Re: BR Maths Corner-1
Posted: 19 Aug 2010 09:13
Here is some serious discussion going on about P Vs NP (Terrence Tao et al.)
Consortium of Indian Defence Websites
Hyderabad, Aug. 20: Centuries before modern European mathematicians formalised the concept of infinity, ancient Indians dabbled in numbers so gigantic that their motivations have left even 21st century mathematicians baffled.
A series of Indian texts dated from about 1000 BC to the 2nd century BC described mystifyingly large numbers, some with no links to reality, delegates at the International Congress of Mathematicians 2010 under way have said.
The Yajur Veda, which some scholars believe was composed around 1400 BC, has a line that invokes successive powers of 10 up to a trillion, Plofker said in a special paper circulated at the congress.
“Such vast numbers could have had no practical application in the context of Vedic culture,” she wrote.
The allure of large numbers is also visible in the epic Mahabharata that describes a cosmic time span over four billion years long, constructed out of 360-year periods, said Plofker, who told The Telegraph she was drawn to ancient Indian mathematics after opting for a Sanskrit course because it had a script unfamiliar to her.
One Buddhist text from the early first millennium refers to what Plofker has described in her paper as “sequences of numbers that could be conceived only through mind-boggling comparisons” such as the number of grains of sand in the beds of a hundred million rivers.
Another Jain text from the early first millennium refers to a time period of two raised to the power of 588.
Some scientists believe observations of nature may have driven the abstract thinking that led to large numbers.
Amber G. wrote:^^^Thx. As said before in this thread, Highly recommend Kim Plofker's book, there was a post by Svenkat.
(I recommended it too) The book is:
Mathematics in India ...
As said before, David Mumford, (winner of the Fields Medal),has written a six-page essay on the book, which captures, concisely, the major achievements of Indian mathematics. There is the use of Pythagoras’ famous theorem, Pāṇini’s rules of Sanskrit grammar and recursion, Pingala, whose study of Sanskrit verses led to the binary notation and the development of Pascal’s famous triangle,binomial coefficients, and Madhava of Sangamagramma and the genius of the Kerala School etc.. etc..very nice
Here is that link again: http://www.ams.org/notices/201003/rtx100300385p.pdf
geeth wrote:I read somewhere that the infinite series was developed in Kerala in the 14th Century and that was the basis for Newton's invention of Calculus
adityaS wrote:geeth wrote:I read somewhere that the infinite series was developed in Kerala in the 14th Century and that was the basis for Newton's invention of Calculus
Nilakantha did some wonderful work on astronomy, and though the pieces are definitely there, I don't think Calculus was explicitly developed.
Here, multiply the repeated summation of the Rsines by the square of the full chord and divide by
the square of the radius. . . . In his manner we get the result that when the repeated summation of
the Rsines up to the tip of a particular arc-bit is done, the result will be the difference between the
next higher Rsine and the corresponding arc. Here the arc-bit has to be conceived as being as minute
as possible. Then the first Rsine difference will be the same as the first arc-bit. Hence, ifmultiplied by
the desired number, the result will certainly be the desired arc.
The discovery of the finite difference equation for sine led Indian mathematicians eventually to the full theory of calculus for polynomials and for sine, cosine, arcsine, andarctangent functions, that is, for verything connectedtothe circle and sphere that might be motivated by the applications to astronomy. This workmatured over the thousandyear period in which the West slumbered, reaching its climax in the work of the Kerala school in the fourteenth to sixteenth centuries. I won’t describe the full evolution but cannot omit a mention of the discovery of the formula for the area and volume of the sphere by Bh¯askara II. Essentially, he rediscovered the derivation found in Archimedes’ On the Sphere and the Cylinder I. That is, he sliced the surface of the sphere by equally spaced lines of latitude and, using this, reduced the calculation of the area to the integral of sine. Now, he knew that cosine differences were sines but, startlingly, he integrates sine by summing his tables! He seems well aware that this is approximate and
that a limiting argument is needed but this is implicit in his work. My belief is that, given his applied orientation, this was the more convincing argument. In any case, the argument using the discrete fundamental theorem of calculus is given a few centuries later by the Kerala school, where one also finds explicit statements on the need for a limiting process, like: “The greater the number
[of subdivisions of an arc], the more accurate the circumference [given by the length of the inscribed
polygon]” and “Here the arc segment has to be imagined to be as small as one wants. . . [but] since
one has to explain [it] in a certain [definite] way, [I] have said [so far] that a quadrant has twenty-four
svenkat wrote:responding very late,though I read up your solution long time back.I felt very dissappointed by the proof.It is so non-constructive in the sense I cannot check the divisibility of 1+n+n^8 +...n^47 by 1+..n^4 for even n=2.Any comments.
May I humbly ask what this corner is doing in the Tech and Economic forum though I guess there must be some reason if you think this deserves publicity but cannot fathom what it is except maybe to show off that we 'Hindoo fanatic' jingoes are also 'intellectually' inclined which does not stand up to scrutiny based on the interesting but noway profound prblems discussed here notwithstanding some formidable minds posting here.
A school has 1,000 students and 1,000 lockers, all in a row. The lockers all start out closed. The first student walks down the line and opens each one. The second student closes the even-numbered lockers. The third student approaches every third locker and changes it state. If it's open, he closes it; if it's closed, he opens it. The fourth student does the same, and so on, through 1,000 students. How many lockers end up open?
Determine all pairs (a,b) of positive integers such that ab^2+b+7 divides a^2b+a+b.
abhishek_sharma wrote:Hey Amber G.
Can you please suggest some magazines which publish articles (or book reviews) about recent changes in Mathematics for non-experts like myself (I am a science graduate but not a Mathematician).
I have noted that the Science and Nature magazines do not publish much about Mathematics.
Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest. The intention of the book is to give a mathematically oriented development of the theory of rare events underlying various applications. This characteristic of the book was strengthened in the second edition by incorporating various new results. In this third edition, the dramatic change of focus of extreme value theory has been taken into account: from concentrating on maxima of observations it has shifted to large observations, defined as excedances over high thresholds. One emphasis of the present third edition lies on multivariate generalized Pareto distributions, their representations, properties such as their peaks-over-threshold stability, simulation, testing and estimation. Reviews of the 2nd edition: "In brief, it is clear that this will surely be a valuable resource for anyone involved in, or seeking to master, the more mathematical features of this field." David Stirzaker, Bulletin of the London Mathematical Society "Laws of Small Numbers can be highly recommended to everyone who is looking for a smooth introduction to Poisson approximations in EVT and other fields of probability theory and statistics. In particular, it offers an interesting view on multivariate EVT and on EVT for non-iid observations, which is not presented in a similar way in any other textbook." Holger Drees, Metrika
This book provides a nice introduction to the mathematical theory of rare events and extreme values. It takes the point process approach to rare events using the empirical point process defined by the crossing of a high or low level to employ an approximate Poisson process. This approach has broad applicability and the authors use it to consider rare and extreme events for stochastic processes including Gaussian processes, Poisson point processes, extreme value models and nonparametric regression.
This book is noteworthy for its mix of theory and application and for the introduction of the software XTREMES. Chapter 6 is devoted to explaining how to analyze data for extremes using the software XTREMES. A diskette is included with the text with version 1.2 of XTREMES that is an interactive program that can run on MS-DOS operated computers. The text is very much up-to-date on the theoretical developments as of 1994.
..one of the leading mathematicians in the United States, was granted the presidential National Medal of Science earlier this month. President Obama will award the medal, which is the highest scientific honor in the country, at a White House ceremony later this fall.
svenkat wrote: also 'intellectually' inclined which does not stand up to scrutiny based on the interesting but noway profound prblems discussed here notwithstanding some formidable minds posting here.