Dattaraya Ramchandra Kaprekar (1905-01-17 - 1986) was an Indian mathematician who discovered many interesting properties in number theory. He was born in the town Devlali, Maharashtra. Having never received any formal postgraduate training, for his entire career (1930-1962) he was a schoolteacher in the small town of Devlali in Maharashtra, India. Yet he became well known in recreational mathematics circles, and has a number, a constant, and a magic square named after him.
Kaprekar received his secondary school education in Thana and studied at Fergusson College in Pune. He attended the University of Bombay, receiving his bachelor"s degree in 1929. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties.
Working largely alone, Kaprekar discovered a number of results that have opened up important avenues of research in number theory. In addition to the Kaprekar constant and the Kaprekar number which were named after him, he also discovered the Self number or Devlali number, and also the important series called the Harshad number. He also constructed certain types of magic squares related to the Copernicus magic square.
Kaprekar"s famous works are below:
Kaprekar constant
The Kaprekar constant, named after him, is a fascinating general property of all number bases and may demonstrate some important but unknown theorem in number theory.The Kaprekar constant, or 6174 (1949). He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from four digits (not all the same). This series converges to 6174 in fewer than seven iterations.
Ex: lets take a non-zero 4 digit number 5, 5, 9, 2,
9552-2559=6993, 9963-3699=6264, 6642-2466=4176, 7641-1467=6174, 7641-1467=6174...this goes on.
Similarly the Kaprekar constant for 3 digits is 495.
Ex: lets take a non-zero 3 digit number 5, 5, 9,
955-559=396, 963-369=594, 954-459=495, 954-495=495...this goes on.
Kaprekar number
The Kaprekar number (also called Kaprekar series, based on the Kaprekar operation). This is a number with the interesting property that if it is squared, then two equal parts of this square also add up to the original number. This operation, of taking the last n digits of a square, and adding it to the number formed by the first (n-1) or n digits, is the Kaprekar operation.
Ex: an example 297
297^2 = 88,209 and 88 + 209 = 297!!!
So, just square the number, split it in half (leave the larger portion to the right) and add the two halves - if you get the original number back, then its a Kaprekar Number.
A few more examples of Kaprekar Numbers are - 9, 45, 297, 4879, 17344, 538461, ...
Devlali or Self number
In 1963, he also defined the property which has come to be known as self numbers, which are integers that cannot be generated by taking some other number and adding its own digits to it. For example, 21 is not a self number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer. He also gave a test for verifying this property in any number. These are sometimes referred to as Devlali numbers (after the town where he lived, Devlali,Maharashtra); though this appears to have been his preferred designation, the term self-number is more widespread. Sometimes these are also designated Colombian numbers after a later designation.
Harshad Number
He also discovered the Harshad numbers which he named harshad, meaning "giving joy" (Sanskrit harsha, joy +da taddhita pratyaya, causative); these have the property that they are divisible by the sum of their digits. Thus 12, which is divisible by 1 + 2 = 3, is a Harshad number. These were later also called Niven numbers after a 1997 lecture on these by the Canadian mathematician Ivan M. Niven. Numbers which are Harshad in all bases (only 1, 2, 4, and 6) are called all-Harshad numbers. Much work has been done on Harshad numbers, and their distribution, frequency, etc. are a matter of considerable interest in number theory today.
In 1927 he won the Wrangler R. P. Paranjpe Mathematical Prize for an original piece of work in mathematics.
Although he was largely unknown outside recreational mathematics circles in his lifetime, D. R. Kaprekar"s work and its impact on number theory has become widely recognized in recent years.
courtesy:internet
Monday, December 17, 2007
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar (19 Oct 1910 - 21 Aug 1995) was known throughout his life as Chandra. Chandra wanted to be a scientist and his mother encouraged him to follow this route. He had a role model in his paternal uncle Sir Chandrasekhara Venkata Raman who went on to win the Nobel prize in 1930 for his 1928 discovery of Raman scattering and the Raman effect, which is a change in the wavelength of light occurring when a beam of light is deflected by molecules.
Chandra studied at Presidency College, University of Madras, and he wrote his first research paper while still an undergraduate there. The paper was published in the Proceeding of the Royal Society where it had been submitted by Ralph Fowler.
Chandra obtained a scholarship from the Indian government to finance his studies in England, and in 1930 he left India to study at Trinity College, Cambridge, England. From 1933 to 1937 he undertook research at Cambridge, but he returned to India in 1936 to marry Lalitha on 11 September.
At first he worked in Yerkes Observatory, part of the University of Chicago in Wisconsin. Later he moved to work on the university campus in the city of Chicago. During World War II he worked in the Ballistic Research Laboratories at the Aberdeen Proving Ground in Maryland.
Two reports, written in 1943, show the type of problems he was working on at this time: the first is On the decay of plane shock waves while the second is The normal reflection of a blast wave. He was honoured with being appointed Morton D Hull distinguished service professor of the University of Chicago in 1952.
Chandrasekhar published around 400 papers and many books.
His research interests were exceptionally broad but we can divide them into topics and rough periods when he was concentrating on these particular topics.
- stellar structure, including the theory of white dwarfs, from 1929 to 1939.
- stellar dynamics from 1939 to 1943.
- the theory of radiative transfer and the quantum theory of the negative ion of hydrogen from 1943 to 1950.
- Hydrodynamic and hydromagnetic stability from 1950 to 1961.
- During most of the 1960s he studied the equilibrium and the stability of ellipsoidal figures of equilibrium but during this period he also began work on topics from general relativity, the radiation reaction process, and the stability of relativistic stars.
- During the period from 1971 to 1983 he undertook research into the mathematical theory of black holes, then for the last period of his life he worked on the theory of colliding gravitational waves.
In 1930 Chandra showed that a star of a mass greater than 1.4 times that of the Sun (now known as the Chandrasekhar's limit) had to end its life by collapsing into an object of enormous density unlike any object known at that time.
Many years later Chandra was awarded the Nobel prize for Physics in 1983 for his theoretical studies of the physical processes of importance to the structure and evolution of the stars.
He found the Mathematical Theory of Black Holes (1983).
His other books include
- An Introduction to the Study of Stellar Structure (1939)
- Principles of Stellar Dynamics (1942)
- Radiative Transfer (1950)
- Plasma Physics (1960)
- Hydrodynamic and Hydromagnetic Stability (1961)
- Ellipsoidal Figures of Equilibrium (1969)
- Truth and Beauty: Aesthetics and Motivations in Science (1987)
- Newton's Principia for the Common Reader (1995).
These texts have played a major role in mathematical astronomy.
From 1952 until 1971 Chandrasekhar was editor of the Astrophysical Journal . This journal was originally a local University of Chicago publication, but it grew in stature to become national publication of the American Astronomical Society, then a leading international journal.
Chandrasekhar received many honours for his outstanding contributions some of which, such as
- the Nobel prize for Physics in 1983
- the Royal Society's Royal Medal of 1962
- the Royal Society's Copley Medal of 1984
- the Bruce medal of the Astronomical Society of the Pacific
- the Henry Draper medal of the National Academy of Sciences (United States)
- the Gold Medal of the Royal Astronomical Society.
Chandra retired in 1980 but continued to live in Chicago where he was made professor emeritus in 1985. He continued to give thought-provoking lectures such as Newton and Michelangelo which he delivered at the 1994 Meeting of Nobel Laureates held in Lindau. Other lectures in a similar vein include Shakespeare, Newton and Beethoven or patterns of creativity and The perception of beauty and the pursuit of science.
Courtesy:Internet
Chandra studied at Presidency College, University of Madras, and he wrote his first research paper while still an undergraduate there. The paper was published in the Proceeding of the Royal Society where it had been submitted by Ralph Fowler.
Chandra obtained a scholarship from the Indian government to finance his studies in England, and in 1930 he left India to study at Trinity College, Cambridge, England. From 1933 to 1937 he undertook research at Cambridge, but he returned to India in 1936 to marry Lalitha on 11 September.
At first he worked in Yerkes Observatory, part of the University of Chicago in Wisconsin. Later he moved to work on the university campus in the city of Chicago. During World War II he worked in the Ballistic Research Laboratories at the Aberdeen Proving Ground in Maryland.
Two reports, written in 1943, show the type of problems he was working on at this time: the first is On the decay of plane shock waves while the second is The normal reflection of a blast wave. He was honoured with being appointed Morton D Hull distinguished service professor of the University of Chicago in 1952.
Chandrasekhar published around 400 papers and many books.
His research interests were exceptionally broad but we can divide them into topics and rough periods when he was concentrating on these particular topics.
- stellar structure, including the theory of white dwarfs, from 1929 to 1939.
- stellar dynamics from 1939 to 1943.
- the theory of radiative transfer and the quantum theory of the negative ion of hydrogen from 1943 to 1950.
- Hydrodynamic and hydromagnetic stability from 1950 to 1961.
- During most of the 1960s he studied the equilibrium and the stability of ellipsoidal figures of equilibrium but during this period he also began work on topics from general relativity, the radiation reaction process, and the stability of relativistic stars.
- During the period from 1971 to 1983 he undertook research into the mathematical theory of black holes, then for the last period of his life he worked on the theory of colliding gravitational waves.
In 1930 Chandra showed that a star of a mass greater than 1.4 times that of the Sun (now known as the Chandrasekhar's limit) had to end its life by collapsing into an object of enormous density unlike any object known at that time.
Many years later Chandra was awarded the Nobel prize for Physics in 1983 for his theoretical studies of the physical processes of importance to the structure and evolution of the stars.
He found the Mathematical Theory of Black Holes (1983).
His other books include
- An Introduction to the Study of Stellar Structure (1939)
- Principles of Stellar Dynamics (1942)
- Radiative Transfer (1950)
- Plasma Physics (1960)
- Hydrodynamic and Hydromagnetic Stability (1961)
- Ellipsoidal Figures of Equilibrium (1969)
- Truth and Beauty: Aesthetics and Motivations in Science (1987)
- Newton's Principia for the Common Reader (1995).
These texts have played a major role in mathematical astronomy.
From 1952 until 1971 Chandrasekhar was editor of the Astrophysical Journal . This journal was originally a local University of Chicago publication, but it grew in stature to become national publication of the American Astronomical Society, then a leading international journal.
Chandrasekhar received many honours for his outstanding contributions some of which, such as
- the Nobel prize for Physics in 1983
- the Royal Society's Royal Medal of 1962
- the Royal Society's Copley Medal of 1984
- the Bruce medal of the Astronomical Society of the Pacific
- the Henry Draper medal of the National Academy of Sciences (United States)
- the Gold Medal of the Royal Astronomical Society.
Chandra retired in 1980 but continued to live in Chicago where he was made professor emeritus in 1985. He continued to give thought-provoking lectures such as Newton and Michelangelo which he delivered at the 1994 Meeting of Nobel Laureates held in Lindau. Other lectures in a similar vein include Shakespeare, Newton and Beethoven or patterns of creativity and The perception of beauty and the pursuit of science.
Courtesy:Internet
Sunday, December 16, 2007
Srinivasa Aiyangar Ramanujan
Srinivasa Aiyangar Ramanujan (22 Dec 1887 - 26 April 1920) was one of India's greatest mathematical geniuses. He was born in Erode, Tamil Nadu state, India. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.
In 1900 he began to work on his own on mathematics summing geometric and arithmetic series. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.
By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions.
By 1908 Ramanujan studied fractions and divergent series.
In 1910 Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations.
In 1911 after publication of a brilliant research paper on Bernoulli numbers in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.
On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920). He had been allowed to enrol in June 1914 despite not having the proper qualifications. Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England.
On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London.
Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function.
Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Other were only proved after Ramanujan's death.
Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death.
Ramanujan's formulae have found applications in the field of crystallography and in string theory.
During the year 1987 (Ramanujan's centennial), the printed form of Ramanujan's Lost Notebook by the Narosa publishing house of Springer-Verlag was released by the late Indian prime minister, Rajiv Gandhi, who presented the first copy to S. Janaki Ammal Ramanujan (Ramanujan's late widow) and the second copy to George Andrews in recognition of his contributions in the field of number theory.
A stamp was issued by the Indian Post Office to celebrate the 75th anniversary of his birth.
A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with the International Mathematical Union, who nominate members of the prize committee.
Courtesy:Internet
In 1900 he began to work on his own on mathematics summing geometric and arithmetic series. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.
By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions.
By 1908 Ramanujan studied fractions and divergent series.
In 1910 Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations.
In 1911 after publication of a brilliant research paper on Bernoulli numbers in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.
On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920). He had been allowed to enrol in June 1914 despite not having the proper qualifications. Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England.
On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London.
Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function.
Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Other were only proved after Ramanujan's death.
Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death.
Ramanujan's formulae have found applications in the field of crystallography and in string theory.
During the year 1987 (Ramanujan's centennial), the printed form of Ramanujan's Lost Notebook by the Narosa publishing house of Springer-Verlag was released by the late Indian prime minister, Rajiv Gandhi, who presented the first copy to S. Janaki Ammal Ramanujan (Ramanujan's late widow) and the second copy to George Andrews in recognition of his contributions in the field of number theory.
A stamp was issued by the Indian Post Office to celebrate the 75th anniversary of his birth.
A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with the International Mathematical Union, who nominate members of the prize committee.
Courtesy:Internet
Vijayanandi
Vijayanandi (940-1010) was an Indian mathematician and astronomer whose most famous work was the Karanatilaka. We should note that there was another astronomer named Vijayanandi who was mentioned by Varahamihira in one of his works. Since Varahamihira wrote around 550 and the Karanatilaka was written around 966, there must be two astronomers both named "Vijayanandi".
The Karanatilaka has not survived in its original form but we know of the text through an Arabic translation by al-Biruni. It is a work in fourteen chapters covering the standard topics of Indian astronomy.
It deals with the topics of:
- units of time measurement;
- mean and true longitudes of the sun and moon;
- the length of daylight;
- mean longitudes of the five planets;
- true longitudes of the five planets;
- the three problems of diurnal rotation;
- lunar eclipses, solar eclipses;
- the projection of eclipses;
- first visibility of the planets;
- conjunctions of the planets with each other and with fixed stars;
- the moon's crescent;
- the patas of the moon and sun.
Like other Indian astronomers, Vijayanandi made contributions to trigonometry and it appears that his calculation of the periods was computed by using tables of sines and versed sines. It is significant that accuracy was need in trigonometric tables to give accurate astronomical theories and this motivated many of the Indian mathematicians to produce more accurate methods of approximating entries in tables.
Courtesy:Internet
The Karanatilaka has not survived in its original form but we know of the text through an Arabic translation by al-Biruni. It is a work in fourteen chapters covering the standard topics of Indian astronomy.
It deals with the topics of:
- units of time measurement;
- mean and true longitudes of the sun and moon;
- the length of daylight;
- mean longitudes of the five planets;
- true longitudes of the five planets;
- the three problems of diurnal rotation;
- lunar eclipses, solar eclipses;
- the projection of eclipses;
- first visibility of the planets;
- conjunctions of the planets with each other and with fixed stars;
- the moon's crescent;
- the patas of the moon and sun.
Like other Indian astronomers, Vijayanandi made contributions to trigonometry and it appears that his calculation of the periods was computed by using tables of sines and versed sines. It is significant that accuracy was need in trigonometric tables to give accurate astronomical theories and this motivated many of the Indian mathematicians to produce more accurate methods of approximating entries in tables.
Courtesy:Internet
Varahamihira
Varahamihira (505-587) was born in Kapitthaka, India. The most famous work by Varahamihira is the Pancasiddhantika (The Five Astronomical Canons) dated 575 AD. This work is important in itself and also in giving us information about older Indian texts which are now lost. The work is a treatise on mathematical astronomy and it summarises five earlier astronomical treatises, namely the Surya, Romaka, Paulisa, Vasistha and Paitamaha siddhantas.
One treatise which Varahamihira summarises was the Romaka-Siddhanta which itself was based on the epicycle theory of the motions of the Sun and the Moon given by the Greeks in the 1st century AD.
Varahamihira made some important mathematical discoveries. Among these are certain trigonometric formulae which translated into our present day notation correspond to
sin x = cos(ð/2 - x),
sin2x + cos2x = 1, and
(1 - cos 2x)/2 = sin2x.
Another important contribution to trigonometry was his sine tables where he improved those of Aryabhata I giving more accurate values. This motivated much of the improved accuracy they achieved by developing new interpolation methods.
The Jaina school of mathematics investigated rules for computing the number of ways in which r objects can be selected from n objects over the course of many hundreds of years. They gave rules to compute the binomial coefficients nCr which amount to
nCr = n(n-1)(n-2)...(n-r+1)/r!
However, Varahamihira attacked the problem of computing nCr in a rather different way. He wrote the numbers n in a column with n = 1 at the bottom. He then put the numbers r in rows with r = 1 at the left-hand side. Starting at the bottom left side of the array which corresponds to the values n = 1, r = 1, the values of nCr are found by summing two entries, namely the one directly below the (n, r) position and the one immediately to the left of it. Of course this table is none other than Pascal's triangle for finding the binomial coefficients despite being viewed from a different angle from the way we build it up today.
Varahamihira's work on magic squares. In particular he examines a pandiagonal magic square of order four which occurs in Varahamihira's work.
Courtesy:Internet
One treatise which Varahamihira summarises was the Romaka-Siddhanta which itself was based on the epicycle theory of the motions of the Sun and the Moon given by the Greeks in the 1st century AD.
Varahamihira made some important mathematical discoveries. Among these are certain trigonometric formulae which translated into our present day notation correspond to
sin x = cos(ð/2 - x),
sin2x + cos2x = 1, and
(1 - cos 2x)/2 = sin2x.
Another important contribution to trigonometry was his sine tables where he improved those of Aryabhata I giving more accurate values. This motivated much of the improved accuracy they achieved by developing new interpolation methods.
The Jaina school of mathematics investigated rules for computing the number of ways in which r objects can be selected from n objects over the course of many hundreds of years. They gave rules to compute the binomial coefficients nCr which amount to
nCr = n(n-1)(n-2)...(n-r+1)/r!
However, Varahamihira attacked the problem of computing nCr in a rather different way. He wrote the numbers n in a column with n = 1 at the bottom. He then put the numbers r in rows with r = 1 at the left-hand side. Starting at the bottom left side of the array which corresponds to the values n = 1, r = 1, the values of nCr are found by summing two entries, namely the one directly below the (n, r) position and the one immediately to the left of it. Of course this table is none other than Pascal's triangle for finding the binomial coefficients despite being viewed from a different angle from the way we build it up today.
Varahamihira's work on magic squares. In particular he examines a pandiagonal magic square of order four which occurs in Varahamihira's work.
Courtesy:Internet
Vijay Kumar Patodi
Vijay Kumar Patodi (12 March 1945-21 Dec 1976) born in Guna, India. Mathematical fame for Patodi came early in his career with papers of great importance coming for the work of his Ph.D. His doctoral thesis, Heat equation and the index of elliptic operators, was supervised by M S Narasimhan and S Ramanan and the degree was awarded by the University of Bombay in 1971.
Patodi's first paper Curvature and the eigenforms of the Laplace operator was part of his thesis. The second paper which came from his thesis was An analytic proof of the Riemann- Roch- Hirzebruch theorem for Kaehler manifolds which extended the methods of his first paper to a much more complicated situation.
The years 1971 to 1973 were ones which Patodi spent on leave at the Institute for Advanced Study at Princeton. There he worked with M F Atiyah and made several visits to work with others in his field at various centres in the United States and England. During this time he also collaborated with R Bott and I M Singer.
On his return to Bombay and the Tata Institute in 1973 Patodi was promoted to associate professor. He was promoted to full professor in 1976.
Patodi's publications, in addition to the two mentioned above, include a number of joint ones with Atiyah and Singer. These papers introduce a spectral invariant of a compact Riemannian manifold. In another paper he studies the relationship between Riemannian structures and triangulations. Other work gives a combinatorial formula for Pontryagin classes.
Courtesy:Internet
Patodi's first paper Curvature and the eigenforms of the Laplace operator was part of his thesis. The second paper which came from his thesis was An analytic proof of the Riemann- Roch- Hirzebruch theorem for Kaehler manifolds which extended the methods of his first paper to a much more complicated situation.
The years 1971 to 1973 were ones which Patodi spent on leave at the Institute for Advanced Study at Princeton. There he worked with M F Atiyah and made several visits to work with others in his field at various centres in the United States and England. During this time he also collaborated with R Bott and I M Singer.
On his return to Bombay and the Tata Institute in 1973 Patodi was promoted to associate professor. He was promoted to full professor in 1976.
Patodi's publications, in addition to the two mentioned above, include a number of joint ones with Atiyah and Singer. These papers introduce a spectral invariant of a compact Riemannian manifold. In another paper he studies the relationship between Riemannian structures and triangulations. Other work gives a combinatorial formula for Pontryagin classes.
Courtesy:Internet
Sripati
Sripati (1019-1066) was born in Rohinikhanda, Maharashtra, India. Sripati was a follower of the teaching of Lalla writing on astrology, astronomy and mathematics. His mathematical work was undertaken with applications to astronomy in mind, for example a study of spheres. His work on astronomy was undertaken to provide a basis for his astrology. Sripati was the most prominent Indian mathematicians of the 11th Century.
Sripati's works are: Dhikotidakarana written in 1039, a work of twenty verses on solar and lunar eclipses; Dhruvamanasa written in 1056, a work of 105 verses on calculating planetary longitudes, eclipses and planetary transits; Siddhantasekhara a major work on astronomy in 19 chapters; and Ganitatilaka an incomplete arithmetical treatise in 125 verses based on a work by Sridhara.
Other mathematics included in Sripati's work includes, in particular, rules for the solution of simultaneous indeterminate equations of the first degree that are similar to those given by Brahmagupta.
He wrote the Jyotisaratnamala which was an astrology text in twenty chapters based on the Jyotisaratnakosa of Lalla. Sripati wrote a commentary on this work in Marathi and it is one of the oldest works to have survived that is written in that language. Another work on astrology written by Sripati is the Jatakapaddhati or Sripatipaddhati which is in eight chapters.
Genethlialogy was the science of casting nativities and it was the earliest branch of astrology which claimed to be able to predict the course of a person's life based on the positions of the planets and of the signs of the zodiac at the moment the person was born or conceived.
There is one other work on astrology the Daivajnavallabha which some historians claim was written by Sripati while other claim that it is the work of Varahamihira. As yet nobody has come up with a definite case to show which of these two is the author, or even whether the author is another astrologer.
Courtesy:Internet
Sripati's works are: Dhikotidakarana written in 1039, a work of twenty verses on solar and lunar eclipses; Dhruvamanasa written in 1056, a work of 105 verses on calculating planetary longitudes, eclipses and planetary transits; Siddhantasekhara a major work on astronomy in 19 chapters; and Ganitatilaka an incomplete arithmetical treatise in 125 verses based on a work by Sridhara.
Other mathematics included in Sripati's work includes, in particular, rules for the solution of simultaneous indeterminate equations of the first degree that are similar to those given by Brahmagupta.
He wrote the Jyotisaratnamala which was an astrology text in twenty chapters based on the Jyotisaratnakosa of Lalla. Sripati wrote a commentary on this work in Marathi and it is one of the oldest works to have survived that is written in that language. Another work on astrology written by Sripati is the Jatakapaddhati or Sripatipaddhati which is in eight chapters.
Genethlialogy was the science of casting nativities and it was the earliest branch of astrology which claimed to be able to predict the course of a person's life based on the positions of the planets and of the signs of the zodiac at the moment the person was born or conceived.
There is one other work on astrology the Daivajnavallabha which some historians claim was written by Sripati while other claim that it is the work of Varahamihira. As yet nobody has come up with a definite case to show which of these two is the author, or even whether the author is another astrologer.
Courtesy:Internet
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