American mathematicians solve Ramanujan's “deathbed” puzzle

[email protected] (The Hindu)
December 30, 2012

sr


London, December 30: American researchers claim to have solved a cryptic formula that renowned mathematician Srinivasa Ramanujan believed came to him in dreams while on his deathbed, the Daily Mail reported on Saturday.

The formula was contained in a letter he wrote to his mentor, the English mathematician G.H. Hardy, from his deathbed in 1920 outlining several new mathematical functions that had never been heard of before, together with a theory about how they worked. It had baffled mathematicians for more than 90 years, but new findings — presented at a conference at the University of Florida last month — reportedly show that Ramanujan's “hunch” about his formula was right — that it could explain the behaviour of black holes.

“We've solved the problems from his last mysterious letters,” said the well-known American mathematician Ken Ono of Emory University.

“For people who work in this area of math, the problem has been open for 90 years … Ramanujan's legacy, it turns out, is much more important than anything anyone would have guessed when Ramanujan died.”

He said the so-called “deathbed puzzle” which, according to Ramanujan, was revealed to him by the goddess Namagiri, may unlock secrets about black holes. “We proved that Ramanujan was right. We found the formula explaining one of the visions that he believed came from his goddess. No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them,” said Professor Ono.

The Mail said that Ramanujan's letter described several new functions that behaved differently from known theta functions, or modular forms, and yet closely mimicked them.

“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. Ramanujan conjectured that his mock modular forms corresponded to the ordinary modular forms earlier identified by Carl Jacobi, and that both would wind up with similar outputs for roots of 1,” it said. Nobody at the time understood what the Indian mathematical genius was talking about.

“It wasn't until 2002, through the work of Sander Zwegers, that we had a description of the functions that Ramanujan was writing about in 1920,” Prof. Ono said.

His team, which used modern mathematical tools to solve the puzzle, was “stunned” to find the function could be used even today.

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Agencies
February 26,2020

Islamabad, Feb 26: Islamabad on Tuesday declared former Prime Minister Nawaz Sharif an absconder while simultaneously denying extending bail to him.

The federal cabinet presided over by Prime Minister Imran Khan, cited that Sharif failed to provide required medical reports and has violated the bail terms.

The government has also decided to freeze gas and electricity tariffs for the next four months, The Dawn reported.

"After Nawaz Sharif failed to submit his medical report of any hospital in London, the medical board rejected a medical certificate sent by him and [the government] declared him an absconder. From today, Nawaz Sharif is an absconder according to the law of land and if he does not return to the country he will be declared a proclaimed offender," said Dr Firdous Ashiq Awan, Special Assistant to the Prime Minister on Information, in a press conference.

She further asserted that the Punjab government, which was authorized by the Islamabad High Court (IHC) to decide Sharif's case on medical grounds, had written several letters asking him to submit his medical report from any hospital in London. However, he failed to do so and only sent a certificate that was not accepted by the medical board.

"If he (Nawaz Sharif) is seriously ill then why a comprehensive medical report is not being submitted to the medical board," Firdous added.

Further, she said that the office of the opposition leader was also waiting for his younger brother and Leader of the Opposition in the National Assembly Shahbaz Sharif, who was also staying in London for 'no reason'.

"He is getting a monthly salary and enjoying luxurious offices and other perks and privileges but not performing his duties required by his office and the people. Shahbaz Sahib, return to the country and justify your salary and other benefits being given from taxpayers' money," Firdous added.

On October 29 last year, the IHC granted bail for eight weeks to Sharif, who was convicted and disqualified in corruption cases, on medical grounds.

Sharif left Pakistan for London along with Shahbaz on November 19, 2019, for his medical treatments there.

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News Network
March 28,2020

Washington, Mar 28: The world is in the face of a devastating impact due to the coronavirus pandemic and has clearly entered a recession, the International Monetary Fund said on Friday, but projected a recovery next year.

"We have reassessed the prospects for growth for 2020 and 2021. It is now clear that we have entered a recession as bad or worse than in 2009. We do project recovery in 2021," IMF Managing Director Kristalina Georgieva told reporters at a news conference.

Georgieva was addressing the press after a meeting of governing body of the IMF, the International Monetary and Financial Committee. Representing 189 members, the body met virtually to discuss the unprecedented challenge posed to the world by COVID-19.

The key to recovery in 2021, she said, is only if the international community succeeds in containing the virus everywhere and prevent liquidity problems from becoming a solvency issue.

"The US is in recession, as is the rest of the advanced economies of the world. And in a big chunk of developed and emerging markets in developing economies. How severe? We are working now on our projections for 2020, Georgieva said in response to a question.

The new projections are expected in the next few weeks.

Stressing that while containment is the main reason for the economy to stand still and get into a recession, she said containment is very necessary to come out of this period and step in to recovery. "Until the virus is not contained, it would be very difficult to go to the lives we love."

"A key concern about a long-lasting impact of the sudden stop of the world economy is the risk of a wave of bankruptcies and layoffs that not only can undermine the recovery. But can erode the fabric of our societies," the IMF chief said.

To avoid this from happening, many countries have taken far-reaching measures to address the health crisis and to cushion its impact on the economy, both on the monetary and on the fiscal side, she said.

The IMF chief said 81 emergency financing requests, including 50 from lower-income countries, have been received. She said current estimate for the overall financial needs of emerging markets is 2.5 trillion dollars.

"We believe this is on the lower end. We do know that their own reserves and domestic resources will not be sufficient," she added.

The G-20, a day earlier, reported fiscal measures totalling some 5 trillion dollars or over 6 per cent of the global GDP.

Responding to another question, Georgieva said the IMF is projecting recession for 2020.

"We do expect it to be quite deep and we are very much urging countries to step up containment measures aggressively so we can shorten the duration of this period of time when the economy is in standstill," she said.

"And also to apply well-targeted measures, primarily focusing on the health system to absorb that enormous stress that comes from coronavirus. And on people, businesses and the financial system, I am very pleased to say that when we went through countries' responses, that sense of targeted fiscal measures is there and are also very impressive to see the size of these measures," she added.

"Countries are doing all they can on the fiscal and on the monetary front. We have heard from our members' very impressive decisions taken over the last days," the IMF chief said.

"We also want to caution that as we are responding now, we want to make the recession as possibly short and not too deep. We also want to think about what is going to follow the recovery and make sure that we are putting forward measures that can be supportive in this regard," she said.

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News Network
June 4,2020

New York, Jun 4: The Minneapolis police officer who used his knee to pin down George Floyd's neck before his death was the most experienced of the four officers involved in the arrest, with a record that included medals for bravery and 17 complaints against him, including one for pulling a woman out of her car during a speeding stop.

New details about Derek Chauvin and the other now-fired officers emerged Wednesday after prosecutors upgraded Chauvin's charge to second-degree murder and charged the others with aiding and abetting in a case that has convulsed the nation with protests over race and police brutality.

Heavily redacted personnel files show that Chauvin, a 19-year veteran of the force, was initially trained as a cook and served in the Army as a military police officer.

Eleven-year veteran and native Hmong speaker Tou Thao began as a community service officer and was the subject of six complaints.

The other two officers were relative newcomers to the department, including Thomas Lane, a former juvenile detention guard who did volunteer work with Somali refugees, and J. Alexander Kueng, who got his start in law enforcement by patrolling his college campus and a department store.

The files were notable for what they didn't include. Only one of the 17 complaints against Chauvin was detailed, none of the six against Thao were mentioned and there was no further detail about a 2017 excessive force lawsuit against Thao.

Records show that the 44-year-old Chauvin initially studied cooking before taking courses in law enforcement and doing two stints in the Army as a military police officer in the late 1990s, serving at Fort Benning, Georgia, and in Germany.

Chauvin became a Minneapolis police officer in 2001 and the lone reprimand in his file involved a 2007 incident when he was accused of pulling a woman out of her car after stopping her for going 10 mph (16 kph) over the speed limit.

Investigators found it was not necessary for Chauvin to remove the woman from the car and noted that his squad car video was turned off during the stop.

But Chauvin was also singled out for bravery. Files show he won two medals of valor, one in 2006 for being part of a group of officers who opened fire on a stabbing suspect who pointed a shotgun at them, and another in 2008 for a domestic violence incident in which Chauvin broke down a bathroom door and shot a suspect in the stomach.

He also won medals of commendation in 2008 after he and his partner tackled a fleeing suspect who had a pistol in his hand, and in 2009 for single-handedly apprehending a group of gang members while working as an off-duty security guard at the El Nuevo Rodeo, a Minneapolis nightclub.

Since his arrest, the former owner of the club, Maya Santamaria, said Chauvin and Floyd both worked as security guards there at various times but that she wasn't sure if they had known one another.

She said Chauvin was unnecessarily aggressive on nights when the club had a black clientele, quelling fights by dousing the crowd with pepper spray and calling in several police squad cars as backup, a tactic she called “overkill.”

Chauvin's wife, Kellie, a Laotian immigrant who became the first Hmong winner of the Mrs. Minnesota pageant, said shortly after his arrest last week that she intends to divorce him.

Before news of the upgraded charges, an attorney for Chauvin said he was not making any statements at this time. Lawyers for the others did not return messages seeking comment.

In cellphone video of the May 25 arrest of Floyd, Chauvin is shown pressing his knee onto Floyd's neck for more than eight minutes while Floyd cries out “I can't breathe” and eventually stops moving.

During much of the arrest, Kueng and Lane were helping Chauvin restrain Floyd. Thao was standing nearby keeping onlookers back.

According the complaint, at one point during the arrest, as Chauvin held Floyd down with his knee, Lane asked Chauvin twice whether they should roll Floyd over.

“No, staying put where we got him,” Chauvin replied, “I am worried about excited delirium or whatever,” Lane said. And Chauvin replied again, “That's why we have him on his stomach.” None of the three officers moved from their positions.

Lane joined the police early last year as a 35-year-old cadet — much older than most rookies — and became a full-fledged officer last December. He had no complaints in his file during his short time on the force.

On employment forms, the University of Minnesota graduate said he done volunteer work tutoring Somali youth and as a mentor helping at-risk elementary school students with reading and homework.”

Kueng, at 26 the youngest of the four officers, was also a recent recruit to the police force. He completed his year's probation just three months before the Floyd arrest.

His personnel file, which notes that he speaks, reads and writes Russian, did not include any commendations or disciplinary actions during his short time on force.

Kueng was a 2018 graduate of the University of Minnesota, where he worked part-time as part of the campus security force. He also worked nearly three years as a theft-prevention officer at Macy's.

Thao joined the police force part-time in 2008 while attending a community college. Before that, he worked as a security guard, a supermarket stocker and trainer at McDonald's.

City records show six complaints were filed against Thao, but there was no mention of that in the records released Wednesday. There also was no mention of a 2017 federal lawsuit accusing him and another officer of excessive force.

According to the lawsuit, Lamar Ferguson claimed that in 2014, Thao and his partner stopped him and beat him up while he was on his way to his girlfriend's house. The lawsuit was settled for $25,000 ___

Richmond reported from Madison, Wisconsin. Associated Press writer Scott Bauer in Madison and researcher Rhonda Shafner in New York contributed to this report.

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