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FAQs
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What is the most asked question on Quora (by the number of questions merged into it)?
As someone who writes mostly technology-related answers, I see the following question so much it makes me want to tear my hair out:“Can iCloud Activation Lock be Bypassed?”For those who don’t know, Apple devices that have an iCloud account active on them with Find My iPhone enabled will lock the device to that Apple ID even if it is restored to factory defaults. This is designed to prevent thievery, since stolen devices (typically iPhones) are useless without the Apple ID password they are locked with to unlock it. It is incredibly common for people to sell devices without removing the lock beforehand (likely because they don’t know it exists, or how to remove it) or because it is stolen. Either way, the lock cannot be bypassed without that password… but that doesn’t stop everyone and their mother from asking if it can be done as if the rules somehow don’t apply to them.Instead of viewing the answers on an existing question, or even asking new people to answer that existing question, they make a new one. Every. Single. Time. Quora is absolutely flooded with these questions, and I get A2A requests for them more than anything else.
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What facts about the United States do foreigners not believe until they come to America?
I came to the US as a student. I had heard about a lot of stuff about Americans and America but much of it I felt was plausible even though I had not set foot in the US...Below are some of the stuff I had to see or live in the US for a while before coming to terms with it.Building houses with wood...Coming from Asia most of the houses, even for middle class folks, are built with bricks. I was told they were built with wood/sheet rock but didn't believe it until I came here.Americans are very private, and value their personal space. ..I am not saying this is true for ALL Americans but feel 80% fall in this category. They won't invite you to their house for a cup of coffee and chit chat unless they know you really well. No inviting co-workers over and stuff like that. Again coming from the East, I found that a bit unbelievable but once I lived here, it dawned on me that this aspect was true. Back home we invited our neighbours and co-workers for all kinds of functions or for just chit-chat/gossip.High school kids and pre-marital sex: I saw and read of this in the American movies, but always thought, eh..the movies just want to sell more of their stuff by peddling nudity and sex scenes. Boy, was I wrong... in college we had designated areas across campus where the students could pick up free condoms, talk freely with resident counselors about birth control, and most of my American friends had lost their virginity by 18, the more ambitious ones losing it by age of 16. And even in their early to late adulthood, they are free to experiment and try out numerous partners sexually before deciding to get married with "The One"Prevalence and easy access to lawyers: Where I grew up either the lawyers were expensive for most households or corrupt or both. Going to a lawyer meant numerous nail biting conversations about what the next steps entailed. It is kind of a big deal back home. Here in the US, lawyers offices are everywhere and easy to find in Yellow Pages and approach and talk to.Hostility of cops towards African Americans: Again much of knowledge of this initially came from American popular culture like movies and rap songs, so I found it a bit specious. But once I came to the US, it was evident that the cops don't have good relations with African Americans, and also understand the reasons.People's love for pets: Americans love their pets, to the point that some feed their cats and dogs organic locally sourced gluten free pet food. My landlady would let her two dogs and cat sleep on her bed with her. That would not be acceptable where I lived back home. Keeping pets is one thing, letting them get on the couch and feeding them Whole Foods type meals would be grounds to have you arrested.Obsession with sports: People knowing the stats of baseball or NFL football or college football players and what year this team did this and that team did that. Entire radio and TV broadcasts are dedicated to sports analysis. I sort of heard of this from friends who went to America earlier than I did but the sheer scale of obsession with their national, regional and local sports teams left me amused.Adults/Seniors doing what is expected of young people mostly: This was just too shockingly funny to me. My 64 year English professor in college had a boyfriend. He would visit her sometimes on campus wearing tie-dye t-shirts, soccer mom jeans and sported a ponytail. She was just one year younger than my grandmother. Back home, boyfriends are for teens and young adults. Old people have husbands/wives or just pass away alone (assuming spouse dies before they do), maybe with a religious book/prayer beads/rosary in hand. No dressing up and hitting the clubs followed by wine and cheese parties. Just a bit bewildering to believe until someone from the East comes here.Returns/returning stuff/refund: This was hard to believe as well. That you can buy something, anything short of undergarments, and if you don't like it, return it for full refund (for the most part), and the cheery sales rep will take it back without any complaint. Back home, once you buy it, it is YOURS! no refund for you. One example: When I was moving from Ohio to Arizona, I decided to drive (about 2 days drive) instead of flying. I needed a new GPS as my old one wasn't working. When I went to Best Buy, I wasn't sure which GPS was the most reliable and cost effective. The one I really liked was really expensive. The sales rep says, "look, just buy the one you like...use it to get to AZ and when you get there, just return it to one of our stores. Make sure you have your receipt." Wow.Superficial wealth status: You can buy any expensive car or house as long as you have the desire (and decent to not-so-decent credit), even if you make McDonald's or janitor wages. I have been thrown off numerous times by people living in small houses, eating meager grilled cheese sandwiches/ramen noodles for lunch and dinner but driving a nice Porsche or Cadillac. The belief is: why shouldn't you own something you desire? This is America and it is your God given right to get what your heart desires. Back home, they will laugh you out of the expensive car showroom if you work as a janitor but desire the expensive car. And no, you won't get a higher interest rate (APR), even if you can afford it.
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As a startup founder of three years our legal housekeeping is a bit of mess, how can I best setup a system to organize and track
As a startup founder of three years myself, I can relate to how legal housekeeping can be messy. Once a year, I have our own lawyers go through and do an audit of all of our legal paperwork (which costs a couple thousand dollars to be extremely thorough, but it’s worth it). Luckily, there are now many ways to easily manage and track all of your legal, financial, and HR documents via third-party sites that specialize in these management proceedings. I wrote a blog post about this awhile back titled “5 Ways to Save Time Dealing With Documents” which highlights certain sites that can be very beneficial depending on what paperwork you’d like to track or manage. They are as follows:1. GroupDocsGroupDocs is a new, comprehensive online service for document creation and management. It has multiple features, including a viewer for reading documents in your browser, an electronic signature service, an online document converter, a document assembly service, a feature for comparing different versions of a document, and an annotation feature. An individual plan is $10 per month for limited storage and 500 documents, while a group plan for up to 9 people is $19 per user per month. Based on the number of features and pricing, GroupDoc is a good-value purchase for a small business. As you’ll see below, GroupDocs can be cheaper than a service that offers only one such feature.2. signNowWhen you’re closing a deal and need to get documents signed, the last thing you need is a slow turnaround due to fax machine problems or the postal service. The solution is to use an electronic signature service such as signNow, which is one of the most popular e-signature companies in the world. This service allows you to email your documents to the person whose signature you need. Next, the recipient undergoes a simply e-signing process, and then signNow alerts you when the process is completed. Finally, signNow electronically stores the documents, which are accessible at any time. As a result, you can easily track the progress of the signature process and create an audit trail of your documents. The “Professional” plan is recommended for sole proprietors and freelancers, and costs $180 per year ($15 per month) for up to 50 requested signatures per month. The “Workgroup” plan is geared towards teams and businesses, and it costs $240 per user per year ($20 per month per user), for unlimited requested signatures.3. signNowsignNow is another e-signature service. Similar to signNow, signNow allows you to upload a PDF file, MS Word file or web application document. Next, you can edit the document, such as by adding initials boxes or tabs, and then email them out for signatures. Once recipients e-sign the document, signNow notifies you and archives the document. signNow offers low rates for these services: a 1-person annual plan with unlimited document sending costs $11 per month. An annual plan for 10 senders with unlimited document sending costs only $39 per month.4. ExariExari is a document assembly and contract management service that assists in automating high-volume business documents, such as sales agreements or NDAs. First, the document assembly service allows authors to create automated document templates. No technical knowledge is required; most authors are business analysts and lawyers. Authors have a variety of options for customizing documents, such as fill-in-the-blank fields, optional clauses, and dynamic updating of topic headings. They also can add questions that the end user must answer. Once you send out the document, the user answers the questionnaire, and Exari uses that data to customize the document. Next, the contract management feature allows you to store and track both the templates and the signed documents. Pricing is based on the size and scope of your planned implementation, so visit their website for more information.5. FillanyPDFIt’s a hassle having to print out PDF forms in order to complete them. Fortunately, FillanyPDF is a service that allows you to edit, fill out and send any PDFs, while entirely online. This “Fill & Sign” plan costs $5 per month, or $50 per year. If you subscribe to the “Professional” plan, you can also create fillable PDFs using your own documents. With this service, any PDF, JPG or GIF file becomes fillable when you upload it to the site. You can modify a form using white-out, redaction and drawing tools. Then, you can email a link to your users, who can fill out and e-sign your form on the website. FillanyPDF also allows you to track who filled out your forms, and no downloads are necessary to access these services. The “Professional” plan costs $49 per month, or $490 per year.Switching firms can be a hassle. As a former startup attorney, I have a bit of advice about finding the right attorney for your business: it’s best to focus on the specific attorney you’ll be working with. He or she should have a solid understanding of the ins and outs of your business industry, a deep knowledge of the legal issues your startup may face, and previous work experience with startups to ensure a quality and efficient work product. This is absolutely key when matching our startup clients at UpCounsel to attorneys on our platform who can perform their legal work and hash out their legal projects in a timely manner. We also allow clients to store any and all of their legal documents directly on UpCounsel so they don’t have to go searching in alternative places for the correct paperwork. It’s proven to be a free and lightweight way to store legal documents that our clients love. Here's what it looks like:As I’ve mentioned, it’s more important to find the right attorney as opposed to the right law firm. And seeing as you’re a startup, our own startup clients typically save an average of 50-60% on their legal work, since the attorneys don't include overhead fees (a.k.a. the fees included for doing business with the firm itself) in their invoices.Hope this gives you a deeper look into what other sites and services are out there. If you have any questions or would like more information on how best to handle your legal housekeeping/ attorney matters, feel free to signNow out to me directly. As a former startup attorney at Latham & Watkins, I’d be happy to give you some guidance.
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What were Apple's 1981 "Awful Bullflap memos"?
The “Awful Bullflap” memos were a satirical, anonymous Apple internal publication made in a “press release” or “internal memo” format similar to the official “Apple Bulletin” memos. The name even fit into the same number of letters as the Bulletin’s distinctive banner.In the imaginary world of the memos, the “Awful Bullflap” reports on the activities of “Awful Computer” on “Badly Drive” in “Cuspiduro, CA” in a way that ‘coincidentally’ corresponds to what the “Apple Bulletin” might report about “Apple Computer” on “Bandley Drive” in “Cupertino, CA.” “Awful” stands for “Apple” and “Bullflap” euphemistically stands in for “Bullshit.” The slang term “flap” also means “agitation, panic, or fluster,” adding an additional layer of meaning.The Apple archives at Stanford have three specimens of “Awful Bullflap” from April 1981. It is not clear if these were the only three memos or if more were made.The first memo is styled as a press release and describes a fictional testing facility being opened at Sing Sing prison, presumably a protest against setting up a low-cost 3rd party facility in place of Apple Employees.The second memo describes a “Proposed Sacrificial Offering” of shareholders, probably mocking stock offering memos of the era.The third memo is of the form of an internal memo (which is how companies communicated before e-mail, you younglings) from “C. A. Drakkula” (Probably a mashup of Count Dracula and Mike Markkula) making fun of an internal reorganization. As in all the Bullflap memos, the thinly veiled “changed” names are pretty indicative of the names they represent.MAKING OF THIS ANSWERI first saw this question two years ago, and I was intrigued. I followed it. When I searched for “Awful Bullflap” I found that Stanford University housed the original Apple Computer, Inc. records, 1977-1998 in the Special Collections & University Archives.When I looked further, I found that members of the public could request access to these archives and read them in the Field Reading Room. Since the special collections live in offsite storage and not in the library itself, Stanford requires at least 48 hours advance notice to transport the materials to the reading room.I posted this information in the comments of the question and said, “hey, someone local should do this!”Crickets.In September 2017, I knew I would have an afternoon appointment near Stanford, and I knew with enough advance notice to request the material. So I did! I blocked the two hours on my calendar before my meeting to park, go to the Green Library at Stanford, and see this mysterious material that is not available in digital form, and then get to my other appointment.Thankfully, Stanford is generous in allowing public access to a lot of rare archives, and they have a straightforward, well documented process that I successfully used to access this rare and unique collection.First, I requested the material through Access to Apple Collections on Tuesday, setting an appointment to view on Friday. After a painless on-line registration process. I got an email which let me track the progress of the material from an offsite location to the Field Reading Room. Before I left for Stanford, I could check that it had made it to the reading room.On Friday afternoon I departed for the labyrinthine alternate universe of Stanford. I meandered among an alphabet soup of vaguely threatening permit parking signs that festoon the campus and reflect the rigorous hierarchy of privilege there. I finally found a visitor lot and paid for parking. The clock began ticking!Using Google Maps to navigate my walk, I found my way to the Green Library where a gentleman pointed me to a computer that registered my existence by scanning my drivers license and ingesting my vital information. Once the computer acknowledged my existence, I checked in to the library and printed an adhesive name tag granting me admission to the hallowed halls.I found my way to the Field Reading Room on the second floor of the Bing Wing and identified myself there. On a separate computer system tracking users of archival material, I registered myself and signed in. They gave me a locker key in return for my ID so that I could leave everything I brought in a secure locker. I was permitted only a laptop and phone, and free Wi-Fi access. I locked up my stuff, washed my hands and headed in.Now that I had cleared all of the administrative hurdles, they gave me this:Then I sat down at their posh, academic-looking desk with a wide, comfy chairAnd found the object in question:This is what research looked like before Google, kids! (EDIT: as Sean Owczarek points out, much modern research looks this way, too, especially for history! Contrary to popular belief, not everything is on Google.)It was fun to research this answer. I hope the question followers enjoy it. There were additional questions in the comments. If they turn into actual questions, I can probably answer them.
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Gmail in 2015: As a techie person, what is in your Gmail?
As a techie person, What is in your gmail ? There's a lot of stuff in my GMail; I run much of my routine business through it. (My regular Comcast E-mail is mostly used for mailing-list traffic and such, and gets auto-collected by a server at home. My phone is configured to access my GMail and Erbosoft mailboxes.)I do make extensive use of E-mail filters to redirect much of the incoming mail to other categories, and out of the inbox. Quora notifications get redirected, for instance, as do those from LinkedIn, Medium, Goodreads, and YouTube. Most advertising messages from companies I do business with get redirected to the label "...
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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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Why is Contrarian Investing a good strategy?
Thanks for the A2A!Before answering the question let's look at the definition of Contrarian according to Wikipedia, "In finance, a contrarian is one who attempts to profit by investing in a manner that differs from the conventional wisdom, when the consensus opinion appears to be wrong." As the definition suggests, such investors buy when everybody is selling and sell when everyone is buying. Now this isn't necessarily a good strategy if it is not backed by thorough research (Sometimes the herd is right :P) but if backed by thorough research, it could really lead to enormous profits. How you ask? Suppose you see a company with a great business being pummeled in the market. You have done your research on the company and know that the present price massively over-rates the risk on the company and under-rates the profit. You start buying. Now this company would be trading at a low point and you get to buy it at a low price. After a few days, the Company releases it's quarterly earnings and lo! to everyone's surprise, the Company has made huge profits. Everyone starts buying and you see that now the Company has been over-valued. You sell your stock at a much higher price. Since you have quoted Warren Buffett, I agree that the principles of Value Investing espoused by Buffett is somewhat similiar to this principle. Buffett's principle relies on buying undervalued companies and selling overvalued ones. Examples:1.) After the 1973-1974 Bear Market crash, Buffett bought shares in Washington Post, an investment which increased in value by over a 100 times over a period of time. (This was a great investment because even the book value of the assets would have been greater than $400 million while the market cap was just $80 million.)2.) The best example I would give would be John Paulson's bets against the subprime-mortgage market in 2006 using credit default swaps. Paulson bought these swaps which were essentially bets against the subprime mortgage market when it was at it's peak in 2006. 2 years later, the subprime mortgage market crashed and John Paulson grew from a relatively-unknown investor to fame and fortune. Gregory Zuckerman has also called this trade "The greatest trade ever". Paulson made $15 billion on an investment of $12.5 billion (an increase of over 100%)Caution points:1.) Don't adopt this strategy without thorough research.2.) This strategy takes time to yield it's often fantastic results. So the returns may take time to materialize. Do have cash to insulate yourself in these times. (As is always said, In times of crisis, cash is king.)
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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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Do you regret having LASIK eye surgery done?
The day that changed my vision. From normal video to HD or 4K.The basic criteria for doing the surgery:The minimum preferable age for doing the surgery is 20 years.The power your eyes should be constant for at least 2 years.You should do a test in which they check whether you are eligible for the surgery by checking the thickness of the cornea and other eye specifications. (Your ophthalmologist can tell you in more detail)These precautions are necessary because you minimize or eliminate the chance of regaining spectacles not wearing contact lenses for a week before the test and 2 weeks before the actual surgery.There are basically 3 types of LASIK surgeries.PRK method (Basic and crude) Costs around 25,000–30,000 rupees. It is the most safest and oldest technique with strong and credible evidence in it’s support. LASER is definitely used.Femto-LASIK (Advanced) Costs around 60,000–80,000.ReleX Smile (Latest) Costs around 1,20,000.Still the mostly done procedure is LASIK>PRK>SMILEFemto-Lasic has many disadvantages compared to ReleX Smile. These were told by my doctor. The disadvantages are:The flap taken is 300 degrees on the cornea so after the surgery, the eye needs to be continuously protected for a period ranging from a month to 3 months.Once there is a flap or the original is cut, it can never become as it was in the first place.Continuous watering from eyes and dryness on the other hand.No form of physical activity for a month.No form of contact sports for 3 months.Swimming and gymming after 3–6 months on the advice of your doctor.Constant itching during the first 3 weeks.Advantages:Less costly but it is your eyes we are talking about.Can be re-operated if the number does not become absolute zero.The eye print does not change so you can apply to the army. No procedure is declared fit by army. There are many other factors. Among all, PRK is the most acceptable.On the other hand Relex Smile has the following advantages (This is the surgery I underwent):The cut taken on the cornea is 2mm.Recovery time is 3 days for normal activities.I could play sports after a week.Contact sports, gymming and swimming can be done after a week.The eye print does not change so you can apply to the army.Watering for the first 3 days.After 1 month you feel like you got back your natural eyes.Disadvantages:Costly.It cannot be re-operated. Patient can be re-operated with PRK.Surgery Day:No conditions need to be met. You can have proper food and water. You are told to sign a form that you are doing it at your own risk. (The only risk is that your power will not become absolute zero.) They tell you to wear a surgical coat, gloves and socks. Then you are told to enter the operation room. No perfumes to be used.The Operation:My operation lasted for 4 minutes for left eye and 7 minutes for right eye with a gap of 2 minutes in between.The operation room looked a bit intimidating. The SMILE procedure is only around since 2014 pretty much, so all the equipment is basically new. The laser is programmed exactly to your eyes, so that it won’t do anything unless your eye is in the exact right position it needs to be in. You’re given anaesthetic eye drops and your eyes are clipped so that your eye remains open, so you cannot blink during the procedure. That already sounds way worse than it was, because of the anaesthetic you don’t even notice.I was moved under the laser machine and instructed to look at a green light in front of you as the machine moved closer. To make sure your eye is in place and absolutely still, there is a suction ring on the machine, but honestly I felt absolutely nothing of that. If I hadn’t been told it was there I wouldn’t have even known. I only saw that green light I was told to focus on. Then the laser did it’s job – the procedure is kind of “narrated” by a voice from the machine “suction on” – “laser operating” – “suction off” – really like in those sci-fi movies where a friendly robotic female voice tells you when the space ship doors are unlocked or something. Nurse and doc counted down from 25 seconds, and that was over. The part following was when the surgeon removed the corneal material from my eye, but I also saw nothing. It was a bright white light, that’s all. There was some pressure on the eye as he told me to look left, right and down, but I saw no tools and there was no pain, maybe slight discomfort but that’s to be expected. This took about 3 minutes. Naturally you’re required to remain absolutely still all the time, and basically this is also over before you know it.After that, they moved on to the left eye, same thing – and I was done. Like I said, less than 20 or even 15 minutes in there and you’re done with the procedure.Post-surgeryImmediate after: My vision was blurry and dim, a bit like without glasses for me, I was escorted to the resting room where I was put in a very comfy chair. I napped for about 30 minutes until the doctor took me for a final check-up before they sent me home. He took a look with the microscope, told me everything went extremely well and my cornea was looking good, and I was free to leave.I was told to leave with a little bag with eye drops, goggle and some info material. The prescription drops I would need to use four times a day for a week, starting the day of my surgery before I’d go to sleep at night.Next morning: I woke up and I could read the label of the book that was on the table at the other end of the room. And then I freaked out a little! I COULD SEE!!! It was all still pretty hazy but STILL!!! DAMN!!!! I was genuinely excited, but at the same time my eyes still felt pretty sore so it was hard to realize. Use as many moisturizing eye drops as you want, those help a lot.I could already go about things but you’re not allowed to go outside though to protect your eyes from the environment, wind and rain are bad ideas at this point. Reading was still difficult because it was all very hazy, but manageable. I just didn’t trust myself writing e-mails or doing anything relevant today. Mainly you have to be careful to not let anything come in your eye and you have to wear sunglasses for protection as well.During the rest of the week: after the surgery, my vision got more stable day by day. At first I wasn’t sure if it was really corrected 100%, but on the third day I lost that thought. During the whole first week I needed to use both prescription eye drops four times a day, the first one right after you get up and the last one before you go to sleep at night. All that healing still is pressure on your eye, and my eyes got heavy and tired quickly.Three months after surgery: You yourself understand that you have healed 100% and can see normally. That confirmation was given to me by doctor when he showed that the power of both my eyes has come to 0.00.Benefits:Your vision actually goes to HD or 4K quality.No need to take care of the spectacles or restrict your vision from that rectangle of your specs.You can see under water.No need to search for spectacles anymore.You can use a head in football!There are many other advantages. GO for it!Hope that helps. All the best!
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