text:
Srinivasa Ramanujan

Srinivasa Ramanujan (22 December 1887 – 26 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 mathematical problems then considered unsolvable.

Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a mail correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that "defeated me completely; I had never seen anything in the least like them before", and some recently proven but highly advanced results.

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 of work and inspired further research. Of his thousands of results, most have been proven correct. The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan, and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death as a source of new mathematical ideas. As late as 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death. He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge.

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 mathematical ideas and theorems. His "lost notebook", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976.

Early life
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 and sang at a local temple. They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam. The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, but recovered, unlike the 4,000 others who died in a bad year in the Thanjavur district around this time. He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). His mother gave birth to two more children, in 1891 and 1894, both of whom died before their first birthdays.

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 grandparents, then living in Madras. He did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.

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 School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, and arithmetic with the best scores in the district. That year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time.

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 certificates and academic awards that continued throughout his school career, and he assisted the school in the logistics of assigning its 1,200 students (each with differing needs) to its approximately 35 teachers. He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902. He would later develop his own method to solve the quartic. In 1903, he tried to solve the quintic, not knowing that it was impossible to solve with radicals.

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 the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places. His peers at the time said they "rarely understood him" and "stood in respectful awe" of him.

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 Arts College, Kumbakonam, but was so intent on mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process. In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, and stayed in Rajahmundry for about a month. He later enrolled at Pachaiyappa's College in Madras. There, he passed in mathematics, choosing only to attempt questions that appealed to him and leaving the rest unanswered, but performed poorly in other subjects, such as English, physiology, and Sanskrit. Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. Without an FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation.

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.

Adulthood in India
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 (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony. As was common at that time, Janaki continued to stay at her maternal home for three years after marriage, until she reached puberty. In 1912, she and Ramanujan's mother joined Ramanujan in Madras. 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.

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.

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 recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram, a city under French control. In 1912, Ramanujan moved with his wife and mother to a house in Saiva Muthaiah Mudali street, George Town, Madras, where they lived for a few months. In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to Triplicane.

Pursuit of career in mathematics
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: 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. 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 doubted that it was his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a fraud. Ramanujan's friend C. V. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to give him another chance, and listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately convinced him of Ramanujan's brilliance. When Rao asked him what he wanted, Ramanujan replied that he needed work and financial support. Rao consented and sent him to Madras. He continued his research with Rao's financial aid. With Aiyer's help, Ramanujan had his work published in the ''Journal of the Indian Mathematical Society. One of the first problems he posed in the journal was to find the value of:

 * $$\sqrt{1+2\sqrt{1+3 \sqrt{1+\cdots}}}$$

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.

 * $$x+n+a = \sqrt{ax+(n+a)^2 +x\sqrt{a(x+n)+(n+a)^2+(x+n) \sqrt{\cdots}}}$$

Using this equation, the answer to the question posed in the Journal was simply 3, obtained by setting $x = 2$, $n = 1$, and $a = 0$. 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 divisible by six. He also devised a method of calculating $B_{n}$ based on previous Bernoulli numbers. One of these methods follows:

It will be observed that if n is even but not equal to zero,
 * 1) $B_{n}$ is a fraction and the numerator of $B_{n}⁄n$ in its lowest terms is a prime number,
 * 2) the denominator of $B_{n}$ contains each of the factors 2 and 3 once and only once,
 * 3) $2^{n}(2^{n} − 1)B_{n}⁄n$ is an integer and $2(2^{n} − 1)B_{n}$ consequently is an odd integer.

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: 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. 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 Accountant of the Madras Port Trust.

In a letter dated 9 February 1912, Ramanujan wrote: Sir,

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 developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.

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 IV accounting clerk, making 30 rupees per month. At his office, Ramanujan easily and quickly completed the work he was given and spent his spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.

Contacting British mathematicians
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", he lacked the necessary educational background and foundation to be accepted by mathematicians. Although Hill did not offer to take Ramanujan on as a student, he gave thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.

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, whom he knew from studying Orders of Infinity (1910). 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 Ramanujan's formulae but others "seemed scarcely possible to believe". One of the theorems Hardy found amazing was on the bottom of page three (valid for $0 < a < b + 1⁄2$):

\int\limits_0^\infty \frac{1+\dfrac{x^2}{(b+1)^2}}{1+\dfrac{x^2}{a^2}} \times\frac{1+\dfrac{x^2}{(b+2)^2}}{1+\dfrac{x^2}{(a+1)^2}}\times\cdots\,dx = \frac{\sqrt \pi}{2} \times\frac{\Gamma\left(a+\frac12\right) \Gamma(b+1) \Gamma(b-a+1)}{\Gamma(a)\Gamma\left(b+\frac12\right)\Gamma\left(b-a + \frac12 \right)}. $$

Hardy was also impressed by some of Ramanujan's other work relating to infinite series:

 * $$1 - 5\left(\frac12\right)^3 + 9\left(\frac{1\times3}{2\times4}\right)^3 - 13\left(\frac{1\times3\times5}{2\times4\times6}\right)^3 + \cdots = \frac{2}{\pi}$$

 * $$1 + 9\left(\frac14\right)^4 + 17\left(\frac{1\times5}{4\times8}\right)^4 + 25\left(\frac{1\times5\times9}{4\times8\times12}\right)^4 + \cdots = \frac{2\sqrt 2}{\sqrt\pi\,\Gamma^2\left(\frac34\right)}.$$

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 Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before", and that they "must be true, because, if they were not true, no one would have the imagination to invent them". Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power". One colleague, E. H. Neville, later remarked that "No one who was in the mathematical circles in Cambridge at that time can forget the sensation caused by this letter... not one [theorem] could have been set in the most advanced mathematical examination in the world".

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. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip. In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land", and his parents were also opposed for the same reason. Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically."

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, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan". The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees for the next two years at the University of Madras.

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 Madras University." Later in November, British Professor Edward B. Ross of Madras Christian College, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived in the day's mail. In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.

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, "Ramanujan needed no converting" and "his parents' opposition had been withdrawn". Apparently, Ramanujan's mother had a vivid dream in which Ramanujan was surrounded by Europeans, and the family goddess, the deity of Namagiri, commanded her "to stand no longer between her son and the fulfilment of his life's purpose". On 17 March 1914, Ramanujan traveled to England by ship, leaving his wife to stay with his parents in India.

Life in England

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, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, a five-minute walk from Hardy's room.

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 left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a Jacobi", while Hardy said he "can compare him only with Euler or Jacobi."

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 mathematics had come into question and the need for mathematically rigorous proofs recognised. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition and insights. Hardy tried his best to fill the gaps in Ramanujan's education and to mentor him in the need for formal proofs to support his results, without hindering his inspiration—a conflict that neither found easy.

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 long and proved various properties of such numbers. Hardy disliked this topic area but remarked that though it engaged with what he called the 'backwater of mathematics', in it Ramanujan displayed 'extraordinary mastery over the algebra of inequalities'.

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 investigation in elliptic functions and the Theory of Numbers." On 13 October 1918, he was the first Indian to be elected a Fellow of Trinity College, Cambridge.

Illness and death
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 deficiency, and confined to a sanatorium. He attempted suicide in late 1917 or early 1918 by jumping on the tracks of a London underground station. Scotland Yard arrested him for attempting suicide (which was a crime), but released him after Hardy intervened. In 1919, Ramanujan returned to Kumbakonam, Madras Presidency, where he died in 1920 aged 32. After his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series and continued fractions. In his last days, though in severe pain, "he continued doing his mathematics filling sheet after sheet with numbers", Janaki Ammal recounts.

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 family. In her later years, she was granted a lifetime pension from Ramanujan's former employer, the Madras Port Trust, and pensions from, among others, the Indian National Science Academy and the state governments of Tamil Nadu, Andhra Pradesh and West Bengal. She continued to cherish Ramanujan's memory, and was active in efforts to increase his public recognition; prominent mathematicians, including George Andrews, Bruce C. Berndt and Béla Bollobás made it a point to visit her while in India. She died at her Triplicane residence in 1994.

A 1994 analysis of Ramanujan's medical records and symptoms by 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 dysentery before he left India. When not properly treated, amoebic dysentery can lie dormant for years and lead to hepatic amoebiasis, whose diagnosis was not then well established. At the time, if properly diagnosed, amoebiasis was a treatable and often curable disease; British soldiers who contracted it during the First World War were being successfully cured of amoebiasis around the time Ramanujan left England.

Personality and spiritual life
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 Mahalakshmi) of Namakkal. He looked to her for inspiration in his work and said he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions of scrolls of complex mathematical content unfolding before his eyes. He often said, "An equation for me has no meaning unless it expresses a thought of God."

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.

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 afford the paper to record more permanently.

Berndt reported that Janaki said in 1984 that Ramanujan spent so much of his time on mathematics that he did not go to the temple, that she and her mother often fed him because he had no time to eat, and that most of the religious stories attributed to him originated with others. However, his orthopraxy was not in doubt.

Mathematical achievements
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 a byproduct of his work, new directions of research were opened up. Examples of the most intriguing of these formulae include infinite series for $\pi$, one of which is given below:

 * $$ \frac{1}{\pi} = \frac{2\sqrt 2}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}.$$

This result is based on the negative fundamental discriminant $d$ = −4 × 58 = −232 with class number $h(d)$ = 2. Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 3962, which is related to the fact that

 * $ e^{\pi\sqrt{58}} = 396^4 - 104.000000177\dots. $

This might be compared to Heegner numbers, which have class number 1 and yield similar formulae.

Ramanujan's series for π converges extraordinarily rapidly and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation $9801√2⁄4412$ for π, which is correct to six decimal places; truncating it to the first two terms gives a value correct to 14 decimal places.

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:

"Imagine that you are on a street with houses marked 1 through $n$. There is a house in between ($x$) such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. If $n$ is between 50 and 500, what are $n$ and $x$?' This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. 'It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind', Ramanujan replied.'"

His intuition also led him to derive some previously unknown identities, such as

\begin{align} & \left ( 1+2\sum_{n=1}^\infty \frac{\cos(n\theta)}{\cosh(n\pi)} \right )^{-2} + \left (1+2\sum_{n=1}^\infty \frac{\cosh(n\theta)}{\cosh(n\pi)} \right )^{-2} \\[6pt] = {} & \frac {2 \Gamma^4 \bigl( \frac34 \bigr)} \pi = \frac{8\pi^3}{\Gamma^4 \bigl( \frac14 \bigr)} \end{align} $$

for all $θ$ such that $$|\Re (\theta)|<\pi$$ and $$|\Im (\theta)|<\pi$$, where $Γ(z)$ is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating coefficients of $θ^{0}$, $θ^{4}$, and $θ^{8}$ gives some deep identities for the hyperbolic secant.

In 1918, Hardy and Ramanujan studied the partition function $P(n)$ 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's work in this area gave rise to a powerful new method for finding asymptotic formulae called the circle method.

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.

The Ramanujan conjecture
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 size of the tau-function, which has a generating function as the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for that work.

In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called $τ(n)$ (the Ramanujan tau function). He proved many congruences for these numbers, such as $τ(p) ≡ 1 + p^{11} mod 691$ for primes $p$. This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture that there is a theory of Galois representations that "explains" these congruences and more generally all modular forms. $Δ(z)$ is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The proof of Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory, there would be no proof of Fermat's Last Theorem.

Ramanujan's notebooks
While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of looseleaf paper. They were mostly written up without any derivations. This is probably the origin of the misapprehension that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to prove most of his results, but chose not to record the proofs in his notes.

This may have been for any number of reasons. Since paper was very expensive, Ramanujan did most of his work and perhaps his proofs on slate, after which he transferred the final results to paper. At the time, slates were commonly used by mathematics students in the Madras Presidency. He was also quite likely to have been influenced by the style of G. S. Carr's book, which stated results without proofs. It is also possible that Ramanujan considered his work to be for his personal interest alone and therefore recorded only the results.

The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material. The second has 256 pages in 21 chapters and 100 unorganised pages, and the third 33 unorganised pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself wrote papers exploring material from Ramanujan's work, as did G. N. Watson, B. M. Wilson, and Bruce Berndt.

In 1976, George Andrews rediscovered a fourth notebook with 87 unorganised pages, the so-called "lost notebook".

Hardy–Ramanujan number 1729
The number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. In Hardy's words:

"I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. 'No', he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'"

Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends."

The two different ways are:

 * $$1729=1^3+12^3=9^3+10^3.$$

Generalisations of this idea have created the notion of "taxicab numbers".

Mathematicians' views of Ramanujan
In his obituary of Ramanujan, written for Nature in 1920, Hardy observed that Ramanujan's work primarily involved fields less known even among other pure mathematicians, concluding: "His insight into formulae was quite amazing, and altogether beyond anything I have met with in any European mathematician. It is perhaps useless to speculate as to his history had he been introduced to modern ideas and methods at sixteen instead of at twenty-six. It is not extravagant to suppose that he might have become the greatest mathematician of his time. What he actually did is wonderful enough… when the researches which his work has suggested have been completed, it will probably seem a good deal more wonderful than it does to-day."

Hardy further said:

"He combined a power of generalisation, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was...'"

As an example, Hardy commented on 15 theorems in the first letter. Of those, the first 13 are correct and insightful, the 14th is incorrect but insightful, and the 15th is correct but misleading.

(14): The coefficient of $$x^n$$ in $$\left(1-2 x+2 x^4-2 x^9+\cdots\right)^{-1}$$ is the integer nearest to$$ \frac{1}{4 n}\left(\cosh (\pi \sqrt{n})-\frac{\sinh (\pi \sqrt{ n})}{\pi \sqrt{n}}\right). $$This "was one of the most fruitful he ever made, since it ended by leading us to all our joint work on partitions".

When asked about the methods Ramanujan used to arrive at his solutions, Hardy said they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account." He also said that he had "never met his equal, and can compare him only with Euler or Jacobi". Hardy thought Ramanujan worked in a 19th-century style, where arriving at correct formulas was more important than systematic formal theories. Hardy thought his achievements were greatest in algebra, especially hypergeometric series and continued fractions.

"It is possible that the great days of formulas are finished, and that Ramanujan ought to have been born 100 years ago; but he was by far the greatest formalist of his time. There have been a good many more important, and I suppose one must say greater, mathematicians than Ramanujan during the last 50 years, but not one who could stand up to him on his own ground. Playing the game of which he knew the rules, he could give any mathematician in the world fifteen."

He discovered fewer new things in analysis, possibly because he lacked the formal education and did not find books to learn it from, but rediscovered many results, including the prime number theorem. In analysis, he worked on the elliptic functions and the analytic theory of numbers. In analytic number theory, he was as imaginative as usual, but much of what he imagined was wrong. Hardy blamed this on the inherent difficulty of analytic number theory, where imagination had led many great mathematicians astray. In analytic number theory, rigorous proof is more important than imagination, the opposite of Ramanujan's style. His "one great failure" is that he knew "nothing at all about the theory of analytic functions".

Littlewood reportedly said that helping Ramanujan catch up with European mathematics beyond what was available in India was very difficult, because each new point mentioned to Ramanujan caused him to produce original ideas that prevented Littlewood from continuing the lesson.

K. Srinivasa Rao has said, "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100. Hardy gave himself a score of 25, J. E. Littlewood 30, David Hilbert 80 and Ramanujan 100. During a May 2011 lecture at IIT Madras, Berndt said that over the last 40 years, as nearly all of Ramanujan's conjectures had been proven, there had been greater appreciation of Ramanujan's work and brilliance, and that Ramanujan's work was now pervading many areas of modern mathematics and physics.

Posthumous recognition
The year after his death, Nature listed Ramanujan among other distinguished scientists and mathematicians on a "Calendar of Scientific Pioneers" who had achieved eminence. Ramanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan's birthday) as 'State IT Day'. Stamps picturing Ramanujan were issued by the government of India in 1962, 2011, 2012 and 2016.

Since Ramanujan's centennial year, his birthday, 22 December, has been annually celebrated as Ramanujan Day by the Government Arts College, Kumbakonam, where he studied, and at the IIT Madras in Chennai. The International Centre for Theoretical Physics (ICTP) has created a prize in Ramanujan's name for young mathematicians from developing countries in cooperation with the International Mathematical Union, which nominates members of the prize committee. SASTRA University, a private university based in Tamil Nadu, has instituted the SASTRA Ramanujan Prize of US$10,000 to be given annually to a mathematician not exceeding age 32 for outstanding contributions in an area of mathematics influenced by Ramanujan.

Based on the recommendations of a committee appointed by the University Grants Commission (UGC), Government of India, the Srinivasa Ramanujan Centre, established by SASTRA, has been declared an off-campus centre under the ambit of SASTRA University. House of Ramanujan Mathematics, a museum of Ramanujan's life and work, is also on this campus. SASTRA purchased and renovated the house where Ramanujan lived at Kumabakonam.

In 2011, on the 125th anniversary of his birth, the Indian government declared that 22 December will be celebrated every year as National Mathematics Day. Then Indian Prime Minister Manmohan Singh also declared that 2012 would be celebrated as National Mathematics Year and 22 December as National Mathematics Day of India.

Ramanujan IT City is an information technology (IT) special economic zone (SEZ) in Chennai that was built in 2011. Situated next to the Tidel Park, it includes 25 acre with two zones, with a total area of 5.7 e6sqft, including 4.5 e6sqft of office space.

Commemorative postal stamps
Commemorative stamps released by India Post (by year):

In popular culture

 * The Man Who Loved Numbers is a 1988 PBS NOVA documentary about Ramanujan (S15, E9).
 * The Man Who Knew Infinity is a 2015 film based on Kanigel's book of the same name. British actor Dev Patel portrays Ramanujan.
 * Ramanujan, an Indo-British collaboration film chronicling Ramanujan's life, was released in 2014 by the independent film company Camphor Cinema. The cast and crew include director Gnana Rajasekaran, cinematographer Sunny Joseph and editor B. Lenin. Indian and English stars Abhinay Vaddi, Suhasini Maniratnam, Bhama, Kevin McGowan and Michael Lieber star in pivotal roles.
 * Nandan Kudhyadi directed the Indian documentary films The Genius of Srinivasa Ramanujan (2013) and Srinivasa Ramanujan: The Mathematician and His Legacy (2016) about the mathematician.
 * Ramanujan (The Man Who Reshaped 20th Century Mathematics), an Indian docudrama film directed by Akashdeep released in 2018.
 * M. N. Krish's thriller novel The Steradian Trail weaves Ramanujan and his accidental discovery into its plot connecting religion, mathematics, finance and economics.
 * Partition, a play by Ira Hauptman about Hardy and Ramanujan, was first performed in 2013.
 * The play First Class Man by Alter Ego Productions was based on David Freeman's First Class Man. The play centres around Ramanujan and his complex and dysfunctional relationship with Hardy. On 16 October 2011 it was announced that Roger Spottiswoode, best known for his James Bond film Tomorrow Never Dies, is working on the film version, starring Siddharth.
 * A Disappearing Number is a British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan.
 * David Leavitt's novel The Indian Clerk explores the events following Ramanujan's letter to Hardy.
 * Google honoured Ramanujan on his 125th birth anniversary by replacing its logo with a doodle on its home page.
 * Ramanujan was mentioned in the 1997 film Good Will Hunting, in a scene where professor Gerald Lambeau (Stellan Skarsgård) explains to Sean Maguire (Robin Williams) the genius of Will Hunting (Matt Damon) by comparing him to Ramanujan.

Selected papers

 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.
 * Posthumously published extract of a longer, unpublished manuscript.

Further works of Ramanujan's mathematics

 * George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part I (Springer, 2005, ISBN 0-387-25529-X)
 * George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part II, (Springer, 2008, ISBN 978-0-387-77765-8)
 * George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part III, (Springer, 2012, ISBN 978-1-4614-3809-0)
 * George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part IV, (Springer, 2013, ISBN 978-1-4614-4080-2)
 * George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook: Part V, (Springer, 2018, ISBN 978-3-319-77832-7)
 * M. P. Chaudhary, A simple solution of some integrals given by Srinivasa Ramanujan, (Resonance: J. Sci. Education – publication of Indian Academy of Science, 2008)
 * M.P. Chaudhary, Mock theta functions to mock theta conjectures, SCIENTIA, Series A : Math. Sci., (22)(2012) 33–46.
 * M.P. Chaudhary, On modular relations for the Roger-Ramanujan type identities, Pacific J. Appl. Math., 7(3)(2016) 177–184.

Selected publications on works of Ramanujan

 * This book was originally published in 1927 after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C. Berndt.
 * This book was originally published in 1927 after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C. Berndt.

 * These books contain photocopies of the original notebooks as written by Ramanujan.
 * These books contain photocopies of the original notebooks as written by Ramanujan.

 * This book contains photo copies of the pages of the "Lost Notebook".
 * This book contains photo copies of the pages of the "Lost Notebook".

 * Problems posed by Ramanujan, Journal of the Indian Mathematical Society.
 * This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.
 * This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.

Media links

 * Feature Film on Mathematics Genius Ramanujan by Dev Benegal and Stephen Fry
 * BBC radio programme about Ramanujan – episode 5
 * A biographical song about Ramanujan's life
 * A biographical song about Ramanujan's life

Biographical links

 * A short biography of Ramanujan
 * "Our Devoted Site for Great Mathematical Genius"
 * A short biography of Ramanujan
 * "Our Devoted Site for Great Mathematical Genius"
 * "Our Devoted Site for Great Mathematical Genius"

Other links

 * A Study Group For Mathematics: Srinivasa Ramanujan Iyengar
 * The Ramanujan Journal – An international journal devoted to Ramanujan
 * International Math Union Prizes, including a Ramanujan Prize
 * Hindu.com: Norwegian and Indian mathematical geniuses, Ramanujan – Essays and Surveys, Ramanujan's growing influence, Ramanujan's mentor
 * Hindu.com: The sponsor of Ramanujan
 * "Ramanujan's mock theta function puzzle solved"
 * Ramanujan's papers and notebooks
 * Sample page from the second notebook
 * Ramanujan on Fried Eye
 * Sample page from the second notebook
 * Ramanujan on Fried Eye

title: Srinivasa Ramanujan


text:
Flagstaff

Flagstaff commonly refers to:

 * Flagpole, a staff for displaying a flag
 * Flagstaff, Arizona, the county seat of Coconino County, Arizona

Flagstaff may also refer to:

United States

 * Flagstaff station, in Flagstaff, Arizona
 * United States Naval Observatory Flagstaff Station, in Flagstaff, Arizona
 * Flagstaff Mountain (Boulder County, Colorado), a peak near Boulder, Colorado
 * Flagstaff Mountain (Stevens County, Washington), a peak near Northport, Washington
 * Flagstaff, Maine, a submerged former town
 * Flagstaff Lake (Maine), on the Dead River
 * Flagstaff (Mandeville, Louisiana), a historic home listed on the National Register of Historic Places in St. Tammany Parish

Australia

 * Flagstaff Gardens, a park in Melbourne
 * Flagstaff railway station, Melbourne
 * Flagstaff, Victoria

Canada

 * Flagstaff County, a municipal district in Alberta

Ghana

 * The Flagstaff House, Accra; residence and office of the President of Ghana

Hong Kong

 * Flagstaff House, the home to the Commander of British Forces in Hong Kong and now home to a museum of teaware

New Zealand

 * Flagstaff (Otago), a hill above Dunedin
 * Flagstaff, Hamilton, a suburb

Pakistan

 * Flagstaff House or Quaid-e-Azam House, Karachi; home of Muhammad Ali Jinnah, Governor General of Pakistan and founder of modern Pakistan

South Africa

 * Flagstaff, Eastern Cape

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Flagstaff


text:
Coulomb

The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). It is equal to the electric charge delivered by a 1 ampere current in 1 second and is defined in terms of the elementary charge e, at about $2,997,924,580 statC$.

Name and history
By 1878, the British Association for the Advancement of Science had defined the volt, ohm, and farad, but not the coulomb. In 1881, the International Electrical Congress, now the International Electrotechnical Commission (IEC), approved the volt as the unit for electromotive force, the ampere as the unit for electric current, and the coulomb as the unit of electric charge. At that time, the volt was defined as the potential difference [i.e., what is nowadays called the "voltage (difference)"] across a conductor when a current of one ampere dissipates one watt of power. The coulomb (later "absolute coulomb" or "abcoulomb" for disambiguation) was part of the EMU system of units. The "international coulomb" based on laboratory specifications for its measurement was introduced by the IEC in 1908. The entire set of "reproducible units" was abandoned in 1948 and the "international coulomb" became the modern coulomb.

Definition
The SI defines the coulomb by taking the value of the elementary charge e to be $6.242 e$, but was previously defined in terms of the force between two wires. The coulomb was originally defined, using the latter definition of the ampere, as 1 A × 1 s. The 2019 redefinition of the ampere and other SI base units fixed the numerical value of the elementary charge when expressed in coulombs and therefore fixed the value of the coulomb when expressed as a multiple of the fundamental charge.

One coulomb is approximately $6.242 e$ (and is thus not an integer multiple of the elementary charge), where the number is the reciprocal of The coulomb is exactly $$ 1 ~ \mathrm{C} = \frac{1}{1.602\,176\,634 \times 10^{-19}} ~ e .$$

SI prefixes
Like other SI units, the coulomb can be modified by adding a prefix that multiplies it by a power of 10.

Conversions

 * The magnitude of the electrical charge of one mole of elementary charges (approximately, the Avogadro number) is known as a faraday unit of charge (closely related to the Faraday constant). One faraday equals In terms of the Avogadro constant (NA), one coulomb is equal to approximately $1.602 C$ × NA elementary charges.
 * Every farad of capacitance can hold one coulomb per volt across the capacitor.
 * One ampere hour equals $6,241,509,074,460,762,607.776 e$, hence $1.036 mol$ = $3,600 C$.
 * One statcoulomb (statC), the obsolete CGS electrostatic unit of charge (esu), is approximately $1 mA⋅h$ or about one-third of a nanocoulomb.

In everyday terms

 * The charges in static electricity from rubbing materials together are typically a few microcoulombs.
 * The amount of charge that travels through a lightning bolt is typically around 15 C, although for large bolts this can be up to 350 C.
 * The amount of charge that travels through a typical alkaline AA battery from being fully charged to discharged is about $3.6 C$ = $3.336 C$ ≈ $5 kC$.
 * A typical smartphone battery can hold $5,000 C$ ≈ $1,400 mA⋅h$.

title: Coulomb


text:
School of Advanced Military Studies

The School of Advanced Military Studies (SAMS) is one of four United States Army schools that make up the United States Army Command and General Staff College (CGSC) at Fort Leavenworth, Kansas. This "enormously rigorous" graduate school comprises three programs: the larger Advanced Military Studies Program (AMSP); the Advanced Strategic Leadership Studies Program (ASLSP), a Joint Military Professional Education II (JPME II) certified senior service college program for senior field-grade officers, and the Advanced Strategic Planning and Policy Program (ASP3), which supports officers in obtaining doctorates from civilian schools.

The school educates future leaders of the United States Armed Forces, its allies, and the Interagency at the graduate level to be agile and adaptive leaders who think critically at the strategic and operational levels to solve complex ambiguous problems. The student body is small but diverse and comprises members of each of the US armed forces, various US government agencies, and allied military forces. Graduates of AMSP are colloquially known as "Jedi Knights".

The school issues a master's degree in Military Art and Science, and provides its graduates with the skills to deal with the disparate challenges encountered in contemporary military and government operations. The modern course produces "leaders with the flexibility of mind to solve complex operational and strategic problems in peace, conflict, and war". Various senior military leaders have recognized the contributions of SAMS graduates in supporting global contingency operations.

The first class began at the school in mid-1983 and 13 students graduated the following year. Due to increasing requirements for SAMS graduates in the US military, the army expanded the school in the 1990s, and in 2010 over 120 students graduated. Since the school's inception, SAMS planners have supported every major US military campaign, providing the army "with many of its top campaign planners for the late twentieth and early twenty-first centuries".

History
The SAMS course was designed to fill a gap in US military education between the CGSC's focus on tactics and the War College's focus on grand strategy and national security policy. In 1981, Colonel Huba Wass de Czege convinced Lieutenant General William R. Richardson, who was serving as Commander of the Combined Arms Center and Commandant of the Command and General Staff College at Fort Leavenworth from 1979 to 1981, that a second year of military education was needed for select officers. After receiving final approval, Wass de Czege helped plan and develop the school, which would open in mid-1983. Although there was some disagreement about the purpose of the course, army leaders and the course designers settled on a plan to provide officers with a "broad, deep military education in the science and art of war."

In June 1983, the first class of 13 US Army students began in the basement of Bell Hall at Fort Leavenworth. Initially, there were some internal problems with facilities and scheduling, and in the school's early years there was uncertainty whether its graduates would be accepted and how they would perform in the force. When the first class graduated in 1984, SAMS had already become "the symbol for intellectual renaissance in the officer corps". When the first director, Wass de Czege, was succeeded by Colonel Richard Sennreich in 1985, the school was already beginning to produce results and the US Army and the college regarded SAMS as a "useful experiment". By 1987, enrollment of high-quality officers had risen and sister services were becoming interested in sending students to SAMS. The program's growing popularity and reputation also began attracting students from allied countries.

SAMS graduates first saw active service in December 1989 during Operation Just Cause in Panama. A core planning cell of seven SAMS graduates "crafted a well rehearsed and well executed plan that simultaneously struck some roughly 50 objectives in a single coordinated blow". According to Colonel Kevin Benson, the tenth director of the school, "The Army and SAMS faced a test of battle and the new group of highly-educated planners appeared to have passed the test with flying colors." After its mission in Panama, the army's leaders began to draw on SAMS to assist in additional ways. In the early 1990s, US Army leaders called upon the school to help develop army doctrine. Lieutenant Colonel Thomas E. Mitchell, Colonel James McDonough (the fifth SAMS director), and other members of the SAMS team helped revise the US Army Doctrinal Manual 100-5 Operations in 1990–1993.

Lieutenant General Guy C. Swan noted that SAMS graduates were indispensable in Europe after the fall of the Berlin Wall and the dissolution of the Warsaw Pact. They were expected to "re-engineer the decades of planning that had gone into the GDP [General Defense Plan] almost overnight". Swan stated that this was "the first true test of SAMS on a large scale". SAMS graduates served in Operation Desert Shield/Desert Storm, and were "remembered most famously in the early days for producing the 'Jedi Knights' employed by Gen. Norman Schwarzkopf in developing the famous 'left hook'". SAMS graduates also served in roles beyond the initial planning, with 82 graduates participating in diverse theater tasks by February 1991. As a result, US Army leadership regarded SAMS as a source of "superb planners".

After Desert Storm, the army struggled with military operations other than war, such as peacekeeping and peace enforcement operations. The school and its graduates examined the situations in Bosnia, Haiti, and Somalia. Graduates also participated in Defense Support of Civil authorities missions. The course continued to change in the 1990s. Under Colonel Gregory Fontenot, the school moved from Fort Leavenworth's Flint Hall to Eisenhower Hall in October 1994. In later years, the school's leadership expanded the number of seminars and the civilian faculty. The military continues to draw heavily on SAMS in the twenty-first century. SAMS planners have played a significant role in the Global War on Terror. Beginning in 2002, the United States Central Command requested planners from SAMS and its sister schools, the United States Air Force's School of Advanced Air and Space Studies (SAASS), which was designed to be similar to SAMS, and the United States Marine Corps's School of Advanced Warfighting (SAW). SAMS students from the 2002 and 2003 classes participated as planners in the preparations for the invasion of Iraq and the plan for the post-combat occupation.

The school continued to change and develop, and an additional faculty expansion occurred in 2005–2006. Also, the Fellows' curriculum shifted further away from that of the AMSP program. To keep pace with increasing demand for SAMS planners, the commander of the army's Training and Doctrine Command directed an expansion that was approved by the Chief of Staff of the Army, and the school's 11th director, Colonel Steve Banach, began a winter-start course in 2007. During this period, SAMS provided planners to help forward-deployed headquarters plan operations and contingencies. The school moved to new premises in the newly renovated Muir Hall at Fort Leavenworth on 30 August 2011.

Contributions
SAMS graduates have supported every major US military campaign between 1984 and 2009. SAMS graduates are known for their "critical thinking skill sets", and are consistently called for by combatant commanders around the world. In 2010, Brigadier General Sean MacFarland said, "In a crisis, the president always asks, 'where are the aircraft carriers?' In the Army, leaders ask, 'Where are the SAMS graduates?' Just as the aircraft carrier was a game changer in naval warfare, SAMS graduates and practitioners of operational art have been game changers in land warfare."

The school has been praised by senior US military leaders. According to Major General David Hogg, "SAMS has a reputation for producing skilled planners that can take complex ideas and develop cohesive plans." In 2010, army Vice Chief of Staff Peter W. Chiarelli said that SAMS was "at the forefront of the effort to remake strategic military planning for the 21st century".

Facilities and student body
, the SAMS teaching facilities are mainly housed in Muir Hall (image right), which was once a stable, and Flint Hall. The AMSP courses are taught mostly in Muir Hall—the current SAMS headquarters—while Flint Hall houses additional AMSP seminars. Both buildings were renovated in 2011 and their classrooms accommodate seminars of about 16–18 students and an instructor. The renovations for Muir Hall cost $12.2 million, including $3 million in information systems that allow students to collaborate digitally, replicating a common practice seen in militaries today.

The application process includes an examination, an interview, and a supervisor assessment. Applicants must also complete the US Army's Command and General Staff School or an equivalent intermediate-level education course offered by another uniformed service. The student body of SAMS comprises mostly US Army field grade officers from combat, combat support, and combat service support branches. However, in 1987 the US Air Force graduated three officers and officers from the US Navy and US Marine Corps graduated the following years. US government agencies began sending students to SAMS in 2007. The Department of State, Federal Bureau of Investigation and United States Agency for International Development (USAID) have sent students to the school. Also, Warrant officers first attended SAMS in 2010.

Allied foreign militaries also provide students. In 1999, the school graduated its first international officers—Norwegian and Canadian. Argentina, Australia, Colombia, France, Germany, Hungary, India, Jordan, Republic of Macedonia, the Netherlands, Pakistan, Romania, South Korea, Brazil, Spain, and the United Kingdom have also sent students through the course.

Curricula
The Command and General Staff College awards a Master of Military Art and Science (M.M.A.S.) professional degree to graduates of the School of Advanced Military Studies. The degree is accredited by the Higher Learning Commission for collegiate institutions in the midwestern United States.

Advanced Military Studies Program
Most students participate in the Advanced Military Studies Program (AMSP). In 2009, the school held eight AMSP seminars. AMSP is intended to educate students in military arts and science, and focuses on operational art and covers a variety of subjects, including military problem solving; military theory and history, military doctrine, operational planning; battle dynamics, operational theory and practice, contemporary military operations, and the application of national elements of power. Besides classroom studies and operational exercises, students must complete a research monograph and an oral examination. After graduating, officers serve on a division, corps, or Army Service Component Command staff, or in a functional area assignment.

Advanced Strategic Leadership Studies Program
ASLSP is the school's senior service college resident program, educating sixteen officers for strategic-level responsibilities; developing them to be senior leaders. Students are senior lieutenant colonels, colonels, and their Naval, Coast Guard, and US interagency civilian equivalents. Classes include students from the UK, Canadian, and German armies. Most military students are former battalion commanders. The faculty consists of four civilian professors, one of whom also serves as the program's Director, plus one military faculty member. Graduates are awarded a Masters of Arts degree in Strategic Studies, Military Education Level (MEL) I, and Joint Professional Military Education (JPME) II. US military officers are also awarded the SAMS Additional Skill Identifier (ASI) 6S.

The program began in 1984 as the Advanced Operational Studies Fellowship (AOSF) by diverting lieutenant colonel War College selectees to Fort Leavenworth for an equivalent education program. The AOSF program had students completing the AMSP coursework, and then serving as the principal instructors of AMSP during their second year. In 1995, the name of the program was changed to the Advanced Operational Art Studies Fellowship (AOASF), and in the early 21st century its curriculum was more closely aligned to the strategic level of war. In 2013, the program was again modified in part to bring it more in alignment with TRADOC policies, and to prepare it for Joint Professional Education II accreditation. It was at that time renamed the Advanced Strategic Leadership Studies Program (ASLSP).

The course focuses on the strategic and political aspects of war and educationally prepares students for assignments as strategic leaders. The academic year runs from late June through late May. The ASLSP curriculum provides a comprehensive, multifaceted focus at the theater-strategic level across the spectrum of Joint and land force operations during peace, crisis, and war. The program includes classroom studies of strategy, regional studies, joint operations, strategic leadership, and twentieth-century conflict. Students also research and write a publishable-quality monograph of 10,000- 12,000 words on a suitable subject. ASLSP includes an extensive field studies program, with approximately eight weeks of TDY, to reinforce and expand classroom studies and meet with senior leaders across JIIM organizations. Field Studies include interactions with several agencies in the National Capital Region, Europe, Asia-Pacific, and CONUS-based combatant commands, and visits to various military and civilian governmental agencies. Normally, six US Army officers and the USMC, Canadian, and German officers remain for a second year to serve as seminar leaders in the AMSP course, while one US Army officer joins the faculty of the ASLSP as the military faculty member. Graduates typically serve in a follow-on command assignment or work for a three- or four-star general officer as a member of his or her staff.

Notable graduates

 * General Vincent K. Brooks (class of 1992)
 * Lieutenant General William B. Caldwell, IV (class of 1988)
 * General Charles C. Campbell (class of 1986)
 * Michael T. Flynn, Former National Security Advisor (Class of 1994)
 * Lieutenant General Mark P. Hertling (class of 1988)
 * Lieutenant General David H. Huntoon (class of 1988)
 * General Charles H. Jacoby Jr. (class of 1991)
 * General David M. Rodriguez (class of 1989)
 * Lieutenant General John E. Sterling, Jr. (class of 1992)
 * General Jonathan Vance, Canadian Chief of Defence Staff
 * Lieutenant General William G. Webster (class of 1984)

title: School of Advanced Military Studies


text:
Mitropa Cup

The Mitropa Cup, officially called the La Coupe de l'Europe Centrale or Central European Cup, was one of the first international major European football cups for club sides. It was conducted among the successor states of the former Austria-Hungary. After World War II in 1951 a replacement tournament named Zentropa Cup was held, but just for one season, the Mitropa Cup name was revived, and again in 1958 the name of the tournament changed to Danube Cup but only for one season. The tournament was discontinued after 1992.

The most successful club is Vasas with six titles.

History

This "International" competition for football clubs was founded in 1897 in Vienna. The Challenge Cup was invented by John Gramlick Sr., a co-founder of the Vienna Cricket and Football-Club. In this cup competition all clubs of the Austro-Hungarian Empire that normally would not meet could take part, though actually almost only clubs from the Empire's three major cities Vienna, Budapest and Prague participated. The Challenge Cup was carried out until the year 1911 and is today seen as the predecessor to the Mitropa Cup and consequently the European Cup and Champions League. The last winner of the cup was Wiener Sport-Club, one of the oldest and most traditional football clubs of Austria where the cup still remains.

The idea of a European cup competition was shaped after World War I which brought the defeat and collapse of the Austro-Hungarian Empire. The centre of this idea were the Central European countries that, at this time, were still leading in continental football. In the early 1920s they introduced professional leagues, the first continental countries to do so. Austria started in 1924, followed by Czechoslovakia in 1925 and Hungary in 1926. In order to strengthen the dominance of these countries in European football and to financially support the professional clubs, the introduction of the Mitropa Cup was decided at a meeting in Venice on 17 July, following the initiative of the head of the Austrian Football Association (ÖFB), Hugo Meisl. Moreover, the creation of a European Cup for national teams – that unlike the Challenge Cup and the Mitropa Cup would not be annual – was also part of the agreement. The first matches were played on 14 August 1927. The competition was between the top professional teams of Central Europe.

Initially two teams each from Austria, Hungary, Czechoslovakia and Yugoslavia entered, competing in a knock-out competition. The countries involved could either send their respective league winners and runners-up, or league winners and cup winners to take part. The first winners were the Czech side, AC Sparta Prague. In 1929 Italian teams replaced the Yugoslavian ones. The competition was expanded to four teams from each of the competing countries in 1934. Other countries were invited to participate – Switzerland in 1936, and Romania, Switzerland and Yugoslavia in 1937. Austria was withdrawn from the competition following the Anschluss in 1938. In 1939, prior to the start of World War II, the cup involved only eight teams (two each from Hungary, Czechoslovakia and Italy and one each from Romania and Yugoslavia). The level of the competing nations is clearly shown by Italy's two World Cup titles (1934 & 1938), Czechoslovakia's (1934) and Hungary's (1938) World Cup final, and Austria's (1934) and Yugoslavia's (1930) semi-finals. Out of the eleven different teams competing in the first three World Cups, five were part of the Mitropa Cup.

A tournament was started in 1940, but abandoned before the final match due to World War II. Again, only eight teams competed, three each from Hungary and Yugoslavia and two from Romania. Hungarian Ferencváros and Romanian Rapid (which had won on lots after three draws) qualified for the final, but did not meet because the northern part of Transylvania (lost shortly after World War I) was ceded to Hungary from Romania.

Finals

 * Notes

Performances
Note: The 1960 edition is not included in the list because it was won by a nation rather than club.

Mitropa Super Cup Final
Additionally, a "Mitropa Super Cup" was contested in 1989 between the winners of 1988 and 1989.

title: Mitropa Cup


text:
Dicaearchus

Dicaearchus of Messana (Δικαίαρχος Dikaiarkhos; c. 370/350), also written Dikaiarchos, was a Greek philosopher, geographer and author. Dicaearchus was a student of Aristotle in the Lyceum. Very little of his work remains extant. He wrote on geography and the history of Greece, of which his most important work was his Life of Greece. Although modern scholars often consider him a pioneer in the field of cartography, this is based on a misinterpretation of a reference in Cicero to Dicaearchus' tabulae, which does not refer to any maps made by Dicaearchus but is a pun on account books and refers to Dicaearchus' Descent into the Sanctuary of Trophonius. He also wrote books on ancient Greek poets, philosophy and politics.

Life
He was the son of one Pheidias, and born at Messana in Sicily, Magna Graecia, though he passed part of his life in Greece, and especially in Athens and the Peloponnesus. He also travelled to make his measurements of mountains. He was a disciple of Aristotle and a friend of Aristoxenus (a letter written to him is attested in Cicero ). Eighteenth- and early nineteenth-century scholarship often considered him a friend of Theophrastus as well, but this is based on the reference to a man named Theophrastus in the spurious Description of Greece, which is transmitted under Dicaearchus' name but actually consists of excerpts from a geographic poem written by Dionysius, son of Calliphon, and from a prose periegesis of Greece, written by Heraclides Criticus. It is uncertain when Dicaearchus died. The only certain terminus post quem is the death of Alexander the Great (323 BC). According to Pliny, Dicaearchus measured mountains "with the support of the kings" (cura regum). Most scholars identify these kings as Cassander and Ptolemy I Soter. If this identification is correct, this would put Dicaearchus' activity between 306 and 287 BC. However, the kings might also refer to Philip III Arrhidaeus and Alexander IV, who were the nominatim kings after the death of Alexander the Great. If that identification is correct, this moves his activity to 323–317 BC.

Writings
Dicaearchus was highly esteemed by the ancients as a philosopher and as a man of most extensive and learned information upon a great variety of things. His work is known only from the many fragmentary quotations of later writers. His works were geographical, political, historical and philosophical; but it is difficult to draw up an accurate list of them, since some that are quoted as distinct works may have been only sections of greater ones, and many titles are only attested once. The fragments extant, moreover, do not always enable us to form a clear notion of the works to which they once belonged. The geographical works of Dicaearchus were, according to Strabo, criticised in many respects by Polybius, though Strabo himself was more forgiving of Dicaearchus' ignorance of western and northern Europe, since – unlike Polybius – Dicaearchus had never visited these places.

Dicaearchus wrote on cultural history, music and literature: Among his geographical works may be mentioned:
 * Life of Greece (Βίος Ἑλλάδος) – The Bios Hellados, in three books is Dicaearchus’ most famous work. It inspired the Bios Hellados of Jason (identified as Jason of Nysa by the Suda but perhaps a different historian with the same name ) and Varro's De vita populi Romani and De gente populi Romani. Only a few fragments actually cite the title, but there are many fragments on cultural history that cite no title and might belong to this work. Dicaearchus apparently attempted to write a biography of the Greek nation from earliest times to the reign of Philip II. The most famous passages are those cited by Varro and Porphyry on early history, which suggest a dualistic view of progress. His anthropological theory combined elements from the Hesiodic tradition of decay with progressivist theories. From Hesiod, he adopted the concept of a "golden race", which led a life of bliss but ultimately degenerated because humans became greedy, which led to war. This is combined with progressivist ideas, such as the harsh life of early man and the gradual invention of the arts. In his reconstruction, Dicaearchus distinguished three stages: the golden race, in which man lived off the spontaneously grown fruits of the earth, the pastoral life, in which man started domesticating and hunting animals, and the agricultural life, in which agriculture was introduced. Dicaearchus apparently also explained the saying, "sharing stops choking", as a reference to how humans learned to distribute surplus fairly. Dicaearchus also discussed the origin of the polis, which he derived from the expansion of families through marriage and the kinship of brothers and sisters. According to Dicaearchus, various stages of this evolution were reflected in patrae, phratries and phylae, which were social organisations into which citizens were grouped in many poleis. Another remarkable feature of the Life of Greece is that the first book discussed Egypt and Babylon, probably as cultural predecessors of the Greeks. In the Life of Greece, Dicaearchus also stated that Euripides' Medea plagiarised the Medea play of the obscure tragedian Neophron.
 * On Musical Contests (Περὶ μουσικῶν ἀγώνων) and On Dionysiac Contests (Περὶ Διονυσιακῶν ἀγώνων) – These works discussed innovations in Greek music and drama. In On Dionysiac Contests, Dicaearchus probably also treated the story that, when competing at the Dionysia with the Oedipus Tyrannus, Sophocles was defeated by the tragedian Philocles, and that Aristophanes' Frogs was staged twice. In this work, Dicaearchus probably also stated that the third actor was introduced to Athenian tragedy by Sophocles.
 * On Alcaeus (Περὶ Ἀλκαίου) – This was a monograph on the Lesbian poet Alcaeus of Mitylene.
 * Works on Homer and Euripides – Plutarch cites Dicaearchus alongside Aristotle and Heraclides Ponticus as men who wrote on Homer and Euripides. Dicaearchus' works were probably either monographs on these poets (like his work On Alcaeus) or belonged to problemata literature (both Aristotle and Heraclides Ponticus wrote works on Homeric problems).
 * Circuit of the Earth (Γῆς περίοδος) – This work was a geographical description of the world as it was then known. Dicaearchus divided the inhabited world into a northern and southern part, separated by a line running from the Strait of Gibraltar through Sicily to the Himalaya. He attempted to calculate the distances between various points on this line. In the Circuit of the Earth, Dicaearchus also treated the famous problem of the flooding of the Nile.
 * Measurements of mountains – The Suda cites the work as Measurements of Mountains in the Peloponnese (Καταμετρήσεις τῶν ἐν Πελοποννήσῳ ὀρῶν), but other fragments show that Dicaearchus also measured mountains in Thessaly and on Rhodes. The work may have been part of his Circuit of the Earth. It was the earliest known attempt to measure the heights of various mountains by triangulation, for which he used a dioptra instrument. Dicaearchus' goal was to show that mountains were not as high as people believed and therefore did not impact the sphericity of the earth. Many of his results were later adopted by Eratosthenes.

Of a political nature was: Among his other philosophical works may be mentioned:
 * Tripolitikos (Τριπολιτικός) – A work which has been the subject of much dispute. It was probably a political dialogue (perhaps to be identified with the "political meeting" mentioned by Cicero ) or a speech. Since the only extant fragment discusses the Spartan public meals, some scholars identify it with the Spartan Constitution. Many scholars consider it a work on the mixed constitution, but this is based on a misinterpretation of the εἶδος δικαιαρχικόν, mentioned in Photius; the word δικαιαρχικός does not refer to Dicaearchus but simply means "relating to a just government".
 * Constitutions – Dicaearchus is said to have written a Spartan Constitution (Πολιτεία Σπαρτιατῶν), Pellenian Constitution (Πελληναίων πολιτεία), Corinthian Constitution (Κορινθίων πολιτεία) and Athenian Constitution (Ἀθηναίων πολιτεία). No fragments survive of these works. The Spartan Constitution was apparently read annually in the council of the ephors in Sparta, in the presence of the Spartan youth.
 * On the Soul (Περὶ ψυχῆς) – Dicaearchus wrote two works on the soul: the Lesbian Dialogue (Λεσβιακός) and the Corinthian Dialogue (Κορινθιακός), both in three books. The titles refer to the scene where the philosophical dialogue was set, viz. at Mytilene in Lesbos and at Corinth, respectively. The general title for these two works was On the Soul (Περὶ ψυχῆς). In the Lesbian Dialogue, Dicaearchus endeavoured to prove that the soul was mortal. In the Corinthian Dialogue, Dicaearchus argued that the soul has no existence outside the body. The first book presented a discussion of learned men about the soul. In the subsequent two books, a man named Pherecrates argued for the non-existence of the soul.
 * On the destruction of men (de interitu hominum) – This work collected the causes of human destruction and concluded that man himself was the greatest threat. It is uncertain whether "de interitu hominum" is a book title or merely describes the content of the work.
 * Descent into Sanctuary of Trophonius (Εἰς Τροφωνίου κατάβασις) – A work which consisted of several books. It appears to have been a work on luxury. The title refers to the oracle of Trophonius in Lebadea, where people who wished to consult the oracle had to descend into a cave. Cicero informs us that he translated a section from this work in his De re publica. The work may have also contained an account of the degenerate and licentious proceedings of the priests in the cave of Trophonius.
 * On Lives (Περὶ βίων) – This was probably a philosophical work on the right way of life. The title is only attested once. The fragments on philosophers (the Seven Sages, Pythagoras and Plato) are usually attributed to this work, though they may belong to other works such as the Life of Greece or the Circuit of the Earth as well. Of particular interest is the fragment on the death of Pythagoras, for which Dicaearchus claims to rely on local oral sources from Southern Italy.
 * On the Sacrifice at Ilium (Περὶ τῆς ἐν Ἰλίῳ ϑυσίας) – The title refers to the sacrifice which Alexander the Great performed at Ilium at the start of his expedition against Darius III. Together with his beloved Hephaestion, he paid honours on the tombs of Achilles and Patroclus at Troy, probably as a proclamation of their own relationship, which was mirrored on these two mythical heroes. The sole fragment that remains of this work discusses the story that, after the march through the Gedrosian Desert, Alexander passionately kissed the eunuch Bagoas in a full theatre, to great applause. This suggests that the work may have treated Alexander's relationships, perhaps as part of a more general discussion on love.

Dicaearchus also wrote speeches:

 * Olympikos (Ὀλυμπικός), Panathenaikos (Παναθηναϊκός) – These are usually considered either works on contests held at the Olympic Games and the Panathenaic Games or philosophical dialogues. However, the most plausible interpretation is that they were speeches. Indeed, the Suda explicitly calls Dicaearchus an orator (ῥήτωρ).

There is lastly one spurious work and one doubtful work:

 * Description of Greece (Ἀναγραφὴ τῆς Ἑλλάδος) – This work is spurious. Although it is transmitted under Dicaearchus' name, it actually consists of excerpts from two separate works. One is a geographic work dedicated to "Theophrastus", and consisting of 150 iambic lines; the acrostic of the first twenty-three lines shows that it was really the work of one "Dionysius, son of Calliphon". The other work is a prose periegesis entitled On the Cities in Greece and was written by Heraclides Criticus.
 * Summaries of the Tales from Euripides and Sophocles (ὑποθέσεις τῶν Εὐριπίδου καὶ Σοφοκλέους μύθων) – This was a collection of plot summaries of the plays of Euripides and Sophocles. Many of these summaries survive in papyri or have been prefaced to the plays of Euripides in medieval manuscripts. This work is unlikely to have been written by the Peripatetic Dicaearchus. It was probably either wrongly attributed to him or was the work of another man named Dicaearchus, a grammarian of Lacedaemon, who, according to the Suda, was a disciple of Aristarchus.

title: Dicaearchus