2nd edition of annual magazine of Mathematics and Computing Department in IIT Guwahati. Designed by: Himanshu Bansal

1. Indian Institute of Technology
Guwahati
A
2nd Edition
April ‘11 - March ’12
see inside
1
$ 3 2
Dimension PHP security Infinity Pagerank Chomp

2. Message from the
HoD
I am pleased to learn about the publica-
tion of the second edition of Anantha, the
magazine of MATRIX.
I am hopeful that the publication of this
edition will play an important role in the
dissemination of information on various
relevant academic activities within and
outside the department.
I invite the faculty, students and alumni to
send their views, opinions and write-ups
of academic nature for the purpose of
publication in the magazine. I hope their
contributions would be of great help to
the readers.
I congratulate the editorial team for their
great effort towards bringing out the
second edition, with the expectation that
this magazine will maintain a high stand-
ard of professionalism commensurate
with the academic goals of the depart-
ment.
Dr. Rajen Kumar Sinha
Head of Department
Department of Mathematics
1

3. About Us - MATRIX
MATRIX is the student association of MATRIX aims at making a mark for
the Department of Mathematics. the Department of Mathematics, IIT
Since its inception in 2008, MATRIX Guwahati in the industry. Dimension,
has actively conducted various activi- a technical confluence was started
ties for the intellectual development last year with the vision to provide a
and technical nourishment of its platform for students to interact with
members. MATRIX has organised the mathematical fraternity around
various lectures and workshops in the globe. This year with the success-
fields like Object Oriented Program- ful conduction of the second edition
ming, Web Designing, Stock Index of Dimension, we are definitely a step
Calculation etc. MATRIX plays a key closer to that goal. This year Dimension
role in fostering a friendly environ- hosted 5 online events, ranging from
ment for both student-faculty and Cryptography to Hacking and from
student-student relationships to Math Modelling to Trading and Arbi-
grow. It undertakes social initiatives trage. The 2 days of Dimension wit-
like organising the department nessed an eminent Lecture Series
picnic, freshers and faculty-student with people from financial institu-
interaction sessions. MATRIX recog- tions like HDFC Life, ICICI Securities
nises the importance of a healthy dis- and from technical and research
cussion in the development of stu- institutions like TCS Innovation Labs.
dents and therefore has its own dis-
cussion forum where faculty and stu- MATRIX strives to promote the student
dents can share knowledge and also interaction between academia and
discuss important issues related to industry. It also tries to link with other
current happenings around the mathematics and financial organiza-
globe. tions in various parts of the world.
2

4. Message from the President
Here At MATRIX, we Science and social
continue our onward entrepreneurship.
march towards pro- The past year has
fessional and intel- been an eventful
lectual development, one. From tough
paving the road to courses and labs to
the future. The key- fun filled parties, we
words remain the have seen it all.
same, opportunity, Among other things
challenge, innova- like the department
tion, support and picnic, we had con-
development but the ducted Dimension
efforts have increased which was a great
manifold. The highly qualified and crea- success. The highlight of the year was
tive professors of our department with of course the placement session. The
their midas touch produce winners. details of the placement session are
We are committed towards providing mentioned in this magazine.
the students with ample opportunities Though not very old we have been
to explore different ideas and fields of able to establish ourselves as one of
Mathematics and Computing. the best in the world. Credit goes to
MATRIX values the fact that the col- the undying spirit of the students. I
lege is the most important phase of a would like to mention the tireless
person’s life where he/she learns to work of the previous and present
work in a team, take responsibility, executive councils of MATRIX, for
make critical decisions and most bringing it where it is today. Last but
importantly manage time. A major not the least, I would like to thank our
step towards achieving our objectives faculty advisor Dr. N Selvaraju and
was Dimension, a technical conflu- the chairman Dr. R. K. Sinha for their
ence. Dimension consisted of 5 support and guidance which has
online events and a lecture series been essential in the growth of
which covered the industrial application of MATRIX through the years.
Mathematical Finance, Computer
Paranjay Srivastava
President
MATRIX
3

5. Message from the Editor
magazine includes articles on PageR-
ank Algorithm, PHP security and
Chomp, a strategic two-player game.
The aim of this magazine is to pre-
sent some interesting and enlighten-
ing facts and discoveries from the
fields of Mathematics, Computer
Science and Finance and surely this
issue of ANANTHA is a step forward
in this direction.
I am extremely grateful to HoD, Dr.
R.K. Sinha and Faculty Advisor Dr. N
Selvaraju, who have always encour-
I take immense pleasure in unveiling aged us by providing their invaluable
before you the second edition of the suggestions. I am elated to acknowl-
annual magazine of the Department edge the invaluable contributions of
of Mathematics, ANANTHA. all the members of the Publication
This issue of ANANTHA, brings to Committee for putting together this
you some exceptional articles. The magazine in its present form. Also, I
cover article of the magazine is a would like to express my profound
research article on Multi fractal temporally sense of gratitude to the President,
weighted detrended fluctuation analy- Paranjay Srivastava, MATRIX for being
sis of arctic sea ice data, which has constantly supportive throughout
been an issue of great interest the session and for his efforts in
recently. It also includes interesting helping with this magazine.
articles about the history of Infinity And for the closing. I, for one, think
and its significance in maths, the that we have done a good job in
appearance of Fibonacci numbers, making something you can cherish,
Golden ratio and Phi in nature, the 2 something you can respect, so dive
envelopes paradox and the Prisoners’ into the world of ANANTHA.
Dilemma. Continuing the trend of pre-
senting technological extravaganza, the Infinity is just the beginning...
Shagun Rawat
Publication Secretary
MATRIX
4

6. Contents
Message from the HoD
5

7. A Walk With the History of Infinity
Before starting with and before that you
Infinity, let me give have to get a quar-
a faulty proof of ter of the way, etc.
astatement which i.e. travel through
may highlight on an infinite number
why guys should of process.
look to Infinity in a different way. From this Zeno concluded that
Statement: There exists an infinite motion is impossible. Now a days, a
decreasing sequence of natural num- satisfactory answer of Zeno’s paradox
bers. can be established- through the con-
Proof. Assume for the sake of con- ception of infinitesimal Process or
tradiction that the longest decreasing Limits which was not there at Zeno’s
sequence of natural numbers is time.
finite. Let S = {a1,a2,...,an} be such a Galileo in 1638 noticed a paradox on
sequence. Then, choose some a0 Infinity. He observed the Natural
larger than max ai, and note that S’ = numbers (N) as numerous as the set
{a0,a1, a2,...,an} forms a length of (n+ of its perfect squares S={1,4,9,…} by
1) decreasing sequence. This contra- one-to-one correspondences which
dicts the maximality of S, and hence produces two contradictory state-
completes the proof . ments
Clearly the statement is wrong. But 1) While some natural numbers are
what is wrong in the above proof? perfect squares, some are clearly not.
The error lies in the first line of the Hence the set N must
proof. One is not allowed to assume be more numerous than the set S, or
that a longest decreasing sequence |N|>|S|.
exists. Observe that Well Ordering 2) Since for every perfect square
principle only ensures the shortest there is exactlyone natural that is its
decreasing. Infact in this case longest square root, and for every
decreasing sequence does not exist. natural there is exactly one perfect
Moral of the story: Be careful square, it follows that S and N are
when dealing with infinity! equinumerous, or |N|=|S|.
History of infinity starts with a mis- With the language of Cardinality
conception of Zeno of Elea (490- (number of elements in the set) there
425BC). He stated that-there is actu- can be two propositions
ally no motion, because to get any- 1) that all infinite sets are equal in
where you first have to get halfway, cardinality, and
6

8. A Walk With the History of Infinity
2) that if one set can be obtained by
deleting members of another, then
they have unequal cardinalities. The
latter verdict can be paraphrased
thus: some infinite sets have a larger
cardinality than other infinite sets, or
not all infinite sets are equal in cardi-
nality. Therefore, these two verdicts
of intuition directly contradict one only, it was denied and disbelieved; it
another and cannot both be true as was hated. In particular, Kronecker
we have seen from the last example. (one of Cantor’s teachers) opposed
Galelio’s paradox is paradoxical only Cantor’s ideas and blocked his career.
in the weak sense: it violates our For the sake of later discussion, let us
intuitions. It is not a contradiction as say that a set which can be put into
we always expecting that something one-to-one correspondence with at
similar will be occurring for infinite least one of its proper subsets is
sets from finite sets. self-nesting.
Later innovations due to Georg Charles Peirce in 1885, and Richard
Cantor (worked on 1870-95), is set Dedekind in 1888, proposed to
theory itself, the theory of infinite define infinity through self-nesting.
sets, and the modern concept of infi- According to this proposal, we don't
nite cardinality. The key to this solu- know that infinite sets are self-
tion is simply to define equal cardi- nesting because of some proof; we
nality through one-to-one corre- know it because infinite sets are
spondence, and then to show that defined as those which are self-
these sets can be put into one-to- nesting.
one correspondence with one It's not hard to prove that all infinite
another. Similarly, we can prove that sets, in fact, are self-nesting. And we
some infinite sets have a larger cardi- already knew that only infinite sets
nality than others by showing that are self-nesting, or that no finite sets
they cannot be put into one-to-one have this property.
correspondence. We might cautiously generalize that
Cantor's theory faced tremendous all infinite sets have the same cardinality.
opposition in the late 19th century, Cantor found an elegant proof that
from mathematicians as well as from the power set of any set, finite or
philosophers and theologians. Not infinite, possesses a greater cardinal-
7

9. A Walk With the History of Infinity
ity than the original set; this impor- proposed a hypothetical scenario
tant result is simply called Cantor's that keenly illustrates Cantor’s
Theorem. counter-intuitive results on infinite
In most of the times we faced diffi- sets which we know as paradox of the
culty to visualize Infinity. Descartes Grand Hotel (or Hotel Infinity) which
asks us to imagine, that is, possible to says it is possible to accommodate
visualize,1,000-sided regular poly- countably infinite passengers to a
gon. Can you do it? Try it right now. hotel with infinitely many rooms, all
Chances are- you are, either visualiz- of which are occupied by a unit.
ing something like a 20 or 30-sided There is another paradox discovered
polygon and pretending it has 1,000 by Bertrand Russell in 1901, known as
sides, or you are visualizing a circle Russell paradox, showed that the naïve
and pretending the sides set theory created by
are too small to see with Georg Cantor leads to a
your mind's eye. We contradiction. The non
know exactly what a constructive nature
polygon is; we can of the formal proofs
even compute the relying on ‘Axiom of
interior angle of its Choice’ leads to
sides and, for a given Banach-Tarski paradox
edge, its area and perim- which is on to decompose
eter. But we cannot visualize a sim- a 3-dimensional unit ball into a finite
pler one. number of disjoint pieces and then
One reason I like Descartes' example is reassemble the pieces to form two
that it is finite. Philosophers who think unit balls.
the infinite utterly beyond human Cantor’s basic approach in developing
understanding often fail to notice that his theory of sets was too systematically
their arguments, once made specific, relate the cardinality of various infi-
also apply to very large finite magni- nite number sets (in particular, the
tudes as well. We cannot visualize infi- sets of integer, rational and real num-
nitely many cherries in a tree, but nei- bers) to that of the naturals. He
ther can we visualize a billion. Does started with the convention that the
that disqualify us from using billions cardinality of naturals equal to N0,
intelligibly and accurately? denotes the first transfinite cardinal
In between around 1900’s, the and then showed the hierarchy of
German mathematician David Hilbert cardinal numbers: {1,2,3,…; N0,N1,N2,…}.
8

10. A Walk With the History of Infinity
This led to an extended transfinite Eventually Cantor’s ideas won out
arithmetic and also to the first con- andbecame part of mainstream
jecture of the continuum hypothesis. mathematics.
Continuum Hypothesis (CH): There David Hilbert, the greatest math-
doesn’t exist a set S satisfying |N|< |S| ematician ofthe early 20th century,
< |R|. said in 1926 that
In other words continuum has “No one can expel us from the
greater cardinality that the naturals. paradise Cantor has created.”
But after |N| is it |R| the next cardinal
numbers? Cantor stumbled into this Here are few quotes on INFINITY.
question. But he failed to resolve it. Alcohol gives you infinite patience
This was finally answered many years for stupidity. ---Sammy Davis, Jr.
later, ina way that Cantor never would
have imagined: A point is not part of a line.The small-
It’s neither provable nor dis- est natural point is larger than all
provable! mathematical points, and this is
Which a mathematician can never proved because the natural point has
think at first… continuity, and anything that is con-
tinuous is infinitely divisible; but the
Kurt G¨odel (1940) showed that mathematical point is indivisible
adding CH to the usual axioms of set because it has no size. --Leonardo
theorydoes not produce a contradiction. da Vinci
Paul Cohen (1960) showed thatad-
ding not-CH to the usual axioms ofset A stupid man’s report of what a
theory does not produce acontradic- clever says can never be accurate,
tion. In other words it is independent because he unconsciously translates
of the continuum hypothesis from what he hears into something that he
Ernst Zermelo and Abraham Fraenkel can understand. ---Bertrand Russell
axiomatic theory (ZFC).
Pratibhamoy Das
Research Scholar
9

11. PHP S ecurity
<?php
set_error_handler('my_error_handler');
function my_error_handler($number,
$string, $file, $line, $context)
{
$error = "=========nPHP
ERRORn =========n";
$error .= "Number: [$number]n";
$error .= "String: [$string]n";
$error .= "File: [$file]n";
$error .= "Line: [$line]n";
$error .= "Context:n" . print_r($context,
TRUE) . "nn";
error_log($error, 3, '/path/to/error_log');
}
This article covers some basic con- ?>
cepts related to PHP security which
should be kept in mind while writing Escaping output to user:
PHP code. Say you have taken some data from
the user and saved it in your data-
Error reporting: base. The data can be anything. For
PHP’s error reporting features can e.g. you are saving user comments in
help PHP developers to identify and the database.
locate the mistakes. But the detailed While showing the data again to the
information that PHP provides can be user, make an habit of escaping the
displayed to a malicious attacker, data using the PHP function htmlen-
which is not desirable as the attacker tities(), so that the special characters
can get a lot of confidential informa- entered by the user and thus saved in
tion from that. the database are not interpreted by
So, try to define your own function the browser.
(as below) to handle errors. <?php
This function (my_error_handler) does $data = “<p>hi</p>”;
not echo the php errors but just log $data = htmlentities($data, ENT _QUOTES,
the errors in the file mentioned so 'UTF-8');
that only the php developers can see echo "{$data}";
the errors and nobody else. // Now the output will be: {<p>hi</p>}
10

12. PHP S ecurity
// If we wouldn’t have used htmlenti-
ties() function then the output would
have been: {hi}
?>
The best way to use htmlentities() is
to specify two optional arguments -
the quote style (second argument)
and the character set (the third argu-
ment). The quote style should always
be ENT_QUOTES in order for the
escaping to be most exhaustive, and Security from session hijacking:
the character set should match the The most common session attack is
character set indicated in the session hijacking. This refers to any
Content-type header that your appli- method that an attacker can use to
cation includes in each response. access another user’s session. The
first step for any attacker is to obtain
Escaping data to be entered in a valid session identifier, and there-
database: fore the secrecy of the session identi-
For MySQL users, the best escaping fier is paramount.
function is mysql_real_escape_string
(). If there is no native escaping func- For defense against session hijacking
tion for your database, addslashes() the following technique can be used:
can be used as a last resort. <?php
<?php session_start();
$username = mysql_real_escape _str- if (isset($_SESSION['HTTP_US ER_AG ENT']))
ing($username); {
$sql = "SELECT * if ($_SESSION['HTTP_USER_AGENT']
FROM profile !=md5($_SERVER['HTTP_USER_AGENT']))
WHERE username = '{$username}'"; { /* Prompt for password. */
$result = mysql_query($sql); exit;
?> }
If we dont use mysql_real_escape_str- }
ing() function and variable username else
contains some quotes then the query { $_SESSION['HTTP_USER_AGENT'] =
would give an error and gives oppor- md5($_SERVER['HTTP_USER_AGENT']);
tunity to the attacker to get informa- }
tion about the database. So, basically ?>
this method is providing security
against SQL Injection. Piyush Bengani
st
B.Tech. 1styear
11
x

13. Internship Experience
It was in March 2011 when I got the
An Experience that Remains
offer for the Amazon internship. With
thoughts juggling between the lures
and fantasies of a foreign internship
like MITACS and of associating with
Amazon but having had the taste of a
research internship after the sopho-
more year, I chose to go for the latter.
The beginning: I worked as an intern
in the Amazon Development Center
at Hyderabad for about two and half
months. The internship started with
preliminaries like introduction to the
various cultures and work ethics of
the company, gifting cool company
t-shirts and tumblers!! I was then introduced
to my team named “Customer Returns”
which dealt with refunding and
exchanging customer orders. a single emailing service and move
After a week or so of induction the templates to the Amazon Cloud
sessions on the various building which would make them easily cus-
tools, frameworks etc. that are tomizable according to the various
used at Amazon, my mentor marketplaces that Amazon currently
introduced me to the project I serves.
was to work on. After some grueling research of what
exactly the current daemons did, I
The project: My team sent vari- realized that these were in every
ous kinds of emails to the cus- sense true to their name “daemons”!!
tomers at various stages of the return My team gave me full liberty to
processing. The email processing was explore into choosing any angle to
being handled by various daemons approach my project from. I talked to
running separately and the email various other teams, especially the
templates that were being populated Amazon Cloud team to decide on the
by these daemons were not customizable exact service to store the email tem-
without deployment. The project was plates from the many cloud storage
to integrate the various daemons into services that were available.
12

14. Internship Experience
By the end of the internship, I was
An Experience that Remains
able to successfully implement the
new service for receiving notification
about the kind of email to be sent,
processing the necessary informa-
tion, fetching the appropriate tem-
plate from the cloud, populating the
required data and finally sending the
email.
Experience: The experience of the
internship, if one was to describe it in
a single word was “fulfilling”. Before
going for the internship, the biggest
conundrum I was facing was the deci-
sion to opt for a job, MBA or MS. I
believe that one’s 3rd year internship
should be chosen according to what
one plans to do after graduation.
Being slightly more inclined towards
a corporate future, interacting with
various people from eclectic back-
grounds at Amazon gave me a
clearer picture. Where I was iffy about
my after-graduation-plans before the
internship, I was definitely certain by
the end of it.
Companies like Amazon offer you a
very open culture where anyone can
talk to anyone. I met people
who had joined there immediately different backgrounds in a whole new way.
after graduation, or after MS, PhD I got a whole new insight into how one
and some who had many many years should decide about one’s career and vari-
of experience. Frequent team lunches ous other important aspects of life.
and dinners and activities like foose The independence I got as an intern for my
ball etc provide another way to project was a novel experience in itself. It
refresh and interact with people from really taught me to trust my instinct,
13

15. Internship Experience
to weigh all the various pros caters to umpteen cultures and won-
An Experience that
and cons of the abundant tech- derfully so. The people, the culture,
nology and methods that are everything fills you with vigor and
available to complete one energy. I discovered the wonderful
single task, to come with tech- city with friends and my team. My
niques to solve any problem at team, most of them hyderabadis
hand, to convince others why proved to be wonderful guides to the
what you think is right and most old sites like Char Minar and Gol-
importantly to process and conda Fort. The history of the city is
absorb what everyone in the breathtakingly beautiful and engross-
team says and find the opti- ing. Necklace road, Paradise ki
mum solution considering all Biryani, Karachi Bakery’s biscuits,
possibilities and to everyone’s Birla Mandir, Ramuji film city etc are
satisfaction!! some of the few out of the myriad
wonderful things of Hyderabad one
Hyderabad: After a demanding should not miss.
6th semester, Hyderabad came
as a refreshing and a rejuvenat- A risky decision against a lucrative
ing respite. It’s a city which foreign internship.
Remains An enriching experience.
And a summer totally worth it!
Nikita Garg
th
B.Tech. 4thyear
14

16. Trends and Multi-Fractality
in Arctic Sea Ice
The modulation of the atmosphere/ocean heat flux,
considered as a bellwether of climate change, can be
attributed to the Earth’s polar oceans since their surfaces
are covered by a thin (several meters) mosaic of high
albedo sea ice floes. As a consequence, sea ice is con-
sidered to be a more sensitive component of the
cryosphere to perturbations and feedbacks, particularly
the ice-albedo feedback which has driven large-scale
climate events over Earth history, instead of the massive
meteoric ice sheets that are several kilometers thick.
The retreat of the Arctic sea ice coverage during
recent decades (Figs. 1 and 2), has captured substantial
interest. The fundamental question concerning the
nature of the decay in ice coverage is; whether is it a
trend associated with greenhouse forcing, or is it a
fluctuation in a quantitative record that is short (~ 30
years) relative to the dynamics of the cryosphere on
climatic epochs (> 10^6 yrs.)?
Figure 1. The Sea Ice
Concentration as on
September 15 2001
and on September 15
2007. The Sea Ice
Extent was observed to
be at its minimum
during the September of
2007. This image is
retrieved from The
Cryosphere Today-
UIUC.
That a sufficiently large increase in greenhouse gas
concentration will drive decay in the ice cover, is indi-
cated by both, the past climate data and the basic
physical arguments.
15

17. Trends and Multi-Fractality
in Arctic Sea Ice
However, Tietsche et al. in an article of the system on time scales longer
“Recovery mechanisms of Arctic than the seasonal record may reside
summer sea ice” (2011), numerically in that record itself.
prescribed ice-free summer states at Most observational studies of the
various times during the projection satellite records of ice coverage
of 21st century climates and found extrapolate in time the annual or
that ice extent typically recovered monthly means. Whilst the observed
within several years. Whilst such declines over this troika of decades,
rapid response times can be captured particularly the last decade are striking,
within the framework of relatively our goal here is to begin a systematic
simple theory both internal and effort to distinguish between long-term
external forcings and their intrinsic time correlations and trends in this finite
scales manifest record. In so
themselves in large doing we examine
scale observations whether there exists
of the geophysical a multiplicity of
state of the system. persistent scales
in the data that
Due to the fact can provide a
that we cannot basis for examin-
a-priori exclude the ing cause and
observed decline effect in the geo-
in the ice cover physical scale
as being an observables of the
intrinsic decadal system.
Figure 2. Equivalent Ice Extent (EIE)
oscillation or non
during the satellite era (blue), shown
stationary relative to the mean (red) with the sea-
The basic approach
influence in the sonal cycle removed. EIE, which differs of relevance is the
climate system, from traditional ice extent or ice area, is multi-fractal gen-
we used the defined as the total surface area, includ- eralization of
finest temporal ing land, north of the zonal–mean ice Detrended Fluc-
resolution in the edge latitude, and thus is proportional tuation Analysis
observed record to the sine of the ice edge latitude. EIE (DFA) aptly called
to examine the was defined by Eisenman in an article Multi-fractal
action of multi- “Geographic muting of changes in the Detrended Fluc-
ple scales. The Arctic sea ice cover” (2010), to deal with tuation Analysis
fingerprints of the geometric muting of ice area associ- (MF-DFA). In the
ated with the seasonal bias of the
the noisy dynamics last decade this
influence of the Arctic basin land mass
boundaries.
16

18. Trends and Multi-Fractality
in Arctic Sea Ice
approach has been developed in many the running sum of the raw data. This
directions, from studying extreme offers several advantages over the
events with nonlinear long-term commonly used MF-DFA. Firstly, in
memory, to examining the influence MF-DFA the profile of the time series
of additive noise on long-term corre- is fit using discontinuous polynomials,
lations. We used a new extension of this which can introduce errors in the
methodology called Multifractal Tem- determination of crossover times for
porally Weighted Detrended Fluctua- new scalings, with a particular
tion Analysis (MF-TWDFA) developed relevance at long time scales. Secondly,
by Zhou and Leung, which exploits for time series of length N, MF-DFA is
the intuition that in any time series typically informative only up to N/4
points closer in time are more likely whereas MF-TWDFA can be carried out
to be related than distant points and to N/2. Finally, the generalized
can provide a rather more clear sig- fluctuation functions Fq(s) for all mo-
nature of long time scales in the ments q as a function of time scale s
fluctuation function and its moments. are substantially smoother for all s and
This is expressed by application of this is markedly so for large values. This
weighted moving windows–points facilitates clear extraction of crossover
times from one scaling to another.
Figure3. The profile for the Abedo after Figure 4. The profile for the EIE after
removing the seasonal cycle from the removing the seasonal cycle from the
original timeseries. original time series.
nearer each other are weighted more We used MF-TWDFA to examine the
than those farther away–to determi- multi-scale structure of two satellite
nethe function used to fit the time based geophysical data sets for Arctic
series profile (Figs. 3 and 4); sea ice; the Equivalent Ice Extent (EIE)
17

19. Trends and Multi-Fractality
in Arctic Sea Ice
and albedo retrievals from the cycle from 1978-present is shown in Fig.
Advanced Very High Resolution Radi- 6. Daily satellite retrievals of the directional
ometer (AVHRR) Polar Pathfinder hemispheric apparent albedo are deter-
(APP) archive. The EIE data derives mined from the APP archive. The apparent
from retrievals of satellite passive albedo is what would be measured by
microwave radiances over the Arctic upward and downward looking radiometers
converted to daily sea ice concentra- and thus varies with the state of the
tion using the NASA Team Sea Ice atmosphere and the solar zenith angle.
Algorithm. The mean EIE seasonal
Figure 5. The albedo histo-
grams shown for days in mid-
March (blue) and-September
(red). If there is ice in a pixel for
the 23 years duration of the
data set then we compute the
albedo for that pixel and aver-
age over all pixels that have
met this criterion.
Figure 6. The mean seasonal
cycle of the Equivalent Ice
Extent (EIE) during the satel-
lite era (Fig. 2).
18

20. Trends and Multi-Fractality
in Arctic Sea Ice
We examined the long-term correla- When removing the seasonal scale
tions and multifractal properties of from the original record, the EIE data
daily satellite retrievals of Arctic sea exhibits a white noise behavior from
ice albedo and ice areal extent, for the seasonal to the bi-seasonal time
periods of ~ 23 years and 32 years scales, but the clear fingerprints of the
respectively, with and without the short (weather) and long (~ 7 and 9
seasonal cycle removed. The general- year) time scales remain, demonstrating
ized Hurst exponents and multiple a reentrant long-term persistence.
crossover timescales were found to Therefore, it is not possible to distinguish
range from the synoptic or weather whether a given ice area minimum
time scale to decadal, with several (maximum) will be followed by a
between. Such multiple time scales minimum (maximum) that is larger or
were exhibited in both data sets and smaller. This means that while it is
hence the approach provides a tempting to use an anomalous excursion
framework to examine ice dynamical associated with a low ice year to predict
and thermo-dynamical responses to the following year’s minimum, or that
climate forcing that goes beyond two year’s henceforth, the data do
treatments that assume a process not justify such a prediction.
involving a single autocorrelation Finally, other methods find solely a
decay, such as a first-order autore- rapid de-correlation and whereas we
gressive process. Indeed, the method find multi-year and decadal transi-
shows that single decay autocorrela- tions as well as the origin of the
tions cannot be meaningfully fitted dominance of the seasonal cycle in
to these geophysical observations. long term persistence. Hence, we
Our most important finding is that believe that combining such multi-
the strength of the seasonal cycle is fractal studies of model output and
such that it dominates the power other observations will substantially
spectrum and “masks” long term improve the acuity with which one
correlations on time scales beyond can disentangle the strength of the
seasonal. seasonal cycle in this highly forced
system from the longer term trends.
th
This work has been contributed by Sahil Agarwal (B.Tech. 4thyear),
Woosok Moon (Yale University) and Professor John Wettlaufer
(Yale University)
19

21. 1
2 Pagerank Algorithm
3
Pagerank is an algorithm developed incoming edge.
by Sergey Brin and Larry Page (and
hence the name Pagerank) which is The number of incoming edges relate
used to assign importance or rank to to the importance of the page as per-
various pages on the World Wide ceived by other pages. If a page has
Web. This algorithm is one of the key been referred by many other pages
foundations of Google search then that page is important and must
engine, but it is not the only one, it’s be given a higher priority.
just one of the many indicators used
by Google to rank various pages and Now let’s see how it works with an
present them when searched for. The example. Suppose for simplicity that
algorithm in general may be applied there are only 5 pages, let them be
to any set of entities with cross refer- named A, B, C, D and E. First of all
ences between the entities. each of the pages is given an equal
pagerank, 1/5. Then for any page
It is a mathematical algorithm based (let’s say A), the pages which have a
on graph theory. A higher rank indi- link to that particular page (A in this
cate more value is associated with case) are considered (let them be C
that page and it will be given priority and D), and the new pagerank of A is
(in terms of position in the search calculated given by
results) to pages with rank lower to it.
Pagerank works on the assumption PR(A) = PR(C) + PR(D)
that a person surfing the net clicks Here PR(i) refers to Pagerank of page i.
on the links randomly without read- This is the most basic architecture.
ing or thinking about what the link Next taking into considerations some
has to offer and based on this it others factors, this basic form is
calculates the probability that such a improved.
random surfer will land on a
page. Suppose that page C has a total
of 3 out links out of which 1 is
The basic idea behind the of Page A (repeated links are
whole algorithm is as ignored) and page D has total
follows. All the pages in two out links, one for A,
the World Wide Web then according to the
are taken as nodes random surfer model the
of a directed graph probability of landing on
and each incom- page A is actually one
ing hyperlink is third from page C and
taken as an half from page D, so
20

22. 1
2 Pagerank Algorithm
3
we redefine our formula as term (1-d)/N (N being the total
number of pages) is added. This
PR(A) = PR(C)/3 + PR(D)/2 addition is done more due to math-
ematical reasons than any intuitive
So now generalising the model for reasons, the mathematics behind it
any number of pages the recursive being that the sum of pageranks of
definition of pagerank given as all the pages must be 1. Hence the
new improved formula
PR(i) = ∑ j PR(j)/O(j)
PR(i)= (1-d)/N + d(∑ j PR(j)/O(j))
Where O( j) represents the total
number of out links from the page j. Various studies have been conducted
to empirically determine the value of d
Now to make the model more realis- and it is generally taken to be around
tic, the following fact is considered. 0.85 as was suggested in the original
The probability of a surfer clicking on paper by Sergey Brin and Larry Page.
links decreases with the number of
clicks. This is a necessity, or else the Now comes the computation part. As
person will never settle down for a is clear from the formula itself, it is
particular page, which is not at all recursive in nature, also with each
realistic. recursion the pagerank of particular
page will keep on increasing. So
So a damping factor, d is introduced where do we stop? The answer
defined as the chance that a person depends on how precise we want the
on a particular page will actually click pageranks to be. So depending on
on one of the links on the page. the needs, a number  is decided
upon and after each iteration, the
So the total pagerank score calcu- difference between the final and the
lated by the above formula is multi- penultimate pageranks is calculated
plied by the factor d to take into and when it is less than the desired
account the above arguments. Also a value of , the iterations are stopped.
Rajat Kateja
B.Tech. 2nd year
nd
21

23. Chomp
Even the simplest of games can pose the first move of a winning strategy
tough mathematical challenges. One for the second player. If that is the
such example is the game of Chomp. case, then the first player could have
Chomp starts with a rectangular array opened with that very move and
of counters arranged neatly in rows been guaranteed a win. Therefore,
and columns. A move consists of the second player could not have a
selecting any counter, then removing winning strategy.
that counter along with all the coun-
ters above and to the right of it. In Winning strategies are known for a
effect, the player takes a rectangular few cases. The first scenario is when
or square "bite" out of the array—just the array is a square. Here the first
as if the array player can win by
were a rectangu- selecting the counter
lar, segmented that is diagonally
chocolate bar. up and to the right
Two players take of the poisoned
turns removing counter. This would
counters.The leave only the last
loser is the one row and column
forced to take the with the poisoned
last "poisoned" piece at the vertex.
counter in the From that point on,
lower left corner. the first player
The chocolate- simply takes from
bar formulation one line whatever
of Chomp is due his or her oppo-
to David Gale, of nent takes from
the University of California, Berkeley. the other line. Eventually, the second
player must take the poisoned piece.
As it turns out, the first player can win The second scenario is when the array
for any rectangular position bigger is two columns or two rows wide. Here
than 1 x 1. To see this, assume that the first player can always win by
the second player has a winning taking the counter at the top right so
strategy for any initial move of the that one column or row is one counter
first player. Now suppose the first
longer than the other. From then on,
player makes the first move by
the first player always plays so as to
chomping just the top right-hand
restore this situation.
cookie. Then there is a reply which is
22

24. Chomp
Chomp belongs to a particular family game played on a partially ordered
of two-player combinatorial games set P with smallest element 0. A move
(games in which nothing is hidden consists of picking an element x of P
from the players and no chance is and removing x and all larger elements
involved), which are described as from P. Whoever picks 0 loses.
poset games. A poset, or partially
ordered set, is a set of elements in While on one hand this is very easy
which some elements are smaller to mathematically prove that such a
than other elements but not every winning strategy exists, finding out
pair of elements can necessarily be that strategy is still a mystery that
compared. Chomp can be seen as a nobody yet has been able to solve.
Tanvi Rai
th
B.Tech. 4thyear
Two envelope paradox
I have two envelopes, and inside each You reason as follows: My envelope
I have put some money. In fact, one has Rs.x, and with probability 1/2 the
envelope contains twice as much other envelope has either x/2 or
money as the other. I'll let you select Rs.2x. Thus the expected value of the
one envelope, which you can have other envelope is (1/2)(x/2) +
after the game is over. But as soon as (1/2)(Rs.2x) which is Rs.1.25x. This is
you select one, I offer you the option greater than the Rs.x in my current
to switch envelopes. Should you envelope. Therefore I should switch
switch? envelopes...
But if you do switch, a similar argu-
ment would instruct you to switch
back... and therefore keep switching!
What's going on here? Is there a flaw
* Get explanation in the appendix in the reasoning?
23

25. The Prisoner's Dilemma
Can cooperation evolve in a society
of egoists? This is an intriguing question
as we know that nature prefers indi-
viduals with selfish motives. With a
limited supply of resources available
to a population, the competition
amongst the organisms increases.
Since it is the selfish behavior of the
organism that ensures its sustenance,
there is no reason for cooperation to played repeatedly (Iterated Prisoner's
occur. Whereas we know of instances Dilemma-IPD), cooperation becomes
where cooperation does take place – possible among rational players.
Symbiosis; people cooperate with The best strategy for this game is 'Tit for
each other. Tat’ i.e. cooperate in the first round and
Mutual cooperation, which benefits then, in the subsequent rounds, make
the cooperators and the lack of which the move made by the opponent in the
is harmful for them, easily persist. But previous round. This strategy cannot be
there are types of cooperation in exploited more than once, but the
which on cooperating one does well player tends to cooperate a lot, generat-
but any one of them would do better ing many reward payoffs.
by failing to cooperate. In such a case Consider the strategies available for
it is difficult to find group coopera- playing the iterated Prisoner's Dilemma
tion, for the organisms cooperating that are deterministic and which use the
are in a worst position than their outcomes of the previous three moves,
counterparts leading to a dying out of say to make a choice in the current
cooperating tendencies in a population. move. Since there are 4 possible out-
The Prisoner's Dilemma is an elegant comes for each move, there are 4x4x4 =
embodiment of such a case. In the 64 different histories of the three previ-
Prisoner's Dilemma, two individuals can ous moves. Therefore to determine its
either cooperate or defect. And the choice of cooperation or defection , a
selfish choice of defection yields a strategy would only need to determine
higher payoff than cooperation. But if what to do in each of the situations that
both defect, both do worse than if both could arise. To get the strategy started,
had cooperated. three hypothetical moves are needed
Game theory tells us that each player which precede the start of the game for
thinks defecting as a better option and which we require six more genes
assumes that the other player will come making a total of 70. This string would
to the same conclusion. Thus, rational specify what the individual would do in
players of Prisoner's Dilemma will always every possible circumstance and therefore
defect. However, when this game is completely define a particular strategy.
24

26. The Prisoner's Dilemma
21
There are precisely 270≈ 1021 strategies.
70 using two genetic operators: crossover
An exhaustive search for good strategies and mutation.
in such a big collection is impossible. If a a. Crossover is a way of constructing
computer had examined these strate- the chromosomes of the two offspring
gies at the rate of 100/s since the begin- from the parent chromosomes. It selects
ning of the universe, less than 1% would one or more places to break the parent
have been checked by now. chromosomes so as to construct the
To find effective strategies in such a offsprings each of whom has some
huge set , a very powerful technique is genetic material from both parents.
needed. The “genetic algorithm” given b. Mutation in the offspring occurs by
by J. Holland is such a technique. These randomly changing a very small proportion
algorithms are inspired by biological of the chromosome.
genetics and adapted by Holland into a 5. This gives a new population which is
general problem solving technique. The again checked for its effectiveness. This
simulation process works in five stages: new population will show behavior
1. An initial population is chosen. more like those of the successful individuals
2. Each individual is run in the current envi- of the previous generation than the
ronment to determine its effectiveness. unsuccessful ones.
3. The relatively successful candidates
are selected to have more offspring.
4. The successful individuals are then Computer programs that “evolve” in
randomly paired off to produce two ways that resemble natural selection
offspring per mating. The strategy of an can solve complex problems even their
offspring is determined from the strate- creators do not fully understand
gies of the two parents. This is done by -John Holland
Salwa Ali Khan
M.Sc. 2nd year
nd
25

27. Campus Placement Result
2007 – 2011 Batch 2008 – 2012 Batch
A ABHINAVA KASHYAP : Infosys ABHILASH PARIDA : Infosys - Engg. , iNautix
ABHISHEK KUMAR : Directi AJAY MEHRA : Verizon
ABHISHEK KUMAR SINGH : Evalueserve, MphasiS ALI HUSSAIN KADAR : Oracle - AD
ABHISHEK MEHRA : Nagarro Software ANKIT AGGARWAL : FICO
ARIHANT JAIN : TCS-CTO GOURAV CHOUDHARY : Infosys - Engg.
BADIDA CHAITANYA : Nagarro Software, MphasiS
HARSH JHA : Paypal/e-Bay
BOLLIBISAI GIRIDHAR : Infosys
NAVDEEP SINGH : Flipkart - MT
DEEPAK DAGAR : Oracle(OFSS)
NIKITA GARG : InMobi
HEMANT KESARWANI : Cisco Systems
KARTHIK V S : Z.S. Associates PALASH GOYAL : Oracle - OFSS
MANJEET SUDHANSHU : Cisco Systems PARANJAY SRIVASTAVA : Capital IQ
MUCHENTHULA ANIL K REDDY : Global Analytics PRANEETH KUMAR RUTHALA : Samsung -SISO
N VIKAS REDDY : Adobe Systems PRATEEK GUPTA : Flipkart - MT
NANDAMURI SRAVAN KUMAR : Amazon PRATIK SARDA : EXL Service - DA
PIYUSH LAKHAWAT : Nagarro Software GOPALAKRISHNA SHARMA : Samsung - SISO
RAJAN GOYAL : 3D Solid Compression, iNautix RAUNAK SINGHI : Verizon
RANGOJU ABILASH : Directi RISHI RAJ DIWAKAR : IBM – GBS, Bank of India
VISHAL KUMAR G : Goldman Sachs SAHIL AGARWAL : Verizon
WAIKOS ASHWIN ARUN: Nagarro Software, Attomic Labs SARADA PRASANNA MISHRA : Future First
YOGENDRA BOYAPATI : Z.S. Associates SHYAM SUNDARA SOUMITRA JOSYULA: Capital IQ
HARSHAL GOEL : Oracle(OFSS)
MAYANK TAKKER : Oracle - AD
GAURAV PAGARIA : Oracle(OFSS)
TANVI RAI : Goldman Sachs
PRANJAL KUMAR : Z.S. Associates
PRAVEEN A : InMobi
GUNUGANTI ADITYA : Microsoft IDC
N.J. AJAY : Deloitee, TCS-CTO KRISHNA KANHAIYA : SCA Tech, Cognizant Engg.
26

28. Phi - 1.618.....
Everyone, meet Examples of Golden
PHI, pronounced Ratio phi in nature
as fee. (row-wise)
Not to be con- 1) Sea shell
fused with PI. 2) Cactus
As we math-
3) Cyclon in ocean
ematicians like
4) Plant
to say: PHI is
one H of a lot 5) Sun flower
cooler than PI! 6) Galaxy
The number PHI
was derived from the Fibonacci on plant stalks, insect
sequence—a progression famous not segmentation—all display astonishing
only because the sum of adjacent obedience to the Divine Proportion.
terms equals the next term, but The human body is literally made of
because the quotients of adjacent building blocks whose proportional
terms possess the astonishing prop- ratios always equal PHI. PHI's ubiquity
erty of approaching the number in nature clearly exceeds coincidence,
1.618—PHI! and so the ancients assumed the
Despite PHI's seemingly mystical number PHI must have been preor-
mathematical origins, the truly mind- dained by the Creator of the universe.
boggling aspect of PHI was its role as Early scientists heralded one-point-six-
a fundamental building block in one-eight as the Divine Proportion.
nature. Plants, animals, and even All of you. Try it. Measure the distance
human beings all possess dimensional from the tip of your head to the floor.
properties that adhere with eerie Then divide that by the distance from
exactitude to the ratio of PHI to 1.
your belly button to the floor. Guess
If we study the relationship between
what number you get. PHI!
females and males in a honeybee com-
Want another example? Measure the
munity, we find that the female bees
distance from your shoulder to your
always outnumber the male bees. If
fingertips, and then divide it by the
you divide the number of female bees
distance from your elbow to your
by the number of male bees in any
fingertips. PHI again.
beehive in the world, you always get
the same number. PHI. Another?
Sunflower seeds grow in opposing Hip to floor divided by knee to floor. PHI
spirals. Can you guess the ratio of each again. Finger joints. Toes. Spinal divisions.
rotation's diameter to the next? PHI. PHI. PHI. PHI. My friends, each of you is a
Spiralled pinecone petals, leaf arrangement walking tribute to the Divine Proportion.
Shubham Luhadia
st
B.Tech. 1s year
27

29. DIMENSION 2012
3-4 March 2012
Lecturer 1
Dr. Rajan M A
Research Scientist, TCS Innovation Labs, Bangalore
Lecture: MATHEMAGIC FOR PHYSICAL WORLD
PROBLEMS- ENGINEER’S PARADISE
Dr. Rajan M A, is a prominent research scientist at TCS
Innovation Labs and is also working as a visiting
academic faculty at SJC Institute of Technology, Chikka-
ballapur and University VCE, Bengaluru. He has around 10
years of experience in IT and space sciences R&D services. He has also worked as a Devel-
opment Manager in TCS and Senior Scientist Engineer in ISRO Satellite Centre. His lecture
on ‘Mathemagic for physical world problems Engineer’s Paradise’ was highly informative.
He discussed about the research of ancient Indian mathematicians and the use of innova-
tive techniques to solve real life problems life like using Graph theory in Wireless adhoc
networks, hashing and Chinese remainder theorem in Large Scale Data Handling, Robust
data storage and Cryptographic schemes.
Lecturer 2
Mr. Sudhir Kumar Jha
Zonal Head, Active Trader Service of East and
Central India, ICICI Securities
Lecture: HOW STOCK MARKETS WORK AND
MUTUAL FUNDS
Mr. Sudhir Kumar Jha is an eminent personality in the
field of Investment Banking and financial services, with an
MBA from the The Institute of Chartered Financial Ana-
lysts of India. He was the Senior Vice-President at IndiaB-
ulls Securities Ltd. Presently he is the Zonal Head, Active Trader Service of East and Central
India at ICICI Securities Ltd. based at Kolkata. In his lecture on ‘How Stock markets work and
Mutual Funds’ he discussed about the dynamics and technicalities of stock markets and
investing strategies. He started by giving a basic view about stocks and options and also
discussed the technicalities of IPO and book building.
28

30. DIMENSION 2012
Lecturer 3
Mr. Pradip Bhattacharyya
Regional Training Manager, HDFC Life, NE Region
Lecture: THE CONCEPT AND BENEFITS OF LIFE
INSURANCE
Mr. Pradip Bhattacharyya, with over 18 years of experience
in the field of investment banking is the Regional Training
Manager, HDFC Life for the North East Region. Mr. Bhattacharyya
has 11 years of experience in the field of Sales and 7 years of experience in Training. His
lecture on ‘The concept of Life-Insurance and the benefits of savings through Life-
Insurance’ gave an insight on the functioning of life-insurance and awareness about the
technicalities pertaining to insurance so that insurance is not considered just a ‘Money
Back policy’.
Audience
29

31. DIMENSION 2012
Online Events
What to buy? What to sell? When to trade to hedge
ETF 2.0
risk? Or just speculate! Want to try something more?
DIMENSION 2012 presented the second edition of its
very own Online Virtual Stock Market, ETF 2.0. With all
the features of a real time stock market and some serious
challenge from fellow traders, ETF 2.0 was the right
place to test trading instinct! The event was a huge success
just like ETF with 8,897 hits over a stretch of 4 days.
Digital Wall
"Know Hacking! But No Hacking!" DIMENSION
presented before you “The Hacker’s World” where
people could battle on the grounds of computer
and prove to be the next tech-genius. This Hacking
event included PHP and Linux knowledge.
The online hacking event had 368 hits.
D-Crypt
This cryptographic event involved using deciphering
skills to break the 3-tier security barrier set up by the
security agencies to prevent burglary. From the simplest
of ciphers to the trickiest of encryption algorithms, it
needed all.
This event was highly appreciated with a total of 1,123
hits over a stretch of 4 days.
30

32. DIMENSION 2012
Online Events
Arbitrage opportunities appear here and there, all the
Invent Money
time in the financial markets. Pure 100% risk-free stock
market arbitrage opportunities are hard to spot, how-
ever it is interesting to understand how the markets
offer these opportunities when theres no risk at all!
This Arbitrage event included skills to spot arbitrage
opportunities to make money...without risks!! This
innovative event had a total of 126 hits.
Model-M
The Mathematical Modeling event comprised of using
mathematical modeling skills to solve the water crisis
issues of Dime Valley. The local population recently
discovered a water source at the hill top that promises
to resolve the water scarcity problem in the valley. The
job was to devise an optimal water pipeline network
that provides water to each and every village in the
valley. The challenge was to find the shortest and
cheapest pathway. This strategic event received 72 hits
and proved the toughest of all.
Over the stretch of 3 days,
www. thedimension. org recorded more
than 90,000 hits and was a huge success!
Over the year it received more than 5
lakh hits!
31

33. Appendix
Explanation of Two envelope paradox
The expected value calculation is However, surprisingly, there are some
flawed because the conditioning on prior probability distributions of money
the relative value Rs.x is incorrect. in the envelopes for which it always
You need to have some idea of what makes sense to switch (whether or not
the prior distribution of money in the you look at what's inside your
envelope is before you can do the envelope)!
calculation. For instance, if you knew For instance, suppose that amount of
that the two amounts were Rs.5 and money in the two envelopes is
Rs.10, then if you took the Rs.5 envelope (Rs.2k,Rs.2k+1) with probability (2/3)k/3,
(i.e., if x=5), there is NO chance that for each integer k>=0. It is a fun
the amount in the other envelope is exercise to check that no matter what
Rs.x/2; it must be Rs.2x=Rs.10. Similarly, you have in your envelope, the other
if you took the Rs.10 envelope, the envelope has higher expected value,
other envelope must be Rs.5. So con- and you should switch!
ditioning on whether you took the How to resolve this paradox is a perplexing
Rs.5 or Rs.10 envelope, the expected philosophical question. (Some of you
value of the other envelope is actually may object that the prior distribution
(1/2)(10) + (1/2)(5) = 7.5. However, if has infinite mean, but this does not
you observe what's in your envelope, fully resolve the paradox, since in
then you can condition on what you theory if such a distribution exists,
see; the expected value of the other one would still have to wrestle with
envelope is Rs.10 if you see Rs.5 in the paradox of continually switching
yours, or vice versa. If you see Rs.5 envelopes!)
you should switch, and if you see The study of mathematical models
Rs.10 you should not. So there is no for decision-making is called game
paradox in this case. theory, and probability theory helps
us understand expected values.
Pavan Kumar
B.Tech. 1st year
32

34. Try this
-> Start from capital ‘M’ taking n=1.
-> Take i=1 if you want to read it as Mathematics or
take i=2 if you want to read it as Matrix.
P.S. :- initially you may find it strange but later you would like it
n++
if (i==2)
if (i==1,n==1)
if (
i==
1,n
==
2)
designed by :- Himanshu Bansal

35. Content
Message from the HoD 1
About Us - MATRIX 2
Message from the President 3
Message from the Editor 4
A Walk With the History of Infinity 6
PHP Security 10
Internship Experience 12
Trends and Multi-Fractalality in Arctic Sea Ice 15
Pagerank Algorithm 20
Chomp 22
The Prisoner's Dilemma 24
Campus Placement Result 26
Phi - 1.618..... 27
DIMENSION 2012 28
Appendix 32
5