Matrix Magazine' 12- Anantha


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2nd edition of annual magazine of Mathematics and Computing Department in IIT Guwahati.
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Matrix Magazine' 12- Anantha

  1. 1. Indian Institute of Technology Guwahati A 2nd Edition April ‘11 - March ’12see inside 1 $ 3 2Dimension PHP security Infinity Pagerank Chomp
  2. 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 Mathematics1
  3. 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. 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 MATRIX3
  5. 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 MATRIX4
  6. 6. Contents Message from the HoD5
  7. 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, and6
  8. 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 dont 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 Its 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 Cantors 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. 9. A Walk With the History of Infinity ity than the original set; this impor- proposed a hypothetical scenario tant result is simply called Cantors 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 minds 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. 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 Scholar9
  11. 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. 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. 1styear11x
  13. 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. 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. 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. 4thyear14
  16. 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. 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. 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. 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. 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. 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, so20
  22. 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 nd21
  23. 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 is22
  24. 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. Ill 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! Whats going on here? Is there a flaw * Get explanation in the appendix in the reasoning?23
  25. 25. The Prisoners 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 Prisoners 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 Prisoners 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 Prisoners 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- Prisoners 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 Prisoners Dilemma will always every possible circumstance and therefore defect. However, when this game is completely define a particular strategy.24
  26. 26. The Prisoners 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 nd25
  28. 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. PHIs ubiquity erty of approaching the number in nature clearly exceeds coincidence, 1.618—PHI! and so the ancients assumed the Despite PHIs 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. rotations 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 year27
  29. 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. 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’. Audience29
  31. 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. 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. 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 whats 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 whats 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 year32
  34. 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 itn++ if (i==2) if (i==1,n==1) if ( i== 1,n == 2) designed by :- Himanshu Bansal
  35. 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 Prisoners Dilemma 24 Campus Placement Result 26 Phi - 1.618..... 27 DIMENSION 2012 28 Appendix 325