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
June 23,2020

New Delhi, Jun 23: The meeting between Indian Army's 14 Corps Commander Lt Gen Harinder Singh and his Chinese counterpart got over after around 11 hours, sources said.

"Today's meeting between the Corps Commander-level officers of India and China is over. The meeting which started at 11:30 am went on for around 11 hours. More details awaited," sources said.

The meeting started at around 11:30 am at Moldo on the Chinese side of Line of Actual Control (LAC) opposite Chushul to defuse the tensions in Eastern Ladakh sector due to Chinese military build-up, the sources said.

This is the second meeting between the two corps commanders. They had met on June 6 and had agreed to disengage at multiple locations. India had asked the Chinese side to go back to pre-May 4 military positions along the LAC.

The Chinese side had not given any response to the Indian proposal and not even shown intent on the ground to withdraw troops from rear positions where they have amassed over 10,000 troops.

India is also likely to discuss the change in rules of engagement on the LAC where the forces have been empowered to use firearms in extraordinary circumstances, sources had said.

They said India will also ask China to honour the commitment given during June 6 talks to disengage in the Galwan valley completely and other places.

The build-up of Chinese air assets including strategic bombers by the PLA Air Force in fields near Indian territory close to Ladakh is also likely to figure in discussions.

India and China have been involved in talks to ease the ongoing border tensions since last month.

However, last week as many as 20 Indian soldiers lost their lives in the face-off in the Galwan Valley after an attempt by the Chinese troops to unilaterally change the status quo during the de-escalation in eastern Ladakh.

The Indian intercepts have revealed that the Chinese side suffered 43 casualties including dead and seriously injured in the violent clash.

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Agencies
June 6,2020

United Nations, Jun 6: The coronavirus disease has not "exploded" in India, but the risk of that happening remains as the country moves towards unlocking its nationwide lockdown that was imposed in March to contain the Covid-19, according to a top WHO expert.

WHO Health Emergencies Programme Executive Director Michael Ryan on Friday said the doubling time of the coronavirus cases in India is about three weeks at this stage.

“So the direction of travel of the epidemic is not exponential but it is still growing,” he said, adding that the impact of the pandemic is different in different parts of India and varies between urban and rural settings.

“In South Asia, not just in India but in Bangladesh and...in Pakistan, other countries in South Asia, with large dense populations, the disease has not exploded. But there is always the risk of that happening,” Ryan said in Geneva.

He stressed that as the disease generates and creates a foothold in communities, it can accelerate at any time as has been seen in a number of settings.

Ryan noted that measures taken in India such as the nationwide lockdown have had an impact in slowing transmission but the risk of an increase in cases looms as the country opens up.

“The measures taken in India certainly had an impact in dampening transmission and as India, as in other large countries, open up and as people begin to move again, there's always a risk of the disease bouncing back up,” he said.

He added that there are specific issues in India regarding the large amount of migration, the dense populations in the urban environment and the fact that many workers have no choice but to go to work every day.

India went past Italy to become the sixth worst-hit nation by the COVID-19 pandemic.

India saw a record single-day jump of 9,887 coronavirus cases and 294 deaths on Saturday, pushing the nationwide infection tally to 2,36,657 and the death toll to 6,642, according to the health ministry.

The lockdown in India, was first clamped on March 25 and spanned for 21 days, while the second phase of the curbs began on April 15 and stretched for 19 days till May 3. The third phase of the lockdown was in effect for 14 days and ended on May 17. The fourth phase ended on May 31.

The country had registered 512 coronavirus infection cases till March 24.

The nation-wide lockdown in containment zones will continue till June 30 in India but extensive relaxations in a phased manner from June 8 are listed in the Union home ministry's fresh guidelines on tackling the Covid-19 pandemic issued last week.

WHO Chief Scientist Soumya Swaminathan said the over 200,000 current coronavirus cases in India, a country of over 1.3 billion people, "look big but for a country of this size, it's still modest.”

She stressed that it is important for India to keep track of the growth rate, the doubling time of the virus and to make sure that that number doesn't get worse.

She said that India is a “heterogeneous and huge country” with very densely populated cities and much lower density in some rural areas and varying health systems in different states and these offer challenges to the control of Covid-19.

Swaminathan added that as the lockdown and restrictions are lifted, it must be ensured that all precautions are taken by people.

“We've been making this point repeatedly that really if you want behaviour change at a large level, people need to understand the rationale for asking them to do certain things (such as) wearing masks,” she said.

In many urban areas in India, it's impossible to maintain physical distancing, she said adding that it then becomes very important for people to wear appropriate face coverings when they are out, in office settings, in public transport and educational institutions.

“As some states are thinking about opening, every institution, organisation, industry and sector needs to think about what are the measures that need to be put in place before you can allow a functioning and it may never be back to normal.”

She said that in many professions working from home can be encouraged but in several jobs, people have to go to work and in such cases measures must be put in place that allow people to protect themselves and others.

“I think communication and behaviour change is a very large part of this whole exercise,” she added.

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News Network
April 26,2020

New Delhi, Apr 26: The total count of coronavirus cases in the country surged to 26,917 on Sunday.

The total COVID-19 cases in the country are inclusive of 5,913 cured and discharged patients, one migrated, and 826 deaths. At present, there are 20,177 active COVID-19 cases in the country.

The Federation of Resident Doctors' Association (FORDA) wrote to the Union Health Minister, Dr Harsh Vardhan, seeking assistance regarding accommodation facilities for resident doctors advised for home quarantine.

In a letter to the Health Minister, the association stated that recently there have been reports of resident doctors from various hospitals who tested positive for COVID-19. As a consequence, their primary contacts who are mostly their colleagues and resident doctors, have been advised to be on home quarantine, the association added.

On the other hand, Maharashtra continues to have the largest number of COVID-19 positive cases at 7,628 . 1,076 persons have been cured in the state while 323 persons have died.

Fresh cases have been reported several states including West Bengal, Maharashtra, Rajasthan and Uttar Pradesh.

The total number of cases in Indore has risen to 1,176, including 57 deaths.

While 133 deaths have been reported from Gujarat where the total number of cases spiked up to 3,071.

Kerala has seen a recovery rate of around 74 per cent as 338 out of the total 457 COVID-19 positive patients recovered in the state with only 4 fatalities.

Delhi has seen 2,625 COVID-19 positive cases and 54 deaths due to the pandemic.

Here's a quick read on the COVID-19 related updates:

1. A five-member Central team visited Telangana Director General of Police (DGP) office here on Sunday, to review the law and order situation in the state and oversee how the state police are ensuring the implementation of the lockdown.

2. Post Graduate Institute of Medical Education and Research (PGIMER) Chandigarh and AIIMS Delhi and Bhopal will study the effectiveness of Mycobacterium w in critical COVID-19 patients.

3. Chandigarh-based Post- Graduate Institute of Medical Education and Research (PGIMER) said that it has assessed the safety of mycobacterial w (Mw), an immunomodulator for leprosy, in four hospitalised patients of COVID-19, and has found no short-term adverse effect.

4. As many as 2,189 cases were lodged and 10,062 people have been arrested so far, for the breach of lockdown norms in Uttarakhand.

5. The Delhi High Court has directed that COVID-19 related tests should be made available to the general public at the lowest cost possible as the country is going through an unprecedented medical crisis affecting public order.

6. Bihar government has ordered two automatic RNA extraction machines, said the Principal Secretary of Health, Sanjay Kumar today.

7. Medical services at Babu Jagjivan Ram Hospital in the Jahangirpuri area have been closed and the hospital is being sanitised after 44 staff members including doctors were tested positive for COVID-19, Delhi Health Department said.

8. In order to support the frontline workers in the fight against COVID-19, Samsung and Google will be offering free phone repairs to health care workers and first responders.

9. Bhopal Division of Indian Railways has converted 74 railways coaches into isolation wards, said Sunil Dhingra, Senior Section Engineer of Bhopal Division.

10. Indian High Commission here on Sunday said two Air India and one Blue Dart flights will send about 78 tonnes of cargo to India as part of the 1 million PPE kits being sourced from Singapore-based company.

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