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3,700 | The main armament of the original model M1 and M1IP was the M68A1 105 mm rifled tank gun firing a variety of armor-piercing fin-stabilized discarding sabot, high-explosive anti-tank, high explosive, white phosphorus rounds and an anti-personnel (multiple flechette) round. This gun used a license-made tube of the Britis... | https://en.wikipedia.org/wiki?curid=37744 |
3,701 | The main armament of the M1A1 and M1A2 is the M256A1 120 mm smoothbore gun, designed by Rheinmetall AG of Germany, manufactured under license in the U.S. by Watervliet Arsenal, New York. The M256A1 is an improved variant of the Rheinmetall 120 mm L/44 gun carried on the German Leopard 2 on all variants up to the Leopar... | https://en.wikipedia.org/wiki?curid=37744 |
3,702 | The M256A1 fires a variety of rounds. The primary APFSDS round of the Abrams is the depleted uranium M829 round, of which four variants have been designed. M829A1, known as the "Silver Bullet", saw widespread service in the Gulf War, where it proved itself against Iraqi armor such as the T-72. The M829A2 APFSDS round w... | https://en.wikipedia.org/wiki?curid=37744 |
3,703 | Later, the M829A3 round was introduced to improve its effectiveness against next generation ERA equipped tanks, through usage of a multi-material penetrator and increased penetrator diameter that can resist the shear effect of K-5 type ERA. As a counter to that, the Russian army introduced Relikt, the most modern Russi... | https://en.wikipedia.org/wiki?curid=37744 |
3,704 | The Abrams also fires high-explosive anti-tank warhead shaped charge rounds such as the M830, the latest version of which (M830A1) incorporates a sophisticated multi-mode electronic sensing fuse and more fragmentation that allows it to be used effectively against armored vehicles, personnel, and low-flying aircraft. Th... | https://en.wikipedia.org/wiki?curid=37744 |
3,705 | The new M1028 120 mm anti-personnel canister cartridge was brought into service early for use in the aftermath of the 2003 invasion of Iraq. It contains 1,098 tungsten balls that spread from the muzzle to produce a shotgun effect lethal out to . The tungsten balls can be used to clear enemy dismounts, break up hasty am... | https://en.wikipedia.org/wiki?curid=37744 |
3,706 | Also in use is the M908 obstacle-reduction round. It is designed to destroy obstacles and barriers. The round is a modified M830A1 with the front fuse replaced by a steel nose to penetrate into the obstacle before detonation. | https://en.wikipedia.org/wiki?curid=37744 |
3,707 | The U.S. Army Research Laboratory (ARL) conducted a thermal analysis of the M256 from 2002 to 2003 to evaluate the potential of using a hybrid barrel system that would allow for multiple weapon systems such as the XM1111 Mid-Range munition, airburst rounds, or XM 1147. The test concluded that mesh density (number of el... | https://en.wikipedia.org/wiki?curid=37744 |
3,708 | In 2013 the Army was developing a new round to replace the M830/M830A1, M1028, and M908. Called the Advanced Multi-Purpose (AMP) round, it will have point detonation, delay, and airburst modes through an ammunition data-link and a multi-mode, programmable fuse in a single munition. Having one round that does the job of... | https://en.wikipedia.org/wiki?curid=37744 |
3,709 | In addition to these, the XM1111 (Mid-Range-Munition Chemical Energy) was also in development. The XM1111 was a guided munition using a dual-mode seeker that combined imaging-infrared and semi-active laser guidance. The MRM-CE was selected over the competing MRM-KE, which used a rocket-assisted kinetic energy penetrato... | https://en.wikipedia.org/wiki?curid=37744 |
3,710 | The Abrams is equipped with a ballistic fire-control computer that uses user and system-supplied data from a variety of sources to compute, display, and incorporate the three components of a ballistic solution—lead angle, ammunition type, and range to the target—to accurately fire the main gun. These three components a... | https://en.wikipedia.org/wiki?curid=37744 |
3,711 | All of these factors are computed into a ballistic solution and updated 30 times per second. The updated solution is displayed in the Gunner's or Tank Commander's field of view in the form of a reticle in both day and Thermal modes. The ballistic computer manipulates the turret and a complex arrangement of mirrors so t... | https://en.wikipedia.org/wiki?curid=37744 |
3,712 | The fire-control system uses this data to compute a firing solution for the gunner. The ballistic solution generated ensures a hit percentage greater than 95 percent at nominal ranges. Either the commander or gunner can fire the main gun. Additionally, the Commander's Independent Thermal Viewer (CITV) on the M1A2 can b... | https://en.wikipedia.org/wiki?curid=37744 |
3,713 | If the primary sight system malfunctions or is damaged, the main and coaxial weapons can be manually aimed using a telescopic scope boresighted to the main gun known as the Gunner's Auxiliary Sight (GAS). The GAS has two interchangeable reticles; one for "high-explosive anti-tank" (HEAT) and "multi-purpose anti-tank" (... | https://en.wikipedia.org/wiki?curid=37744 |
3,714 | The commander's M2HB .50 caliber machine gun on the M1 and M1A1 is aimed by a 3× magnification sight incorporated into the Commander's Weapon Station (CWS), while the M1A2 uses the machine gun's own iron sights, or a remote aiming system such as the Common Remotely Operated Weapon Station (CROWS) system when used as pa... | https://en.wikipedia.org/wiki?curid=37744 |
3,715 | In late 2017, the 400 USMC M1A1 Abrams were to be upgraded with better and longer-range sights on the Abrams Integrated Display and Targeting System (AIDATS) replacing the black-and-white camera view with a color sight and day/night thermal sight, simplified handling with a single set of controls, and a slew to cue but... | https://en.wikipedia.org/wiki?curid=37744 |
3,716 | The M1 Abrams's powertrain consists of a Honeywell AGT1500 (originally made by Lycoming) multifuel gas turbine capable of at 30,000 rpm and at 10,000 rpm and a six-speed (four forward, two reverse) Allison X-1100-3B Hydro-Kinetic automatic transmission. This gives it a governed top speed of on paved roads, and cross-co... | https://en.wikipedia.org/wiki?curid=37744 |
3,717 | The tank was built around this engine and it is multifuel–capable, including diesel, kerosene, any grade of motor gasoline, and jet fuel (such as JP-4 or JP-8). For logistical reasons, JP-8 is the U.S. military's universal fuel powering both aircraft and vehicle fleets. The Australian M1A1 AIM SA burns diesel fuel, sin... | https://en.wikipedia.org/wiki?curid=37744 |
3,718 | The gas turbine propulsion system has proven quite reliable in practice and combat, but its high fuel consumption is a serious logistic issue. The engine burns more than per mile ( per hour) when traveling cross-country and per hour when idle. | https://en.wikipedia.org/wiki?curid=37744 |
3,719 | The high speed, high temperature jet blast emitted from the rear of M1 Abrams tanks makes it hazardous for infantry to take cover or follow behind the tank in urban combat. The turbine is very quiet when compared to diesel engines of similar power output and produces a significantly different sound from a contemporary ... | https://en.wikipedia.org/wiki?curid=37744 |
3,720 | The Army received proposals, including two diesel options, to provide the common engine for the XM2001 Crusader and Abrams. In 2000, the Army selected the gas turbine engine LV100-5 from Honeywell and subcontractor General Electric. The new LV100-5 engine was lighter and smaller (43% fewer parts) with rapid acceleratio... | https://en.wikipedia.org/wiki?curid=37744 |
3,721 | An Auxiliary Power Unit (APU) was designed by the Army's TARDEC, replacing an existing battery pack that weighs about . It uses a high power density Wankel rotary engine modified to use diesel and military grade jet fuel. The new APU will also be more fuel efficient than the tank's main engine. Testing of the first APU... | https://en.wikipedia.org/wiki?curid=37744 |
3,722 | Although the M1 tank is not designed to carry riders easily, provisions exist for the Abrams to transport troops in tank desant with the turret stabilization device switched off. A battle equipped infantry squad may ride on the rear of the tank, behind the turret. The soldiers can use ropes and equipment straps to prov... | https://en.wikipedia.org/wiki?curid=37744 |
3,723 | Strategic mobility is the ability of the tanks of an armed force to arrive in a timely, cost effective, and synchronized fashion. The Abrams can be carried by a C-5 Galaxy or a C-17 Globemaster III. The limited capacity (two combat-ready tanks in a C-5, one combat-ready tank in a C-17) caused serious logistical problem... | https://en.wikipedia.org/wiki?curid=37744 |
3,724 | Marines transport their Marine Air Ground Task Force (MAGTF)-attached Abrams tanks by combat ship. A "Wasp"-class Landing Helicopter Dock (LHD) typically carries a platoon of 4 to 5 tanks attached to the deployed Marine Expeditionary Unit, which are then amphibiously transported to shore by Landing Craft Air Cushion (L... | https://en.wikipedia.org/wiki?curid=37744 |
3,725 | The Abrams is also transportable by truck, namely the Oshkosh M1070 and M1000 Heavy Equipment Transporter System (HETS) for the US Military. The HETS can operate on highways, secondary roads, and cross-country. It accommodates the four tank crew members. The Australian Army uses customised MAN trucks to transport its A... | https://en.wikipedia.org/wiki?curid=37744 |
3,726 | The first instance of the Abrams being airlifted directly into a battlefield occurred in October 1993. Following the Battle of Mogadishu, 18 M1 tanks were airlifted by C-5 aircraft to Somalia from Hunter Army Airfield, Georgia. | https://en.wikipedia.org/wiki?curid=37744 |
3,727 | ISO 4217 is a standard published by the International Organization for Standardization (ISO) that defines alpha codes and numeric codes for the representation of currencies and provides information about the relationships between individual currencies and their minor units. This data is published in three tables: | https://en.wikipedia.org/wiki?curid=15403 |
3,728 | The first edition of ISO 4217 was published in 1978. The tables, history and ongoing discussion are maintained by SIX Group on behalf of ISO and the Swiss Association for Standardization. | https://en.wikipedia.org/wiki?curid=15403 |
3,729 | The ISO 4217 code list is used in banking and business globally. In many countries, the ISO 4217 alpha codes for the more common currencies are so well known publicly that exchange rates published in newspapers or posted in banks use only these to delineate the currencies, instead of translated currency names or ambigu... | https://en.wikipedia.org/wiki?curid=15403 |
3,730 | In 1973, the ISO Technical Committee 68 decided to develop codes for the representation of currencies and funds for use in any application of trade, commerce or banking. At the 17th session (February 1978), the related UN/ECE Group of Experts agreed that the three-letter alphabetic codes for International Standard ISO ... | https://en.wikipedia.org/wiki?curid=15403 |
3,731 | Over time, new currencies are created and old currencies are discontinued. Such changes usually originate from the formation of new countries, treaties between countries on shared currencies or monetary unions, or redenomination from an existing currency due to excessive inflation. As a result, the list of codes must b... | https://en.wikipedia.org/wiki?curid=15403 |
3,732 | In the case of national currencies, the first two letters of the alpha code are the two letters of the ISO 3166-1 alpha-2 country code and the third is usually the initial of the currency's main unit. So Japan's currency code is : "JP" for Japan and "Y" for yen. This eliminates the problem caused by the names "dollar, ... | https://en.wikipedia.org/wiki?curid=15403 |
3,733 | In some cases, the third letter of the alpha code is not the initial letter of a currency unit name. There may be a number of reasons for this: | https://en.wikipedia.org/wiki?curid=15403 |
3,734 | In addition to codes for most active national currencies ISO 4217 provides codes for "supranational" currencies, procedural purposes, and several things which are "similar to" currencies: | https://en.wikipedia.org/wiki?curid=15403 |
3,735 | The use of an initial letter "X" for these purposes is facilitated by the ISO 3166 rule that no official country code beginning with X will ever be assigned. | https://en.wikipedia.org/wiki?curid=15403 |
3,736 | The inclusion of EU (denoting the European Union) in the ISO 3166-1 reserved codes list allows the euro to be coded as EUR rather than assigned a code beginning with X, even though it is a supranational currency. | https://en.wikipedia.org/wiki?curid=15403 |
3,737 | ISO 4217 also assigns a three-digit numeric code to each currency. This numeric code is usually the same as the numeric code assigned to the corresponding country by ISO 3166-1. For example, USD (United States dollar) has numeric code which is also the ISO 3166-1 code for "US" (United States). | https://en.wikipedia.org/wiki?curid=15403 |
3,738 | A number of currencies had official ISO 4217 currency codes and currency names until their replacement by another currency. The table below shows the ISO currency codes of former currencies and their common names (which do not always match the ISO 4217 names). That table has been introduced end 1988 by ISO. | https://en.wikipedia.org/wiki?curid=15403 |
3,739 | Examples for the ratios of :1 and :1 include the United States dollar and the Bahraini dinar, for which the column headed “Minor unit” shows “2” and “3”, respectively. , two currencies have non-decimal ratios, the Mauritanian ouguiya and the Malagasy ariary; in both cases the ratio is 5:1. For these, the “Minor unit” c... | https://en.wikipedia.org/wiki?curid=15403 |
3,740 | The ISO standard does not regulate either the spacing, prefixing or suffixing in usage of currency codes. According however to the European Union's Publication Office, in English, Irish, Latvian and Maltese texts, the ISO 4217 code is to be followed by a hard space and the amount: | https://en.wikipedia.org/wiki?curid=15403 |
3,741 | In Bulgarian, Croatian, Czech, Danish, Dutch, Estonian, Finnish, French, German, Greek, Hungarian, Italian, Lithuanian, Polish, Portuguese, Romanian, Slovak, Slovene, Spanish and Swedish the order is reversed; the amount is followed by a hard space and the ISO 4217 code: | https://en.wikipedia.org/wiki?curid=15403 |
3,742 | Note that, as illustrated, the order is determined not by the currency but by the native language of the document context. | https://en.wikipedia.org/wiki?curid=15403 |
3,743 | The US dollar has two codes assigned: USD and USN ("US dollar next day"). The USS (same day) code is not in use any longer, and was removed from the list of active ISO 4217 codes in March 2014. | https://en.wikipedia.org/wiki?curid=15403 |
3,744 | A number of active currencies do not have an ISO 4217 code, because they may be: (1) a minor currency pegged at par (1:1) to a larger currency, even if independently regulated, (2) a currency only used for commemorative banknotes or coins, or (3) a currency of an unrecognized or partially recognized state. These curren... | https://en.wikipedia.org/wiki?curid=15403 |
3,745 | Despite having no presence or status in the standard, three letter acronyms that resemble ISO 4217 coding, are sometimes used locally or commercially to represent currencies or currency instruments. | https://en.wikipedia.org/wiki?curid=15403 |
3,746 | Minor units of currency (also known as currency subdivisions or currency subunits) are often used for pricing and trading stocks and other assets, such as energy, but are not assigned codes by ISO 4217. Two conventions for representing minor units are in widespread use: | https://en.wikipedia.org/wiki?curid=15403 |
3,747 | A third convention is similar to the second one but uses an upper-case letter, e.g. ZAC for the South African Cent. | https://en.wikipedia.org/wiki?curid=15403 |
3,748 | Cryptocurrencies have "not" been assigned an ISO 4217 code. However, some cryptocurrencies and cryptocurrency exchanges use a three-letter acronym that resemble an ISO 4217 code. | https://en.wikipedia.org/wiki?curid=15403 |
3,749 | Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to... | https://en.wikipedia.org/wiki?curid=47717 |
3,750 | During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations). Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas... | https://en.wikipedia.org/wiki?curid=47717 |
3,751 | In 1919, ill health—now believed to have been hepatic amoebiasis (a complication from episodes of dysentery many years previously)—compelled Ramanujan's return to India, where he died in 1920 at the age of 32. His last letters to Hardy, written in January 1920, show that he was still continuing to produce new mathemati... | https://en.wikipedia.org/wiki?curid=47717 |
3,752 | A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said his family goddess, Namagiri Thayar, revealed his mathematical knowledge to him. He once said, "An equation for me has no meaning unless it expresses a thought of God." | https://en.wikipedia.org/wiki?curid=47717 |
3,753 | Ramanujan (literally, "younger brother of Rama", a Hindu deity) was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, in present-day Tamil Nadu. His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a clerk in a sari shop. His mother, Komalatammal, was a housewife ... | https://en.wikipedia.org/wiki?curid=47717 |
3,754 | On 1 October 1892, Ramanujan was enrolled at the local school. After his maternal grandfather lost his job as a court official in Kanchipuram, Ramanujan and his mother moved back to Kumbakonam, and he was enrolled in Kangayan Primary School. When his paternal grandfather died, he was sent back to his maternal grandpare... | https://en.wikipedia.org/wiki?curid=47717 |
3,755 | Since Ramanujan's father was at work most of the day, his mother took care of the boy, and they had a close relationship. From her, he learned about tradition and puranas, to sing religious songs, to attend pujas at the temple, and to maintain particular eating habits—all part of Brahmin culture. At Kangayan Primary Sc... | https://en.wikipedia.org/wiki?curid=47717 |
3,756 | A child prodigy by age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book written by S. L. Loney on advanced trigonometry. He mastered this by the age of 13 while discovering sophisticated theorems on his own. By 14, he received merit certifica... | https://en.wikipedia.org/wiki?curid=47717 |
3,757 | In 1903, when he was 16, Ramanujan obtained from a friend a library copy of "A Synopsis of Elementary Results in Pure and Applied Mathematics", G. S. Carr's collection of 5,000 theorems. Ramanujan reportedly studied the contents of the book in detail. The next year, Ramanujan independently developed and investigated th... | https://en.wikipedia.org/wiki?curid=47717 |
3,758 | When he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum. He received a scholarship to study at Government ... | https://en.wikipedia.org/wiki?curid=47717 |
3,759 | In 1910, after a meeting between the 23-year-old Ramanujan and the founder of the Indian Mathematical Society, V. Ramaswamy Aiyer, Ramanujan began to get recognition in Madras's mathematical circles, leading to his inclusion as a researcher at the University of Madras. | https://en.wikipedia.org/wiki?curid=47717 |
3,760 | On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994), a girl his mother had selected for him a year earlier and who was ten years old when they married. It was not unusual then for marriages to be arranged with girls at a young age. Janaki was from Rajendram, a village close to Marudur... | https://en.wikipedia.org/wiki?curid=47717 |
3,761 | After the marriage, Ramanujan developed a hydrocele testis. The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac, but his family could not afford the operation. In January 1910, a doctor volunteered to do the surgery at no cost. | https://en.wikipedia.org/wiki?curid=47717 |
3,762 | After his successful surgery, Ramanujan searched for a job. He stayed at a friend's house while he went from door to door around Madras looking for a clerical position. To make money, he tutored students at Presidency College who were preparing for their Fellow of Arts exam. | https://en.wikipedia.org/wiki?curid=47717 |
3,763 | In late 1910, Ramanujan was sick again. He feared for his health, and told his friend R. Radakrishna Iyer to "hand [his notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College." After Ramanujan rec... | https://en.wikipedia.org/wiki?curid=47717 |
3,764 | In 1910, Ramanujan met deputy collector V. Ramaswamy Aiyer, who founded the Indian Mathematical Society. Wishing for a job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled: | https://en.wikipedia.org/wiki?curid=47717 |
3,765 | I was struck by the extraordinary mathematical results contained in [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department. | https://en.wikipedia.org/wiki?curid=47717 |
3,766 | Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras. Some of them looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society. Rao was impressed by Ramanujan's research but d... | https://en.wikipedia.org/wiki?curid=47717 |
3,767 | He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied an incomplete solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem. | https://en.wikipedia.org/wiki?curid=47717 |
3,768 | Using this equation, the answer to the question posed in the "Journal" was simply 3, obtained by setting , , and . Ramanujan wrote his first formal paper for the "Journal" on the properties of Bernoulli numbers. One property he discovered was that the denominators of the fractions of Bernoulli numbers are always divisi... | https://en.wikipedia.org/wiki?curid=47717 |
3,769 | In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures. His writing initially had many flaws. As "Journal" editor M. T. Narayana Iyengar noted: | https://en.wikipedia.org/wiki?curid=47717 |
3,770 | Mr. Ramanujan's methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him. | https://en.wikipedia.org/wiki?curid=47717 |
3,771 | Ramanujan later wrote another paper and also continued to provide problems in the "Journal". In early 1912, he got a temporary job in the Madras Accountant General's office, with a monthly salary of 20 rupees. He lasted only a few weeks. Toward the end of that assignment, he applied for a position under the Chief Accou... | https://en.wikipedia.org/wiki?curid=47717 |
3,772 | I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and devel... | https://en.wikipedia.org/wiki?curid=47717 |
3,773 | Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics". Three weeks after he applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade ... | https://en.wikipedia.org/wiki?curid=47717 |
3,774 | In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. M. J. M. Hill of University College London commented that Ramanujan's papers were riddled with holes. He said that although Ramanujan had "a taste for mathematics, and some ability", h... | https://en.wikipedia.org/wiki?curid=47717 |
3,775 | The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment. On 16 January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible fraud. Hardy recognised some of Ram... | https://en.wikipedia.org/wiki?curid=47717 |
3,776 | The first result had already been determined by G. Bauer in 1859. The second was new to Hardy, and was derived from a class of functions called hypergeometric series, which had first been researched by Euler and Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals. After seeing Ramanuj... | https://en.wikipedia.org/wiki?curid=47717 |
3,777 | On 8 February 1913, Hardy wrote Ramanujan a letter expressing interest in his work, adding that it was "essential that I should see proofs of some of your assertions". Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secr... | https://en.wikipedia.org/wiki?curid=47717 |
3,778 | To supplement Hardy's endorsement, Gilbert Walker, a former mathematical lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge. As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college,... | https://en.wikipedia.org/wiki?curid=47717 |
3,779 | While he was engaged as a research student, Ramanujan continued to submit papers to the "Journal of the Indian Mathematical Society." In one instance, Iyer submitted some of Ramanujan's theorems on summation of series to the journal, adding, "The following theorem is due to S. Ramanujan, the mathematics student of Madr... | https://en.wikipedia.org/wiki?curid=47717 |
3,780 | Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England. Neville asked Ramanujan why he would not go to Cambridge. Ramanujan apparently had now accepted the proposal; Neville said, "R... | https://en.wikipedia.org/wiki?curid=47717 |
3,781 | Ramanujan departed from Madras aboard the S.S. "Nevasa" on 17 March 1914. When he disembarked in London on 14 April, Neville was waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ra... | https://en.wikipedia.org/wiki?curid=47717 |
3,782 | Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems in the notebooks. Hardy saw that some were wrong, others had already been discovered, and the rest were new breakthroughs. Ramanujan... | https://en.wikipedia.org/wiki?curid=47717 |
3,783 | Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood, and published part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous few decades, the foundations of ... | https://en.wikipedia.org/wiki?curid=47717 |
3,784 | Ramanujan was awarded a "Bachelor of Arts by Research" degree (the predecessor of the PhD degree) in March 1916 for his work on highly composite numbers, sections of the first part of which had been published the preceding year in the "Proceedings of the London Mathematical Society." The paper was more than 50 pages lo... | https://en.wikipedia.org/wiki?curid=47717 |
3,785 | On 6 December 1917, Ramanujan was elected to the London Mathematical Society. On 2 May 1918, he was elected a Fellow of the Royal Society, the second Indian admitted, after Ardaseer Cursetjee in 1841. At age 31, Ramanujan was one of the youngest Fellows in the Royal Society's history. He was elected "for his investigat... | https://en.wikipedia.org/wiki?curid=47717 |
3,786 | Ramanujan had numerous health problems throughout his life. His health worsened in England; possibly he was also less resilient due to the difficulty of keeping to the strict dietary requirements of his religion there and because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and a severe vitamin d... | https://en.wikipedia.org/wiki?curid=47717 |
3,787 | Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay. In 1931, she returned to Madras and settled in Triplicane, where she supported herself on a pension from Madras University and income from tailoring. In 1950, she adopted a son, W. Narayanan, who eventually became an officer of the State Bank of India and raised a ... | https://en.wikipedia.org/wiki?curid=47717 |
3,788 | A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D. A. B. Young concluded that his medical symptoms—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an illness then widespread in Madras, than tuberculosis. He had two episodes of d... | https://en.wikipedia.org/wiki?curid=47717 |
3,789 | Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners. He lived a simple life at Cambridge. Ramanujan's first Indian biographers describe him as a rigorously orthodox Hindu. He credited his acumen to his family goddess, Namagiri Thayar (Goddess Mahalaksh... | https://en.wikipedia.org/wiki?curid=47717 |
3,790 | Hardy cites Ramanujan as remarking that all religions seemed equally true to him. Hardy further argued that Ramanujan's religious belief had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict vegetarianism. | https://en.wikipedia.org/wiki?curid=47717 |
3,791 | Similarly, in an interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers to Ramanujan's mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks," further speculating that Ramanujan worked out intermediate results on slate that he could not a... | https://en.wikipedia.org/wiki?curid=47717 |
3,792 | In mathematics, there is a distinction between insight and formulating or working through a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As... | https://en.wikipedia.org/wiki?curid=47717 |
3,793 | This result is based on the negative fundamental discriminant with class number . Further, and , which is related to the fact that | https://en.wikipedia.org/wiki?curid=47717 |
3,794 | Ramanujan's series for converges extraordinarily rapidly and forms the basis of some of the fastest algorithms currently used to calculate . Truncating the sum to the first term also gives the approximation for , which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 de... | https://en.wikipedia.org/wiki?curid=47717 |
3,795 | One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which P. C. Mahalanobis posed a problem: | https://en.wikipedia.org/wiki?curid=47717 |
3,796 | for all such that formula_9 and formula_10, where is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating coefficients of , , and gives some deep identities for the hyperbolic secant. | https://en.wikipedia.org/wiki?curid=47717 |
3,797 | In 1918, Hardy and Ramanujan studied the partition function extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In 1937, Hans Rademacher refined their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy'... | https://en.wikipedia.org/wiki?curid=47717 |
3,798 | In the last year of his life, Ramanujan discovered mock theta functions. For many years, these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms. | https://en.wikipedia.org/wiki?curid=47717 |
3,799 | Although there are numerous statements that could have borne the name "Ramanujan conjecture," one was highly influential on later work. In particular, the connection of this conjecture with conjectures of André Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the ... | https://en.wikipedia.org/wiki?curid=47717 |
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