Indian Mathematicians and Their Contributions
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Three Puzzles on Mathematics, Computation, and Games
P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (551–606) THREE PUZZLES ON MATHEMATICS, COMPUTATION, AND GAMES G K Abstract In this lecture I will talk about three mathematical puzzles involving mathemat- ics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible? 1 Introduction The theory of computing and computer science as a whole are precious resources for mathematicians. They bring up new questions, profound new ideas, and new perspec- tives on classical mathematical objects, and serve as new areas for applications of math- ematics and mathematical reasoning. In my lecture I will talk about three mathematical puzzles involving mathematics and computation (and, at times, other fields) that have preoccupied me over the years. The connection between mathematics and computing is especially strong in my field of combinatorics, and I believe that being able to person- ally experience the scientific developments described here over the past three decades may give my description some added value. For all three puzzles I will try to describe in some detail both the large picture at hand, and zoom in on topics related to my own work. Puzzle 1: What can explain the success of the simplex algorithm? Linear program- ming is the problem of maximizing a linear function subject to a system of linear inequalities. The set of solutions to the linear inequalities is a convex polyhedron P . -
Brahmagupta, Mathematician Par Excellence
GENERAL ARTICLE Brahmagupta, Mathematician Par Excellence C R Pranesachar Brahmagupta holds a unique position in the his- tory of Ancient Indian Mathematics. He con- tributed such elegant results to Geometry and Number Theory that today's mathematicians still marvel at their originality. His theorems leading to the calculation of the circumradius of a trian- gle and the lengths of the diagonals of a cyclic quadrilateral, construction of a rational cyclic C R Pranesachar is involved in training Indian quadrilateral and integer solutions to a single sec- teams for the International ond degree equation are certainly the hallmarks Mathematical Olympiads. of a genius. He also takes interest in solving problems for the After the Greeks' ascendancy to supremacy in mathe- American Mathematical matics (especially geometry) during the period 7th cen- Monthly and Crux tury BC to 2nd century AD, there was a sudden lull in Mathematicorum. mathematical and scienti¯c activity for the next millen- nium until the Renaissance in Europe. But mathematics and astronomy °ourished in the Asian continent partic- ularly in India and the Arab world. There was a contin- uous exchange of information between the two regions and later between Europe and the Arab world. The dec- imal representation of positive integers along with zero, a unique contribution of the Indian mind, travelled even- tually to the West, although there was some resistance and reluctance to accept it at the beginning. Brahmagupta, a most accomplished mathematician, liv- ed during this medieval period and was responsible for creating good mathematics in the form of geometrical theorems and number-theoretic results. -
Indian Mathematics
Indian Mathemtics V. S. Varadarajan University of California, Los Angeles, CA, USA UCLA, March 3-5, 2008 Abstract In these two lectures I shall talk about some Indian mathe- maticians and their work. I have chosen two examples: one from the 7th century, Brahmagupta, and the other, Ra- manujan, from the 20th century. Both of these are very fascinating figures, and their histories illustrate various as- pects of mathematics in ancient and modern times. In a very real sense their works are still relevant to the mathe- matics of today. Some great ancient Indian figures of Science Varahamihira (505–587) Brahmagupta (598-670) Bhaskara II (1114–1185) The modern era Ramanujan, S (1887–1920) Raman, C. V (1888–1970) Mahalanobis, P. C (1893–1972) Harish-Chandra (1923–1983) Bhaskara represents the peak of mathematical and astro- nomical knowledge in the 12th century. He reached an un- derstanding of calculus, astronomy, the number systems, and solving equations, which were not to be achieved any- where else in the world for several centuries...(Wikipedia). Indian science languished after that, the British colonial occupation did not help, but in the 19th century there was a renaissance of arts and sciences, and Indian Science even- tually reached a level comparable to western science. BRAHMAGUPTA (598–670c) Some quotations of Brahmagupta As the sun eclipses the stars by its brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more, if he solves them. Quoted in F Cajori, A History of Mathematics A person who can, within a year, solve x2 92y2 =1, is a mathematician. -
Integrity, Authentication and Confidentiality in Public-Key Cryptography Houda Ferradi
Integrity, authentication and confidentiality in public-key cryptography Houda Ferradi To cite this version: Houda Ferradi. Integrity, authentication and confidentiality in public-key cryptography. Cryptography and Security [cs.CR]. Université Paris sciences et lettres, 2016. English. NNT : 2016PSLEE045. tel- 01745919 HAL Id: tel-01745919 https://tel.archives-ouvertes.fr/tel-01745919 Submitted on 28 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT de l’Université de recherche Paris Sciences et Lettres PSL Research University Préparée à l’École normale supérieure Integrity, Authentication and Confidentiality in Public-Key Cryptography École doctorale n◦386 Sciences Mathématiques de Paris Centre Spécialité Informatique COMPOSITION DU JURY M. FOUQUE Pierre-Alain Université Rennes 1 Rapporteur M. YUNG Moti Columbia University et Snapchat Rapporteur M. FERREIRA ABDALLA Michel Soutenue par Houda FERRADI CNRS, École normale supérieure le 22 septembre 2016 Membre du jury M. CORON Jean-Sébastien Université du Luxembourg Dirigée par -
Rationale of the Chakravala Process of Jayadeva and Bhaskara Ii
HISTORIA MATHEMATICA 2 (1975) , 167-184 RATIONALE OF THE CHAKRAVALA PROCESS OF JAYADEVA AND BHASKARA II BY CLAS-OLOF SELENIUS UNIVERSITY OF UPPSALA SUMMARIES The old Indian chakravala method for solving the Bhaskara-Pell equation or varga-prakrti x 2- Dy 2 = 1 is investigated and explained in detail. Previous mis- conceptions are corrected, for example that chakravgla, the short cut method bhavana included, corresponds to the quick-method of Fermat. The chakravala process corresponds to a half-regular best approximating algorithm of minimal length which has several deep minimization properties. The periodically appearing quantities (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, etc.) are correctly understood only with the new theory. Den fornindiska metoden cakravala att l~sa Bhaskara- Pell-ekvationen eller varga-prakrti x 2 - Dy 2 = 1 detaljunders~ks och f~rklaras h~r. Tidigare missuppfatt- 0 ningar r~ttas, sasom att cakravala, genv~gsmetoden bhavana inbegripen, motsvarade Fermats snabbmetod. Cakravalaprocessen motsvarar en halvregelbunden b~st- approximerande algoritm av minimal l~ngd med flera djupt liggande minimeringsegenskaper. De periodvis upptr~dande storheterna (jyestha-m~la, kanistha-mula, ksepaka, kuttakara, os~) blir forstaellga0. 0 . f~rst genom den nya teorin. Die alte indische Methode cakrav~la zur Lbsung der Bhaskara-Pell-Gleichung oder varga-prakrti x 2 - Dy 2 = 1 wird hier im einzelnen untersucht und erkl~rt. Fr~here Missverst~ndnisse werden aufgekl~rt, z.B. dass cakrav~la, einschliesslich der Richtwegmethode bhavana, der Fermat- schen Schnellmethode entspreche. Der cakravala-Prozess entspricht einem halbregelm~ssigen bestapproximierenden Algorithmus von minimaler L~nge und mit mehreren tief- liegenden Minimierungseigenschaften. Die periodisch auftretenden Quantit~ten (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, usw.) werden erst durch die neue Theorie verst~ndlich. -
The Brahmagupta Triangles Raymond A
The Brahmagupta Triangles Raymond A. Beauregard and E. R. Suryanarayan Ray Beauregard ([email protected]) received his Ph.D. at the University of New Hampshire in 1968, then joined the University of Rhode Island, where he is a professor of mathematics. He has published many articles in ring theory and two textbooks. Linear Algebra (written with John Fraleigh) is currently in its third edition. Besides babysitting for his grandchild Elyse, he enjoys sailing the New England coast on his sloop, Aleph One, and playing the piano. E. R. Suryanarayan ([email protected]) taught at universities in India before receiving his Ph.D. (1961) at the University of Michigan, under Nathaniel Coburn. He has been at the University of Rhode Island since 1960, where is a professor of mathematics. An author of more than 20 research articles in applied mathematics, crystallography, and the history of mathematics, he lists as his main hobbies music, languages, and aerobic walking. The study of geometric objects has been a catalyst in the development of number theory. For example, the figurate numbers (triangular, square, pentagonal, . ) were a source of many early results in this field [41.Measuring the length of a diagonal of a rectangle led to the problem of approxin~atingfi for a natural number N. The study of triangles has been of particular significance. Heron of Alexandria (c. A.D. 75)-gave the well-known formula for the area A of a triangle in terms of its sides: A = Js(s - a)(s- b)(s- c),where s = (a + b + c)/2 is the semiperimeter of the triangle having sides a,b, c [41.He illustrated this with a triangle whose sides are 13,14,15 and whose area is 84. -
Astronomy in India
TRADITIONSKnowledg & PRACTICES OF INDIA e Textbook for Class XI Module 1 Astronomy in India CENTRAL BOARD OF SECONDARY EDUCATION Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India TRADITIONSKnowledg & PRACTICESe OF INDIA Textbook for Class XI Module 1 Astronomy in India CENTRAL BOARD OF SECONDARY EDUCATION Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India No part of this publication may be reproduced or stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical photocopying, recording or otherwise, without the prior permission of the Central Board of Secondary Education (CBSE). Preface India has a rich tradition of intellectual inquiry and a textual heritage that goes back to several hundreds of years. India was magnificently advanced in knowledge traditions and practices during the ancient and medieval times. The intellectual achievements of Indian thought are found across several fields of study in ancient Indian texts ranging from the Vedas and the Upanishads to a whole range of scriptural, philosophical, scientific, technical and artistic sources. As knowledge of India's traditions and practices has become restricted to a few erudite scholars who have worked in isolation, CBSE seeks to introduce a course in which an effort is made to make it common knowledge once again. Moreover, during its academic interactions and debates at key meetings with scholars and experts, it was decided that CBSE may introduce a course titled ‘Knowledge Traditions and Practices of India’ as a new Elective for classes XI - XII from the year 2012-13. It has been felt that there are many advantages of introducing such a course in our education system. -
Three Puzzles on Mathematics Computations, and Games
January 5, 2019 17:43 icm-961x669 main-pr page 551 P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (551–606) THREE PUZZLES ON MATHEMATICS, COMPUTATION, AND GAMES G K Abstract In this lecture I will talk about three mathematical puzzles involving mathemat- ics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible? 1 Introduction The theory of computing and computer science as a whole are precious resources for mathematicians. They bring up new questions, profound new ideas, and new perspec- tives on classical mathematical objects, and serve as new areas for applications of math- ematics and mathematical reasoning. In my lecture I will talk about three mathematical puzzles involving mathematics and computation (and, at times, other fields) that have preoccupied me over the years. The connection between mathematics and computing is especially strong in my field of combinatorics, and I believe that being able to person- ally experience the scientific developments described here over the past three decades may give my description some added value. For all three puzzles I will try to describe in some detail both the large picture at hand, and zoom in on topics related to my own work. Puzzle 1: What can explain the success of the simplex algorithm? Linear program- ming is the problem of maximizing a linear function subject to a system of linear inequalities. -
Review of Research Impact Factor : 5.7631(Uif) Ugc Approved Journal No
Review Of ReseaRch impact factOR : 5.7631(Uif) UGc appROved JOURnal nO. 48514 issn: 2249-894X vOlUme - 8 | issUe - 2 | nOvembeR - 2018 __________________________________________________________________________________________________________________________ ANCIENT INDIAN CONTRIBUTIONS TOWARDS MATHEMATICS Madhuri N. Gadsing Department of Mathematics, Jawahar Arts, Science and Commerce College, Anadur (M.S.), India. ABSTRACT Mathematics having been a progressive science has played a significant role in the development of Indian culture for millennium. In ancient India, the most famous Indian mathematicians, Panini (400 CE), Aryabhata I (500 CE), Brahmagupta (700 CE), Bhaskara I (900 CE), Mahaviracharya (900 CE), Aryabhata II (1000 CE), Bhaskara II (1200 CE), chanced to discover and develop various concepts like, square and square roots, cube and cube roots, zero with place value, combination of fractions, astronomical problems and computations, differential and integral calculus etc., while meditating upon various aspects of arithmetic, geometry, astronomy, modern algebra, etc. In this paper, we review the contribution of Indian mathematicians from ancient times. KEYWORDS: Mathematics , development of Indian , astronomical problems and computations. INTRODUCTION: Mathematics having been a progressive science has played a significant role in the development of Indian culture for millennium. Mathematical ideas that originated in the Indian subcontinent have had a profound impact on the world. The aim of this article is to give a brief review of a few of the outstanding innovations introduced by Indian mathematicians from ancient times. In ancient India, the most famous Indian mathematicians belong to what is known as the classical era [1-8]. This includes Panini (400 CE), Aryabhata I (500 CE) [9], Brahmagupta (700 CE), Bhaskara I (900 CE) [5, 6], Mahavira (900 CE), Aryabhata II (1000 CE), Bhaskaracharya or Bhaskara II (1200 CE) [10-13]. -
Equation Solving in Indian Mathematics
U.U.D.M. Project Report 2018:27 Equation Solving in Indian Mathematics Rania Al Homsi Examensarbete i matematik, 15 hp Handledare: Veronica Crispin Quinonez Examinator: Martin Herschend Juni 2018 Department of Mathematics Uppsala University Equation Solving in Indian Mathematics Rania Al Homsi “We owe a lot to the ancient Indians teaching us how to count. Without which most modern scientific discoveries would have been impossible” Albert Einstein Sammanfattning Matematik i antika och medeltida Indien har påverkat utvecklingen av modern matematik signifi- kant. Vissa människor vet de matematiska prestationer som har sitt urspring i Indien och har haft djupgående inverkan på matematiska världen, medan andra gör det inte. Ekvationer var ett av de områden som indiska lärda var mycket intresserade av. Vad är de viktigaste indiska bidrag i mate- matik? Hur kunde de indiska matematikerna lösa matematiska problem samt ekvationer? Indiska matematiker uppfann geniala metoder för att hitta lösningar för ekvationer av första graden med en eller flera okända. De studerade också ekvationer av andra graden och hittade heltalslösningar för dem. Denna uppsats presenterar en litteraturstudie om indisk matematik. Den ger en kort översyn om ma- tematikens historia i Indien under många hundra år och handlar om de olika indiska metoderna för att lösa olika typer av ekvationer. Uppsatsen kommer att delas in i fyra avsnitt: 1) Kvadratisk och kubisk extraktion av Aryabhata 2) Kuttaka av Aryabhata för att lösa den linjära ekvationen på formen 푐 = 푎푥 + 푏푦 3) Bhavana-metoden av Brahmagupta för att lösa kvadratisk ekvation på formen 퐷푥2 + 1 = 푦2 4) Chakravala-metoden som är en annan metod av Bhaskara och Jayadeva för att lösa kvadratisk ekvation 퐷푥2 + 1 = 푦2. -
Brahmagupta's 18 Laws of Mathematics
EDUCATION “Brahmagupta’s 18 laws of mathematics are completely missing from India’s present mathematics curriculum.” ndia is known to be the birthplace of modern mathematics. Yet, many Indian Ischool-going children lack proper understanding of mathematical concepts. Australian mathematics historian Jonathan J. Crabtree was apprehensive of mathematics as a student like many of his classmates. He felt there was a need to better explain the laws and rules of mathematics. Over the years, Crabtree has retraced the origin of mathematics and linked it to ancient Indian wisdom. He found that India’s definition of zero never made it to Europe. The disconnect between western mathematical explanations from the original teachings of ancient Indian mathematicians like Brahmagupta, Mahāvira (c. 850) or Bhāscara (c. 1150) was identified by Crabtree as a major contributing factor. BE’s Isha Chakraborty spoke to him. Jonathan J. Crabtree . Why and when did you feel that there were some Brahmagupta was an astronomer. Today, children are told that Qmistakes in basic mathematical concepts? negative numbers are defined as being less than zero, yet that is mathematically and historically incorrect. The Chinese were A. Fifty years ago, in 1968, my Class 2 teacher gave me the using negative and positive numbers for around 1400 years wrong explanation of multiplication. People have said aaa × bbb before they adopted India’s zero. So the Chinese could never equals aaa added to itself bbb times for centuries. Yet this leads to 1 have considered negatives numbers less than zero. Instead, × 1 = 2. When asked ‘what is two added to itself three times’, they just viewed negatives as equal and opposite to positives, I said 8. -
Påˆini and Euclid: Reflections on Indian Geometry* (Published In: Journal of Indian Philosophy 29 (1-2; Ingalls Commemoration Volume), 2001, 43-80)
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Serveur académique lausannois Påˆini and Euclid 1 JOHANNES BRONKHORST Påˆini and Euclid: reflections on Indian geometry* (published in: Journal of Indian Philosophy 29 (1-2; Ingalls Commemoration Volume), 2001, 43-80) Professor Ingalls — in an article called "The comparison of Indian and Western philosophy" — made the following interesting observation (1954: 4): "In philosophizing the Greeks made as much use as possible of mathematics. The Indians, curiously, failed to do this, curiously because they were good mathematicians. Instead, they made as much use as possible of grammatical theory and argument." This observation should not — as goes without saying in our day and age — be read as a description of the Indian “genius” as opposed to that of the Greeks (at least not in some absolute sense), but as a reminder of the important roles that mathematics and linguistics have played as methodical guidelines in the development of philosophy in Greece and in India respectively. Ingalls appears to have been the first to draw attention to this important distinction. He was not the last. Ingalls's observation has been further elaborated by J. F. (= Frits) Staal in a few articles (1960; 1963; 1965).1 Staal focuses the discussion on two historical persons in particular, Euclid and Påˆini, both of whom — as he maintains — have exerted an important, even formative, influence on developments in their respective cultures. Staal also broadens the horizon by drawing other areas than only philosophy into the picture. To cite his own words (1965: 114 = 1988: 158): "Historically speaking, Påˆini's method has occupied a place comparable to that held by Euclid's method in Western thought.