Schwartz–Zippel lemma

Template:Use British English 30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí. Template:Infobox person Madhava of Sangamagrama (Template:Circa), was an Indian mathematician-astronomer from the town of Sangamagrama (present day Irinjalakuda) near Thrissur, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. He was the first in the world to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity". His discoveries opened the doors to what has today come to be known as Mathematical Analysis. One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra.

Some scholars have also suggested that Madhava's work, through the writings of the Kerala school, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it may have had an influence on later European developments in analysis and calculus.

Name

Madhava was born as Irińńaŗappiļļy or Iriññinavaļļi Mādhava Namboodiri. He had written that his house name was related to the Vihar where a plant called "bakuļam" was planted. According to Achyuta Pisharati, (who wrote a commentary on Veņwarõham written by Madhava) bakuļam was locally known as "iraňňi". Dr. K.V. Sarma, an authority on Madhava has the opinion that the house name is either Irińńāŗappiļļy or Iriññinavaļļy'.

Irinjalakuda was once known as 'Irińńāţikuţal'. Sangamagrāmam (lit. sangamam = union, grāmam = village) is a rough translation to Sanskrit from Dravidian word 'Irińńāţikuţal', which means 'iru (two) ańńāţi (market) kǖţal (union)' or the union of two markets.

Historiography

Although there is some evidence of mathematical work in Kerala prior to Madhava (e.g., Sadratnamala c. 1300, a set of fragmentary results), it is clear from citations that Madhava provided the creative impulse for the development of a rich mathematical tradition in medieval Kerala. However, most of Madhava's original work (except a couple of them) is lost. He is referred to in the work of subsequent Kerala mathematicians, particularly in Nilakantha Somayaji's Tantrasangraha (c. 1500), as the source for several infinite series expansions, including sinθ and arctanθ. The 16th-century text Mahajyānayana prakāra cites Madhava as the source for several series derivations for π. In Jyeṣṭhadeva's Yuktibhāṣā (c. 1530), written in Malayalam, these series are presented with proofs in terms of the Taylor series expansions for polynomials like 1/(1+x²), with x = tanθ, etc.

Thus, what is explicitly Madhava's work is a source of some debate. The Yukti-dipika (also called the Tantrasangraha-vyakhya), possibly composed Sankara Variyar, a student of Jyeṣṭhadeva, presents several versions of the series expansions for sinθ, cosθ, and arctanθ, as well as some products with radius and arclength, most versions of which appear in Yuktibhāṣā. For those that do not, Rajagopal and Rangachari have argued, quoting extensively from the original Sanskrit, that since some of these have been attributed by Nilakantha to Madhava, possibly some of the other forms might also be the work of Madhava.

Others have speculated that the early text Karanapaddhati (c. 1375–1475), or the Mahajyānayana prakāra might have been written by Madhava, but this is unlikely.

Karanapaddhati, along with the even earlier Keralese mathematics text Sadratnamala, as well as the Tantrasangraha and Yuktibhāṣā, were considered in an 1834 article by Charles Matthew Whish, which was the first to draw attention to their priority over Newton in discovering the Fluxion (Newton's name for differentials). In the mid-20th century, the Russian scholar Jushkevich revisited the legacy of Madhava, and a comprehensive look at the Kerala school was provided by Sarma in 1972.

Lineage

Explanation of the sine rule in Yuktibhāṣā

There are several known astronomers who preceded Madhava, including Kǖţalur Kizhār (2nd century), Vararuci (4th century), Sankaranarayana (866 AD). It is possible that other unknown figures may have preceded him. However, we have a clearer record of the tradition after Madhava. Parameshvara Namboodri was a direct disciple. According to a palmleaf manuscript of a Malayalam commentary on the Surya Siddhanta, Parameswara's son Damodara (c. 1400–1500) had both Nilakantha Somayaji as his disciples. Jyeshtadevan was the disciple of Nilakanda. Achyuta Pisharati of Trikkantiyur is mentioned as a disciple of Jyeṣṭhadeva, and the grammarian Melpathur Narayana Bhattathiri as his disciple.

Contributions

If we consider mathematics as a progression from finite processes of algebra to considerations of the infinite, then the first steps towards this transition typically come with infinite series expansions. It is this transition to the infinite series that is attributed to Madhava. In Europe, the first such series were developed by James Gregory in 1667. Madhava's work is notable for the series, but what is truly remarkable is his estimate of an error term (or correction term). This implies that the limit nature of the infinite series was quite well understood by him. Thus, Madhava may have invented the ideas underlying infinite series expansions of functions, power series, trigonometric series, and rational approximations of infinite series.

However, as stated above, which results are precisely Madhava's and which are those of his successors, are somewhat difficult to determine. The following presents a summary of results that have been attributed to Madhava by various scholars.

Infinite series

Main article : Madhava series

Among his many contributions, he discovered the infinite series for the trigonometric functions of sine, cosine, tangent and arctangent, and many methods for calculating the circumference of a circle. One of Madhava's series is known from the text Yuktibhāṣā, which contains the derivation and proof of the power series for inverse tangent, discovered by Madhava. In the text, Jyeṣṭhadeva describes the series in the following manner: Benefits of Living in a Rental or Landed property in Singapore Property New Launches & Undertaking Showcase In Singapore Many residential Singapore property gross sales contain shopping for property in Singapore at new launches. These are often properties below development, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, in accordance with personal property transactions in Could Cellular Apps FREE Signal Up Log in Property Agents Feedback

Banks usually engage skilled valuers to conduct valuations of properties. The valuers are required to abide by industry requirements and guidelines for property valuation prescribed by the Singapore Institute of Surveyors and Valuers. With Singapore conserving its greenery and with town sky view being the attraction to many, it is no wonder that increasingly condos are being developed on the island. Many future traders are placing extra value on units with a view than these with out one. Thus, this may be one of the reasons too as to why your unit of selection is either with a view or with out. How to Spot Abroad Property Scams (at Propwise.sg) New Condominium Eco Sanctuary Launch 01Dec 2012 Coming Quickly !!! New launch rental COCO Palm at Pasir Ris

Association charges has to be paid by each condominium proprietor, weather you are dwelling in it or not, or even in case you have rented the property. So, whether you employ the amenities and amenities at the condominium, you'll any how must pay for them. JOHOR BAHARU, Jan 22 (Bernama) — The property market in Johor Baharu and Iskandar Malaysia has been negatively affected by the cooling measures introduced by the authorities to address rising property prices, says the Actual Property and Housing Developers' Association of Malaysia (Rehda). District 18, 99 years LH Executive Apartment District 25, ninety nine years LH Executive Rental District 19, ninety nine years LH Government Apartment pipelines. Property homeowners haven't any different but to pay up. There's no different sensible Hottest Launch D'Leedon

Dr Victor Lee is a surgeon gazetted to carry out multi-organ transplant surgical procedure. He underwent his fellowship coaching in transplant surgery on the Royal Infirmary of Edinburgh, United Kingdom in 2008 and 2009. He is presently a consultant surgeon at the Singapore Normal Hospital, and visiting guide to the National College Hospital. His scientific interest and skills are in the areas of transplant surgery, hepatobiliary pancreatic surgery and laparoscopic surgical procedure.

Lush Acres Government Condominium goes to launch quickly at Sengkang West Method By Metropolis Developments Limited. Lush Acres reflects the large expense of house, environmentally-inexperienced constructing and plush panorama of this low density development with solely 22% of website protection. There have been students, retirees and property agents napping, snacking and taking part in cards within the queue to kill time. Such a scene made passers-by wonder if the gang really consisted of genuine consumers, or just part-timers being paid to fake how sizzling the new undertaking was. The model was filed in Switzerland in October 2009 with a model r egistration dated March 25, 2010 A deposit was also recorded in France at the INPI (National Institute of Intellectual Property) Coastal commercial spaceports

Individuals all wish to be seen having the identical foresight because the specialists in property investment or the massive names in their own fields. Thus the discharge of these tales works to motivate other patrons to follow go well with. Bartley Ridge is the hottest new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-yr leasehold new condo at Mount Vernon highway, excellent subsequent to Bartley MRT station (CC12). If you wish to get more Rehda Johor chairman Koh Moo Hing stated potential property patrons within the two areas at the moment are adopting a wait-and-see angle. How can I get the ebrochure and flooring plans of the new launch initiatives ? The Current Mortgage to your HDB District thirteen, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris

The simplest way is to contact a real estate agent who's a Advertising Workforce Member The agent will doubtless meet to show you details of the undertaking supplied by the developer. Attending a preview doesn't suggest you are obliged to purchase a unit. You can select not to and your cheque will likely be returned to you. No cost concerned. Your cheque will likely be returned to you in the event you decide to not proceed to purchase. Early hen low cost (up to 10%) at the preview earlier than the official launch. The purpose of that is to stimulate demand and to create a buying house in singapore momentum. As soon as this function is achieved, the developer would then take away the low cost so as to maximize his profitability. 2/3/4 bedrooms units obtainable ! GREATEST PURCHASE NOW !!! AVG $1900psf The Key Commercial Cluster in the North Area This yields:

$r\theta ={\frac {r\sin \theta }{\cos \theta }}-(1/3)\,r\,{\frac {\left(\sin \theta \right)^{3}}{\left(\cos \theta \right)^{3}}}+(1/5)\,r\,{\frac {\left(\sin \theta \right)^{5}}{\left(\cos \theta \right)^{5}}}-(1/7)\,r\,{\frac {\left(\sin \theta \right)^{7}}{\left(\cos \theta \right)^{7}}}+\cdots$

or equivalently:

$\theta =\tan \theta -{\frac {\tan ^{3}\theta }{3}}+{\frac {\tan ^{5}\theta }{5}}-{\frac {\tan ^{7}\theta }{7}}+\cdots$

This series was traditionally known as the Gregory series (after James Gregory, who discovered it three centuries after Madhava). Even if we consider this particular series as the work of Jyeṣṭhadeva, it would pre-date Gregory by a century, and certainly other infinite series of a similar nature had been worked out by Madhava. Today, it is referred to as the Madhava-Gregory-Leibniz series.

Trigonometry

Madhava also gave a most accurate table of sines, defined in terms of the values of the half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle. It is believed that he may have found these highly accurate tables based on these series expansions:

sin q = q – q³/3! + q⁵/5! – ...

cos q = 1 – q²/2! + q⁴/4! – ...

The value of π (pi)

Madhava's work on the value of π is cited in the Mahajyānayana prakāra ("Methods for the great sines").Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. While some scholars such as Sarma feel that this book may have been composed by Madhava himself, it is more likely the work of a 16th-century successor. This text attributes most of the expansions to Madhava, and gives the following infinite series expansion of π, now known as the Madhava-Leibniz series:

${\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{n}}{2n+1}}+\cdots$

which he obtained from the power series expansion of the arc-tangent function. However, what is most impressive is that he also gave a correction term, R_n, for the error after computing the sum up to n terms. Madhava gave three forms of R_n which improved the approximation, namely

R_n = 1/(4n), or

R_n = n/ (4n² + 1), or

R_n = (n² + 1) / (4n³ + 5n).

where the third correction leads to highly accurate computations of π.

It is not clear how Madhava might have found these correction terms. The most convincing is that they come as the first three convergents of a continued fraction which can itself be derived from the standard Indian approximation to π namely 62832/20000 (for the original 5th-century computation, see Aryabhata).

He also gave a more rapidly converging series by transforming the original infinite series of π, obtaining the infinite series

$\pi ={\sqrt {12}}\left(1-{1 \over 3\cdot 3}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right)$

By using the first 21 terms to compute an approximation of π, he obtains a value correct to 11 decimal places (3.14159265359). The value of 3.1415926535898, correct to 13 decimals, is sometimes attributed to Madhava, but may be due to one of his followers. These were the most accurate approximations of π given since the 5th century (see History of numerical approximations of π).

The text Sadratnamala, usually considered as prior to Madhava, appears to give the astonishingly accurate value of π =3.14159265358979324 (correct to 17 decimal places). Based on this, R. Gupta has argued that this text may also have been composed by Madhava.

Algebra

Madhava also carried out investigations into other series for arclengths and the associated approximations to rational fractions of π, found methods of polynomial expansion, discovered tests of convergence of infinite series, and the analysis of infinite continued fractions. He also discovered the solutions of transcendental equations by iteration, and found the approximation of transcendental numbers by continued fractions.

Calculus

Madhava laid the foundations for the development of calculus, which were further developed by his successors at the Kerala school of astronomy and mathematics. (It should be noted that certain ideas of calculus were known to earlier mathematicians.) Madhava also extended some results found in earlier works, including those of Bhāskara II.

Madhava developed some components of calculus such as differentiation, term-by-term integration, iterative methods for solutions of non-linear equations, and the theory that the area under a curve is its integral.

Madhava's works

K.V. Sarma has identified Madhava as the author of the following works:

1. Golavada
2. Madhyamanayanaprakara
3. Mahajyanayanaprakara
4. Lagnaprakarana (लग्नप्रकरण)
5. Venvaroha (वेण्वारोह)
6. Sphutacandrapti (स्फुटचन्द्राप्ति)
7. Aganita-grahacara (अगणित-ग्रहचार)
8. Chandravakyani (चन्द्रवाक्यानि)

Kerala School of Astronomy and Mathematics

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

The Kerala school of astronomy and mathematics flourished for at least two centuries beyond Madhava. In Jyeṣṭhadeva we find the notion of integration, termed sankalitam, (lit. collection), as in the statement:

ekadyekothara pada sankalitam samam padavargathinte pakuti,

which translates as the integration a variable (pada) equals half that variable squared (varga); i.e. The integral of x dx is equal to x² / 2. This is clearly a start to the process of integral calculus. A related result states that the area under a curve is its integral. Most of these results pre-date similar results in Europe by several centuries. In many senses, Jyeshthadeva's Yuktibhāṣā may be considered the world's first calculus text.

The group also did much other work in astronomy; indeed many more pages are developed to astronomical computations than are for discussing analysis related results.

The Kerala school also contributed much to linguistics (the relation between language and mathematics is an ancient Indian tradition, see Katyayana). The ayurvedic and poetic traditions of Kerala can also be traced back to this school. The famous poem, Narayaneeyam, was composed by Narayana Bhattathiri.

Influence

Madhava has been called "the greatest mathematician-astronomer of medieval India", or as "the founder of mathematical analysis; some of his discoveries in this field show him to have possessed extraordinary intuition." O'Connor and Robertson state that a fair assessment of Madhava is that he took the decisive step towards modern classical analysis.

Possible propagation to Europe

The Kerala school was well known in the 15th and 16th centuries, in the period of the first contact with European navigators in the Malabar Coast. At the time, the port of Muziris, near Sangamagrama, was a major center for maritime trade, and a number of Jesuit missionaries and traders were active in this region. Given the fame of the Kerala school, and the interest shown by some of the Jesuit groups during this period in local scholarship, some scholars, including G. Joseph of the U. Manchester have suggested that the writings of the Kerala school may have also been transmitted to Europe around this time, which was still about a century before Newton.

See also

• Madhava's sine table
• Madhava series
• Venvaroha
• Indian mathematics
• List of Indian mathematicians
• Kerala school of astronomy and mathematics
• History of calculus
• Ganita-yukti-bhasa

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.