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FAQs
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Was Taj Mahal really a Hindu temple or it is just a propoganda theory spreaded by Hindutva leader?
Thanks Virendra for A2A.Right from childhood, I was fascinated with this story and have read about this extensively. Also, heavily influenced from RSS in childhood, I was biased towards the P.N. Oak’s narrative.However, ,my conclusion in short: Taj Mahal in the current form was never a Hindu temple. Whether or not a temple stood before Taj Mahal is difficult to answer until and unless you dig the ground up under Taj or some other technology comes up.My inferences step by step (for the current structure)P.N. Oak claims that the carbon dating of the door proves that the structure is older than common narrative. Well, the door might have been installed after dismantling it from somewhere else. May be some temple’s door was tore down to make it the door for Taj Mahal. But that doesnt prove anything about Taj Mahal.P.N. Oak claims that there are numerous symbols of Hindus on TajSign Om: I have personally gone to Taj Mahal and tried to identify that sign. The sign Om is part of the flower and was only observed only in few flowers. All the other flowers had different patterns. And there is a possibility that artisans being Hindu, did have Om in mind. And even if you agree that this is indeed OM. there is no parallel of making OMs in flower across whole Rajasthan. If you still have doubts, do visit Taj Mahal and judge yourself.b. Sign of Kalash over the top: Well, Mughals used to borrow deigns from Indic architecture and there seems to be similar case for Humayun tomb and many other tombs constructed later.Also, it is resting on half crecent moon, definitely not a Hindu design.c. Vedic quarters: Well the quarters are not necessarily Vedic in design. they resembled the one in that age. And Taj Mahal seemed to be garden for Shah Jahan. And there used to be security around the Taj Mahal and even to the opposite side, where gardens used to be there. Hence, these houses seems to be belong to the security of that time.d. Marble theory (on which aayaats are written): I went to taj Myself, but i didnt notice such a difference. May be I am not expert but this argument of PN Oak did not convince me. Further, P.N. Oak only captured black and white photos- too difficult to understand them.e. Rajputs made this temple: First of all, it is very difficult to built such an extravagant temple, and quite visible from Agra Fort so easily under Sultanate or Mughal rule. Even if they wanted to create a temple, it could have been quite modest or toned down. For ex. Jain temples in Chandni Chowk were without any shikhars. Bright White grand temple looks odd on Agra.f. Documentation for Taj Mahal: is very extensive and detailed. But there is no account of Taj Mahal temple/Tejo Mahalaya/Agra Shiva temple even in Rajput history. it looks odd that none of the Rajput history mentions this temple. Rajput records are quite extensive in both details and signing paens of their kings. Thus, ommision of Tejo Mahalya seems to be very odd.g. As per P.N. Oak, if Tejo Mahalaya was so important, then it should have been impossible to erase that from public memory. Kashi was rebuilt during Marathas. Mathura during British times and Ayodhya is still a burning topic. Agra temple never appears in any of the scripture/king’s record/forklore in north India or south India or even South East Asian history.Conclusion: To me P.N. Oak’s theory seems delusional at best.
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How do I delete my Quora account?
Quora allows users to delete their account if they choose to do so. Deleting your Quora account means that the following content will be removed from public view: your profile including photos and bio, your answers, comments, blog posts, votes, endorsements, and messages. Questions you may have asked will remain, since questions on Quora are community owned, but will not be associated with your name publicly. Deletion of your account is not reversible once the process is complete. Alternatives to deletion include: 1. Deactivation [ https://www.quora.com/How-do-I-deactivate-my-Quora-account-Can-it-be-reactivated-later/answer/Quora-Official-Account ] 2. Edit your Quora Privacy Settings [ https://www.quora.com/How-can-I-edit-my-Quora-Privacy-Settings/answer/Quora-Official-Account ] 3. Deleting individual pieces of content, such as answers, comments, or posts If you are certain you wish to delete your account, visit your account privacy settings and choose “Delete Account”. Once you confirm, your account will be deactivated immediately and the deletion process will begin. If you login during the next 14 days, the account will be reactivated and deletion will be canceled. Once the 14-day grace period has expired and your account has been deleted, your content and profile will be permanently deleted, and personal data associated with your account will be removed from Quora’s databases. Keep in mind that your content may have been republished or shared by others outside of Quora. Account deletion here does not remove any links or data hosted by others. If you have further questions regarding account deletion, contact us using our contact form [ https://www.quora.com/contact ].
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Why is Ramanujan considered one of the great mathematicians?
He is in fact, I am certain, one of the greatest mathematicians according to any criteria, let it be because he had no formal education, or because of the 3000 odd identities and theorems he came up with.anyway i don’t want to compare but i am just trying to say the striking things i saw in his life.Anyway I would first show the marklist Ramanujan acquired in the 1st year examinations.Roughly speaking, for these things,Ramanujan’s name is seen everywhere around the world, even if some might disagree.•Magic Square•Brocard – Ramanujan Diophatine equation•Dougall – Ramanujan identity•Hardy – Ramanujan number•Landau – Ramanujan constant•Ramanujan’s congruences•Ramanujan – Nagell equation•Ramanujan – Peterssen conjecture•Ramanujan – Skolem’s theorem•Ramanujan – Soldner constant•Ramanujan summation•Ramanujan theta function•Ramanujan graph•Ramanujan’s tau function•Ramanujan’s ternary quadratic form•Ramanujan’s prime•Ramanujan’s costant•Ramanujan’s sum•Rogers – Ramanujan’s identityNow, let us see a quote of an English Mathematician“Srinivasa Ramanujan was a mathematician so great that his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last thousand years.”He continued thus: “His leaps of intuition confound mathematicians even today, a century after his death. His papers are still plumbed for their secrets. His theorems are being applied in areas- polymer chemistry, computers, astrophysics, molecular physics, even (it has been recently suggested) cancer – scarcely imaginable during his lifetime. And always the nagging question: What might have been, had he been discovered a few years earlier, or lived a few years longer?”Now just see Ramanujan’s childhood prodigy:Teacher: n/n = 1. Any number divided by itself is one. If there are 3 apples and there are three students, each one will get one apple. Likewise if there are 1000 children and 1000 pens, each will get one pen.Ramanujan: What about 0/0? If there are 0 apples and 0 students, will each still get one?Teacher got perplexed!Ramanujan’s Explanation: 0/0 can be anything, the zero in the numerator could be many times 0 in the denominator, and vice versa.Just before the age of 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmeticWith his scores, he stood first in the district. That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his house.He was later lent a book on advanced trigonometry written by S. L. LoneyHe completely mastered this book by the age of 13 and discovered sophisticated theorems on his own.Now Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method. Its like this:It is easy to solve simple equation of the first degree, e.g., 3a = 15. And we are taught how to solve second degree equations with the power of x as 2.Ramanujan found his own method in solving not only cubic equations but also equations of fourth degree.Next year not knowing that quintic equations, or equations with power of x as 5, cannot be solved, he tried and failed in his attempt.In 1903 when he was16, Ramanujan came across the book by G. S. Carr on A Synopsis of Elementary Results in Pure and Applied Mathematics, a collection of 4865 formula and theorems without proofThe book is generally acknowledged as a key element in awakening the genius of RamanujanThe next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal placesWhen he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics as an outstanding student who deserved scores higher than the maximum possible marksHe received a scholarship to study at Government Arts College, Kumbakonam, However, Ramanujan could not focus on any other subjects and failed most of them, losing his scholarship in the processHe later enrolled at Pachaiyappa' College in Madras. He again excelled in mathematics but performed poorly in other subjectsRamanujan failed his Fine Arts degree exam in December 1906 and again a year laterWithout a degree, he left college and continued to pursue independent research in mathematics. At this point of his life, he lived in extreme poverty and was suffering from starvation.Deplorable Condition of Ramanujan is expressed in his own words:“When food is the problem, how can I find money for paper? I may require four reams of paper every month.”On 14 July 1909, Ramanujan was married to a nine-year old girl, Janaki Ammal (21 March 1899 - 13 April 1994)After the marriage, Ramanujan developed a hydrocele problemsHis family did not have the money for the operation, but in January 1910, a doctor volunteered to do the surgery for freeAfter his successful surgery, Ramanujan searched for a jobHe stayed at friends' houses while hewent door to door around the city of Chennai looking for a clerical positionTo make some money, he tutored some students at Presidency College who were preparing for their examRamanujan met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical SocietyRamanujan, wishing for a job at the revenue department where Ramaswamy Aiyer worked, showed him his mathematics notebooksAs Ramaswamy Aiyer later recalled:“I had no mind to smother his genius by an appointment in the lowest level as clerk in the revenue department.”Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends.Some of these friends looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector of Nellore and the secretary of the Indian Mathematical SocietyRamachandra Rao was impressed by Ramanujan's research but doubted that it was actually his own work !Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to clear any doubts over Ramanujan's academic integrityRao listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, through which Rao was convinced of Ramanujan's mathematical brilliance . When Rao asked him what he wanted, Ramanujan replied that he needed some work and financial supportRamanujan continued his mathematical research with Rao's financial aid taking care of his daily needsWith the help of Ramaswamy Aiyer, Ramanujan had his work published in the Journal of Indian Mathematical SocietyOne of the first problems he posed in the journal was to evaluate:He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied the solution to the problem himselfHe formulated an equation that could be used to solve the infinitely nested radicals problem. Using this equation, the answer to the question posed in the Journal was simply 3In early 1912 he got a job in the Madras Accountant Generals office with a salary of Rs 20 per month.Later he applied for a position under the Chief Accountant of the Madras Port TrustHe was Accepted as a Class III, Grade IV accounting clerk making 30 rupees per monthHe used to Spend spare time doing Mathematical ResearchIn the spring of 1913, Narayana Iyer and Ramachandra Rao tried to present Ramanujan's work to British mathematiciansOne mathematician, M. J. M. Hill of University College London, commented that although Ramanujan had "a taste for mathematics, and some ability", he lacked the educational background and foundation needed to be accepted by mathematiciansOn 16 January 1913, Ramanujan wrote to G. H. HardyComing from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible "fraud“ !Hardy recognized some of Ramanujan's formulae but others "seemed scarcely possible to believe"G.H. Hardy was an academician at Cambridge UniversityHe was a prominent English mathematician, known for his achievements in number theory and mathematical analysis.Later on Ramanujan wrote to G.H.HardyHardy recognised some of his formulae but other “seemed scarcely possible to believe”. Some of them were –Initially, G. H. Hardy thought that the works of Ramanujan were fraud because most of them were impossible to believe.But eventually ,he was convinced and interested in his talent.This is one approximation formula of Pi mentioned in Ramanujan’s letters:Hardy was also impressed by some of Ramanujan's other work relating to infinite series:This second one was new to Hardy, and was derived from a class of functions called hypergeometric series which had first been researched by L. Euler and Carl F. Gauss.After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that the "[theorems] defeated me completely; I had never seen anything like them before”He figured that Ramanujan's theorems "must be true”Hardy asked a colleague, J. E. Littlewood, to take a look at the papersLittlewood was amazed by the mathematical genius of RamanujanRamanujan’s notebook referring calculus and number theory:Ramanujan boarded the S.S.Nevasa on 17 March 1914 and arrived in London on 14th AprilRamanujan began working with Hardy and LittlewoodHardy received 120 theorems from him in 1st 2 letters but there were many more results in his notebookRamanujan spent nearly 5 years in CambridgeRamanujan was awarded the B.A degree by Research in March 1916 at an age of 28 years for his work on Highly Composite Numbers.He was elected a Fellow of the Royal Society of London in February 1918 at an age of 30 years.He was the second Indian to become FRS.( First one was in 1841).He was elected to a Trinity College Fellowship as the FIRST INDIAN.During his five years stay in Cambridge he published twenty one research papers containing theorems.A few words regarding the 1729, Ramanujan NumberHardy arrived in a cab numbered 1729He commented that the number was uninteresting or dull.Instantly Ramanujan claimed that it was the smallest natural number which can be written as sum of cubes in 2 ways1729 = sum of cubes of 12 and 1/ sum of cubes of 10 and 9.Actually only this much is available in the popular version of the story.But Ramanujan had worked extensively on this number and made some simple reuslts along with other startling contributions.1729 = 7 x 13 x 19 product of primes in A.P1729 divisible by its sum of digits.1729 = 19 x 911729 is a sandwich number or HARSHAD number."Ramanujan was using 1729 and elliptic curves to develop formulas for a K3 surface," Ono says. "Mathematicians today still struggle to manipulate and calculate with K3 surfaces. So it comes as a major surprise that Ramanujan had this intuition all along."Ono had worked with K3 surfaces before and he also realized that Ramanujan had found a K3 surface, long before they were officially identified and named by mathematician André Weil during the 1950s.Just as K2 is an extraordinarily difficult mountain to climb, the process of generalizing elliptic curves to find a K3 surface is considered an exceedingly difficult math problem.And in Ramanujan’s writing he was relying on this number 1729 in order to arrive at some combination of numbers which could prove that Fermat’s last conjecture could be counter exampled.there are some popular misconceptions regarding ramanujan:Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper (About 4000 theorems)These results written up without any derivations.Since paper was very expensive, He would do most of his work (derivations) on SLATE and transfer just the results to paper.Hence the perception that he was unable to prove his results and simply thought up the final result directly is NOT CORRECTProfessor Bruce C.Berndt of University of Illinois, who worked on Ramanujan note books, stated that “Over the last 40 years, nearly all of Ramanujan’s theorems have been proven right”.Also Mathematicians agreed unanimously on the point that it was not possible for someone to imagine those results without solving / proving.I think I will complete this answer tomorrow, because I feel sleepy: Good Night!Edited in Later:I am extremely sorry for not turning up yesterday to finish the answer I started, because I had gone for an outing to Hoggenakkal in Tamil Nadu.I think I would say something more about the GENIUS before I complete.Well, once G. H. Hardy rated his contemporary mathematicians based on pure talent.Hardy rated himself a score of 25 out of 100,J.E. Littlewood 30, David Hilbert 80 andRamanujan 100 !Hardy also said that Ramanujan’s solutions were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account”Ramanujan’s genius was recognized by TN Government andNow, Tamil Nadu celebrates 22 December as ‘State IT Day’A Stamp was released by the Govt. in 196222nd December started to be celebrated as Ramanujan Day in Govt Arts College, Kumbakonam. Now on 22nd December 2011, Then prime minister Manmohan Singh said that the 125th birth year of Ramanujan will be celebrated as National Mathematics Year and from that year onwards, December 22 is National Mathematics Day.There is a National Symposium On Mathematical Methods and Applications on his name (NSMMA)And there is SASTRA Ramanujan Prize which is given under the auspices of National Mathematics Society and the society for Physics.Let me tell something about the Hardwork of Ramanujan:Once P.C. Mahalanobis, the founder of Indian Statistical Institute visited Ramanujan while in Cambridge and said to him: “ Ramanju, these English Mathematicians say that you are a Genius, A real incomparable Genius.Immediately, showing his thickly black elbow Ramanujan replied, dear friend, everything owes to this elbow.Shocked by the answer, P.C. Asked: How Can it be so?????Ramanujan replied with a smile: “During my childhood days, while using a slate for calculations, repeated erasing used to leave remnants of chalk in it, then I stopped using duster for rubbing.”“This meant that every few minutes I had to rub my slate using my elbow, it means I owe everything to this elbow.”Regarding the spiritual dimension of Ramanujan’s life, all will agree that he was a sort of a mystic, and in fact, Ramanujan was a person with a somewhat shy and quiet dispositionHe was absolutedly a dignified man with pleasant mannersRamanujan credited his success to his family Goddess, Namagiri of Namakkalin fact, He claimed to receive visions of scrolls of complex mathematical content unfolding before his eyes. And we have no idea to contradict his words.And this could be in one way regarded as his Dictom"An equation for me has no meaning, unless it represents a thought of God.”We get amazed the more we know about Ramanujan’s spiritual understanding of many mathematical concepts, I will brief just one.For example, 2n – 1 will denote the primordial GOD.When n is zero, the expression denotes ZERO.He spoke of “ZERO” as the symbol of the absolute (Nirguna – Brahmam) of the extreme monistic school of philosophy)The reality to which no qualities can be attributed,of which no qualities can be there.When n is 1, it denotes UNITY, the Infinite GOD.When n is 2, it denotes TRINITY.When n is 3, it denotes SAPTHA RISHIS and so on.Crazy isn’t it, but all such craziness constituted Ramanujan.He looked “infinity” as the totality of all possibilities which was capable of becoming manifest in reality and which was inexhaustible.According to Ramanujan, The product of infinity and zero would supply the whole set of finite numbers.Each act of creation, could be symbolized as a particular product of infinity and zero, and from each product would emerge a particular individual of which the appropriate symbol was a particular finite number.If you want to go through the life of Srinivasa Ramanujan in its fullness, I humbly refer to you my guide, the book which opened my eyes towards realizing the pearl of Indian Mathematics, and that is:“The man who knew infinity: A life of the Genius Ramanujan”It was written by Robert Kanigel.In that book Kanigel claims some very amazing facts about Ramanujan.Sheer intuitive brilliance coupled to long, hard hours on his slate made up for most of his educational lapse.This ‘poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe’ as Hardy called him, had rediscovered a century of mathematics and made new discoveries that would captivate mathematicians for next century.S.Chandrasekhar, Indian Astrophysicist, Nobel laureate 1983, told thus:“I think it is fair to say that almost all the mathematicians who signNowed distinction during the three or four decades following Ramanujan were directly or indirectly inspired by his example.Even those who do not know about Ramanujan’s work are bound to be fascinated by his life.”“The fact that Ramanujan’s early years were spent in a scientifically sterile atmosphere, that his life in India was not without hardships that under circumstances that appeared to most Indians as nothing short of miraculous. He had gone to Cambridge, supported by eminent mathematicians, and had returned to India with very assurance that he would be considered, in time as one of the most original mathematicians of the century.The words of Hardy himelf speak volumes of Ramanujan:“I have to form myself, as I have never really formed before and try to help you to form, some of the reasoned estimate of the most romantic figure in the recent history of mathematics, a man whose career seems full of paradoxes and contradictions, who defies all cannons by which we are accustomed to judge one another andabout whom all of us will probably agree in one judgement only, that he was in some sense a very great mathematician.”Bertrand arthur william russell, British philosopher & mathematician, Nobel laureate and almost contemporary to Ramanujan, stated thus:“I found Hardy and Littlewood in a state of wild excitement because they believe, they have discovered a second Newton, a Hindu Clerk in Madras… He wrote to Hardy telling of some results he has got, which Hardy thinks quite wonderful.”The life of Ramanujan is actually a textbook from which many things could be conceived. Despite the hardship faced by Ramanujan, he rose to such a scientific standing and reputation no Indian has ever enjoyed.It should be enough for youngsters like us to comprehend that if we can work hard with indomitable determination, sheer perseverance and sincere commitment, we too can perhaps soar the way like Srinivasa Ramanujan.Even today in India, Ramanujan cannot get a lectureship in a school / college because he had no degree.Many researchers / Universities will pursue studies / researches on his work but he will have to struggle to get even a teaching job.Even after more than 90 years of the death of Ramanujan, the situation is not very different as far the rigidity of the education system is concerned. Today also a ‘Ramanujan’ has to clear all traditional subjects’ exams to get a degree irrespective of being genius in one or more different subjects.He was offered a chair in India only after becoming a Fellow of the Royal Society.But it is disgraceful that India’s talent has to wait for foreign recognition to get acceptance in India or else immigrate to other places.Many of those won international recognition including noble prizes had no other option but to migrate for opportunities & recognition.(Ex. Karmerkar)The process of this brain drain is still continuing.Here is a pic of Ramanujan with his colleagues in Cambridge University.Talking about certain contributions of Ramanujan which shook me off my feet.As we all know we use the notation P(n) to represent the number of partitions of an integer n. Thus P(4) = 5, similarly, P(7) = 15.I don’t need to explain that If we were to start enumerating the partitions for larger numbers, even for small numbers such as 10 we start seeing that there is a combinatorial explosion! To illustrate this consider P(30) = 5604 and P(50) = 204226 and so on. (btw, partitions can be visualized by Young tableau!).A similar search was on for asymptotic formulae for the partition number P(n) and because of the combinatorial explosion an accurate formula was considered difficult. Ramanujan believed that he could come up with an accurate formula even though it was considered extremely hard, and he came close.One work of Ramanujan (done with G. H. Hardy) is his formula for the number of partitions of a positive integer n, the famous Hardy-Ramanujan Asymptotic Formula for the partition problem. The formula has been used in statistical physics and is also used (first by Niels Bohr) to calculate quantum partition functions of atomic nuclei.The formula he proposed gives a very close value to that of the true value, and it is a mouth-watering feat considering its very pattern less nature.I had written another answer in quora regarding how Ramanujan provided a rapidly converging series as the value of Pi. I will just copy and paste it here.For a long time, the series used for finding the value of Pi was given by the Leibniz-Gregory Series.π = (4/1) - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) + (4/13) - (4/15) ...But in order to give the value of Pi correctly upto 5 decimal places, this series required around 500000 terms.Now, in the Indian tradition, another formula was given by Nilakantha, a mathematician of Kerala School of Mathematics who lived couple of centuries before Leibniz and the series converged much rapidly.π = 3 + 4/(2*3*4) - 4/(4*5*6) + 4/(6*7*8) - 4/(8*9*10) + 4/(10*11*12) - 4/(12*13*14) ...And in order to give the value of Pi upto 5 decimal places, this series required only 6 terms. And thats a great thing but which failed to catch the eye of westerners until the nineteenth century.Now, take into consideration all these and what Ramanujan did. Ramanujan simply penned down an infinite series, looking so horrendous, which would be equal to the reciprocal of Pi.And this is the most rapidly converging series ever given for the value of Pi and the algorithm based on this have actually been used in computers.Now the most beautiful factor. In order to have the value of Pi upto 6 decimal places the infinite series of Ramanujan needed only ONE SINGLE TERM.And you take the second term and there you have suddenly the value of Pi upto 11 terms in your hands.I think it speaks something Great, and Ramanujan was indeed Great!!!Ramanujan has done extensive works in finding out highly composite numbers, and he has written down a long list of similar numbers which had more factors than any of the previous number.The highest highly composite number listed by Ramanujan is 6746328388800Having 10080 factorsHe received his degree from the university (later named Ph.D) for his work of highly composite numbers.I would just say another thing which caught my eye and unleashed an array of thoughts.Ramanujan while sick and dying in India, mentioned some very peculiarly behaving functions which mimicked the original moldular functions.The mock theta functions remained a mystery for most part of the last century and only the Great Ono made inroads towards their reality.In fact, no one at the time understood what Ramanujan was talking about.It wasn’t until 2002, through the work of Sander Zwegers, that we had a description of the functions that Ramanujan was writing about in 1920,' Ono said.Ono and his colleagues drew on modern mathematical tools that had not been developed before Ramanujan’s death to prove this theory was correct.Ramanujan actually wrote those functions claiming that he saw it in a scroll in the hands of A Goddess.Anyway now they are used to calculate the entropy of Black Holes ( A concept which developed years after his death.)Ono’s team was stunned to find the function could be used today.'No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them,' Ono says.Ramanujan’s Intuition Stands OUT!I think, just for a fun I would show the Mock Theta FunctionsNow I think I shoudl mention atleast something about the impact of Ramanujan’s work on statistical physics.For example imagine studying the statistics of a gas made of electrons confined to 2D. You could do something complicated like model the exact positions and momenta of many of electrons along with the force between them. Or you can simplify by imagining that the electrons can only occupy positions on a discrete triangular lattice, and instead of a repulsive force you can make the simple approximation that two electrons aren't allowed to be next to each other.The result is the Hard hexagon model and some work of Ramanujan's appears when you try to model it. Even if it's not physically realistic, these models share characteristics with more realistic physical models and give useful insight.In fact a whole bunch of different identities related to Ramanujan's work can appear when you study these kinds of simple physical models, especially 2-dimensional models. Eg. Hard Hexagon ModelI think I will conclude with a simple assumption of Ramanujan, I think it deserves mention:The mock theta functions which we mentioned earlier looked unlike any known modular forms, but he stated that their outputs would be very similar to those of modular forms when computed for the roots of 1, such as the square root -1. Characteristically, Ramanujan offered neither proof nor explanation for this conclusion.It was only 10 years ago that mathematicians formally defined this other set of functions, now called mock modular forms. But still no one fathomed what Ramanujan meant by saying the two types of function produced similar outputs for roots of 1.Ono and his colleagues have exactly computed one of Ramanujan’s mock modular forms for values very close to -1. They discovered that the outputs rapidly balloon to vast, 100-digit negative numbers, while the corresponding modular form balloons in the positive direction.Ono’s team found that if you add the corresponding outputs together, the total approaches 4, a relatively small number. In other words, the difference in the value of the two functions, ignoring their signs, is tiny when computed for -1, just as Ramanujan said. Incredible Intuition !I am just adding some pictures I came across.his notebooks, the last three,His handwritings and works mentioned without calculation:I think I can say nothing more, but if at all someone asks me, I would say if I know!By the way, I have actally spoken nothing regarding the complex mathematical contributions of this great mathematician,even without that I think you are thrilled and that is why, even if the statement is wrong in itself.“ Ramanujan is the greatest Mathematician of all time, atleast I believe so.”
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Should I be worried about these birthmarks/moles?
If they are the rare form of malignant melanomas then yes … if they are common moles, then no. I am not a dermatologist but I was fortunate enough to have a skin cancer (malignant melanoma) identified early enough to have it effective treated through surgery.Malignant melanomas are the moles that can kill you with 1,600 of my fellow Australians dying from it each year. Sadly, my country Australia is the skin cancer capital of the world. Melanoma facts and statistics.The common mole is benign and should cause you no concern but my malignant melanoma looked nothing like a common mole but rather looked a lot like this one displayed below, with a shiny black raised bubble surrounded by non-symmetrical edges :After having the scare with this malignant melanoma, I make it a practice to visit a skin cancer specialist every 6–12 month and I suggest that you should do the same if you are concerned about any of your moles changing.These doctors are experienced at identifying malignant melanomas which have a particular look as you can see by Googling “malignant melanoma” and reviewing the images.Even though I am covered in moles like the ones you have displayed, thanks to the Australian sun incessantly frying my English heritage skin, I have not had another malignant melanoma in over 20 years.What I have been able to gather from my skin cancer specialists is that there are markers that typically identify a malignant melanoma, with the self-check being as easy a ABCDE - How do I check myself for melanoma?:A is for ASYMMETRY: One-half of a mole or birthmark does not match the other.B is for BORDER irregularity: The edges are irregular, ragged, notched, or blurred.C is for COLOUR variation: The colour is not the same all over, but may have differing shades of brown or black, sometimes with patches of red, white, or blue.D is for DIAMETER: The area is larger than 6 mm (about the size of a pencil eraser) or is growing larger.E is for EVOLVING : Growing for more than a month plus changes in size, shape, colour, elevation, or another trait (such as itching, bleeding, crusting or firming). (This last point is likely the strongest of all of the warning signs)
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What are some “Wait, what?” moments that you’ve had with your child’s school?
My daughter's teacher sent out a sheet before the beginning of the school year with $150 worth of “required supplies” that included pencils, erasers, notebook paper, folders, glue, crayons, markers, Kleenex, sanitizer, chalk, and so on. Many of these items were duplicated and it ran up the amount.Okay … this seemed excessive and some of the items on the list seemed to me to be the school's responsibility, but I dutifully picked them up.First day: My daughter comes home after school and tells me that the first thing her teacher did was have all the kids place all their “required” items in the middle of the floor and the teacher then divided everything equally, so that nearly half of my child’s supplies went to other students whose parents had not sent anything.Needless to say, I was miffed by this. Suffice to say I’d had other issues with this instructor and others in her “clique” at the school in the past. I was half expecting something like this to occur.Second day: I took the day off and went into school with my daughter. I sent her to class and went to the Principal's office. I informed her that I was livid about what this teacher had done, and I demanded the return of the property I had purchased for my child, and I demanded a written apology from the instructor be sent out to all parents with an explanation of why what she had done was wrong. I insisted that this would happen or my next three phone calls would be to: 1. The local police to have the instructor brought up on charges of robbery; 2. The Superintendent of Schools to let them know I would be filing suit in civil court; 3. The local newspapers to ensure that there was media coverage of everything.Of course she initially didn’t take me seriously. She tried to brush me off claiming I was overreacting and that this was no big deal. So I called the police department and had a car dispatched. I held off on the other calls for the moment.Keep in mind that I come from a family of cops and at the time was Safety Commissioner of the City this occurred in. I knew most of the cops in the city. When the officer arrived and I explained the situation, she turned to the Principal and explained that this was not a “petty theft” issue at worst (as the Principal had been asserting). Almost quoting from the statute she said something like: robbery, in contrast to theft, is a taking of property that does involve person-to-person interaction with force, intimidation, and/or coercion. That robbery of any amount was considered a felony and that the principal could be considered an accessory after the fact if I chose to press charges.This immediately changed the course of the conversation. I got my apology, other parents got their explanation and apology (turns out this instructor had her own welfare system going where she would tell certain families that the supplies were taken care of), my child got her items back, the instructor was forced to end her little Socialist experiment and ended up leaving the District halfway through the second trimester. Lists had to be approved from then on and were about $30.***EDIT***Many comments bring up the point that these kids “had nothing”. Not so. There are already several programs in place that get school supplies to kids and the PTA sends out requests for donations which we had already given to. My point was that it wasn’t the job of the teacher to do this, or to spread the lesson that kids should believe it is right to have things that their parents had given them taken from them without the knowledge or permission of the parents.My tax dollars are already paying the third highest per-pupil funding in the nation. Education takes up 28% of all spending in Minnesota, and we are the 4th highest tax rate state. This does not include local referendums that are voted on (and usually passed) on the ballot by school district. Those things are paid for by the schools here.If “extra” is needed there are requests that go out at the beginning of every trimester. This list was “required” not “requested”.***EDIT Part Deux***Thanks for the 1000+ upvotes! Long live parental rights!***EDIT: The Final Chapter***First answer with 2000 upvotes! I didn’t expect that. Thanks!
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Why Indian scientist Ramanujan (Mathematitian) famous?
Roughly speaking, for these things,Ramanujan’s name is seen everywhere around the world, even if some might disagree.•Magic Square•Brocard – Ramanujan Diophatine equation•Dougall – Ramanujan identity•Hardy – Ramanujan number•Landau – Ramanujan constant•Ramanujan’s congruences•Ramanujan – Nagell equation•Ramanujan – Peterssen conjecture•Ramanujan – Skolem’s theorem•Ramanujan – Soldner constant•Ramanujan summation•Ramanujan theta function•Ramanujan graph•Ramanujan’s tau function•Ramanujan’s ternary quadratic form•Ramanujan’s prime•Ramanujan’s costant•Ramanujan’s sum•Rogers – Ramanujan’s identityNow, let us see a quote of an English Mathematician“Srinivasa Ramanujan was a mathematician so great that his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last thousand years.”He continued thus: “His leaps of intuition confound mathematicians even today, a century after his death. His papers are still plumbed for their secrets. His theorems are being applied in areas- polymer chemistry, computers, astrophysics, molecular physics, even (it has been recently suggested) cancer – scarcely imaginable during his lifetime. And always the nagging question: What might have been, had he been discovered a few years earlier, or lived a few years longer?”Now just see Ramanujan’s childhood prodigy:Teacher: n/n = 1. Any number divided by itself is one. If there are 3 apples and there are three students, each one will get one apple. Likewise if there are 1000 children and 1000 pens, each will get one pen.Ramanujan: What about 0/0? If there are 0 apples and 0 students, will each still get one?Teacher got perplexed!Ramanujan’s Explanation: 0/0 can be anything, the zero in the numerator could be many times 0 in the denominator, and vice versa.Just before the age of 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmeticWith his scores, he stood first in the district. That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his house.He was later lent a book on advanced trigonometry written by S. L. LoneyHe completely mastered this book by the age of 13 and discovered sophisticated theorems on his own.Now Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method. Its like this:It is easy to solve simple equation of the first degree, e.g., 3a = 15. And we are taught how to solve second degree equations with the power of x as 2.Ramanujan found his own method in solving not only cubic equations but also equations of fourth degree.Next year not knowing that quintic equations, or equations with power of x as 5, cannot be solved, he tried and failed in his attempt.In 1903 when he was16, Ramanujan came across the book by G. S. Carr on A Synopsis of Elementary Results in Pure and Applied Mathematics, a collection of 4865 formula and theorems without proofThe book is generally acknowledged as a key element in awakening the genius of RamanujanThe next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal placesWhen he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics as an outstanding student who deserved scores higher than the maximum possible marksHe received a scholarship to study at Government Arts College, Kumbakonam, However, Ramanujan could not focus on any other subjects and failed most of them, losing his scholarship in the processHe later enrolled at Pachaiyappa' College in Madras. He again excelled in mathematics but performed poorly in other subjectsRamanujan failed his Fine Arts degree exam in December 1906 and again a year laterWithout a degree, he left college and continued to pursue independent research in mathematics. At this point of his life, he lived in extreme poverty and was suffering from starvation.Deplorable Condition of Ramanujan is expressed in his own words:“When food is the problem, how can I find money for paper? I may require four reams of paper every month.”On 14 July 1909, Ramanujan was married to a nine-year old girl, Janaki Ammal (21 March 1899 - 13 April 1994)After the marriage, Ramanujan developed a hydrocele problemsHis family did not have the money for the operation, but in January 1910, a doctor volunteered to do the surgery for freeAfter his successful surgery, Ramanujan searched for a jobHe stayed at friends' houses while hewent door to door around the city of Chennai looking for a clerical positionTo make some money, he tutored some students at Presidency College who were preparing for their examRamanujan met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical SocietyRamanujan, wishing for a job at the revenue department where Ramaswamy Aiyer worked, showed him his mathematics notebooksAs Ramaswamy Aiyer later recalled:“I had no mind to smother his genius by an appointment in the lowest level as clerk in the revenue department.”Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends.Some of these friends looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector of Nellore and the secretary of the Indian Mathematical SocietyRamachandra Rao was impressed by Ramanujan's research but doubted that it was actually his own work !Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to clear any doubts over Ramanujan's academic integrityRao listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, through which Rao was convinced of Ramanujan's mathematical brilliance . When Rao asked him what he wanted, Ramanujan replied that he needed some work and financial supportRamanujan continued his mathematical research with Rao's financial aid taking care of his daily needsWith the help of Ramaswamy Aiyer, Ramanujan had his work published in the Journal of Indian Mathematical SocietyOne of the first problems he posed in the journal was to evaluate:He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied the solution to the problem himselfHe formulated an equation that could be used to solve the infinitely nested radicals problem. Using this equation, the answer to the question posed in the Journal was simply 3In early 1912 he got a job in the Madras Accountant Generals office with a salary of Rs 20 per month.Later he applied for a position under the Chief Accountant of the Madras Port TrustHe was Accepted as a Class III, Grade IV accounting clerk making 30 rupees per monthHe used to Spend spare time doing Mathematical ResearchIn the spring of 1913, Narayana Iyer and Ramachandra Rao tried to present Ramanujan's work to British mathematiciansOne mathematician, M. J. M. Hill of University College London, commented that although Ramanujan had "a taste for mathematics, and some ability", he lacked the educational background and foundation needed to be accepted by mathematiciansOn 16 January 1913, Ramanujan wrote to G. H. HardyComing from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible "fraud“ !Hardy recognized some of Ramanujan's formulae but others "seemed scarcely possible to believe"G.H. Hardy was an academician at Cambridge UniversityHe was a prominent English mathematician, known for his achievements in number theory and mathematical analysis.Later on Ramanujan wrote to G.H.HardyHardy recognised some of his formulae but other “seemed scarcely possible to believe”. Some of them were –Initially, G. H. Hardy thought that the works of Ramanujan were fraud because most of them were impossible to believe.But eventually ,he was convinced and interested in his talent.This is one approximation formula of Pi mentioned in Ramanujan’s letters:Hardy was also impressed by some of Ramanujan's other work relating to infinite series:This second one was new to Hardy, and was derived from a class of functions called hypergeometric series which had first been researched by L. Euler and Carl F. Gauss.After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that the "[theorems] defeated me completely; I had never seen anything like them before”He figured that Ramanujan's theorems "must be true”Hardy asked a colleague, J. E. Littlewood, to take a look at the papersLittlewood was amazed by the mathematical genius of RamanujanRamanujan’s notebook referring calculus and number theory:Ramanujan boarded the S.S.Nevasa on 17 March 1914 and arrived in London on 14th AprilRamanujan began working with Hardy and LittlewoodHardy received 120 theorems from him in 1st 2 letters but there were many more results in his notebookRamanujan spent nearly 5 years in CambridgeRamanujan was awarded the B.A degree by Research in March 1916 at an age of 28 years for his work on Highly Composite Numbers.He was elected a Fellow of the Royal Society of London in February 1918 at an age of 30 years.He was the second Indian to become FRS.( First one was in 1841).He was elected to a Trinity College Fellowship as the FIRST INDIAN.During his five years stay in Cambridge he published twenty one research papers containing theorems.A few words regarding the 1729, Ramanujan NumberHardy arrived in a cab numbered 1729He commented that the number was uninteresting or dull.Instantly Ramanujan claimed that it was the smallest natural number which can be written as sum of cubes in 2 ways1729 = sum of cubes of 12 and 1/ sum of cubes of 10 and 9.Actually only this much is available in the popular version of the story.But Ramanujan had worked extensively on this number and made some simple reuslts along with other startling contributions.1729 = 7 x 13 x 19 product of primes in A.P1729 divisible by its sum of digits.1729 = 19 x 911729 is a sandwich number or HARSHAD number."Ramanujan was using 1729 and elliptic curves to develop formulas for a K3 surface," Ono says. "Mathematicians today still struggle to manipulate and calculate with K3 surfaces. So it comes as a major surprise that Ramanujan had this intuition all along."Ono had worked with K3 surfaces before and he also realized that Ramanujan had found a K3 surface, long before they were officially identified and named by mathematician André Weil during the 1950s.Just as K2 is an extraordinarily difficult mountain to climb, the process of generalizing elliptic curves to find a K3 surface is considered an exceedingly difficult math problem.And in Ramanujan’s writing he was relying on this number 1729 in order to arrive at some combination of numbers which could prove that Fermat’s last conjecture could be counter exampled.there are some popular misconceptions regarding ramanujan:Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper (About 4000 theorems)These results written up without any derivations.Since paper was very expensive, He would do most of his work (derivations) on SLATE and transfer just the results to paper.Hence the perception that he was unable to prove his results and simply thought up the final result directly is NOT CORRECTProfessor Bruce C.Berndt of University of Illinois, who worked on Ramanujan note books, stated that “Over the last 40 years, nearly all of Ramanujan’s theorems have been proven right”.Also Mathematicians agreed unanimously on the point that it was not possible for someone to imagine those results without solving / proving.I think I will complete this answer tomorrow, because I feel sleepy: Good Night!Edited in Later:I am extremely sorry for not turning up yesterday to finish the answer I started, because I had gone for an outing to Hoggenakkal in Tamil Nadu.I think I would say something more about the GENIUS before I complete.Well, once G. H. Hardy rated his contemporary mathematicians based on pure talent.Hardy rated himself a score of 25 out of 100,J.E. Littlewood 30, David Hilbert 80 andRamanujan 100 !Hardy also said that Ramanujan’s solutions were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account”Ramanujan’s genius was recognized by TN Government andNow, Tamil Nadu celebrates 22 December as ‘State IT Day’A Stamp was released by the Govt. in 196222nd December started to be celebrated as Ramanujan Day in Govt Arts College, Kumbakonam. Now on 22nd December 2011, Then prime minister Manmohan Singh said that the 125th birth year of Ramanujan will be celebrated as National Mathematics Year and from that year onwards, December 22 is National Mathematics Day.There is a National Symposium On Mathematical Methods and Applications on his name (NSMMA)And there is SASTRA Ramanujan Prize which is given under the auspices of National Mathematics Society and the society for Physics.Let me tell something about the Hardwork of Ramanujan:Once P.C. Mahalanobis, the founder of Indian Statistical Institute visited Ramanujan while in Cambridge and said to him: “ Ramanju, these English Mathematicians say that you are a Genius, A real incomparable Genius.Immediately, showing his thickly black elbow Ramanujan replied, dear friend, everything owes to this elbow.Shocked by the answer, P.C. Asked: How Can it be so?????Ramanujan replied with a smile: “During my childhood days, while using a slate for calculations, repeated erasing used to leave remnants of chalk in it, then I stopped using duster for rubbing.”“This meant that every few minutes I had to rub my slate using my elbow, it means I owe everything to this elbow.”Regarding the spiritual dimension of Ramanujan’s life, all will agree that he was a sort of a mystic, and in fact, Ramanujan was a person with a somewhat shy and quiet dispositionHe was absolutedly a dignified man with pleasant mannersRamanujan credited his success to his family Goddess, Namagiri of Namakkalin fact, He claimed to receive visions of scrolls of complex mathematical content unfolding before his eyes. And we have no idea to contradict his words.And this could be in one way regarded as his Dictom"An equation for me has no meaning, unless it represents a thought of God.”We get amazed the more we know about Ramanujan’s spiritual understanding of many mathematical concepts, I will brief just one.For example, 2n – 1 will denote the primordial GOD.When n is zero, the expression denotes ZERO.He spoke of “ZERO” as the symbol of the absolute (Nirguna – Brahmam) of the extreme monistic school of philosophy)The reality to which no qualities can be attributed,of which no qualities can be there.When n is 1, it denotes UNITY, the Infinite GOD.When n is 2, it denotes TRINITY.When n is 3, it denotes SAPTHA RISHIS and so on.Crazy isn’t it, but all such craziness constituted Ramanujan.He looked “infinity” as the totality of all possibilities which was capable of becoming manifest in reality and which was inexhaustible.According to Ramanujan, The product of infinity and zero would supply the whole set of finite numbers.Each act of creation, could be symbolized as a particular product of infinity and zero, and from each product would emerge a particular individual of which the appropriate symbol was a particular finite number.If you want to go through the life of Srinivasa Ramanujan in its fullness, I humbly refer to you my guide, the book which opened my eyes towards realizing the pearl of Indian Mathematics, and that is:“The man who knew infinity: A life of the Genius Ramanujan”It was written by Robert Kanigel.In that book Kanigel claims some very amazing facts about Ramanujan.Sheer intuitive brilliance coupled to long, hard hours on his slate made up for most of his educational lapse.This ‘poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe’ as Hardy called him, had rediscovered a century of mathematics and made new discoveries that would captivate mathematicians for next century.S.Chandrasekhar, Indian Astrophysicist, Nobel laureate 1983, told thus:“I think it is fair to say that almost all the mathematicians who signNowed distinction during the three or four decades following Ramanujan were directly or indirectly inspired by his example.Even those who do not know about Ramanujan’s work are bound to be fascinated by his life.”“The fact that Ramanujan’s early years were spent in a scientifically sterile atmosphere, that his life in India was not without hardships that under circumstances that appeared to most Indians as nothing short of miraculous. He had gone to Cambridge, supported by eminent mathematicians, and had returned to India with very assurance that he would be considered, in time as one of the most original mathematicians of the century.The words of Hardy himelf speak volumes of Ramanujan:“I have to form myself, as I have never really formed before and try to help you to form, some of the reasoned estimate of the most romantic figure in the recent history of mathematics, a man whose career seems full of paradoxes and contradictions, who defies all cannons by which we are accustomed to judge one another andabout whom all of us will probably agree in one judgement only, that he was in some sense a very great mathematician.”Bertrand arthur william russell, British philosopher & mathematician, Nobel laureate and almost contemporary to Ramanujan, stated thus:“I found Hardy and Littlewood in a state of wild excitement because they believe, they have discovered a second Newton, a Hindu Clerk in Madras… He wrote to Hardy telling of some results he has got, which Hardy thinks quite wonderful.”The life of Ramanujan is actually a textbook from which many things could be conceived. Despite the hardship faced by Ramanujan, he rose to such a scientific standing and reputation no Indian has ever enjoyed.It should be enough for youngsters like us to comprehend that if we can work hard with indomitable determination, sheer perseverance and sincere commitment, we too can perhaps soar the way like Srinivasa Ramanujan.Even today in India, Ramanujan cannot get a lectureship in a school / college because he had no degree.Many researchers / Universities will pursue studies / researches on his work but he will have to struggle to get even a teaching job.Even after more than 90 years of the death of Ramanujan, the situation is not very different as far the rigidity of the education system is concerned. Today also a ‘Ramanujan’ has to clear all traditional subjects’ exams to get a degree irrespective of being genius in one or more different subjects.He was offered a chair in India only after becoming a Fellow of the Royal Society.But it is disgraceful that India’s talent has to wait for foreign recognition to get acceptance in India or else immigrate to other places.Many of those won international recognition including noble prizes had no other option but to migrate for opportunities & recognition.(Ex. Karmerkar)The process of this brain drain is still continuing.Here is a pic of Ramanujan with his colleagues in Cambridge University.Talking about certain contributions of Ramanujan which shook me off my feet.As we all know we use the notation P(n) to represent the number of partitions of an integer n. Thus P(4) = 5, similarly, P(7) = 15.I don’t need to explain that If we were to start enumerating the partitions for larger numbers, even for small numbers such as 10 we start seeing that there is a combinatorial explosion! To illustrate this consider P(30) = 5604 and P(50) = 204226 and so on. (btw, partitions can be visualized by Young tableau!).A similar search was on for asymptotic formulae for the partition number P(n) and because of the combinatorial explosion an accurate formula was considered difficult. Ramanujan believed that he could come up with an accurate formula even though it was considered extremely hard, and he came close.One work of Ramanujan (done with G. H. Hardy) is his formula for the number of partitions of a positive integer n, the famous Hardy-Ramanujan Asymptotic Formula for the partition problem. The formula has been used in statistical physics and is also used (first by Niels Bohr) to calculate quantum partition functions of atomic nuclei.The formula he proposed gives a very close value to that of the true value, and it is a mouth-watering feat considering its very pattern less nature.I had written another answer in quora regarding how Ramanujan provided a rapidly converging series as the value of Pi. I will just copy and paste it here.For a long time, the series used for finding the value of Pi was given by the Leibniz-Gregory Series.π = 3 + 4/(2*3*4) - 4/(4*5*6) + 4/(6*7*8) - 4/(8*9*10) + 4/(10*11*12) - 4/(12*13*14) ...But in order to give the value of Pi correctly upto 5 decimal places, this series required around 500000 terms.Now, in the Indian tradition, another formula was given by Nilakantha, a mathematician of Kerala School of Mathematics who lived couple of centuries before Leibniz and the series converged much rapidly.π = 3 + 4/(2*3*4) - 4/(4*5*6) + 4/(6*7*8) - 4/(8*9*10) + 4/(10*11*12) - 4/(12*13*14) ...And in order to give the value of Pi upto 5 decimal places, this series required only 6 terms. And thats a great thing but which failed to catch the eye of westerners until the nineteenth century.Now, take into consideration all these and what Ramanujan did. Ramanujan simply penned down an infinite series, looking so horrendous, which would be equal to the reciprocal of Pi.And this is the most rapidly converging series ever given for the value of Pi and the algorithm based on this have actually been used in computers.Now the most beautiful factor. In order to have the value of Pi upto 6 decimal places the infinite series of Ramanujan needed only ONE SINGLE TERM.And you take the second term and there you have suddenly the value of Pi upto 11 terms in your hands.I think it speaks something Great, and Ramanujan was indeed Great!!!Ramanujan has done extensive works in finding out highly composite numbers, and he has written down a long list of similar numbers which had more factors than any of the previous number.The highest highly composite number listed by Ramanujan is 6746328388800Having 10080 factorsHe received his degree from the university (later named Ph.D) for his work of highly composite numbers.I would just say another thing which caught my eye and unleashed an array of thoughts.Ramanujan while sick and dying in India, mentioned some very peculiarly behaving functions which mimicked the original moldular functions.The mock theta functions remained a mystery for most part of the last century and only the Great Ono made inroads towards their reality.In fact, no one at the time understood what Ramanujan was talking about.It wasn’t until 2002, through the work of Sander Zwegers, that we had a description of the functions that Ramanujan was writing about in 1920,' Ono said.Ono and his colleagues drew on modern mathematical tools that had not been developed before Ramanujan’s death to prove this theory was correct.Ramanujan actually wrote those functions claiming that he saw it in a scroll in the hands of A Goddess.Anyway now they are used to calculate the entropy of Black Holes ( A concept which developed years after his death.)Ono’s team was stunned to find the function could be used today.'No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them,' Ono says.Ramanujan’s Intuition Stands OUT!I think, just for a fun I would show the Mock Theta FunctionsNow I think I shoudl mention atleast something about the impact of Ramanujan’s work on statistical physics.For example imagine studying the statistics of a gas made of electrons confined to 2D. You could do something complicated like model the exact positions and momenta of many of electrons along with the force between them. Or you can simplify by imagining that the electrons can only occupy positions on a discrete triangular lattice, and instead of a repulsive force you can make the simple approximation that two electrons aren't allowed to be next to each other.The result is the Hard hexagon model and some work of Ramanujan's appears when you try to model it. Even if it's not physically realistic, these models share characteristics with more realistic physical models and give useful insight.In fact a whole bunch of different identities related to Ramanujan's work can appear when you study these kinds of simple physical models, especially 2-dimensional models. Eg. Hard Hexagon ModelI think I will conclude with a simple assumption of Ramanujan, I think it deserves mention:The mock theta functions which we mentioned earlier looked unlike any known modular forms, but he stated that their outputs would be very similar to those of modular forms when computed for the roots of 1, such as the square root -1. Characteristically, Ramanujan offered neither proof nor explanation for this conclusion.It was only 10 years ago that mathematicians formally defined this other set of functions, now called mock modular forms. But still no one fathomed what Ramanujan meant by saying the two types of function produced similar outputs for roots of 1.Ono and his colleagues have exactly computed one of Ramanujan’s mock modular forms for values very close to -1. They discovered that the outputs rapidly balloon to vast, 100-digit negative numbers, while the corresponding modular form balloons in the positive direction.Ono’s team found that if you add the corresponding outputs together, the total approaches 4, a relatively small number. In other words, the difference in the value of the two functions, ignoring their signs, is tiny when computed for -1, just as Ramanujan said. Incredible Intuition !I am just adding some pictures I came across.his notebooks, the last three,His handwritings and works mentioned without calculation:I think I can say nothing more, but if at all someone asks me, I would say if I know!By the way, I have actally spoken nothing regarding the complex mathematical contributions of this great mathematician,even without that I think you are thrilled and that is why, even if the statement is wrong in itself.“ Ramanujan is the greatest Mathematician of all time, at least I believe so.”His life and contributions ought to be known and famous, I suppose!But if you ask the second question whether he could be compared to other mathematicians, well as the other answers explicitly point out, we can never do a comparison. Because, they are all unique and their contributions remain unique for this field. Its like having a room full of good artworks, the world of mathematics. Each one is perfect in its own regard, but comparing them won’t do any help because the absence of one would leave a vacuum and the same is the case here, we cannot talk about the world of mathematics excluding one of these, or else, we would be terribly wrong!!!!! And our knowledge, just incomplete, however beautiful it may be.
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What is the wisest thing your best friend has said?
Surprisingly enough, I’ve many friends who’ve said this to me in one form or the other, which sounds like the wisest piece of advice. This story can summarize it:There was a businessman who was deep in debt and trapped.He sat on the park bench, head in hands, wondering if anything could save his company from bankruptcy.Suddenly an old man appeared before him. “I can see that something is troubling you,” he said.After listening to the businessman’s story, the old man said, “I believe I can help you.”He asked the man his name, wrote out a cheque, and pushed it into his hand saying, “Take this money. Meet me here exactly one year from today, and you can pay me back at that time.”Then he turned and disappeared.The businessman saw in his hand a cheque for $500,000, signed by John D. Rockefeller, then one of the richest men in the world!“I can erase my money worries in an instant!” he realized. But instead, he decided to put the uncashed check in his safe. Just knowing it was there might give him the strength to work out a way to save his business, he thought.With renewed optimism, he closed several big sales. Within a few months, he was out of debt and making money once again.Exactly one year later, he returned to the park with the uncashed cheque. At the agreed-upon time, the old man appeared. But just as he was about to hand back the cheque and share his success story, a nurse came running up and grabbed the old man.“I’m so glad I caught him!” she cried. “I hope he hasn’t been bothering you. He’s always escaping from the rest home and telling people he’s John D. Rockefeller.”And she led the old man away by the arm.The astonished businessman just stood there, stunned. All year long he’d been wheeling and dealing, buying and selling, convinced he had half a million dollars behind him.Suddenly, he realized that it wasn’t the money, real or imagined, that had turned his life around. It was his newfound self-confidence that gave him the power to achieve anything he went after.One of my friends is a cricket player. I asked him once, “What if you aren’t able to become a cricketer? Have you thought of any backups?”This is what he told me, “I don’t want to become a cricketer, I am already one. I just haven't been recognized yet. Cricket is my passion. The worst that can happen is, I fail to get the recognition. It’s okay, I’ll sell fruits. But the fear of failing is too meager in front of my enthusiasm for what I love the most!”Another friend of mine is preparing for a competitive exam. One day, I casually shared with her about the competition that exists today in any field. This is what she said, “I really don’t understand why competition doesn’t bother me. I just have this strange belief that if I’m putting in sincere efforts and I keep doing that, favourable results will come, why won’t they?”These people have motivated me in life at several times. When I look at their persistence and passion, it strengthens my self-belief.Amazingly true, it is one of the best qualities to be possessed and nurtured in life. So many people have emphasized on the importance of self belief:"Confidence comes not from always being right but from not fearing to be wrong." Peter T. Mcintyre"People are like stained-glass windows. They sparkle and shine when the sun is out, but when the darkness sets in their true beauty is revealed only if there is light from within." Elisabeth Kubler-Ross"Talk to yourself like you would to someone you love." Brene Brown"You can have anything you want if you are willing to give up the belief that you can't have it." Dr. Robert Anthony"Inaction breeds doubt and fear. Action breeds confidence and courage. If you want to conquer fear, do not sit home and think about it. Go out and get busy." Dale Carnegie"Nothing can stop the man with the right mental attitude from achieving his goal; nothing on earth can help the man with the wrong mental attitude." Thomas Jefferson"You wouldn't worry so much about what others think of you if you realized how seldom they do." Eleanor Roosevelt"If you are insecure, guess what? The rest of the world is, too. Do not overestimate the competition and underestimate yourself. You are better than you think." T. Harv Eker"To anyone that ever told you you're no good ... They're no better." Hayley Williams"I've finally stopped running away from myself. Who else is there better to be?" Goldie HawnEven Wile E. Coyote didn’t fall till he looked beneath and realized that gravity existed. Then, why would you? Believe in yourself!Story Source - A Short Story on Self ConfidenceImages - Google Images Reference - The Road Runner Show
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Do we have a collective paradigm? Else, is it fragmented?
I answer a definite yes, we do have a collective paradigm, which the vast majority of people within and outside of science subscribe to. They are not doing so consciously, it is experienced as "simply the way things are". I will first illustrate the existence of the paradigm via quotes then briefly describe the structure of the paradigm, followed by a brief examination of its core beliefs and limitations, as well as the reasons for these. Once this is understood it then becomes possible to comprehend a glimpse of the emerging paradigm.Revealing the collective paradigmThe paradigm is so deeply engrained and unconscious that it is essentially invisible to those who subscribe to it. Anything that cannot be understood within that paradigm is assumed to be fundamentally incomprehensible. Any problems that lie outside of its scope are avoided. The few who do not subscribe to it are assumed to be stupid, charlatans or crazy. So how can it be revealed?"The essence of science: ask an impertinent question, and you are on the way to a pertinent answer." (Jacob Bronowski)Core features of that paradigm have been brought into question, most notably by quantum mechanics. Hence that 'invisible' paradigm is beginning to come into view and people are beginning to question its validity."There exists a cognitive repression of the interpretation problem by the majority of physicists. For that majority the questions concerning the meaning of quantum mechanics are answered once and for all by the Copenhagen interpretation, and all further inquiry is rejected as a sign that the inquirer does not understand the topic. Further questions are called "only philosophical" and thus not befitting a physicist. But if one inquires in depth what the Copenhagen interpretation says one gets a variety of different answers. According to Fox-Keller this, too, is a sign for evasion, whereby what is evaded is the necessity of a new cognitive structure which differs radically from the existing one. Fox-Keller calls the old structure classical objectivism." (Anton Zeilinger, On the Interpretation and Philosophical Foundation of Quantum Mechanics)"discussions about the meaning of quantum mechanics remain stymied as a result of the failure of physicists to formulate a cognitive paradigm adequate to their theory... implicitly, they retain one or the other of the two basic tenets of classical physics, the objectivity or the knowability of nature." (Evelyn Fox Keller, Cognitive repression in contemporary physics) How can one recognise when one is caught within a self-reinforcing delusion?"The most exciting phrase to hear in science, the one that heralds the most discoveries, is not ‘Eureka!’ (I found it!) but ‘That’s funny...’" (Isaac Asimov)"The paradox is only a conflict between reality and your feeling what reality ought to be." (Richard Feynman)For an example of the paradoxical nature of quantum systems see John Ringland's answer to In simple terms, what does the Stern-Gerlach experiment imply about the nature of quantum systems and observable phenomena?A good way to reveal an all-pervasive, unconscious collective paradigm is to observe the cognitive dissonance that arises when that paradigm clashes with the paradigm that is emerging from quantum mechanics.Here are some quotes that illustrate the prevailing cognitive dissonance:"He has a grad student who is thinking about the meaning of quantum mechanics – he's doomed." (John von Neumann)"If [quantum theory] is correct, it signifies the end of physics as a science." (Albert Einstein)"Einstein said that if quantum mechanics is right, then the world is crazy. Well, Einstein was right. The world is crazy." (Daniel Greenberger)"Everything we call real is made of things that cannot be regarded as real." (Niels Bohr)"I do not like [quantum mechanics], and I am sorry I ever had anything to do with it." (Erwin Schrödinger)"Those who are not shocked when they first come across quantum theory cannot possibly have understood it." (Niels Bohr)"It is safe to say that nobody understands quantum mechanics." (Richard Feynman)"Quantum mechanics makes absolutely no sense." (Roger Penrose)"If you are not completely confused by quantum mechanics, you do not understand it." (John Wheeler)"Do not keep saying to yourself, if you can possibly avoid it, ‘but how can it be like that?’ because you will get ‘down the drain,’ into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that." (Richard Feynman)"Of all the theories proposed in the 20th century, the silliest is quantum theory... The only thing quantum theory has going for it is that it is unquestionably correct." (Michio Kaku)"Whoever endows the wave function with more meaning than is needed for computing observable phenomena is responsible for the consequences." (Nico van Kampen)"Electrons seem to have modes of being, or modes of moving, available to them which are quite unlike what we know how to think about." (David Z. Albert)"Quantum mechanics is magic." (Daniel Greenberger)"If you think you understand quantum mechanics, you don’t understand quantum mechanics." (Richard Feynman)"Had I known that we were not going to get rid of this damned quantum jumping, I never would have involved myself in this business!" (Erwin Schrödinger)"[Quantum mechanics is] the intellectual scandal of the century." Rene Thom"After more than 50 years (now over 80 years) of unquestionable success as a theory, questions about the interpretation of quantum mechanics continue to plague both physicists and philosophers." Evelyn Fox Keller"There is a major 'dangerous' scientific idea in contemporary physics, with a potential impact comparable to Copernicus or Darwin. It is the idea that what the physics of the 20th century says about the world might in fact be true." (C. Rovelli)Structure of the collective paradigmIt originates from Naive Realism, see John Ringland's answer to What is naïve realism? This is its unconscious roots that permeate almost the entire human population.It manifests within science as classical objectivism and empiricism, see: John Ringland's answer to Naive Realism: Can it ever be said that Scientific realism takes off from the springboard of commonsense or naive realism?Core beliefsThe affect of naive realism means that the resulting paradigm inevitably leads to an unconscious, unshakeable, unquestionable belief in materialism and an objective physical universe, as well as a strong aversion to subjective experience and anything related to it.“some perceptions are interrelated or associated to form other perceptions which are then projected onto a world putatively outside the mind.” (David Hume)The reason why these beliefs arise is because naive realism leads us to unconsciously ignore the role of experience in the apprehension of that which is experienced and to assume objective existence for the objects that are portrayed by experience. This is why empiricism was such an obvious and compelling methodology for science to initially adopt. It also explains why the role of the objective observer was the obvious role for naive realist scientists to adopt.In the early formulation of the scientific method people were assumed to be capable of studying nature "with minds washed clean from opinions" (Francis Bacon) thus able to apprehend the natural world as it is. Thus the influence of naive realism was not recognised and it became enshrined in the scientific method in the form of empiricism.“Empiricists claim that sense experience is the ultimate source of all our concepts and knowledge” (Rationalism vs. Empiricism)Thus the scientific discourse became entangled within a closed loop of self-reinforcing hidden assumptions, which necessitated the postulation of more and more beliefs to explain the distorted view of reality that was being examined.To explain the many objects that we apprehend we long ago postulated the existence of 'matter', which serves as the substance in which the observable properties inhere. Not realising that those observations were actually arising in our experiences and not inhering in external objects."materialism is the philosophy of the subject who forgets to take account of himself." (Schopenhauer)This further required us to postulate an objective 'fabric' of space and an objective 'arrow' of time. And much more, analogous to the proliferation of epicycles in the Ptolemaic system of astronomy. These assumptions underlie all of classical physics and they served us well right up until the advent of relativity theory and quantum mechanics which revealed how inadequate these ideas really are. In the face of energy / mass conversion, relativistic inertial frames, quantum non-locality, etc the ideas of objective matter, space and time break down. "We have no satisfactory reason for ascribing objective existence to physical quantities as distinguished from the numbers obtained when we make the measurements which we correlate with them... On the contrary, we get into a maze of contradiction as soon as we inject into quantum mechanics such concepts as carried over from the language and philosophy of our ancestors... It would be more exact if we spoke of ‘making measurements’ of this, that, or the other type instead of saying that we measure this, that, or the other ‘physical quantity’." (E. C. Kemble, The Fundamental Principles of Quantum Mechanics)Core limitationThe whole idea of an objective external world is undermined by modern science and the world-view that modern science forces upon us seems paradoxical because the belief system upon which classical science was formed had, from the outset, consistently ignored the role of subjective experience in the apprehension of phenomena. “Useful as it is under everyday circumstances to say that the world exists 'out there' independent of us, that view can no longer be upheld.” (John Wheeler)Overlooking the role of subjective experience in the apprehension of phenomena leads directly to the core limitation of the current paradigm. That is its complete inability to coherently approach the topic of consciousness, which is thus known as the Hard Problem of Consciousness.Science has "looked through" consciousness whilst pretending to be objectively apprehending an external world. It has studied those allegedly external objective phenomena and then much later it attempts to explain consciousness in terms of those phenomena. That is why consciousness is such a mystery to science.This is signNowly compounded by the entrenched belief in inanimate matter that engages in inanimate mechanical processes that are mysteriously guided by universal laws. This belief arose from the focus on the outer appearances of things whilst overlooking the inner experiences of ourselves and totally denying that other systems could also have inner experiences of some kind appropriate to their nature. Animals were once believed to be automatons and behavioural psychology in the 20th century even garnered signNow support for its claims that humans also had no inner experiences. That is the extreme of the objectivist approach, to utterly deny the existence of subjectivity.“The old foundations of scientific thought are becoming unintelligible. Time, space, matter, material, ether, electricity, mechanism, organism, configuration, structure, pattern, function, all require reinterpretation. What is the sense of talking about a mechanical explanation when you do not know what you mean by mechanics? The truth is that science started its modern career by taking over ideas derived from the weakest side of the philosophies of Aristotle's successors. In some respects it was a happy choice. It enabled the knowledge of the seventeenth century to be formulated so far as physics and chemistry were concerned, with a completeness which lasted to the present time. But the progress of biology and psychology has probably been checked by the uncritical assumption of half-truths. If science is not to degenerate into a medley of ad hoc hypotheses, it must become philosophical and must enter upon a thorough criticism of its own foundations.” (Alfred North Whitehead)However quantum mechanics, due to the fact that it is a rationalist science and not an empiricist science could overcome the limitations of naive realism. See John Ringland's answer to Can it ever be said that Scientific realism takes off from the springboard of commonsense or naive realism?Thus in quantum mechanics the role of the observer is central."Wheeler labels the individual quantum phenomenon an elementary act of creation. We as observers play a signNow role in this process... We have now gradually brought the role of the observer into the center of our discussion, a role which is expressed by Clauser in his joint analysis with Shimony of the present EPR-Bell situation as follows: "perhaps an unheard tree falling in the forest makes no sound after all"". (Anton Zeilinger, On the Interpretation and Philosophical Foundation of Quantum Mechanics)“I had come to suspect, and now felt compelled to acknowledge, that science and the physical world were products of human imagining; that we were not the cool observers of that world, but its passionate creators. We were all poets and the world was our metaphor.” (Roger S Jones)“Quantum theory essentially erased the difference between matter and fields, making reality a unit that exhibits the properties of both. This single, unitary stuff gave rise to the fantastically successful algorithm now used by physicists in all calculations involving quantum theory. But nobody knows what this unitary stuff really is. Most quantum physicists, of course, stop short of calling this unitary substance consciousness.” (Norman Friedman)These tensions between what science is saying about reality and the current paradigm, with its beliefs about how reality ought to be result in avoidance and cognitive repression: see John Ringland's answer to Despite having evidence that contradicts someone's belief, why can't they come to believe something new?This inevitably produces a degree of dogmatism in contemporary science: see John Ringland's answer to Has science become too dogmatic?Glimpse of the emerging paradigmFor a discussion on the paradigm emerging from quantum mechanics and how to understand it see John Ringland's answer to Will we ever be able to truly understand Quantum Mechanics?Instead of recoiling from the apparent paradoxes and simply stating that quantum mechanics makes no sense it is possible to shift to a different paradigm from which these paradoxes can be seen to be sensible and necessary features of reality. For example, see John Ringland's answer to What is light made up of, particles or waves?Once the emerging paradigm is understood it becomes possible to begin to address other outstanding issues, such as the question of what is consciousness. For some insights into this see John Ringland's answer to What is consciousness?It also becomes possible to consider the broader issue of Now that naive realism has been disproven by quantum mechanics, how will this impact our collective paradigm?
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