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## Indians lacks the cultural drive to honor science

This curious note on currency notes is inspired by the talks of my favorite astronomer Dr. Neil deGrasse Tyson, whose lectures fill my iPod and hard-disk and backup hard-disk.

To

The Prime Ministers of India and,

The Governor, Reserve Bank of India.

Dear Sir,

This note is inspired by the public lectures of astronomer Dr. Neil deGrasse Tyson.

Yesterday, the 20th of July, on the one hand I was celebrating the greatest scientific achievement of all times, the 45th anniversary of the, Neil Armstrong’s Apollo 11 landing on the moon; on the other hand I was lamenting the state of Indian science. A government report revealed that India spends only 0.88% of its GDP on scientific research against 8% for the US. And we aspire to become a superpower? Whom are we kidding? I am yet to see a superpower that spends so less on science. As I reflected over this, I had a revelation.

Not only is our science lacking money, but our money is also lacking science.”

The real issue is not that India spends very little on science. The real issue is that we lack the cultural drive to honor science in the first place. Our low spending on science is an outcome of this cultural deficiency.

How do nations honor their heroes? By featuring them on their money. Why? Because we all value and use money, the persons depicted on currency notes are individuals who have made extraordinary contributions to the nation. This allows the citizens to draw inspiration from their heroes. When I looked at the currency notes around the world, I found that every country that had a scientist was proud to display on their money the iconography of their scientific contribution which has led to advancement of the frontier of knowledge not only for their nation but for the entire humanity. Let me quote three examples:

• Sir Isaac Newton (1643 – 1727 AD) is considered the most brilliant person that ever was. He discovered the laws of motion, the universal law of gravity, the composition of light and invented calculus all before he was 26. His portrait was British 1 Pound.

•  Ibn al-Haitham (965 – 1040 AD) the father of modern optics, ophthalmology, experimental physics and scientific methodology and was first theoretical physicist of the world. His image was on the 10 and 10,000 Iraqi Dinari notes.

• Abu Nasr al-Farabi (872 – 950 AD) conducted the first experiments concerning the existence of vacuum. His image was on the 1 Kazakhstani Tenge.

What gets printed on the money is important because it conveys the nation’s message to its citizens and to the rest of the world. Which country do you think make the best engineers in the world? Germany. German engineering is a metaphor for engineering excellence. And how do the Germans convey this fact? They wrote a mathematical distribution function on their money next to image of Carl Friedrich Gauss, the greatest mathematician of all times.

This tradition of honoring scientists is followed not just by developed nations but also by nations far worse off than India. England and Germany are scientific powerhouses but Iraq and Kazakhstan are not. Yet they all had the cultural drive to honor their scientists, going back hundreds of years into history to find the most deserving scientists to grace their current money. Several nations have maintained this tradition. Einstein was on the Israeli currency; Tesla on the Serbian; and Marie Curie was on both the Polish and the French currency. Many countries have featured more than one scientist. Newton, Faraday, Darwin and Kelvin were on the British; and Galileo, Volta and Marconi were on the Italian currency.

What about India? On the one hand our Mangalyaan is on its way to Mars, on the other hand we are trailing the world in the cultural adoption of science. We do not lack scientists; Bhramagupta and Aryabhatta are no lesser scientists than  Al-Haithamor Al-Farabi; Ramanujan is no lesser a mathematician than Gauss. From Baudhayana in the 8th century BC to Chandrashekhar  in the 20th century AD, India has produced dozens of extraordinary scientists that any foreign nation would be proud to feature on their money. But not India; we don’t value our own jewels. Our culture is yet to reach a state of maturity where scientific progress is celebrated in everyday life.   Therefore we haven’t honored a single scientist on our currency even though we proclaim that we have been doing science since Vedic Ages.

After independence, Gandhi has monopolized our currency notes. What message does this convey? It says three things; we value the Gandhian principles, we were British colonial slaves and, nothing else is valuable to this nation. Yes, we value Gandhian principles but do we really need to show this it across all our currency denominations? Nations that have featured scientists on their currency have also featured their other greats including but not limited to rulers and politicians. People from diverse fields have contributed at the highest level of excellence to rise of India. Can we not spare a few denominations of our currency for them? Will it be disrespectful to Gandhi if one of our currency notes features C. V. Raman or Homi Jehangir Bhabha?

Now that we almost completing our 67th orbit around the sun as an independent nation, it is time we catch up with the rest of world on intellectual maturity. We should respect our past and have a vision for the future; then we can become a superpower. The way to do this is by embracing science in our culture. A scientist on our money would convey a healthy and heartwarming message to the nation because scientists point towards the future.

List of scientists in the currencies of countries

Abu Ali al-Hasan Ibn al-Haitham, 10 Iraqi Dinar (1982)

Abu Ali al-Hasan Ibn al-Haitham, 10000 Iraqi Dinar (2005)

Abu Nasr al-Farabi, 1 Kazakhstani Tenge (1993)

Adam Smith, 20 British Pounds (2007)

Adam Smith, 50 British Pounds, Clydesdale (2003)

Albert Einstein, 5 Israeli Lirot (1968)

Alessandro Volta, 10000 Italian Lire (1984)

Alexander von Humboldt, 5 East German Marks (1964)

Benjamin Franklin, 100 United States Dollars (1985)

Blaise Pascal, 500 French Francs (1977)

Carl Friedrich Gauss, 10 Deutsch Marks (1991)

Carl Linne (Linnaeus), 100 Swedish Kroner (2003)

Charles Darwin, 10 British Pounds (2005)

Christian Huygens, 25 Dutch Guilder (1955)

Christopher Polhem, 500 Swedish Kroner (2003)

Democritus of Abdera, 100 Greek Drachma (1967)

Erwin Schrödinger, 1000 Austrian Schilling (1983)

Fryderyk Franciszek Chopin, 5000 Polish Zloty (1982)

Galileo Galilei, 2000 Itialian Lire (1973)

George Stephenson, 5 British Pounds (1990)

Guglielmo Marconi, 2000 Italian Lire (1990)

Hans Christian Ørsted, 100 Danish Kroner (1970)

Issac Newton, 1 British Pound

Janez Vajkard Valvasor, 20 Slovenian Tolarjev (1992)

Johann Balthasar Neumann, 50 Deutsche Marks (1991)

Jovan Jovanovic Zmaj, 500000000000 Yugoslavian Dinar (1993)

Jurij Vega, 50 Slovenian Tolars (1992)

Kristian Birkeland, 200 Norwegian Kroner (1994)

Leonhard Euler, 10 Swiss Francs (1997)

Loius Pasteur, 5 French Francs (1966)

Lord Ernest Rutherford, 100 New Zealand Dollars (current)

Lord Kelvin, 100 British Pounds, Clydesdale (1996)

Marie Curie, 20000 old Polish Zloty (1989)

Marie and Pierre Curie, 500 French Francs (1998)

Marius Mercator, 1000 Belgian Francs (1965)

Michael Faraday, 20 British Pounds (1993)

Nicolaus Copernicus, 1000 old Polish Zloty (1965)

Nicolaus Copernicus, 1000 old Polish Zloty (1982)

Niels Bohr, 500 Danish Kroner (current)

Nikola Tesla, 100 Serbian Dinar (2003)

Nikola Tesla, 100 Yugoslavian Dinar (1994)

Nikola Tesla, 10000000000 Yugoslavian Dinar (1993)

Nikola Tesla, 5 Yugoslavian Dinar (1994)

Nikola Tesla, 5000000 Yugoslavian Dinar (1993)

Ole Rømer, 50 Danish Kroner (1970)

Oswaldo Cruz, 50 Brazilian Cruzados (1986-8?)

Pedro Nunes, 100 Portuguese Escudos (1957)

René Descarts, 100 French Francs (1942)

Ruggero Boscovich, 1 Croatian Dinar (1991)

Ruggero Boscovich, 10 Croatian Dinar (1991)

Ruggero Boscovich, 100000 Croatian October Dinar (1993)

Ruggero Boscovich, 5 Croatian Dinar (1991)

Ruggero Boscovich, 50000 Croatian October Dinar (1993)

Sejong the Great, 10000 South Korean Won (2007)

Sigmund Freud, 50 Austrian Schilling (1986)

Sir Isaac Newton, 1 British Pound (c. 1984)

Thomas Jefferson, 2 United States Dollars (1976)

Urbain Jean Joseph Le Verrier, 50 French Francs (1947)

Viktor Ambartsumian, 100 Armenian Dram (1998)

Voltaire (François-Marie Arouet), 10 French Francs (1964)

Yours scientifically,

Nilotpal Kanti Sinha,

Citizen, India.

This article was published in Abraxas Lifestyle magazine: http://www.abraxaslifestyle.com and http://www.abraxasnu.com

## The analytics of social compatibility

India is the land of arranged marriages and the protagonist story had recently met five prospective brides who were equally eligible and evenly matched. All the girls have at least one unique attribute which he wanted his spouse to posses and each of these attribute was equally important to him; one of them was very beautiful; another was highly educated; another had a good sense of humour and so on. It was difficult for him to choose one girl over another. The protagonist met a data scientist, to discuss his problem of choices.

Data Scientist: This is similar to social choice theory, a framework for weighting individual interests, values, or welfares as an aggregate towards collective decision using symbolic logic. Let’s make an algorithm to evaluate the social compatibility between people. Then we will use it to find your best match form your prospective brides.

Protagonist: Why bother about evaluating social compatibility?

DS: Do you realize its immense business potential? The top two Indian matrimonial sites draw 2 million visitors accessing over 15 million pages daily. If these two websites implement the algorithm then assuming that only 5% of the visitors actually use it, you have a 100,000 daily user in India alone. In the future, if the top matrimonial and dating websites across world implement the algorithm then you can hit half a million daily users. On a pay per use or fixed month rate revenue model, look at the expected income.

P: So how can we quantify compatibility?

DS: A simple approach is to rank the girls in order of each attribute and then combine the individual ranks into a composite rank using the known methods of combining ranks. The top composite ranked girl is your first choice.

DS: Unfortunately a ranking based approach is conceptually flawed. Economics Nobel Laureate Dr. Kenneth Arrow proved the Arrow’s Impossibility Theorem, a pioneering theorem of social choice theory, which states that no rank-order voting system satisfies all fairness criterions. Moreover for critical social decisions psychology could prevail over statistics. The ideal methodology should be able to quantify the physiological aspect of human behaviour.

DS: Ideally you would want all the desired attributes of a dream spouse in one person. But in reality, the desired attributes will be distributed across different girls. So you have to give up on one attribute to gain on another. Thus the attributes are competing against each other so you have to make competitive choices.

DS: Assume that you have a total of twenty points to allocate across the attributes. How much are you willing to give up on the beauty to gain on the educational qualification of your spouse? If you allocate 15 points to beauty, you have only 5 points to allocate to education. When faced with scarce resources (points) you will be much more judicious in spending. Hence competitive choices are a better quantification of your actual psychological preference.

P: And how do we quantify competitive choices?

DS: By using conjoint analysis. It is based on mathematical psychology and is widely used in psychophysics, perception, decision-making and the quantitative analysis of behaviour. I will create a social compatibility algorithm and use your data to see what your actual psychological preferences; then we will find your most suitable match.

P: Really? You can build such a algorithm?

DS: Rest assured; I have learned conjoint analysis from one of the pioneers of the subject, Dr. V. Srinivasan. Give me two days.

(Two days later)

DS: The social compatibility algorithm is ready; and based on your competitive choices, it suggests that your most suitable match is the second girl. Hmmm … she is a teacher but you didn’t tell me what she teaches?

P: Well, she teaches statistics in a college.

DS: Statistics! I knew the algorithm was right.

Claimer: Based on a true incident. Both the protagonist and the data scientist work in the analytics industry. The protagonist and the lady statistician are now seeing each other.

## On power free numbers

Entry 1. Let $f(r)$ be any divergent series of positive terms, $q_{r,k}$ be the $r^{th}$ k-power free number, $\zeta(k)$ be the Riemann Zeta function. Let $S_f(r) = \sum_{i=1}^{r} f(i)$. If  $g(x)$ is Riemann integrable in $(0, \infty)$ then,

$\displaystyle{ \sum_{r=1}^{n} f(q_{r,k}) g \Big(\frac{S_f(r)}{\zeta(k)}\Big) \sim \int_{f(1)}^{\frac{S_f(n)}{\zeta(k)}} g(x)dx.}$

Corollary 1. As $k \rightarrow \infty, \zeta(k) \rightarrow 1$. Also every natural number is k-power free when $k \rightarrow \infty$. Hence the above result reduces to

$\displaystyle{ \sum_{r=1}^{n} f(r) g(S_f(r))\sim \int_{f(1)}^{S_f(n)} g(x)dx}$.

Entry 2. Further let $q'_{k,n}$ be the $n^{th}$ k-power containing number and $f$ be any function Riemann integrable in $(1,\infty)$; then,

$\displaystyle{ \sum_{k=2}^{\infty}\frac{1}{k} \Big\{\frac{f(q'_{k,1}) + f(q'_{k,2}) + f(q'_{k,3}) +\ldots}{f(q_{k,1}) + f(a_{k,2}) + f(q_{k,3}) +\ldots}\Big\}= 1 - \gamma}$.

## General theory of regression

The first documented work on regression 1 2 was published in the year 1898 by Schuster and since then, several regression models have been proposed. Regression is a active area of research because of the wide spread use of regression analysis in scientific, statistical, industrial and commercial applications. Ideally we would want a regression method which gives a perfect between the regression curve and the actual values of the data points. However the existing regression methods suffer form two major drawbacks:

(i) A regression method may be suitable for one type of data and unsuitable for another type of data. For example ordinary linear square is suitable for linear data but data in the real world are not necessarily linear therefore linear regression would not a good choice if the data is not roughly linear.

(ii) Unless there is a perfect match between the actual values of the data and the values given by the regression model, there will always be an error.

We present a new method of regression which works for all type of data; linear, polynomial, logarithmic or erratic random data that shows no particular trend. Our method is an iterative process based the application of sinusoidal series analysis in non linear least square. Each iteration of this method reduces the sum of the square of the residuals and therefore by successive iteration of this method, we show that every finite set of co-planar points can be expanded as a sinusoidal series in in_nitely many ways. In other words, given a set of co-planar points, we can fit infinitely many curves that pass through all these points. By setting a convergence criteria in terms of acceptable an error we can stop the iteration after a finite number of steps. Thus in the limiting case, we obtain a function that gives a perfect fit for the data points.

The regression method is published in ArXiv: Click here

## On the half line: K. Ramachandra

Kanakanahalli Ramachandra (1933-2011) was perhaps the real successor of Srinivas Ramanujan in contemporary Indian mathematics. There would be no exaggeration in saying that without the efforts of Ramachandra, analytical number theory could have been extinct in India back in the mid 1970s. I was fortunate to get the opportunity to learn mathematics from the master himself. ‘On the half line: K. Ramachandra‘ is a short biography on the life and works of K. Ramachandra.

The complete article in PDF format is published in Vol 21, September 2011 issue of the Mathematics Newsletter, Ramanujan Mathematical Society. Click here to read the full article: MNL-Sep11-CRC

Some photographs of K. Ramachandra

## A conjecture on consecutive primes – II

Continuing with the previous post in this topic, a stronger form of conjecture on the lower bound of Corollary 1 is as follows

ConjectureLet $p_n$ be the n-th prime. The for $n \ge 32$,

$\displaystyle{p_n^{\frac{1}{n}} > \Big(1+\frac{1}{n^2}\Big) p_{n+1}^{\frac{1}{n+1}} }$.

The above conjecture implies that for all sufficiently large $n$,

$p_{n+1} - p_n < (\ln p_n - 1)(\ln p_n - \ln\ln n)$.

Prof. Marek Wolf, Institute of Theoretical Physics, Wroclaw, Poland has verified the above conjecture for primes up to $2^{44}$.

## Some series on Fibonacci and Lucas numbers

In this post, we shall see several curious summation formulas of the Fibonacci numbers $F_n$ and the related Lucas numbers $L_n$ that involves the Bell polynomials $B_n(x)$ and the incomplete gamma function $\Gamma(a,x)$. The Fibonacci numbers are defined as $F_0=0, F_1=1, F_{n+2}=F_{n+1} + F_n, n \ge 0$ and the Lucas numbers are defined as $L_0=2, L_1=1, L_{n+2}=L_{n+1} + L_n, n \ge 0$.

Bell polynomials

The Bell polynomials $B_n(x)$ of polynomials are defined by the exponential generating function

$\displaystyle{\sum_{n=0}^{\infty}\frac{B_n(x)t^n}{n!}=e^{(e^t-1)x}}$.

The first few Bell polynomials are

$B_0(x) = 1$

$B_1(x) = x$

$B_2(x) = x^2 + x$

$B_3(x) = x^3 + 3x^2 + x$

$B_4(x) = x^4 + 6 x^3 + 7x^2 + x$

$B_5(x) = x^5 +10x^4 + 25 x^3 + 15x^2 + x$.

Incomplete gamma function

The incomplete gamma function $\Gamma(a,x)$ is defined as

$\displaystyle{\Gamma(a,x) = \int_{x}^{\infty}t^{a-1}e^{-t}dt}$.

Entry 1-6 involve summation identities involving the  Bell polynomials or the incomplete gamma function.

As usual, $\displaystyle{\phi=\frac{1+\sqrt5}{2}}$ denotes golden ratio. We have the following results:

Entry 1.

$\displaystyle{\sum_{r=1}^{\infty}\frac{F_r r^n}{r!} = \frac{e^{-1/\phi}}{\sqrt5}\{e^{\sqrt5} B_n(\phi)-B_n(-1/\phi)\}}$.

Entry 2.

$\displaystyle{\sum_{r=1}^{\infty}\frac{L_r r^n}{r!} = e^{-1/\phi}\{e^{\sqrt5} B_n(\phi)+B_n(-1/\phi)\}}$.

Entry 3.

$\displaystyle{\sum_{k=0}^{n-1}\frac{F_{k+m}x^k}{k!} = \frac{(-1)^me^{-x/\phi}}{\phi^m\sqrt5(n-1)!}\{(-\phi^2)^m e^{\sqrt5x}\Gamma(n,\phi x) - \Gamma(n,-x/\phi)\}}$

Entry 4.

$\displaystyle{\sum_{k=0}^{n-1}\frac{F_{k+m}(-x)^k}{k!} = \frac{(-1)^m e^{-x\phi}}{\phi^m\sqrt5(n-1)!}\{(-\phi^2)^m \Gamma(n,-\phi x) - e^{\sqrt5x}\Gamma(n,x/\phi)\}}$

Entry 5.

$\displaystyle{\sum_{r=0}^{\infty}\frac{F_{r+m} x^r}{r!} = \frac{e^{-x/\phi}}{\sqrt5}\{\phi^m e^{\sqrt5x}-(-\phi)^{-m}\}}$

Entry 6.

$\displaystyle{\sum_{r=0}^{\infty}\frac{F_{r+m} (-x)^r}{r!} = \frac{e^{-x\phi}}{\sqrt5}\{\phi^m -e^{\sqrt5x}(-\phi)^{-m}\}}$

Entry 7.

$\displaystyle{\sum_{r=0}^{\infty}\frac{F_r x^r}{r!} = -e^x \sum_{r=1}^{\infty}\frac{F_r (-x)^r}{r!}}$

Entry 8.

$\displaystyle{\sum_{k=0}^{n-1}\frac{L_{k+m}x^k}{k!} = \frac{(-1)^m e^{-x/\phi}}{\phi^m (n-1)!}\{(-\phi^2)^m e^{\sqrt5x}\Gamma(n,\phi x) + \Gamma(n,-x/\phi)\}}$

Entry 9.

$\displaystyle{\sum_{k=0}^{n-1}\frac{L_{k+m}(-x)^k}{k!} = \frac{(-1)^m e^{-x\phi}}{\phi^m (n-1)!}\{(-\phi^2)^m \Gamma(n,-\phi x) + e^{\sqrt5x}\Gamma(n,x\phi)\}}$

Entry 10.

$\displaystyle{\sum_{r=0}^{\infty}\frac{L_{r+m} x^r}{r!} = e^{-x/\phi}\{\phi^m e^{\sqrt5x}+(-\phi)^{-m}\}}$

Entry 11.

$\displaystyle{\sum_{r=0}^{\infty}\frac{L_{r+m} (-x)^r}{r!} = e^{-x\phi}\{\phi^m + e^{\sqrt5x}(-\phi)^{-m}\}}$

Entry 12.

$\displaystyle{\sum_{r=0}^{\infty}\frac{L_r x^r}{r!} = e^x \sum_{r=1}^{\infty}\frac{L_r (-x)^r}{r!}}$