applications that gave impetus to game theory.
Business negotiations of various types have been
successfully modeled, because similar constraints
apply whether two people are negotiating the
price of a car, that of a corporation, or that of oil in
the international market. A simple model for a
negotiation is a “Take-it-or-leave-it” game, where
one player makes a one-time offer only. A deal is
struck if the other player accepts it, else both
players walk away. The critical parameter to model
is the value attached by each player to the
outcome, and each player's expectation of a
minimum value he / she would have to get from the
deal.
A serious real-life negotiation will rarely stop with a
one-time decision as above; it is likely to go backand-forth, as each party would like to reach
some workable solution. We model this with an
alternating-offer game, where each player can
make a counter-offer. The time value now comes
into the picture – as the game continues, the value
to each player typically reduces due to real-life
deadlines [3]. The outcome of the game also
depends upon the amount of information available
to the players about each-other's valuation of the
deal. A player who knows the value of the deal to
the other player can negotiate the best deal for
themselves. Our intuition supports this principle,
26
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
and the extent of commercial and political
espionage activities confirms it.
Subsidy Decisions
Subsidies for local industries are a hotly debated
topic in international trade agreements. The effect
of subsidies on the industries, the consumers, and
the national economies has been modeled [4]. A
properly-built model can estimate the impact on all
parties. The model chosen in the reference leads to
the conclusion that subsidies help the industry but
hurt the consumers. A what-if analysis can
determine the optimal subsidy levels, once the
country's goals are defined.
Auctions
Auctions have long been used for selling goods with
fluctuating demand (e.g. food grains) or goods
without a fixed value (e.g. fine art). They are
popular because they can maximize a seller's profit
by driving up prices to the highest valuation among
buyers. Common auction types are English auctions
(bidders raise the price until a single bidder remains
– often used for selling artworks), Dutch auctions
(seller drops a high price until one bidder agrees –
used for selling flowers), first-price sealed-bid
auctions (used for selling oil exploration licenses),
and second-price sealed-bid auctions (used for
selling foreign exchange).
Auction rules have a big impact on the outcome,
because bidders change their strategies in different
ways. There have been several instances where the
auction outcome was highly in favor of the buyers
and the seller lost potential revenue. For example,
satellite TV licenses were auctioned in Australia in
1993 [5]. The rules allowed buyers to overbid, and
did not penalize them for defaulting on winning bids.
As a result, buyers placed multiple bids at different
values for the same licenses. They won the auction
with their high bids, defaulted on them, and then
automatically won the same license at their lower
bids. This continued through several sequentially
lower bids. They finally reached a low final winning
bid that was less than half their first bid. The
government naturally lost significant fees. Spectrum
license auctions in other countries, including India,
have been controversial as well [6]. Such problems
can be prevented through systematic and foolproof
design of auction rules using game theory.
Computer Vision
An interesting and novel application of game
theory is in computer vision. When analyzing noisy
images by computer, it is difficult to accurately
identify and distinguish objects. Research has led to
the development of several object identification
techniques, such as color-based techniques and
edge-based techniques. These algorithms have
different strengths and weaknesses. It would be
better if we could use them together to get best
overall results. Game theory can be used to make
these algorithms “compete” against each other in
such a way that they eventually converge jointly
onto a single solution. This has been shown to yield
more accurate results as compared to a single
algorithm [7].
There are many more uses of game theory in the
design of networks, reputation systems, and other
complex multi-player interactions. Game theory
has been used to analyze social phenomena, animal
behavior, and even philosophical questions, e.g. to
determine if rationality implies self-interest, and if
self-interest leads to morality. This clearly brings
out the wide-ranging potential of the central ideas
of game theory: individual choice, incentives, and
interconnected systems.
V. Future Trends
The power of game theory continues to be
harnessed for growing uses. One such sub-field of
economics is mechanism design. A mechanism can
take the form of an auction, a vote, or a game. It is
defined by a set of rules to be followed by the
participants. The mechanism designer's job is to
construct the rules carefully such that a particular
goal is achieved, regardless of player preferences
or manipulation attempts. For example, an online
auction site may wish to maximize its profits in the
shortest time. A government contract tendering
process may need to achieve the maximum “social
good”. One example of a robust mechanism is
Vickrey's second-price auction, which has been
proven to achieve the best overall result for all
parties, including the seller and the buyers [2,8].
Metagames look at mechanism design as
another higher-level game.
The goal of a metagame is to design an optimal game
that leads independent-thinking game players to the
desired decisions. (This is curiously similar to how
current-day C compilers are themselves written in
C!)
Another field bridging game theory and biology is
evolutionary game theory. Preferences of players
can change across generations, and their
evolutionary outcomes can be explained. For
example, the reason males of many bird species
(such as the peacock) develop highly decorative
plumes is explained mathematically by their
competitive advantage in finding mates. Coevolution between species has been similarly
explained. As one example, a certain butterfly
species and an ant species have co-evolved because
the ants protects the caterpillar from parasites, and
the caterpillar rewards the ants with sugar- and
protein-rich secretions.
Accurate predictions have a tremendous universal
appeal – just ask the angel investor trying to select
the right company to fund, or the government
trying to negotiate an international arms limitation
agreement. Game theory is the latest technique
being used to guess the future. Early indications are
promising: experts have made accurate predictions
about government spectrum auctions, national
politics, court cases, military tactics, and even
terrorist behavior [9].
Game theory-based software shall continue to
become more powerful through experience and
new ideas. And the crowds that today seek out
astrologers and godmen for a peek into the future
may well line up tomorrow outside the offices of
“Game Theory Prediction Software Inc”. My guess
is that we shall soon be able to model religion using
game theory. Indeed, can we not look at religion as a
cleverly designed “game” with defined rules that
attempt to achieve the greatest individual and social
good? Despite the good intentions of religion
founders, our society is still far from perfect. Game
theory may enable us to design improved social
mechanisms to encourage good behaviors such as
recycling, cooperation, and altruism.
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
27
VI. Conclusion
[5] “Auctioning Spirits Distribution in West
Virginia and Pennsylvania,” Report byProf.
Andrew J. Buck, February 1997. Available:
http://courses.temple.edu/economics/wkpap
ers/auctions/auctions.htm
Going back to the game we began with, what did
you decide to offer me? I will open up and tell you
that I would have settled for anything from $50 up. I
would be content to recover at least part of my
money, but I would be insulted with an offer any
lower.
[6] “Spectrum auctions in India: lessons from
experience,” R.S. Jain, Telecommunications
Policy, Elsevier, volume 25, issues 10–11, pp.
653-780, November 2001. Available:
http://rru.worldbank.org/Documents/Papers
Links/spectrum_auctions_india.pdf
References
[7] “Game-theoretic integration for image
segmentation,” A. Chakraborty and J.S.
Duncan, IEEE Transactions on Pattern
Analysis and Machine Intelligence, volume 21,
issue 1, pp. 12-30, Jan 1999.
[1] Game Theory, Theodore L. Turocy and
Bernhard von Stengel, CDAM Research
Report LSE-CDAM-2001-09, October 2001.
Available:
http://www.cdam.lse.ac.uk/Reports/Files/cda
m-2001-09.pdf
[2] Algorithmic Game Theory, edited by Noam
Nisan, Tim Roughgarden, Eva Tardos, Vijay V.
Vazirani, Cambridge University Press, 2007.
Available:
http://www.cambridge.org/journals/nisan/do
wnloads/Nisan_Non-printable.pdf
[3] “Bargaining Games – An Application of
Sequential Move Games,” Lecture notes by
Prof. John Duffy, January 2012.
Available:http://www.pitt.edu/~jduffy/econ1
200/Lect03_Slides.pdf
28
[4] “An Introduction to Game Theory with
Economic Applications,” Lecture notes by
Prof. Andrew J. Buck. Available:
http://courses.temple.edu/economics/Econ_
92/Game_Lectures/Static_Intern_Trade/heck
sher.htm
Game theory is a very interesting branch of
mathematics that seeks to model – well, you and me
and the whole world around us. That is indeed a
lofty and exhilarating goal. It does have its
limitations – assumptions such as “players always
make rational choices” may not always hold true.
Nevertheless, it brings us tantalizingly close to
answering, “Why are things the way they are?”,
“What will happen next?”,and “What can we do to
make things better?” So let's keep on playing …
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
[8] Multiagent Systems - Algorithmic, GameTheoretic, and Logical Foundations,
YoavShoham and Kevin Leyton-Brown,
2009.Available:http://www.masf
oundations.org/download.html
[9] “Modellingbehavior: Game theory in
practice,” Article in The Economist,
September 3, 2011.Available:
http://www.economist.com/node/21527025
The math professor just accepted a new position at a university in another
city and has to move. He and his wife pack all their belongings into cardboard
boxes and have them shipped off to their new home. To sort out some family
matters, the wife stays behind for a few more days while her husband has
already left for their new residence.
The boxes arrive when the wife still hasn't rejoined her husband. When they
talk on the phone in the evening, she asks him to count the boxes, just to
make sure the movers didn't loose any of them.
"Thirty nine boxes altogether", says the Prof on the phone.
"That can't be", the wife exclaims. "The movers picked up forty boxes at our
old place."
The Prof counts once again, but again his count only reaches 39.
The next morning, the wife calls the moving company and complains. The
company promises to check; a few hours later, someone calls back and reports
that all forty boxes did arrive.
In the evening, when the Prof and his wife are on the phone again, she asks:
"I don't understand it. When you count, you get 39, and when they do, they
get 40. That's more than strange..."
"Well", the Prof says. "This is a cordless phone, so you can stay on the line
and count with me: zero, one, two, three,..."
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
29
This Figure is 100 year old patent
30
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Hybrid Automotive Applications
About the Author
Tarun Kancharla
Vision Systems,
CREST,
KPIT Cummins Infosystems Ltd.,
Pune, India.
Areas of Interest
Signal Processing and
Pattern Recognition
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
31
I. Introduction:
With rising petrol and gas prices and increased
environmental awareness, consumers are looking
for alternate technology for automobiles.
Manufacturers are making significant changes in
car design to meet the consumer expectations.
Hybrid and Electric vehicles are a step in this
direction, as they give better fuel efficiency and low
emissions [4].
Hybrid Vehicles represent a revolutionary change
in vehicle design. The fundamental difference is the
addition of a large-scale energy storage system to
the vehicle. In the Hybrid vehicles, the drive train is
powered alternately by the gasoline/fuel and the
electric energy obtained from the energy storage
system. By using the energy storage system as a
buffer, the engine can be operated at its most
efficient condition and reduced in size while
maintaining the overall performance of the vehicle.
Because of the reduced size and more efficient
operating condition, the fuel usage and emissions
of the vehicle are dramatically lower than
conventional vehicles.
There is a lot of research going on to develop and
improve the efficiency of the Hybrid Vehicles.
Mathematical modeling plays a major role in this
research. The most important use of mathematics
is its ability to provide an abstract language for
studying underlying relationships. These
relationships can be used to connect seemingly
unrelated observations and gain insight into the
system. In the current article, we discuss how
Mathematical modeling is being used in the
development of different parts/systems of a hybrid
vehicle.
II. Battery Management System (BMS):
Battery modeling, or the mathematical modeling
of batteries, plays an important role in the design
and use of batteries [3]. Since the hybrid vehicle
uses electric power stored in a battery as
alternative to fuel, it is extremely important to
monitor the battery characteristics and
performance. A simple model of battery is as
shown in Fig1. The field of battery modeling can be
divided into the following two areas.
Fig1. Simple Model of a battery
32
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Estimation of battery performance:
Given a constructed battery, the problem is to
estimate how that battery will perform under the
driving conditions. This means, estimating the SOC
(State of Charge) and SOH(State of health). SOC
represents the amount of charge left in the battery,
analogous to the amount of fuel in the fuel tank
while SOH represents the health of battery,
analogous to the wear and tear of any equipment.
Battery Design:
Here, the problem is to estimate how the design of
a battery influences its performance. Since the
mechanisms involved in many battery chemistries is
still not clear, the modeling of battery design is still
being studied. As we understand more about the
exact nature of the mechanisms, we will be better
able to design a required battery.
The SOC of battery can be determined by the
following methods
· Voltage equation: Voltage equation uses battery
characteristics to determine the SOC
Vt is the voltage at load, I is the current, OCV is the
open circuit voltage and R is the net resistance of the
circuit.
· Current equation: The integration of c u r r e n t
passing through the battery gives an estimate of
the SOC.
Where Δt is the time between samples
A combination of both the above equations is used
to get an accurate estimation of SOC. By modeling
the parameters of the battery as shown in the above
equations, the SOC of the battery can be easily
determined.
The problem of SOC estimation is also important in
portable computers, UPS, Aerospace batteries etc.
III. Hybrid Control System:
The efficiency of a HEV depends on the switching of
the power from the gasoline engine to electric and
vice versa. Based on the mechanical architecture,
HEVs can be divided into three categories: parallel
hybrids, series hybrids, and power-split hybrids.
How to determine which technology to use? Using
mathematical modeling one can test which
architecture is suitable for a specific HEV design.
Some engineers [6] have presented a model for
power-split hybrid vehicle. The flow of energy is as
given in Fig 3.It uses a planetary gear set shown in
Fig. 2, to divide the power. By mathematically
modeling the gear set, we will be able to clearly
understand the torques provided by the motors
and how much power is used to charge the battery.
The equations that are used to describe the
dynamics are given in [6] and are listed below for
convenience.
components of a hybrid vehicle are modeled.
ADVISORS, PSAT, VTB are a few examples of
simulation platforms. The input parameters are
set according to the vehicle and a drive cycle is
given as input to the model. Various parts of the
battery can be monitored to understand their
performance. Fig 4 shows the output of a
simulation in ADVISOR [5]. The output tabs can
be changed dynamically to view different
outputs.
By using the above equations, the net power
supplied to the battery can be calculated.
Therefore, the actual experiment need not be
performed but a simulation will tell us the charge
supplied to the battery. A comparison is done
between the simulation and experimental results
in [6] and the difference is found to be minimal.
This is the importance and convenience of using
mathematics for analyzing the control system.
Fig4. Simulation Result Window
With the help of such simulation platforms, the
performance of various components of the
system can be known without actually building a
prototype.
Fig2: Planetary gear system [6]
Fuel tank
Wheel
Engine
Generator
Planetary
gear
Battery
Motor
Fig3: Energy flow of dual hybrid electric vehicle [2]
IV. Simulation Platforms:
In most of the applications, developing a complete
system and testing it for various aspects is time
consuming and a costly process. Using simulation
platforms we can avoid the building of expensive
prototypes as the testing of the complete system in
various test conditions can be carried out using the
simulation platforms.
Since the Hybrid vehicles' technology is in a
development phase, simulation platforms can be a
great way of testing the concept model before the
complete prototype is built. To develop the
simulation platforms, the engineers do not need
the actual prototype of the system but just the
mathematical model of its behavior. The model of
the vehicle can be loaded onto the simulation
platform and the vehicles drive ability can be
checked in various real life scenarios. D.W. Gao, C.
Mi, A. Emadi in [1] present how different
V. Summary:
While it is possible to perform an experiment and
obtain the data, it would be very hard to interpret
the results and understand the underlying
concepts without the help of mathematics. We
have seen how mathematical modeling is used in
developing and testing of some components of a
hybrid vehicle. The major application of
mathematics in present day technology is in
modeling of systems to study its performance.
References
[1] D.W. Gao, C. Mi, A. Emadi, “Modeling and
Simulation of Electric and Hybrid Vehicles”,
Proceedings of IEEE, April 2007.
[2] D.H.Kim,Y. Park, “Modeling and Design of
Hybrid Control System for Dual Hybrid
Electric Vehicle Drive trains”, FISITA World
Automotive Congress, June 2000.
[3] R. M. Spotnitz, “Battery Modeling”, Interface,
The Electrochemical Society,2005.
[ 4 ] h t t p : / / w w w. c a r b o n - m o n o x i d e p o i s o n i n g . c o m / a r t i c l e 5 - h y b r i d - c a remissions.html
[5] Yuliang Leon Zhou, ”Modeling and Simulation
of Hybrid Electric Vehicles”,ME Thesis,
Department of Mechanical Engineering,
University of Victoria 2007.
[6] J. Liu and H.Peng, “Modeling and Control of a
Power-SplitHybrid Vehicle”, IEEE Trans. on
Control Systems Technology, 2008.
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
33
34
Network Operating System (NOS)
About the Author
Arun S Nair
Automotive & Allied Engineering, Bangalore
Areas of interest
In-vehicle Networking &
Embedded Systems
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
35
I. Introduction
In-vehicle Network domain is one of the fast
growing domains in automotive industry. By
knowing the enormous scope in this and
understanding the global recognition for
development of NOS components to the Original
Equipment Manufacturer (OEM), KPIT Cummins
decided to contribute to automotive industry for
its growth by developing in-vehicle Network
Operating Systems products (NOS). In-vehicle
Network Operating Systems products consists of
the four major components viz., Boot-loader,
Communication, Diagnostic Kernel and Network
Management Components. NOS products have
changed the paradigm of automotive electronics
by reducing the weight, providing the flexibility,
removing the redundant sensors and avoiding
inconsistency between suppliers.
The basic building blocks of engineering are
mathematics. Automotive domain or an in-vehicle
networking product is not an exception. There is
multiple usage of mathematics across NOS
components, namely, pointer arithmetic, search
algorithms, security algorithms, decompression
algorithms and checksum algorithms.
This article gives a glimpse through few instances
on linkage of NOS and mathematics. In addition,
there is a separate section on 'Measurement
techniques' used for timing measurements.
II. NOS and Arithmetic Operations
Arithmetic operations are part of NOS
components. This section describes the usage of
pointer arithmetic that makes NOS
communication configurable after the build of
application i.e., post build configuration.
The crux of this implementation is to gather the
configurable information through pointer access.
Signal database at Flash is divided into 2 groups and
the first group provides the relevant index access
pointers for both flash database and RAM
database. The second group contains the variable
flash memory, which in turn will have the access to
RAM pointers as well as flash pointers. The gist of
RAM access pointer calculation is as shown below.
Frame Data Primary
pPrimaryData pRam_Start. Then first pPrimaryData + 8 (next pPrimaryData)
Frame Data Secondary
pSecondaryData of the first multiplexed frame pPrimaryData of Last Frame + 8
Frame Data Dummy
Dummy pSecondaryData of last Multiplexed frame + 8 Where, Dummy
pData of unused signals)
Frame Status Bitmaps
pStatusData (for the first CAN Network Interface) FrameDataDummy + 8
pStatusData of the Next NWI pStatusData of the previous NWI + (UcNoOfMaskBytes * 4)
36
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
III. NOS and Algorithms
This section provides the details of NOS features,
where the usage of polynomial, linear algebra and
various algorithms considered.
Checksum Algorithms
Checksum algorithm is used to verify the
consistency and correctness of data. CAN and LIN
protocols mandates to have the data (information)
correctness verified using CRC and module256
checksum algorithms respectively. For proper
functioning NOS COM component, signal database
stored at ECU memory should be correct and
consistent. In addition, this is to make sure the usage
of correct subset of input configuration files during
the signal database creation and during the
application build.
The following steps are carried out for as part of
signal database verification.
Step 1: Using PC based tool, generate the signal
database along with calculate the checksum of
database
Step 2: This calculated checksum will be stored as
part of database
Step 3: While executing the checksum value
measured using same algorithm
Step 4: Compare the stored value with the
measured checksum value
Step 5: If both the values are same, the execution of
program takes off, otherwise stops the execution
As part of this, CRC Polynomial, x16+x12+x5+1
(1021 hex) polynomial with initial value: FFFF (hex)
was implemented. To have engineering tradeoff of
memory size versus speed two types APIs
implemented.
l
CRC Checksum using 4Bit Lookup Table: The 4bit table was prepared with pre-cooked values of
the given polynomial and the size of table was
with 32 bytes.
l
CRC Checksum using 8Bit Lookup Table: The 8
bit table of the given polynomial with the
database having 256 bytes of pre-cooked data.
This API provides higher performance.
Security Algorithm
The Security Access service was implemented in all
programmable ECUs to restrict access from
unapproved tools and security algorithm unlocks
the ECU for download and upload of data from
authenticated tools. There are couples of popular
Security Algorithms include:
l
Hash Message Authentication Code (HMAC)
Message Digest version (Md5)
l
HMAC Secure Hash Algorithm (SHA)
l
Data Encryption Standard (DES)
DES-Cipher Block Chaining (CBC)
l
Triple DES (3DES)
l
Digital Signature Algorithm (DSA)
l
EESE Common Security Algorithm
l
NOS Software download component uses such
security algorithms to make sure the authenticity
of tester tools while flashing ECUs.
Encryption & Decryption
There are two broad classifications of encryption
techniques—symmetric and asymmetric.
In
symmetric encryption, the same key (private key)
is used for both encryption and for decryption. In
asymmetric encryption, public-private key pair will
be used. This procedure is also known as PKI or
Public Key Infrastructure where data user sends
the public key to sender and send uses this key for
encryption. User shall decrypt the data using it
with its corresponding private key.
Decoding, the compressed stream results
back to the original string.
One symmetric encryption technique widely used
is by XO Ring with same pattern twice to
reproduce the data. For example, data value 0xFF
with private key is 0x55 and the resulted
encrypted data is 0xAA. Publisher sends this data
'0xAA' and while reception incoming data XORedwith 0x55 results to reproduce 0xFF. As part of
NOS SWDL component, such a simple encryption
technique was used to avoid any accidental
mistakes by unwanted execution of stored flashing
routines at flash.
Note that the pointer is output only if it points to a
match longer than the pointer itself; otherwise,
explicit characters are considered. NOS software
downloads component use one of these
decompression techniques to reduce the data
download time.
Compression & Decompression
Software compression is used to minimize the size
of the actual data to transmit to in-vehicle ECUs
during software download. The compression
performed outside the vehicle before the data
transmitted for the flashing process and the
software decompression at vehicle ECU unpacks
the received data during software download.
There are a couple of compression techniques
used in this domain. The Lempel–Ziv (LZ)
compression methods are among the most
popular algorithms for lossless storage. LempelZiv-Storer-Szymanski (LZSS) is another lossless
data compression method, a derivative of LZ77 [3]
that was created in 1982 by James Storer and
Thomas Szymanski. LZSS is a dictionary encoding
technique. It attempts to replace a string of
symbols with a reference to a dictionary location of
the same string. An encoding of the sample string
“cabracadabrarrarrad” depicted in [2].
Search Algorithms
To retrieve the configured frames for particular
ECU, both linear and binary search algorithm was
implemented as part of NOS COM kernel. Binary
search or half-interval search algorithm finds the
position of a specified value within a sorted array.
If N element exists in an array, binary search
algorithm takes exactly [log2 (N) + 1] iterations.
However, in linear search, whose worst-case
behavior is N iterations. Hence, the searching time
was reduced drastically for binary search with
respect to the linear search when there were more
entries in array.
IV. Timing Measurements Techniques in NOS
This section of article explains the timing
measurement techniques used as part of NOS
component development. Those measurement
techniques predominantly use arithmetic
operations.
Worst-case execution timing
This is to measure and reach out equation for
worst-case execution time calculation of APIs. The
following steps will be included as part of this.
Step 1: Factorize API into smaller function blocks
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
37
Step 2: Execute the function block and measure
the time taken by each function block using a
typical clock frequency (E.g. 8 MHz)
Step 3: Create the timing equation for each API as
below.
CPU load =( Input CPU time + Output CPU time )/ Total
Cycle time
For example, the time taken for Input and output
API is around 1.2ms and 0.8ms respectively with the
total cycle is 5 ms and hence the CPU load is going
to be 40%.
Jitter Measurement
Jitter indicates the variation of invocation of APIs
and its is the undesired deviation from true
periodicity. commonly used metrics: absolute jitter,
period jitter, and cycle to cycle jitter.
Step 4: Worst-case timing of API is calculated
using above set of equations
Step 5: Repeat the similar procedure of all APIs
V. Conclusion
Thus the calculation of worst case API timing will
be carried out using simple arithmetic operations
for a given clock frequency.
Mathematics is the queen of science and it rules the
world. This study article describes the usage of
mathematics as part of NOS components, which
Ideal-case polling rate
includes algorithms, pointer arithmetic and various
The ideal-case polling rate is schedule rate that of
scheduler APIs without loss of messages. The
equation for ideal case polling rate shall be as
below.
mathematic techniques of timing measurements. It
Ideal-case polling rate = Number of Slots/message
buffers serviced * 1/baudrate * Number of bits;
For calculation, without stuffing the 'number of
bits' shall be considered as 44 + 8 * N for 11 bit
frames and 64 + 8 * N for 29 bit frames. With an
implementation of two slots for multiplexed
reception, the polling rate should be twice the
minimum time required for one complete message
reception is considered. Based on these
constraints, the ideal polling rates at various baud
rates will be as below.
confirms the supremacy of mathematics on
automotive embedded systems; especially on invehicle networking domain.
References
[1] KPIT NOS Component Specifications
[2] www.ee.unlv.edu/~regent/SPACE/Lossless_4.ppt
[3] Jacob Ziv, Abraham Lempel (May 1977), "A
Universal Algorithm for Sequential Data
Baudrate
Std Id with DLC 1 Std Id with DLC 8 Extd Id with DLC 1 Extd Id with DLC 8
(in micro second) (in micro second) (in micro second) (in micro second)
Compression"; IEEE Transactions on
125 KBPS
832
1728
1152
2048
Information Theory 23 (3): 337–343.
250 KBPS
416
864
576
1024
doi:10.1109/TIT.1977.1055714.
500 KBPS
208
432
288
512
CPU Load
CPU load Measurement indicates the percentage
of CPU time (or CPU usage, process time) used by
NOS scheduler functions with respect to that of
total execution cycle time.
38
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Book Review
A book about mathematics? One can imagine how
dry and unexciting it can be! Well, if you are one of
the many who would open this book with this
attitude, you are going to be pleasantly shocked!
The book, Fermat's Last Theorem, is no less than a
suspense novel complete with stories of courage,
sacrifices, tragedy and heroism! It takes the reader
on a thrilling journey about the mystery
surrounding the Greek mathematician, Pierre de
Fermat's last theorem. Not only does it evoke
excitement and curiosity about the most profound
riddle in mathematical history, left merely as a note
transmitted as on edition of BBC Horizon series. It
by Fermat, it provides a deep insight into what
was also aired in America as part of the NOVA
drives mathematics and the mathematicians
series.
around the world.
The author takes the reader through a journey,
starting right from Pythagoras theorem ,which has
The Fermat's last theorem states that
a direct connection to Fermat's theorem, through
over three centuries of research. He describes,
with a novel-like style, the life and character of
Fermat and some of his extraordinary discoveries.
Where a,b,c are integers and n is also an integer; is
He also explains some fundamental and advanced
valid only for n=1 and n=2.
concepts in mathematics, without resorting to
With n=2 it becomes Pythagoras's theorem.
droning equations beyond the very simple x, y and z
Euler spent for 15 years trying to prove it. Many
basics.
others spent good amount of time on coming up
It is simply intriguing to see the passions and
with a proof. However, it was not until 1992 when
emotions involved in the field of mathematics and
Andrew Wiles Proved the theorem.
the undying urge in search of truth. A must read for
Fermat's Last Theorem is a comprehensive
mathematics students and enthusiasts, but no less
account of a proof that took over three hundred
exciting for a those who have phobia for math.
and sixty years, enormous courage sacrifices and
toil to prove. It is a search of proof for a theorem
that was scribbled in a margin of a book by Fermat
with just a passing mention of an ingenious proof.
Simon Lehna Singh is a British author who is known
to write about mathematics and scientific accounts
in a way that is interesting to a common reader.
Some of his other works include ‘The Code Book’,
‘Big Bang’ and ‘Trick or Treatment? Alternative
Medicine on Trial’. Singh, a graduate from Imperial
Sanjyot Gindi
college, joined the BBC's Science and Features
Vision Systems, CREST
Department as a producer and director in 1990.
Areas of interest
He directed 'The Proof', a BAFTA winning
Digital Signal Processing,
documentary on Fermat's last theorem, which was
Computer Vision
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
39
40
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Mathematical Modeling of
Human Arm Impedance
About the Author
Daniel Ruschen
Nalanda Intern,
RWTH University, Germany
Areas of Interest
Controls Engineering,
Signal Processing.
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
41
I. Introduction
III. Human Arm Model
Today, undertaking several tasks is reasonable
using robots instead of humans. The robot is
superior in jobs that demand high accuracy,
repeatability and endurance. To assure these
characteristics, the robot usually needs to have
complete knowledge of the environment.
Unfortunately, dynamic interaction means
handling impacts and unknown contact forces.
Therefore, compliant systems are active topics of
research in the field of robotics. When it comes to
interaction with an unknown environment,
humans are still more than one-step ahead. Thus
mimicking the behavior of the human arm is an
auspicious field of research, and leads to what we
call biologically inspired robotics.
In this article, new methods to identify and
measure human arm impedance are presented.
The focus is on how mathematical modeling helps
understand the human way of handling interaction
with an uncertain knowledge about the
environment.
A well-known quote from Galileo Galilei is “The
Book of Nature is written in the language of
mathematics”. As a scientist, you first need to
describe the subject you want to investigate; and
your language of communication is mathematics.
According to this, the human arm is modeled as a
multibody system linked by rotational joints.
Thereby, the shoulder is assumed a ball joint, the
connection between the upper arm and the lower
arm is modeled as a two degree of freedom joint.
This mathematical model allows us to describe the
configuration of the human arm unambiguously by
the five joint angles.
IV. EMG – Electromyography
EMG is a method to measure activity of skeletal
muscles. The voltage signal measured by the EMG
sensors is related to muscle force and movement.
Figure 2 shows a sample signal of a biceps
contraction. The x-coordinate is the time in
seconds, the y-coordinate the measured voltage on
the skin over the biceps in volts.
II. Mechanical Impedance
The mechanical impedance relates velocities to
restoring forces. It describes the system's
resistance against motion. The inverse of this
measure tells you how hard you need to push to,
for example, move an object in case of translation.
This example also holds for the rotational case.
Figure 1 depicts the stiffness for both cases; d is the
applied displacement and F is the resulting force.
Note that the terms 'impedance' and 'stiffness' are
used interchangeably here.
Figure 2: Sample EMG signal
V. Stiffness Identification
The joints between the parts of the human body are
assumed to have a rotational stiffness. In case of an
unexpected displacement of the arm, this stiffness
produces an opposing torque in the joints. That
results in an opposing force exerted by the arm.
Figure1 : Translational and rotational stiffness
42
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
This behavior provides advantages during
manipulation such as robustness against external
disturbances and task adaptability.
During the whole experiment, subjects were
seated on a special chair depicted in Figure 3 while
a seat belt restrained the upper body. At the wrist,
a plastic cuff supported the connection to the
robot's “end-effector”. Force and torque sensors
were placed at the interconnection (between
robot and limb) and at the subject's mounting base
(under the seat). To estimate the kinematic
configuration, optical tracking markers were
placed at the upper body, upper arm and forearm
respectively. To map EMG to stiffness, EMG signals
were recorded from eight sources on the arm. The
sources are the muscles that are mainly
responsible for the movement of the arm.
Experimental instructions and visual feedback
were given to the subject via a display. For the
stiffness identification procedure, subjects had to
fulfill a force task. The subject's wrist was coupled
to the lightweight robot's “end-effector” and
desired and actual interaction forces and torques
were displayed. After holding a certain force and
torque level for a random time between 1.5 and
2.5 seconds, the robot perturbs the limb in one
direction randomly chosen from the 10
possibilities (two for each joint degree of
freedom). The stiffness of the joints is identified by
solving a linear system of equations, that relates
the by the robot applied displacement to the
observed forces and torques. The stiffness is
determined by a simple least squares solution of
the system; an example of this is depicted in Figure
4.
Figure 4: Example of a least squares solution
VI. Stiffness Determination from
EMG
The EMG data gathered during the stiffness
identification procedure as well as the identified
stiffness is used to train a neuronal network. With
the trained network, it is possible to predict joint
stiffness from EMG data for a certain subject. The
interconnection between the subject and the robot
is no longer needed. This makes it possible to study
human stiffness modulation while carrying out
various tasks without the need of mechanical
perturbations
VII. Conclusion
In this article, a new and unique method to measure
the stiffness of the human arm is introduced. With a
mapping from EMG data to stiffness, combined with
a detailed kinematic model, an accurate estimate of
the arm impedance without the need of mechanical
perturbations is possible. This is essential in order to
determine human arm impedance not only in static
positions but also along a trajectory during task
execution, without the need of perturbation
measurements. Given this framework, it is now
possible to investigate how humans modulate arm
impedance in any task. The resulting measurements
can be used to derive methods of impedance
modulation for robotic arms.
Reference
D. Lakatos, D. Rueschen, J. Bayer, J. Vogel and P. van
der Smagt, “Identification of human limb stiffness in
5 DoF and estimation via EMG”, Extended abstract
submitted for review, 2012
Figure 3: Experimental setup
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
43
Magical Mathematics from Vedas
Introduction
Calculate 998 x 996. How long does it
take?
At least 30 seconds? Vedic
mathematics suggests methods that can
solve such problems and many others in
just 5 seconds flat! Sounds complex? On
the contrary, it is a very simplified,
systematic and unified version of
conventional mathematics. What makes
it so different and useful is its ability to
create interest in mathematics through
fun and satisfaction. It is all about mental
calculation that in turn encourages
development, intuition and ultimately
innovation. Vedic mathematics was
founded by Swami Sri Bharati Krishna
Tirthaji Maharaja who was the
Sankaracharya (Monk of the Highest
Order) of Goverdhan Matha in Puri,
Orissa. It is called “Vedic” because the
sutras are contained in the “Atharva
Veda” – a branch of architecture,
mathematics and engineering in the
ancient Indian scriptures.
According to Vedic mathematics, there
are just 16 Sutras or formulae that solve
all known mathematics problems in the
branches of arithmetic, algebra,
geometry and calculus. The following
table enlists the 16 Sutras-
44
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Let us go through some of the interesting and
magical methods one by one
1. Square of a number containing 1's:
e.g.: (1111)2
· Add number of 1's present in the number, In this
example, it is 4
· Start writing 4 at center and decreasing digit to
right and left as well, till 1
· The answer is 1234321
2. Multiplication of numbers by series of 9:
a) 654 x 999
· Subtract 1 from 654 and write it as the left half of
the answer i.e. 653___
· Now subtract each of the digit of 653(654-1)
from 9 giving 346
· Final answer will be 653346
b) 45 x 999
· Rewrite as 045 x 999, simple!
· The answer is 044955
3. Base Method to find Square of any number:
a) (96)2
· The nearest power of 10 is 100.
· The difference 100-96 = 4, so we further
subtract 4 from 96 and put 92 on the LHS (Left
Hand Side)
· We square 4, make it 16 and put it on RHS (Right
Hand Side)
· Final answer is 9216
b) (17)2
· Take 10 as base and 7 as surplus
· Add 7 to 17 and make it 24
· Take square of 7 which is 49
· As base is 10, RHS can be only one digit
· Hence, carry over extra digit to LHS
· The answer is 289
4. Multiplication of two numbers:
a) 89 x 92
· Subtract the numbers from nearest 10 power as
100
· 89 à
11 and 92 à
8
· Now multiply 11 x 8 = 88, put it as __88
· And do the cross subtraction, either 92-11 =
81 or 89-8 = 81, giving 81__
· Final answer is 8188
b) 997 x 995
· Subtract the numbers from nearest 10 power
as 1000
· 997 à
3 and 995 à
5
· Now multiply 3 x 5 = 15, append 0 to the left
of 15 as 1000 has 3 zeros so put it as ___015
· Do the cross subtraction, either 997-5 = 992
or 995-3 = 992, giving 992___
· Final answer is 992015
c) 106 X 108
· Subtract nearest 10 power, here 100 from
number
· Make them as 106-100 = 6 and 108-100 = 8
· Now 6 x 8 = 48 simple, put it as ___48
· Add both the numbers and put it to the LHS
· 106 + 108 = 114
· The answer is 11448
Quotient = 84 and Remainder = 6
Magical, is it not? These are just few of the
numerous interesting methods that Vedic
5. Multiplication by 11:
a) 56479 x 11
· Append zeros to both ends as 0564790
· Start adding two consecutive digits as 0+9 = 9
and so on
· If carry addition is of two digit then take carry
over
· As 9+7 = 6 and 1 carry over similarly 7+4+1
= 2 and 1 carry over
· Final answer is 621269
b) 777 x11
mathematics has to offer. Fact that is even more
interesting is that all the methods are devised
assuming that the student knows high school tables
only up to nine!
Apart from faster calculation, it helps in number of
ways too. It cultivates an interest for numbers,
sharpens one's mind and increases mental agility
and intelligence. Moreover, it is easy to understand,
easy to apply and easy to remember. It has also been
proved that it develops left and right sides of the
brain by increasing visualization and concentration.
So go on, explore and be pleasantly surprised. From
0
7 7 7
0
(7+0=7)
(7+7=14)
(7+7=14+1=15)
(0+7=7+1=8)
now on you won't need a calculator. Happy
calculating!
Gives us our final answer :- 8 5 4 7
6. Division by 9:
a)
71 / 9
· Take 7 as a Quotient(Q)
· And 7+1 = 8 as remainder(R)
· Finally answer is Q = 7 and R = 8
b)
1202 / 9
· Start taking 1 as quotient 1__ next is 1+2 = 3
and 13_ as quotient
· Ultimately 3+0 = 3. Finally the quotient will
be 133
· And remainder is 3+2 = 5
Anuradha Dhumal
IDEC ODC, Auto SBU
Area of interest
Mechatronics, Control systems
· Answer is Q = 133 and R = 5
c)
762 / 9
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
45
46
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Life without Mathematics
About the Author
Krishnan Kutty
Vision Systems,
CREST
Areas of interest
Computer Vision,
Image Processing, Pattern Recognition
47
“The great book of nature can be read only by
those who know the language in which it was
written. And that language is mathematics.” –
Galileo
There are these terms: addition, subtraction,
percentage, average, calculus, geometry,
trigonometry, and many more of the like; some of
which we are well versed with; most of which, we
feel, are only for the geeks. All these are
techniques, as we all know, have their roots deeply
ingrained in mathematics. However, let us step
back and think if we really know when math had
originated. In order to answer this, let us rewind
big time and throw some light on our ancestors
from the Stone Age. We might be of the notion
that the Stone Age man had nothing to do with
mathematics. The oldest mathematical artifact
currently known is a piece of baboon fibula with 29
notches, dated 35,000 BC[1]. Some of the cave
paintings that depict patterns have been dated
15,000 BC. With time, the Stone Age human learnt
to use naturally available objects and their
combinations for his own benefit. He learnt to
hunt and to rear cattle. However, there wasn't a
way to keep count of his cattle, because he simply
did not know to count. Slowly he started farming
as well. But there again, he had no clue of how
much land he ought to cultivate, how many trees
he had etc. He had to obtain water for drinking and
for cultivation purposes. He probably knew after
much wandering that there is water somewhere
down there. However, after multiple attempts
spanning multiple paths to reach the water body,
he did figure a route that was the shortest in terms
of distance or which took the least amount of time.
Both 'time' and 'distance' are concepts, that we are
aware of, as a part of math, today. He then wanted
to build a hut to live in– something very contrasting
to the life in caves. It was to be made of stones and
some of them had to be cut into a specific shape for
good fit. 'Shape' again is a concept in math.
Fig 2: Some of the earliest known counting
techniques [4]
It is evident from these examples that though
unknowingly and without a name, mathematics was
being used by our predecessors thousands of years
ago. Things are not too different when we try to
visualize the current scenario. It is remarkable that
even today, a majority of mankind count with their
fingers or with stones; they have no real language
for numbers, and moreover do not appreciate the
general concept of numbers beyond specific
examples.
Today we lead a social life. Being social means that
one needs to communicate. Imagine a
communication that happens which is void of
numbers, time, shapes, transactions, money,
distance, etc. - a communication that is void of
anything that is integral to mathematics or exists
because of some strong roots in mathematics. All
kinds of jobs, every household responsibility, and
almost every personal interest involves
mathematics to some degree. Every aspect of life
has something to do with math. Therefore, one
cannot live without it. Nowadays, we use math
without even realizing it. It goes without saying that
without math, many of the mundane day-to-day
activities become difficult and, in many cases,
meaningless. Believe it or not, through pattern
recognition and application of basic skills, the entire
world becomes mathematical and more
importantly meaningful.
Let us further look into some examples in our dayto-day lives for further illustration. Imagine the
buildings and all the other man-made structures
that we see around us.
1.The structures around us:
Fig 1: Some of the earliest cave paintings with
patterns [1]
48
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
Every manmade structure around us is constructed
to specifications based on mathematics. The design
of the entire structure, calculating stresses,
estimating the strength of materials used – all of it
involves math. How could a civil engineer or an
architect plan and design buildings if there were no
math? If it were not for math, mankind would
TechTalk@KPITCummins, Volume 5, Issue 1, 2012
29
probably be erecting structures on completely
adhoc basis, with no importance given to the
safety/integrity of the structure, since there was
no way to quantify the same. If that were the fate
of stationary structures, one could easily argue
whether multiple of the modes of transport that
we take for granted today would have ever existed
at all.
Another very important area is our economic &
monetary system.
II. Our monetary system:
All business transactions that happen today rely on
a highly advanced and complex monetary system.
The currency we use is a form of mathematics. In
today's world, all business and accounting that
happens are about money, which is just numbers.
Math, in its simplest form, is the manipulation of
numbers. Therefore, it would be reasonable to
conclude that math is necessary for any business. If
it were not for math – what difference would
winning a million dollar business mean as against
winning a multi-billion dollar one? The best one
could imagine here is a bigger business in the
second case. Million, billion, zetta, and beyond are
just terms that mean increasing in that order –
nothing more than that. One would not even
appreciate the significance of numbers if it were
not for math.
Without math, one could not calculate interest,
tender exact fare, or even pay a bill. To live in
today's society, it is important to support self and
the family financially, and that is math as well.
Today's businesses report revenue, profits,
margins etc. on a periodic basis. All of this is based
completely on math. The share markets and all the
trading that happens is again all possible because of
math. The trends, bar charts, pie charts,
predictions etc. make sense – thanks to math
again.
At a macro scale, the world's economic system is
driven by math. Countries plan their growth,
exports, imports, budgets, expenses and so on,
which is again all math.
Let us probe a little more on what it would be like if
there were no math in our governance
mechanisms.
III. Governance:
There are many countries in the world that are
democratic in nature. How would the citizens of
these countries elect their representatives if
counting were not known? How would the elected
leaders run the country by planning budgets,
welfare schemes, costs, exports, imports etc.,
since all this is math again? How would the tax
structure be planned? How to keep track of the
population of the country?
For smooth functioning of the government, how
would one decide on municipalities and other
governing bodies? How to decide on the funds that
would be required for these? How would one
decide quantitatively which of these bodies is
making profits and which are working at loss?
How would one plan to lay road-lines or rail-lines,
how much distance should it be laid for, how much
raw material is required, what would be the cost of
this raw material?
Imagine this scenario. If credit card companies
knew that they were dealing with someone who
knows nothing about mathematics; they could
easily show inaccurate calculations of what the
customer owes them (by jacking up the bill).
One major aspect to be looked into is the
development of science and technology, if it were
not for math.
IV. Technology :
George Boole, in the early 1800's had developed a
branch of algebra for logic manipulations, which is
commonly called Boolean algebra. Every computer
in existence relies on it. It is difficult to even imagine
a world today without computers. Progress in
science, commerce and technology are all linked to
math. Mathematics is undoubtedly the mother of all
sciences. More so, many a times proper
mathematical reasoning needs to be developed
before a technological advent/invention.
Quality control and on-time defect free delivery is
the mantra of any business today. Tools like sixsigma, lean six-sigma, etc. are used extensively for
quality assessment and control of projects.
However, the very basis of these tools lies in math –
thanks to the well known Gaussian curve and
associated statistics.
There would have been no possibility to do secure
transactions for any business today, if it were not for
the invention of encryption algorithms; that again is
math. Nations cannot develop their missiles and
other warfare technology if they cannot plan on
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
49
time of flight, trajectory of missile etc.; that is math
again.
Nature, by virtue of its existence, holds the key to
myriads of theories and phenomena that are yet
unexplored. Let us look into what nature has to
offer as far as math is concerned.
V. Math in nature :
We have seen some illustration of how math is
required in our day-to-day life on various fronts.
One important aspect here is that math is also
hidden in Mother Nature. The branching of trees,
placement of leaves, pineapples and pine cones,
the spiral of a conch shell -- all reflect Fibonacci
numbers, which we learn in mathematics.
Likewise, many patterns also occur in nature – in
the ripples on the surface of a pond, in the sun's
path across the sky, and even in snowflakes and in
the stripes of a zebra. Many a times we overlook
what nature beautifully presents to us on the
beautiful earth. It is math that undoubtedly gives
order and structure to what would otherwise be
random and chaotic.
Fig 4: Pineapple scales following
Fibonacci series [2]
VI. Introspection:
Math is present in everyday life. People sometimes
do not realize the amount of math that surrounds
them and the amount of math that they use every
day. The universal language of the world is math,
and people have been using it for thousands of years
across the world. Today's society would not be in
existence without the application of mathematics.
In today's world, it is math that induces analytical
thinking into an individual. The concepts of logic and
reasoning are also derived from math.
Mathematics is not just the study of the
When it comes to music, from the time of Plato
(~400 BC), harmony was considered a
fundamental branch of physics, now known as
musical acoustics.
The aerodynamics that is exhibited by the shapes
of birds and flying mammals are mathematically
studied and used for creating man made flying
machines.
The very fact that the Mother Earth that we are
staying on is round in shape can be explained only
measurement, properties, and relationships of
quantities and sets, using numbers and symbols.
Math is not boring algebra or geometry or calculus.
Math is undoubtedly the language of nature and
each one of us needs to strive to understand it to the
extent possible. Once we start understanding the
language, nature will start opening up myriad of its
yet untold secrets and will pave the way for living in
harmony with it.
References:
1. http://www.math.tamu.edu/~dallen/masters
/origins/origins.pdf
2. http://britton.disted.camosun.bc.ca/fibslide
/fib11.gif
3. http://en.wikipedia.org/wiki/Music_and_
mathematics
Fig 3: Branches and leaves of a plant following
Fibonacci series [2]
50
TechTalk@KPITCummins, Volume 5, Issue 2, 2012
4. http://nrich.maths.org/2472
Q: Why do you rarely find mathematicians spending time at the
beach?
A: Because they have sine and cosine to get a tan and don't need the
sun!
Q: What is the difference between a mathematician and a
philosopher?
A: The mathematician only needs paper, pencil, and a trash bin for
his work - the philosopher can do without the trash bin...
The chef instructs his apprentice: "You take two thirds of water, one
third of cream, one third of broth..."
The apprentice: "But that makes four thirds already!"
"Well - just take a larger pot!“
Two trains 200 miles apart are moving toward each other; each one is going at a speed
of 50 miles per hour. A fly starting on the front of one of them flies back and forth
between them at a rate of 75 miles per hour. It does this until the trains collide and
crush the fly to death. What is the total distance the fly has flown?
(The fly actually hits each train an infinite number of times
before it gets crushed, and one could solve the problem the
hard way with pencil and paper by summing an infinite series
of distances.
The easy way is as follows:
Since the trains are 200 miles apart and each train is going
50 miles an hour, it takes 2 hours for the trains to collide.
Therefore the fly was flying for two hours. Since the fly was
flying at a rate of 75 miles per hour, the fly must have flown
150 miles. That's all there is to it.)
When this problem was posed to John von Neumann, he immediately replied, "150
miles."
"It is very strange," said the poser, "but nearly everyone tries to sum the infinite series."
"What do you mean, strange?" asked Von Neumann. "That's how I did it!”
About KPIT Cummins Infosystems Limited
KPIT Cummins partners with global automotive and semiconductor
corporations in bringing products faster to their target markets. We help
customers globalize their process and systems efficiently through a unique blend
of domain-intensive technology and process expertise. As leaders in our space,
we are singularly focused on co-creating technology products and solutions to
help our customers become efficient, integrated, and innovative manufacturing
enterprises. We have filed for 38 patents in the areas of Automotive Technology,
Hybrid Vehicles, High Performance Computing, Driver Safety Systems, Battery
Management System, and Semiconductors.
About CREST
Center for Research in Engineering Sciences and Technology (CREST) is focused
on innovation, technology, research and development in emerging technologies.
Our vision is to build KPIT Cummins as the global leader in selected technologies
of interest, to enable free exchange of ideas, and to create an atmosphere of
innovation throughout the company. CREST is now recognized and approved
R & D Center by the Dept. of Scientific and Industrial Research, India.. This
journal is an endeavor to bring you the latest in scientific research and technology.
Invitation to Write Articles
Our forthcoming issue to be released in October 2012 will be based on
“Algorithms” We invite you to share your knowledge by contributing to this
journal.
Format of the Articles
Your original articles should be based on the central theme of “Algorithms” The
length of the articles should be between 1200 to 1500 words. Appropriate
references should be included at the end of the articles. All the pictures should be
from public domain and of high resolution. Please include a brief write-up and a
photograph of yourself along with the article. The last date for submission of
articles for the next issue is May 15, 2012.
To send in your contributions, please write to [email protected] .
To know more about us, log on to www.kpitcummins.com .
SM
KPIT Cummins
Infosystems Limited
initiative
Innovation for customers
You can make a difference
TechTalk@KPITCummins April - June 2012
Srinivas Ramanujan
(1887 - 1920)
35 & 36, Rajiv Gandhi Infotech Park,
Phase - 1, MIDC, Hinjawadi, Pune - 411 057, India.
y
“It is his insight into algebraic formulae, transformations of
infinite series and so forth, that was most amazing. ….most certainly
I have never met his equal, and I can compare him only with Euler or Jacobi.”
Mathematician and colleague of Ramanujan, G. H. Hardy