Ramanujan's 'genius' finally proven by mathematicians

November 10, 2012

ramanujam

London, November 10: Hence proved! A US scientist has finally proven the mathematical relation - discussed by great Indian mathematician Srinivasa Ramanujan with his mentor G H Hardy in one of his last letters.

Ramanujan, the young genius who died at the age of 32, often managed to leap from insight to insight without formally proving the logical steps in between.

Now proof has been found for a connection that he seemed to mysteriously intuit between two types of mathematical functions, the 'New Scientist' reported.

"His ideas as to what constituted a mathematical proof were of the most shadowy description," G H Hardy, Ramanujan's mentor and one of his few collaborators, had once said.

Ken Ono of Emory University in Atlanta, Georgia, who has previously unearthed hidden depths in Ramanujan's work, was prompted by Ramanujan's 125th birth anniversary, to look once more at his writings.

"I wanted to go back and prove something special," Ono said.

Ono settled on a discussion in the last known letter that Ramanujan wrote to Hardy, concerning a type of function now called a modular form.

Functions are equations that can be drawn as graphs on an axis, like a sine wave, and produce an output when computed for any chosen input or value

The functions looked unlike any other modular forms, but Ramanujan wrote that their outputs would be very similar to those of modular forms when computed for the roots of 1 like the square root -1.

Characteristically, Ramanujan offered neither proof nor explanation for this conclusion.

It was only 10 years later that mathematicians formally defined this other set of functions, now known as mock modular forms.

However, still no one fathomed what Ramanujan meant by saying the two types of function produced similar outputs for roots of 1.

Now Ono and colleagues have exactly calculated one of Ramanujan's mock modular forms for values very close to -1, and said the difference in the value of the two functions, ignoring the functions signs, is tiny when computed for -1, just like Ramanujan said.

The result confirms Ramanujan's incredible intuition, says Ono.

Born in 1887 in Erode, Tamil Nadu, Ramanujan was self-taught and worked in almost complete isolation from the mathematical community of his time, the report said.


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News Network
July 24,2020

New Delhi, Jul 24: India reported the highest single-day spike of 49,310 coronavirus cases on Friday, according to the Union Ministry of Health and Family Welfare.

The total COVID-19 positive cases stand at 12,87,945 including 4,40,135 active cases, 8,17,209 cured/discharged/migrated.
With 740 deaths in the last 24 hours, the cumulative toll reached 30,601.

Maharashtra has reported 3,47,502 coronavirus cases, the highest among states and Union Territories in the country. A total of 1,92,964 cases have been reported from Tamil Nadu till now, while Delhi has recorded 1,27,364 coronavirus cases.

According to the Indian Council of Medical Research (ICMR), 3,52,801 samples were tested for coronavirus on Thursday and overall 1,54,28,170 samples have been tested so far. 

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News Network
July 21,2020

Lucknow, Jul 21: Madhya Pradesh Governor Lalji Tandon, a veteran political figure in Uttar Pradesh where he had served as a cabinet minister, died at a hospital here early Tuesday.

The 85-year-old was admitted to the hospital on June 11 with breathing problems, fever and difficulty in urination.

He died at 5:35 am in Medanata Hospital, according to his son Ashutosh Tandon, a UP cabinet minister.

Lalji Tandon is survived by wife and three sons.

His body will be kept at his official residence in Hazratganj and later at his Sindhi Tola residence in Chowk to enable people to pay their last respects.

The last journey will start at 4 in the evening for the Gulala Ghat where his last rites will be performed later in the day, Ashutosh Tandon said in a statement.

The UP government has announced three days mourning as a mark of respect to Lalji Tandon, a former cabinet minister, a government spokesman said.

Belonging to the Atal Bihari Vajpayee and L K Advani era of BJP leaders, Lalji Tandon proved himself as an able administrator during his decades-long political career in Uttar Pradesh.

A former Lok Sabha MP, he was later given gubernatorial responsibility.

He took oath as Madhya Pradesh governor on July 29, 2019, when the Congress was in power in the state, after serving in the same post in Bihar for nearly 11 months. 

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News Network
January 1,2020

Jammu, Jan 1: As many as 160 terrorists were killed in Jammu and Kashmir this year, while 250 terrorists, including 102 of Pakistani origin, were active in the Valley, Director General of Police (DGP) Dilbag Singh said on Tuesday, noting that terror incidents and the number of local youths turning towards terrorism have decreased.

"250 terrorists have been active in Jammu and Kashmir. There is a decrease in number of active terrorists as compared to last year," Mr Singh said at the annual press conference at police headquarters in Jammu.

The DGP said that there is 30 per cent fall in terrorist incidents, less civilian killings and 36 per cent decrease in law and order incidents as compared to 2018.

"218 such (local) youths joined militant outfits in 2018 but only 139 joined in 2019," he said. Out of these new recruits only 89 have survived.

"The rest have been eliminated as their shelf life is between 24 hours to 2-3 months after joining militancy. There are hardly few old terrorists surviving, which include Jehangir Saroori and Riyaz Nayikoo", he said.

There have been only 481 law and order incidents this year as compared to 625 last year, he said.

There were 80 per cent successful anti-terror operations in which 160 terrorists, including foreigners, have been killed during the year.

Mr Singh said 102 terrorists have been arrested and 10 terrorists surrendered during the year.

He said that 102 Pakistan origin terrorists are still operating in Kashmir.

"Eleven valiant police personnel from Jammu and Kashmir besides 72 from other other security forces have been martyred," he said.

There was no collateral damage during anti-terror operation as people fully cooperated. "There was zero law and order problem this year (during anti terror operations)," he added.

The DGP said that "there has been a high degree of incidents of infiltration attempts from across the border this year and also ceasefire violations. But security forces have successfully foiled these attempted as 130 infiltrators have entered in 2019 as compared to 143 last year".

He said Jammu and Kashmir Police has set an example by handling the law and order situation in the most "exemplary" way following the abrogation of Article 370 provisions.

It was the biggest challenge faced by the force in 2019, but "we handled the most critical phase in the best way" and there was no civilian casualty during the period, he said.

Dismissing claims of minors being arrested by police in Kashmir, he said that it is being used as propaganda by some people and asserted that the J-K police has acted within the limits of law.

"We are open to scrutiny. The issue reached the Supreme Court which referred it to Jammu and Kashmir high court. The matter was inquired by the HC committee. The SC said that there is no misuse of law by law enforcement agency. J-K police has acted within the limits of law," he said.

Replying to queries on restoration of internet, the DGP said it is under consideration. "I think J-K is moving towards such a situation (on law and order front). Very soon you will hear positive announcement," Mr Singh said.

He said that though some people will try to misuse internet, "in the past, we took care of them and we will take care of such people in the future too".

Internet services in all government-run hospitals and SMS to all mobile phones will be restored from December 31 midnight in the Kashmir Valley, Jammu and Kashmir official spokesman Rohit Kansal said on Tuesday.

On December 10, some short message service (SMS) were enabled on mobile phones in order to facilitate students, scholarship applicants, traders and others. It has now been decided to fully restore the service throughout Kashmir from midnight of December 31, Kansal said.

Mobile Internet services were restored in Kargil district of Ladakh on Friday after remaining suspended for 145 days in the wake of the Centre abrogating provisions of Article 370 of the Constitution, officials said.

Internet services were suspended on August 4, a day before the Centre announced abrogation of Article 370 and division of the state into the union territories of Jammu and Kashmir, and Ladakh.

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