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
April 4,2020

Mumbai, Apr 4: As many as six Central Industrial Security Force (CISF) personnel stationed at Mumbai airport in Maharashtra have tested positive for coronavirus, taking the total number of positive cases among the central force to 11. The first case of a CISF jawan being diagnosed with the viral disease was reported on March 28. 

After the first case, the armed police force reported four more cases of COVID-19 among the personnel stationed at the airport on Thursday. On the same day, the CISF collected samples of 146 staff and sent them to Kasturba hospital for testing. The results, which arrived on Friday, recorded six more COVID-19 cases among, reported news agency.

The personnel were posted at Kharghar adjoining Mumbai, a senior official told news agency.

As of now, there are 14 COVID-19 cases in Panvel Municipal Corporation (PMC) area in Mumbai. Kharghar comes under the civic body's jurisdiction.

All the 146 CISF personnel were shifted to a quarantine centre at a facility at Kamothe reported the Times of India.

Maharashtra reported 67 new COVID-19 cases, taking the total tally to 490. A total of 26 deaths have been reported in the state.  

In the meantime, the Centre on Friday said there is no shortage of medical supplies across the country to fight COVID-19 outbreak.

"The government of India is making sure that all the essential medical supplies are in place to fight COVID-19. Sixty-two lifeline Udan flights transported over 15.4 tons of essential medical supplies in the last five days," Union Minister for Chemical and Fertilisers DV Sadanada Gowda said in a tweet.

The government is also paying full attention to the manufacturing activities of essential items like pharmaceuticals and hospital devices. For this, over 200 units in Special Economic Zones (SEZs)  are operational, he added.

"A Central Control Room has also been set up for close monitoring of the distribution of essential medical items and to address logistic related issues," Gowda said.

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News Network
February 2,2020

Feb 2: Prime Minister Narendra Modi’s second budget in seven months disappointed investors who were hoping for big-bang stimulus to revive growth in Asia’s third-largest economy.

The fiscal plan -- delivered by Finance Minister Nirmala Sitharaman on Saturday -- proposed tax cuts for individuals and wider deficit targets but failed to provide specific steps to fix a struggling financial sector, improve infrastructure and create jobs. Stocks slumped as a proposal to scrap the dividend distribution tax for companies failed to impress investors.

"Far from being a game changer, the budget provides little in terms of short-term growth stimulus,” said Priyanka Kishore, head of India and South East Asia economics at Oxford Economics Ltd. in Singapore. “While income tax cuts will provide some relief on the consumption front, the multiplier effect is low and the overall stance of the budget is not expansionary."

India has gone from being the world’s fastest-growing major economy three years ago, expanding at 8%, to posting its weakest performance in more than a decade this fiscal year, estimated at 5%.

While the government has taken a number of steps in recent months to spur growth, they’ve fallen short of spurring demand in the consumption-driven economy. Saturday’s budget just added to the glum sentiment.

Okay Budget

“It’s an okay budget but not firing on all cylinders that the market was hoping for,” said Andrew Holland, chief executive officer at Avendus Capital Alternate Strategies in Mumbai.

The government had limited scope for a large stimulus given a huge shortfall in revenues in the current year. The slippage induced Sitharaman to invoke a never-used provision in fiscal laws, allowing the government to exceed the budget gap by 0.5 percentage points. The result: the deficit for the year ending March was widened to 3.8% of gross domestic product from a planned 3.3%.

On Friday, India’s chief economic adviser Krishnamurthy Subramanian said reviving economic growth was an “urgent priority” and deficit goals could be relaxed to achieve that. The adviser’s Economic Survey estimated growth will rebound to 6%-6.5% in the year starting April.

The fiscal gap will narrow to 3.5% next year, as the government budgeted for gross market borrowing to rise marginally to 7.8 trillion rupees from 7.1 trillion rupees in the current year. A plan to earn 2.1 trillion rupees by selling state-owned assets in the year starting April will also help plug the deficit.

Total spending in the coming fiscal year will increase to 30.4 trillion rupees, representing a 13% increase from the current year’s budget, according to latest data.

Key highlights from the budget:

* Tax on annual income up to 1.25 million rupees pared, with riders

* Dividend distribution tax to be levied on investors, instead of companies

* Farm sector budget raised 28%, transport infrastructure gets 7% more

* Spending on education raised 5%

* Fertilizer subsidy cut 10%

Analysts said the muted spending plan to keep the deficit in check will lead to more downside risks to growth in the coming months.

“It is very doubtful that the increase in expenditure will push demand much,” Chakravarthy Rangarajan, former governor at the Reserve Bank of India told BloombergQuint, adding that achieving next year’s budget deficit goal of 3.5% of GDP was doubtful.

With the government sticking to a conservative fiscal path, the focus will now turn to central bank, which is set to review monetary policy on Feb. 6. Given inflation has surged to a five-year high of 7.35%, the RBI is unlikely to lower interest rates.

What Bloomberg’s Economists Say:

The burden of recovery now falls solely on the Reserve Bank of India. With inflation breaching RBI’s target at present, any rate cuts by the central bank are likely to be delayed and contingent upon inflation falling below the upper end of its 2%-6% target range.

-- Abhishek Gupta, India economist

Governor Shaktikanta Das may instead focus on unconventional policy tools such as the Federal Reserve-style Operation Twist -- buying long-end debt while selling short-tenor bonds -- to keep borrowing costs down.

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News Network
January 13,2020

Jan 13: India lost more than $1.33 billion to internet restrictions in 2019 as Prime Minister Narendra Modi’s government pushed ahead with his party’s Hindu nationalist agenda, raising tensions and sparking nationwide protests.

The worst shutdown has been in Kashmir, where after intermittent closures in the first half of the year, the internet has been cut off since Aug. 5 following the government’s decision to revoke the special autonomous status of the country’s only Muslim-majority state, a study said. The prologued closure was criticized by India’s highest court, which ruled Friday that the “limitless” internet shutdown enforced by the government for the last five months was illegal and asked that it be reviewed.

India imposed more internet restrictions than any other large democracy, according to the Cost of Internet Shutdowns 2019 report released by Top10VPN, a U.K.-based digital privacy and security research group. The South Asian nation recorded the third-highest losses after Iraq and Sudan, which lost $2.31 billion and $1.86 billion respectively to disruptions. Worldwide internet restrictions caused losses worth $8.05 billion, the report said.

The cost of internet blackouts was calculated using indicators from groups including the World Bank, International Telecommunication Union, and the Delhi-based Software Freedom Law Center. It includes social media shutdowns in its calculations.

India’s ministry of information and technology didn’t respond to an email seeking a response to the report’s findings.

‘Conservative Estimates’

Through 2019, India shut access to the internet for over 4,000 hours. The report added shutdowns in India were often narrowly targeted, down to the level of blocking city districts for a few hours to allow security forces to restore order. Many of these incidents were not included in the report.

“These are conservative estimates,” said Simon Migliano, head of research at U.K.-based Top10VPN. “Internet shutdowns are increasing and it shows a damaging trend.”

India’s other major internet disruptions coincided with two moves by the government that affect India’s Muslim minority. The first disruption took place in November in the states of Uttar Pradesh and Rajasthan after the Supreme Court handed a victory to Hindu groups over Muslim petitioners in a long-simmering dispute over a plot of land.

There were further disruptions in December when protests erupted against the introduction of a religion-based law that allows undocumented migrants of all faiths except Islam from neighbouring countries to seek Indian citizenship. The government enforced shutdowns across Uttar Pradesh and some Northeastern states in order to quell the protests, the report said.

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