DOCSLIB.ORG

Indians Among the Exalted at ICM 2010 He Presence of Indian Mathematicians Number Theory, Combinatorial Number Theory, of Current Interest in Statistics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Indians Among the Exalted at ICM 2010 He Presence of Indian Mathematicians Number Theory, Combinatorial Number Theory, of Current Interest in Statistics August 27, 2010 History of Mathe- On the footprints of Opinions Emmy Noether 3 5 matics in India 7 Indians Among the Exalted at ICM 2010 he presence of Indian mathematicians number theory, combinatorial number theory, of current interest in statistics. His work in Tamong the plenary and invited speakers elliptic curves and cryptography. In 1986, he multivariate analysis related to developing at ICM 2010 has been significantly more than solved the Waring problem for fourth powers geometric notions of multivariate quantiles is in the past. To be invited to deliver a plenary in collaboration with J-M. Deshouillers and F. considered a major breakthrough in the area or an invited lecture at an ICM is considered Dress. of multivariate nonparametric inference. More highly prestigious. recently, he has developed methods that en- Prof. Venkataramana has made significant able one to discover statistically significant While Prof. R. Balasubramanian of the Insti- contributions to algebraic groups and discrete features in the income distribution in different tute of Mathematical Sciences (IMSc), Chen- subgroups. He proved the positive character- populations. nai, delivered a Plenary Talk, Profs. V. istic version of the Margulis super-rigidity the- Srinivas and T. N. Venkataramana of the Tata orem and obtained important results on Prof. Arup Bose has made significant re- Institute of Fundamental Research (TIFR, cohomology of locally symmetric varieties. search contributions in resampling plans, se- Mumbai), Probal Chaudhuri and Arup Bose of quential analysis and asymptotics, statistics the Indian Statistical Institute (ISI), Kolkata, Prof. Srinivas has done significant work in as- of diffusion processes, rates of convergence, and V. Suresh of the University of Hyderabad pects of algebraic geometry and commutative multinomial selection problems and strong delivered Invited Talk. In fact, Prof. Balasub- algebra. His interests include algebraic cycles laws. His work in resampling provides a deep ramanian's Plenary Talk is the first ever at an on singular algebraic varieties, projective and powerful impact on areas such as time ICM by an Indian mathematician. Prof. Suresh modules, Hilbert functions and multiplicity and series, linear models, general dependent is from a university and the last Indian math- on aspects of algebraic geometry in positive models and nonparametric estimates. ematician from a university to a give an In- characteristic. An important work by him, done vited Talk in an ICM was Minakshisundaram in collaboration with D. Cutkosky, is the solu- Prof. Venapally Suresh has made outstanding in 1958. tion of Zariski's problem (Riemann-Roch prob- contributions to the theory of quadratic forms. lem for surfaces). His work on Galois cohomology and u-invari- Prof. Balasubramanian is a world-renowned ants over function fields of p-adic curves is es- number theorist. His research interests en- Prof. Probal Chaudhuri has made outstanding pecially important and regarded quite compass a wide spectrum of topics in analytic contributions in a number of important areas fundamental. !"#"$%&%'(!)*%+% ,$!(-"./ R. Balasubramanian T. N. Venkataramana V. Srinivas Probal Chaudhuri Arup Bose V. Suresh REFLEXIONS August 27,Friday India and Evolution of Mathematics !"#"$%&'!()%*% +,!'-"./ im Leslie Plofker is a well-known historian a place value system that is also fully inte- Kof mathematics. Currently a Visiting Pro- grated with arithmetic, is definitely Indian. It is fessor of Mathematics at the Union College, part of this trend, this mainstream in Indian USA, she is the author of the recently pub- mathematics, which is producing the sophis- lished Mathematics in India. Plofker has also ticated arithmetic and algebra that we are contributed a chapter on Indian mathematics seeing in the medieval period. And as to ac- in The Mathematics of Egypt, Mesopotamia, tually when that happened is a tough call. I China, India and Islam, which she has also think we can see that this place value system co-edited. She has also contributed to A De- of zero is clearly fully established at least in scriptive Catalogue of the Sanskrit Astronom- the scientific use in India by maybe around ical Manuscripts preserved at the Maharaja 3rd century of this era. Man Singh II Museum in Jaipur, India. She co- edited the volume Studies in the History of the Prof. S. R. Sharma has made some interest- Exact Sciences in Honour of David Pingree. ing arguments in favour of the idea that the place value system, including zero, may have Prof.Kim Leslie Plofker completed her PhD from Brown Uni- been developed in that era. So somewhere versity on ‘Mathematical Approximation by of course the remarkable methods that you around then is when integration of zero into a Transformation of Sine Functions in Medieval see by the Kerala School with infinitesimal number system gets going in the Indian tradi- Sanskrit Astronomical Texts’ under the super- quantities and various operations with infinite tion. vision of David Pingree. Much of the work series. Plofker and Pingree did was founded on the When did Indian mathematics become vis- great early Indian textual scholars of Sanskrit I think one of the things that we may have to ible and how? tradition like Sudhakar Trivedi and Bapudevan thank Indian mathematics for more than we Certainly mathematics is something involving Sastry. realise is the structure of the organisation of various types of calculations. There is a lot of mathematical knowledge; this distinction be- awareness about it in Indian contexts, all the She gave a Plenary Talk at the ICM 2010 titled tween arithmetic as the computation of known way back, when you have hints to the num- ‘Indian rules, Yavana rules: foreign identity quantities and algebra as a parallel type of bers in tens to hundreds to thousands and so and the transmission of mathematics’. In her computation but with unknown quantities is on, and the importance of calculation is defi- talk she discussed the encounter of two cul- something that appears very early in the San- nitely recognized. The other two cultures start tures of mathematics, namely Yavanas and skrit tradition. to understand something about Indian math- Indians, and the reactions of the cultures to ematics (well that’s something I talked about each other. Yavanas (In Sanskrit means Which do you think is the most remarkable in my Plenary presentation), and certainly as Greeks or any Western visitor) came into con- development? far as the West is concerned, after the incur- tact with Indians by crossing the Himalayas The work on the infinitesimal series in the Ker- sions of Alexander. So, the Alexander expedi- and travelling across the Arabian Sea. ala School about fourteenth century is just a tion gets as far as India and gets home but really amazing synthesis of so much work -- that means that there is a sense of link be- In a conversation with Richa Malhotra, not just infinite mathematical concepts they tween Greek culture and Indian culture. From Plofker threw light on ancient Indian mathe- evolved but also the astronomical problems then on Greek and other Western sources -- matics and her engagement with it. Excerpts by people who were masters of all aspects of first in the classical theory -- then in the Mus- from the interview: mathematical sciences and thought very orig- lim and then the modern European started to inally of it. report back on what they understand as “In- What are the iconic contributions of an- dian mathematics” and what things these for- cient Indian mathematics? What is the correct story about India and eign people – the Indians -- are doing in Well, first of all, what everybody knows about the origin of zero? Indian mathematics. is the decimal place value number system, It is a very long story. It is hard to know exactly which is now the universal number system. all the details of the origin of zero. Prof. R. C. Do you think it is important to recognise Though it wasn’t the first place value system Gupta, who will be receiving the [Kenneth O.] “who invented what” in mathematics? to be devised, it was the first that is definitely May Medal for history of mathematics has Well of course we don’t want it just to degrade the ancestor of the universal decimal place written a very good article called ‘Who in- into a priority dispute and argue about a few value numbers. That has really made a differ- vented the zero?’ In that he makes some cru- centuries or years here and there. But like ence to not just mathematics in general and cial distinctions between different concepts of everything else what we want to know or un- practical mathematics but the progress of sci- zero. Are we thinking of zero as just a place derstand is how concepts developed, how ence by enabling calculations more effec- holder in a place value system as there is an things were transmitted, how they evolved. I tively. The place value system had a great empty digit of zero? Or we are still thinking of think a thoughtful adaptation of foreign con- impact on Indian science and then, as it zero as something that plays a certain role in cept is historically just as interesting as the spread, had a great impact on doing science computations? Or are we thinking of zero fully original development. So it is important to elsewhere in the world. integrated as a number, something on which know who did what and when in the sense it you can perform arithmetic operations? If we is important to get all of the history right. Per- But that’s one of the best known examples but go back to the first idea of zero as a place haps we were little over-influenced by the some others, I think, are equally interesting.
Load more

Recommended publications
  • Karen Uhlenbeck Awarded the 2019 Abel Prize
    RESEARCH NEWS Karen Uhlenbeck While she was in Urbana-Champagne (Uni- versity of Illinois), Karen Uhlenbeck worked Awarded the 2019 Abel with a postdoctoral fellow, Jonathan Sacks, Prize∗ on singularities of harmonic maps on 2D sur- faces. This was the beginning of a long journey in geometric analysis. In gauge the- Rukmini Dey ory, Uhlenbeck, in her remarkable ‘removable singularity theorem’, proved the existence of smooth local solutions to Yang–Mills equa- tions. The Fields medallist Simon Donaldson was very much influenced by her work. Sem- inal results of Donaldson and Uhlenbeck–Yau (amongst others) helped in establishing gauge theory on a firm mathematical footing. Uhlen- beck’s work with Terng on integrable systems is also very influential in the field. Karen Uhlenbeck is a professor emeritus of mathematics at the University of Texas at Austin, where she holds Sid W. Richardson Foundation Chair (since 1988). She is cur- Karen Uhlenbeck (Source: Wikimedia) rently a visiting associate at the Institute for Advanced Study, Princeton and a visiting se- nior research scholar at Princeton University. The 2019 Abel prize for lifetime achievements She has enthused many young women to take in mathematics was awarded for the first time up mathematics and runs a mentorship pro- to a woman mathematician, Professor Karen gram for women in mathematics at Princeton. Uhlenbeck. She is famous for her work in ge- Karen loves gardening and nature hikes. Hav- ometry, analysis and gauge theory. She has ing known her personally, I found she is one of proved very important (and hard) theorems in the most kind-hearted mathematicians I have analysis and applied them to geometry and ever known.
    [Show full text]
  • Twenty Female Mathematicians Hollis Williams
    Twenty Female Mathematicians Hollis Williams Acknowledgements The author would like to thank Alba Carballo González for support and encouragement. 1 Table of Contents Sofia Kovalevskaya ................................................................................................................................. 4 Emmy Noether ..................................................................................................................................... 16 Mary Cartwright ................................................................................................................................... 26 Julia Robinson ....................................................................................................................................... 36 Olga Ladyzhenskaya ............................................................................................................................. 46 Yvonne Choquet-Bruhat ....................................................................................................................... 56 Olga Oleinik .......................................................................................................................................... 67 Charlotte Fischer .................................................................................................................................. 77 Karen Uhlenbeck .................................................................................................................................. 87 Krystyna Kuperberg .............................................................................................................................
    [Show full text]
  • Copyright by Magdalena Czubak 2008 the Dissertation Committee for Magdalena Czubak Certifies That This Is the Approved Version of the Following Dissertation
    Copyright by Magdalena Czubak 2008 The Dissertation Committee for Magdalena Czubak certifies that this is the approved version of the following dissertation: Well-posedness for the space-time Monopole Equation and Ward Wave Map. Committee: Karen Uhlenbeck, Supervisor William Beckner Daniel Knopf Andrea Nahmod Mikhail M. Vishik Well-posedness for the space-time Monopole Equation and Ward Wave Map. by Magdalena Czubak, B.S. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN May 2008 Dla mojej Mamy. Well-posedness for the space-time Monopole Equation and Ward Wave Map. Publication No. Magdalena Czubak, Ph.D. The University of Texas at Austin, 2008 Supervisor: Karen Uhlenbeck We study local well-posedness of the Cauchy problem for two geometric wave equations that can be derived from Anti-Self-Dual Yang Mills equations on R2+2. These are the space-time Monopole Equation and the Ward Wave Map. The equations can be formulated in different ways. For the formulations we use, we establish local well-posedness results, which are sharp using the iteration methods. v Table of Contents Abstract v Chapter 1. Introduction 1 1.1 Space-time Monopole and Ward Wave Map equations . 1 1.2 Chapter Summaries . 7 Chapter 2. Preliminaries 9 2.1 Notation . 9 2.2 Function Spaces & Inversion of the Wave Operator . 10 2.3 Estimates Used . 12 2.4 Classical Results . 16 2.5 Null Forms . 18 2.5.1 Symbols .
    [Show full text]
  • Karen Keskulla Uhlenbeck
    2019 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2019 to Karen Keskulla Uhlenbeck University of Texas at Austin “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” Karen Keskulla Uhlenbeck is a founder of modern by earlier work of Morse, guarantees existence of Geometric Analysis. Her perspective has permeated minimisers of geometric functionals and is successful the field and led to some of the most dramatic in the case of 1-dimensional domains, such as advances in mathematics in the last 40 years. closed geodesics. Geometric analysis is a field of mathematics where Uhlenbeck realised that the condition of Palais— techniques of analysis and differential equations are Smale fails in the case of surfaces due to topological interwoven with the study of geometrical and reasons. The papers of Uhlenbeck, co-authored with topological problems. Specifically, one studies Sacks, on the energy functional for maps of surfaces objects such as curves, surfaces, connections and into a Riemannian manifold, have been extremely fields which are critical points of functionals influential and describe in detail what happens when representing geometric quantities such as energy the Palais-Smale condition is violated. A minimising and volume. For example, minimal surfaces are sequence of mappings converges outside a finite set critical points of the area and harmonic maps are of singular points and by using rescaling arguments, critical points of the Dirichlet energy. Uhlenbeck’s they describe the behaviour near the singularities major contributions include foundational results on as bubbles or instantons, which are the standard minimal surfaces and harmonic maps, Yang-Mills solutions of the minimising map from the 2-sphere to theory, and integrable systems.
    [Show full text]
  • Surveys in Differential Geometry
    Surveys in Differential Geometry Vol. 1: Lectures given in 1990 edited by S.-T. Yau and H. Blaine Lawson Vol. 2: Lectures given in 1993 edited by C.C. Hsiung and S.-T. Yau Vol. 3: Lectures given in 1996 edited by C.C. Hsiung and S.-T. Yau Vol. 4: Integrable systems edited by Chuu Lian Terng and Karen Uhlenbeck Vol. 5: Differential geometry inspired by string theory edited by S.-T. Yau Vol. 6: Essays on Einstein manifolds edited by Claude LeBrun and McKenzie Wang Vol. 7: Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer edited by S.-T. Yau Vol. 8: Papers in honor of Calabi, Lawson, Siu, and Uhlenbeck edited by S.-T. Yau Vol. 9: Eigenvalues of Laplacians and other geometric operators edited by A. Grigor’yan and S-T. Yau Vol. 10: Essays in geometry in memory of S.-S. Chern edited by S.-T. Yau Vol. 11: Metric and comparison geometry edited by Jeffrey Cheeger and Karsten Grove Vol. 12: Geometric flows edited by Huai-Dong Cao and S.-T. Yau Vol. 13: Geometry, analysis, and algebraic geometry edited by Huai-Dong Cao and S.-T.Yau Vol. 14: Geometry of Riemann surfaces and their moduli spaces edited by Lizhen Ji, Scott A. Wolpert, and S.-T. Yau Vol. 15: Perspectives in mathematics and physics: Essays dedicated to Isadore Singer edited by Tomasz Mrowka and S.-T. Yau Vol. 16: Geometry of special holonomy and related topics edited by Naichung Conan Leung and S.-T. Yau Vol. 17: In Memory of C. C.
    [Show full text]
  • Karen Uhlenbeck Awarded Abel Prize
    COMMUNICATION Karen Uhlenbeck Awarded Abel Prize The Norwegian Academy of Science and Letters has awarded the Abel Prize for 2019 to Karen Keskulla Uhlenbeck of the University of Texas at Austin, “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” The Abel Prize recognizes contributions of extraordinary depth and influence in the mathematical sciences and has been awarded annually since 2003. It carries a cash award of six million Norwegian krone (approximately US$700,000). Citation detail what happens when the Palais–Smale condition is Karen Keskulla Uhlenbeck is a violated. A minimizing sequence of mappings converges founder of modern geometric outside a finite set of singular points, and, by using resca- analysis. Her perspective has ling arguments, they describe the behavior near the singu- permeated the field and led larities as bubbles or instantons, which are the standard to some of the most dramatic solutions of the minimizing map from the 2-sphere to the advances in mathematics in the target manifold. last forty years. In higher dimensions, Uhlenbeck in collaboration with Geometric analysis is a field Schoen wrote two foundational papers on minimizing of mathematics where tech- harmonic maps. They gave a profound understanding niques of analysis and differ- of singularities of solutions of nonlinear elliptic partial ential equations are interwoven differential equations. The singular set, which in the case Karen Keskulla Uhlenbeck with the study of geometrical of surfaces consists only of isolated points, is in higher and topological problems.
    [Show full text]
  • Spring 2019 Fine Letters
    Spring 2019 • Issue 8 Department of MATHEMATICS Princeton University From the Chair Professor Allan Sly Receives MacArthur Fellowship Congratulations to Sly works on an area of probability retical computer science, where a key the Class of 2019 theory with applications from the goal often is to understand whether and all the finishing physics of magnetic materials to it is likely or unlikely that a large set graduate students. computer science and information of randomly imposed constraints on a Congratulations theory. His work investigates thresh- system can be satisfied. Sly has shown to the members of olds at which complex networks mathematically how such systems of- class of 2018 and suddenly change from having one ten reach a threshold at which solving new Ph. D.s who set of properties to another. Such a particular problem shifts from likely are reading Fine Letters for the first questions originally arose in phys- or unlikely. Sly has used a party invi- time as alumni. As we all know, the ics, where scientists observed such tation list as an analogy for the work: Math major is a great foundation for shifts in the magnetism of certain As you add interpersonal conflicts a diverse range of endeavors. This metal alloys. Sly provided a rigorous among a group of potential guests, it is exemplified by seventeen '18's who mathematical proof of the shift and can suddenly become effectively im- have gone to industry and seventeen a framework for identifying when possible to create a workable party. to grad school; ten to advanced study such shifts occur.
    [Show full text]
  • Copy of Cont2003-2004
    ContinuUM Newsletter of the Department of Mathematics at the University of Michigan 2003-2004 Virginia Young Named to Nesbitt Professorship In 2003 the department welcomed cases, but otherwise require an extensive Virginia (Jenny) Young, who became the analysis to even prove a solution exists. inaugural holder of the Cecil J. and Ethel M. “I am very excited to be in a mathemat- Nesbitt Professorship in Actuarial Math- ics department again; it was one of the ematics. Young is both a Ph.D. and an FSA most attractive aspects of the position,” and is one of the preeminent actuarial re- says Young. “I am looking forward to searchers in the world today. working with the AIM (Applied and Inter- After receiving her Bachelor’s in math- disciplinary Mathematics) program, and ematics from Cumberland College, Young helping to establish a stronger graduate entered the graduate program in mathemat- program presence in financial and actuarial ics at the University of Virginia. She com- mathematics. I want to contribute to the pleted her Ph.D. in just three years with a growth of the actuarial math program here dissertation in algebraic topology. “When I at Michigan, and broaden the scope to in- look back, I now realize that my graduate ca- clude topics in casualty actuarial math. I am reer was too short,” says Young. “How- also looking forward to continuing my re- ever, I had an excellent advisor and mentor search with others in the department like in Robert Strong.” After a two-year Kristen Moore.” postdoctoral fellowship at the Institute for Actuarial Assumptions” with Frees, et al.
    [Show full text]
  • Xssociation for Goomen in (At Matics
    ,_Xssociationfor gOomenin (at matics Volume 20, Number 4 NEWSLETTER July-August 1990 ****************************************************************** PRESIDENT'S REPORT I am looking forward to a lively and interesting panel, "Enrichment programs in urban public schools," this summer in Columbus. The panelists will include Jacqueline Rivers, who runs The Algebra Project in Boston, Paul Sally, who directs the University of Chicago summer program for mathematically talented youth, and Harvey Keynes, who has been involved in an effort to increase the number of girls participating in the University of Minnesota Talented Youth Mathematics Program (UMTYMP). An article by Keynes describing the intervention program appeared in the last issue of the AWM Newsletter. While the latter two programs deal with mathematically talented youth, The Algebra Project, started by Bob Moses in Cambridge, Massachusetts, uses mathematics as a vehicle to raise the level of academic performance in general of inner city youth. Remember, there will be an award ceremony for the Schafer Prize winners at the beginning of the Business Meeting in Columbus, so please come to meet and applaud these talented young women. We would like to increase the endowment so that the prize can be self-supporting, and I hope that you will all help us in this task. I am delighted to congratulate two of our members: Cathleen Morawetz on her election to the National Academy of Sciences, and Lee Lorch on the receipt of an honorary degree from the City University of New York. Rogers Newman, President of the National Association of Mathematicians, sent me a very nice note thanking the AWM for our congratulatory letter on the occasion of their 20th anniversary.
    [Show full text]
  • Nptel Course on Mathematics in India: from Vedic Period
    NPTEL COURSE ON MATHEMATICS IN INDIA: FROM VEDIC PERIOD TO MODERN TIMES Lecture 40 Mathematics in Modern India 2 M. D. Srinivas Centre for Policy Studies, Chennai 1 Outline I Rediscovering the Tradition (1900-1950) I Rediscovering the Tradition (1950-2010) I Modern Scholarship on Indian Mathematics (1900-2010) I Development of Higher Education in India (1900-1950) I Development of Scientific Research in India (1900-1950) I Development of Modern Mathematics in India (1910-1950) I Development of Modern Mathematics in India (1950-2010) I Development of Higher Education in India (1950-2010) I Halting Growth of Higher Education and Science in India (1980-2010) I Halting Growth of Mathematics in India (1980-2010) 2 Rediscovering the Tradition (1900-1950) Several important texts of Indian mathematics and astronomy were published in the period 1900-1950. Harilal Dhruva published the Rekh¯agan. ita, translation of Euclid from Tusi’s Persian version (Bombay 1901). Vindhyesvari Prasad Dvivedi published some of the ancient siddh¯antas in Jyotis.asiddh¯anta-sa_ngraha (Benares 1912). Babuaji Misra edited the Khan. d. akh¯adyaka of Brahmagupta with Amarja¯ ’s commentary (Calcutta 1925) and Siddh¯anta´sekhara of Sr¯ıpati´ with Makkibhat.t.a’s commentary (Calcutta 1932, 47). Padmakara Dvivedi, edited Gan. itakaumud¯ı of N¯ar¯ayan. a Pan. d. ita in two volumes (1936, 1942). Gopinatha Kaviraja edited the Siddh¯antas¯arvabhauma of Mun¯ı´svara, 2 Vols. (Benares 1933, 3); 3rd Vol. Ed. by Mithalal Ojha (Benres 1978) Kapadia edited the Gan. itatilaka of Sr¯ıdhara´ with commentary (Gaekwad Oriental Series 1935) 3 Rediscovering the Tradition (1900-1950) Several important works were published from the Anand¯a´srama¯ Pune: Karan.
    [Show full text]
  • Karen Uhlenbeck Wins Abel Prize Aren Uhlenbeck Is the 2019 Recipient of the Abel Prize
    Karen Uhlenbeck wins Abel Prize aren Uhlenbeck is the 2019 recipient of the Abel Prize. Already in the case dim M = 2, the problem posed considerable The citation awards the prize ‘for her pioneering achieve- challenges: the Palais–Smale condition fails for F, moreover, Kments in geometric partial differential equations, gauge the Euler–Lagrange equation does not enter the framework of theory and integrable systems, and for the fundamental impact classical elliptic partial differential equations. of her work on analysis, geometry and mathematical physics.’ In the 70s, while at Urbana–Champaign, Uhlenbeck devised, She was one of the founders of geometric analysis: how did this in a ground-breaking paper [1] co-authored with J. Sacks, come about? an alternative method to detect harmonic maps (in the case In the mid 60s, Uhlenbeck (then Karen Keskulla) began dim M = 2) by approximation: the key was to consider a se- graduate school at Brandeis University, with Palais as her quence of energies F (u)= u 2. |∇ | adviser; this naturally led her to the calculus of variations. A M fundamental question in this field (among Hilbert’s famous 2 α problems from 1904) is to prove the existence of critical points Fα(u)= (1 + u ) , |∇ | of certain energies (area, elastic energy, etc.). A critical point is M an equilibrium configuration with respect to the given energy; the equilibrium condition is expressed mathematically via the for a > 1; these energies satisfy the Palais–Smale condition Euler–Lagrange equation. and, as a ® 1, they approximate F. The Sacks–Uhlenbeck A classical example is a geodesic on a Riemannian mani- paper constructed a sequence ua of critical points of Fa and fold and, in order to prove the existence of geodesics, it proves effective to consider the energy as a functional on the (infinite-dimensional) ‘space of curves’.
    [Show full text]
  • An Indian Form of Third Order Taylor Series
    HISTORIA MATHEMATICA 1 /1974), 287-289 AN INDIAN FORMOF THIRD ORDERTAYLOR SERIES APPROXIMATIONOF THE SINE BY RADHA CHARAN GUPTA, BIRLA INSTITUTE OF TECHNOLOGY, RANCHI, INDIA SUMMARIES The paper describes an approximation formula for sine (x + h) that differs from the first four terms of the Taylor expansion only by having 4 in place of 6 in the denominator of the fourth term. It appears in Sanskrit stanzas quoted in a work of about the fifteenth century and given here with translation and explanation. Approximation formulas for the sine of x + h equivalent to the first two terms of the Taylor series expansion and to the first three terms were known from at least as early as the tenth century A.D. and the fourteenth century A.D. to Indian mathemati- cians [Sengupta 1932, 5; Gupta 1969, 92-941. An approximation formula equivalent to the first four terms, except that the denominator in the fourth term is 4 instead of 6, appears in a literary form in three stanzas quoted by Parameivara (circa 1360-1455) in his supercommentary, called Siddhlnta-dipikz, on Govindasvtiin's gloss (circa 800-850) on the Mahabhaskariya (about early seventh century) [Kuppanna Sastri 1957, 2051. Below we give the text in transliteration from the Siddhlnta- dipik;i and an almost literal translation of the Sanskrit stanzas. Then follows an explanation of the various steps in the rule laid down in the text. (The Indian sine, or Sine, of an angular arc $I is equal to R sin 9, where sin I+ is the modern sine of the angle 4 and R is the radius of the circle of reference).
    [Show full text]