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 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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News Network
July 3,2020

Islamabad, Jul 3: The US embassy here in a statement on Friday said the Trump administration through the US Agency for International Development (USAID) “donated a shipment of 100 brand-new, state-of-the-art ventilators” to Pakistan.

The ventilators arrived in Karachi on July 2 and will be sent to hospitals across Pakistan.

“This donation delivers on President Donald Trump’s generous offer of these critically-needed supplies and supports Pakistan’s urgent response to the pandemic,” the embassy said.

Made in America, the ventilators are valued at about $3 million and reflect the latest in cutting-edge medical design and technology, it said.

They are compact, easily deployable, and will enable Pakistan to more effectively treat patients suffering from Covid-19.

The US-Pakistan health partnership to curb the spread of the novel coronavirus is helping to improve and expand laboratory testing, disease monitoring, case tracking, infection prevention and control and patient care, the embassy said.

The US has contributed nearly $27 million in new funding so far to this vital partnership that is growing every day. "We are also thankful for Pakistan's contribution of medical supplies to help fight coronavirus in the US," the embassy said in the statement.

Ambassador Paul Jones said, “The US stands with Pakistan in its fight against the coronavirus. These American-made ventilators will help Pakistani patients in the most acute need of medical care."

The announcement comes days after Pakistan said it had started producing locally designed ventilators.

Pakistan reported 78 more deaths from the coronavirus in the past 24 hours, raising virus-related fatalities to 4,551 while the total number of confirmed cases has increased to 221,896.

On Friday, the health authorities said 1,13,623 persons have recovered from the coronavirus, surpassing the number of active Covid-19 infections in the country for the first time.

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Agencies
January 9,2020

Washington, Jan 9: The U.S. and Iran stepped back from the brink of possible war Wednesday as President Donald Trump signaled he would not retaliate militarily for Iran's missile strikes on Iraqi bases housing U.S. troops. No one was harmed in the strikes, but U.S. forces in the region remained on high alert.

Speaking from the White House, Trump seemed intent on deescalating the crisis, which spiralled after he authorized the assassination of Iran's top general, Qassem Soleimani. Iran responded overnight by firing more than a dozen missiles at two installations in Iraq, its most direct assault on America since the 1979 seizure of the U.S. Embassy in Tehran.

Trump's takeaway was that “Iran appears to be standing down, which is a good thing for all parties concerned and a very good thing for the world.”

The region remained on edge, however, and American troops including a quick-reaction force dispatched over the weekend were on high alert. Hours after Trump spoke, an ‘incoming’ siren went off in Baghdad's Green Zone after what seemed to be small rockets “impacted” the diplomatic area, a Western official said. There were no reports of casualties.

Iran's Supreme Leader Ayatollah Ali Khamenei said the overnight strike was not necessarily the totality of Iran's response. “Last night they received a slap,” Khamenei said. “These military actions are not sufficient (for revenge). What is important is that the corrupt presence of America in this region comes to an end.”

The strikes had pushed Tehran and Washington perilously close to all-out conflict and left the world waiting to see whether the American president would respond with more military force. Trump, in his nine-minute, televised address, spoke of a robust U.S. military with missiles that are “big, powerful, accurate, lethal and fast.'' But then he added: “We do not want to use it."

Iran for days had been promising to respond forcefully to Soleimani's killing, but its limited strike on two bases--one in the northern Iraqi city of Irbil and the other at Ain al-Asad in western Iraq--appeared to signal that it too was uninterested in a wider clash with the U.S. Foreign minister Mohammad Javad Zarif tweeted that the country had “concluded proportionate measures in self-defence.”

Trump said the U.S. was “ready to embrace peace with all who seek it.” That marked a sharp change in tone from his warning a day earlier that “if Iran does anything that they shouldn't be doing, they're going to be suffering the consequences, and very strongly.”

Trump opened his remarks at the White House by reiterating his promise that “Iran will never be allowed to have a nuclear weapon.” Iran had announced in the wake of Soleimani's killing that it would no longer comply with any of the limits on uranium enrichment in the 2015 nuclear deal crafted to keep it from building a nuclear device.

The president, who had earlier pulled the U.S. out of the deal, seized on the moment of calm to call for negotiations toward a new agreement that would do more to limit Iran's ballistic missile programmes and constrain regional proxy campaigns like those led by Soleimani.

Trump spoke of new sanctions on Iran, but it was not immediately clear what those would be.

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