American mathematicians solve Ramanujan's “deathbed” puzzle

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

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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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News Network
February 22,2020

Washington, Feb 22: US President Donald Trump will raise the issue of religious freedom with Prime Minister Narendra Modi during his visit to India next week, the White House said on Friday, noting that the US has great respect for India's democratic traditions and institutions.

"President Trump will talk about our shared tradition of democracy and religious freedom both in his public remarks and then certainly in private. He will raise these issues, particularly the religious freedom issue, which is extremely important to this administration," a senior official told reporters in a conference call.

The official was responding to a question on whether the president was planning to speak to Modi on the Citizenship (Amendment) Act or the National Register of Citizens.

"We do have this shared commitment to upholding our universal values, the rule of law. We have great respect for India's democratic traditions and institutions, and we will continue to encourage India to uphold those traditions," the official said, requesting anonymity.

"And we are concerned with some of the issues that you have raised," the senior administration official said, in response to the question on CAA and NRC.

"I think the President will talk about these issues in his meetings with Prime Minister Modi and note that the world is looking to India to continue to uphold its democratic traditions, respect for religious minorities," the official said.

"Of course, it's in the Indian constitution -- religious freedom, respect for religious minorities, and equal treatment of all religions. So this is something that is important to the president and I'm sure it will come up," said the official.

Pointing out that India has a strong democratic foundation, the official said India is a country rich in religious, linguistic, and cultural diversity.

"In fact, it's the birthplace of four major world religions," the official noted.

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News Network
March 25,2020

Hubei, Mar 25: As a bus departed from its terminus at Hankou Railway Station at 5:25 am Wednesday morning, Wuhan started to resume bus service after nine weeks of lockdown.

Apart from a driver, a safety supervisor was also on each bus, whose duty was to make sure all passengers are healthy.
"For those who do not use smartphones, they should bring with them a health certificate issued by the health authorities," said Zhou Jingjing, a safety supervisor aboard bus No. 511 departing from the Wuchang Railway Station complex.
The once hardest-hit city in central China's Hubei Province during the COVID-19 outbreak took unprecedented traffic restrictions on Jan 23. All of its public transport and all outbound flights and trains had been suspended in an attempt to contain the virus within the region.

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News Network
February 6,2020

Washington, Feb 6: U.S. president Donald Trump drew on staunch Republican support to defeat the gravest threat yet to his three-year-old presidency on Wednesday, winning acquittal in the Senate on impeachment charges of abuse of power and obstruction of Congress.

Only the third U.S. leader ever placed on trial, Trump readily defeated the Democratic-led effort to expel him from office for having illicitly sought help from Ukraine to bolster his 2020 re-election effort.

Trump immediately claimed "victory" while the White House declared it a full "exoneration" for the president -- even as Democrats rejected the acquittal as the "valueless" outcome of an unfair trial.

Despite being confronted with strong evidence, Republicans stayed loyal and mustered a majority of votes to clear the president of both charges -- by 52 to 48 on abuse of power and 53 to 47 on obstruction of Congress -- falling far short of the two-thirds supermajority required for conviction.

"Two thirds of the senators present not having found him guilty of the charges contained therein, it is therefore ordered and adjudged that the said Donald John Trump be, and he is hereby, acquitted," said Supreme Court chief justice John Roberts, who presided over the trial.

The months-long impeachment of the 45th US leader shone a harsh light on America's political divide, with Trump's core support base united behind him in rejecting it as a "hoax."

One Republican, senator Mitt Romney, a longtime Trump foe, risked White House wrath to vote alongside Democrats on the first count, saying Trump was "guilty of an appalling abuse of public trust." He voted not guilty on the second charge.

But the verdict was never truly in question since the House of Representatives formally impeached Trump in December, and has now cleared out a major hurdle for the president to fully plunge into his campaign for re-election in November.

Trump to speak Thursday

Responding to the verdict, Trump announced he would deliver a formal statement Thursday from the White House "to discuss our Country's VICTORY on the Impeachment Hoax!"

Shortly before, the president tweeted a montage depicting a fake cover of Time magazine declaring him president for all eternity.

The White House declared that Trump had obtained "full vindication and exoneration."

But Nancy Pelosi, the House Speaker and top Democrat in Congress, said that by clearing Trump, the Republicans had "normalized lawlessness."

"There can be no acquittal without a trial, and there is no trial without witnesses, documents and evidence," she said.

"Sadly, because of the Republican Senate's betrayal of the Constitution, the president remains an ongoing threat to American democracy, with his insistence that he is above the law and that he can corrupt the elections if he wants to."

Senate minority leader Chuck Schumer said the acquittal was "virtually valueless" since Republicans refused witnesses at his trial.

'Forever impeached'

The Democrats' intense 78-day House investigation faced public doubts and high-pressure stonewalling from the White House.

Concerned about the political risk for the party, Pelosi rejected a call early last year to impeach Trump on evidence compiled by then-special counsel Robert Mueller that he had obstructed the Russia election meddling investigation.

But her concerns melted after new allegations surfaced in August that Trump had pressured Ukraine for help for his 2020 campaign.

Though doubtful from the outset that they would win support from Republicans, an investigation amassed with surprising speed strong evidence to support the allegations.

The evidence showed that from early in 2019, Trump's private lawyer Rudy Giuliani and a close political ally, Ambassador to the European Union Gordon Sondland, were scheming to pressure Kiev to help smear Democrats, including Trump's potential 2020 rival Joe Biden, by opening investigations into them.

"We must say enough -- enough! He has betrayed our national security, and he will do so again," Adam Schiff, who led the House investigation, argued on the Senate floor this week.

"He has compromised our elections, and he will do so again," Schiff said.

'Colossal' mistake

In the trial, Trump's defence was not seen as having undermined the facts compiled by Schiff's probe, and several Republican senators acknowledged he did wrong.

But his lawyers and Senate defenders argued, essentially, that Trump's behaviour was not egregious enough for impeachment and removal.

And, pointing to the December House impeachment vote, starkly along party lines, they painted it as a political effort to "destroy the president" in an election year and insisted voters should be allowed to decide Trump's fate.

Senate majority leader Mitch McConnell said impeachment will benefit Republicans.

"Right now this is a political loser for them. They initiated it. They thought this was a great idea. At least for the short term, it has been a colossal political mistake."

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