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
May 19,2020

New Delhi, May 19: In a fresh blow to saffronite journalist Arnab Goswami, the Supreme Court of India today rejected his plea seeking transfer of the investigation of a case, filed against him for defaming Congress interim president Sonia Gandhi, to the CBI. The court also refused to quash the FIRs filed against him.

Goswami, editor-in-chief of Republic TV, has been booked in connection with a TV show on the gathering of migrants outside Bandra railway station on April 14. This apart, multiple FIRs have been filed against him for his show on Palghar lynching. In that show, he had posed certain questions on the incident to Congress President Sonia Gandhi, following which Congress workers lodged complaints against him in various states.

Extending Goswami’s interim protection from arrest by three weeks, the Supreme Court said, “Right of a journalist under 19 1 (a) higher…Free citizens can’t exist if news media can’t speak.”

During the earlier hearing, Senior Advocate Harish Salve, appearing for Goswami, had urged the court to transfer the probe to an agency like CBI. He said the “nature of the” second FIR against Goswami over a show on the migrant gathering outside Bandra station on April 14 “shows that it’s arm-twisting tactic”. 

“They are trying to stifle an unpleasant voice. This is a political party targeting a journalist. All complainants are members of one political party. They have a problem with the government. They want to teach this journalist a lesson,” he added.

Objecting to Salve’s plea to transfer the case to the CBI, Maharashtra government counsel, Senior Advocate Kapil Sibal, had said, “CBI investigation will go into your hands”. 

Sibal denied that Goswami was being harassed and said he was only asked relevant questions. He said Goswami should “stop this communal violence and communal mongering”.

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Agencies
June 29,2020

From March through May, around 1 crore migrant workers fled India’s megacities, afraid to be unemployed, hungry and far from family during the world’s biggest anti-Covid-19 lockdown.

Now, as Asia’s third-largest economy slowly reopens, the effects of that massive relocation are rippling across the country. Urban industries don’t have enough workers to get back to capacity, and rural states worry that without the flow of remittances from the city, already poor families will be even worse off -- and a bigger strain on state coffers.

Meanwhile, migrant workers aren’t expected to return to the cities as long as the virus is spreading and work is uncertain. States are rolling out stimulus programs, but India’s economy is hurtling for its first contraction in more than 40 years, and without enough jobs, a volatile political climate gets more so.

“This will be a huge economic shock, especially for households of short-term, cyclical migrants, who tend to come from vulnerable, poor and low-caste and tribal backgrounds,” said Varun Aggarwal, a founder of India Migration Now, a research and advocacy group based in Mumbai.

In the first 15 days of India’s lockdown, domestic remittances dropped by 90%, according to Rishi Gupta, chief executive officer of Mumbai-based Fino Paytech Ltd., which operates the country’s biggest payments bank.

By the end of May, remittances were back to around 1750 rupees ($23), about half the pre-Covid average. Gupta’s not sure how soon it’ll fully recover. “Migrants are in no hurry to come back,” Gupta said. “They’re saying that they’re not thinking of going back at all.”

If workers stay in their home states long term, policymakers will have more than remittances to worry about. If consumption falls and the new surplus of labor drives wages down, Agarwal said, “there will also be a second-order shock to the local economy. Overall, not looking good.”

India announced a $277 billion stimulus package in May and followed it up with a $7 billion program aimed at creating jobs for 125 days for migrants in villages across 116 districts. Separately, local authorities are also looking for solutions.

Officials in Bihar have identified 2,500 acres of land that could be made available to investors, said Sushil Modi, deputy chief minister of Bihar, a state in east India. “We can use this crisis as an opportunity to speed up reforms,” he said.

The investors haven’t materialised yet, and in the meanwhile, state governments are relying on the national cash-for-work program that guarantees 100 days worth of wages per household.

Skilled workers don’t want to do manual labor offered through the program, and even if they did, says Amitabh Kundu of RIS, many think of it as beneath their station. “There will be an increase in social tensions,” he predicts. “Caste may again start playing a role. It’s absolute chaos.”

For skilled workers, initiatives vary:

* Uttar Pradesh, which received 3.2 million people, is compiling lists of skilled workers who need employment and trying to place them with local manufacturing and real estate industry associations. So far, the government says, it’s placed 300,000 people with construction and real estate firms.

* Bihar has placed returners in state-run infrastructure projects and hired others to stitch uniforms and make furniture for government-run schools, even as they waited in quarantine centres, said Pratyay Amrit, head of the state’s disaster management department.

* The eastern state of Odisha announced an urban wage employment program aimed at putting as many as 450,000 day labourers to work through September. Some 25,000 people have been employed, so far, under the scheme, G. Mathivathanan, principal secretary for housing and urban development said.

Attracting Investments

It’s not clear any of this will be enough to make a dent, says Ravi Srivastava, professor at New Delhi-based Institute of Human Development, adding that the states don’t have much of a track record on economic development.

“It was the failure of these states to improve governance and put development plans in place that led to the out-migration in the first place,” he said.

But officials and workers’ rights advocates see opportunity. Uttar Pradesh has established liaisons to encourage companies from the US, Japan and South Korea to establish manufacturing in the state. There and in Madhya Pradesh and Rajasthan, the government has made labour laws more friendly to employers, making it easier to hire and fire workers.

Modi, the minister from Bihar, said the migration may also give workers--historically a disenfranchised group--new power, particularly as urban centres struggle. “The way industries treated workers during the lockdown -- didn’t pay them, the living conditions were poor -- now these industries will realize the value of this force,” Modi said.

“In the days to come, labour will emerge as a force that can’t be ignored anymore,” he added. “That’s the new normal. We will work out how to ensure dignity, rights to our people who are going to work in other states.”

Bihar is due for elections by November, a vote that could be an early test of the mass migration’s political consequences. The state is currently governed by a coalition that includes Prime Minister Narendra Modi’s Bharatiya Janata Party. Amitabh Kundu, a fellow at the Research and Information System for Developing Countries, a New Delhi-based government think-tank, said migrant workers are likely to be angry voters.

“Chief ministers are telling these migrants that they will not have to go back for work,” he said. “But their capacity to do something miraculous in the next four to five months is doubtful. If they can retain even one-fourth of the migrants, I would call it a success.”

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News Network
June 5,2020

New Delhi, Jun 5: On World Environment Day, Prime Minister Narendra Modi on Friday urged citizens to ensure that the flora and fauna thrive and take a pledge to preserve our planet's rich biodiversity.

"On #WorldEnvironmentDay, we reiterate our pledge to preserve our planet's rich biodiversity. Let us collectively do whatever possible to ensure the flora and fauna with whom we share the Earth thrive. May we leave an even better planet for the coming generations," the Prime Minister tweeted.

He also shared a clip from his last 'Mann Ki Baat' programme where he mentioned about conservation of rainwater and protecting the rich diversity of nature.

"On June 5, the entire world will celebrate 'World Environment Day'. The theme for this year's 'World Environment Day' is bio-diversity. This theme is especially pertinent in the current circumstances. During lockdown in the last few weeks the pace of life may have slowed down a bit but it has also given us an opportunity to introspect upon the rich diversity of nature or biodiversity around us," the Prime Minister had said.

"Much of the avian fauna had sort of disappeared due to sound and air pollution, and now after years people can once again listen to their melodic chirping in their homes," he said.

The Prime Minister also said that water conservation, in particular the conservation of rainwater, is essential and exhorted everyone to strive for saving rainwater in the upcoming monsoons.

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