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Math Student-Part-1-2017 ISSN: 0025-5742 THE MATHEMATICS STUDENT Volume 86, Numbers 1-2, January-June (2017) (Issued: May, 2017) Editor-in-Chief J. R. PATADIA EDITORS Bruce C: Berndt George E: Andrewscopy- M: Ram Murty N: K: Thakare Satya Deo Gadadhar Misra B: Sury Kaushal Verma Krishnaswami Alladi S: K: Tomar Subhash J: Bhatt L: Sunil Chandran M: M: Shikare C: S: Aravinda A: S: Vasudeva Murthy Indranil Biswas Timothy Huber T: S: S: R: K: Rao Clare D0Cruz Atul Dixit circulation for Member's not PUBLISHED BY THE INDIAN MATHEMATICAL SOCIETY www.indianmathsociety.org.in THE MATHEMATICS STUDENT Edited by J. R. PATADIA In keeping with the current periodical policy, THE MATHEMATICS STUDENT will seek to publish material of interest not just to mathematicians with specialized interest but to the postgraduate students and teachers of mathematics in India. With this in view, it will ordinarily publish material of the following type: 1. the texts (written in a way accessible to students) of the Presidential Addresses, the Plenary talks and the Award Lectures delivered at the Annual Conferences. 2. general survey articles, popular articles, expository papers, Book-Reviews. 3. problems and solutions of the problems, 4. new, clever proofs of theorems that graduate / undergraduate students might see in their course work, 5. research papers of interest to larger readership, and 6. articles that arouse curiosity and interest for learning mathematics among readers and motivate them for doing mathematics. Articles of the above type are invited for publication in THE MATHEMATICS STUDENT. Manuscripts intended for publication should be submitted online in the LATEX and .pdf file including figures and tables to the Editor J. R. Patadia on E-mail: [email protected] Manuscripts (including bibliographies, tables, etc.) should be typed double spaced on A4 size paper with 1 inch (2.5 cm.) margins on all sides with font size 10 pt. in LATEX. Sections should appear in the following order: Title Page, Abstract, Text, Notes and References. Comments or replies to previously published articles should also follow this format with the exception of abstracts. In LATEX the following preamble be used as is required by the Press: copy- n documentclass[10 pt,a4paper,twoside,reqno]famsartg n usepackage famsfonts, amssymb, amscd, amsmath, enumerate, verbatim, calcg n renewcommandfn baselinestretchgf1.2g n textwidth=12.5 cm n textheight=20 cm n topmargin=0.5 cm n oddsidemargin=1 cm n evensidemargin=1 cm circulation n pagestylefplaing The details are available on Indian Mathematical Society website: www.indianmath society.org.in Authors of articles / researchfor papers printed in the the Mathematics Student as well as in the Journal shall be entitled to receive a soft copy (PDF file with watermarked \Author's copy")Member'sof the paper published. There are no page charges. However, if author(s) (whose paper is accepted for publication in any of the IMS periodicals) is (are) unable to send the LATEX file of the accepted paper, then a charge Rs. 100 (US $ 10) per page will be levied for LAnotTEX typesetting charges. All business correspondence should be addressed to S. K. Nimbhorkar, Treasurer, Indian Mathematical Society, Dept. of Mathematics, Dr. B. A. M. University, Aurangabad - 431 004 (Maharashtra), India. E-mail: [email protected] Copyright of the published articles lies with the Indian Mathematical Society. In case of any query, the Editor may be contacted. ISSN: 0025-5742 THE MATHEMATICS STUDENT Volume 86, Numbers 1-2, January-June, (2017) (Issued: May, 2017) Editor-in-Chief J. R. PATADIA EDITORScopy- Bruce C: Berndt George E: Andrews M: Ram Murty N: K: Thakare Satya Deo Gadadhar Misra B: Sury Kaushal Verma Krishnaswami Alladi S: K: Tomar Subhash J: Bhatt L: Sunil Chandran M: M: Shikare C: S: AravindacirculationA: S: Vasudeva Murthy Indranil Biswas Timothy Huber T: S: S: R: K: Rao Clare D0Cruz Atul Dixit for Member's not PUBLISHED BY THE INDIAN MATHEMATICAL SOCIETY www.indianmathsociety.org.in ISSN: 0025-5742 ii c THE INDIAN MATHEMATICAL SOCIETY, 2017. This volume or any part thereof may not be reproduced in any form without the written permission of the publisher. This volume is not to be sold outside thecopy- Country to which it is consigned by the Indian Mathematical Society. Member's copy is strictly for personal use. It is not intended for sale or circular. circulation for Member's Published by Prof. N. K. Thakare for the Indian Mathematical Society, type set by J. R. Patadianot at 5, Arjun Park, Near Patel Colony, Behind Dinesh Mill, Shivanand Marg, Vadodara - 390 007 and printed by Dinesh Barve at Parashuram Process, Shed No. 1246/3, S. No. 129/5/2, Dalviwadi Road, Barangani Mala, Wadgaon Dhayari, Pune 411 041 (India). Printed in India. The Mathematics Student ISSN: 0025-5742 Vol. 86, Nos. 1-2, January-June, (2017) CONTENTS 1. D. V. Pai Road to Mathematical Sciences in India - a relook 01-10 2. D. V. Pai Some highlights of our research contributions 11-29 3. Ravindra K. On power contractions and discontinuity at fixed 31-37 Bisht point 4. Kannappan Norm or exception? 39-50 Sampath and B. Sury 5. Diyath Nelaka Cantor-like sets constructed in Tremas 51-54 Pannipitiya 6. Saranya G. Nair Fibonacci sequence with applications and extensions 55-62 and T. N. Shorey 7. B. V. Nathwani Generalized Mittag-Leffler function and its proper- 63-76 and B. I. Dave ties 8. Akash Jena and Revisiting Eisenstein-type criterion over integers 77-86 Binod Kumar Sahoo copy- 9. Saranya G. Nair Generalized Laguerre polynomials with applications 87-101 and T. N. Shorey 10. Arpita Kar Representations of numbers 103-113 11. George E. Eulers Partition Identity and Two Problems of 115-119 Andrews George Beck 12. M. Ram Murty The Art of Researchcirculation121-131 13. Tim Huber and Weierstrass interpolation of Hecke Eisenstein series 133-138 Matthew Levine 14. M. Ram Murty Evaluation of the quadratic Gauss sum 139-150 and Siddhi Pathak for Member's15. - Problem Section 151-157 not ******* copy- circulation for iv Member's not The Mathematics Student ISSN: 0025-5742 Vol. 86, Nos. 1-2, January-June (2017), 01-10 ROAD TO MATHEMATICAL SCIENCES IN INDIA - A RELOOK* D. V. PAI Abstract. By mathematical sciences, we would understand here a broad spectrum of knowledge that encompasses pure mathematics, applied math- ematics, statistics, mathematics of OR, computational mathematics, math- ematical physics, mathematical biology, mathematical economics, etc. One aims in this talk to take a relook at the road to mathematical sciences tra- versed in India since antiquity, with a view to try to gain an understanding about the current status of this domain in modern times. 1. Early Mathematics in India-Some Historical Observations The road to mathematical sciences in India began from antiquity. India has every reason to feel proud of its rich heritage in mathematics and astronomy. The roots of mathematics in India are visible in the vedic literature which is nearly 3500-4000 years old. The sulba sutras - the vedic texts for consruction of ritual altars contain a lot of geometric results andcopy- constructions which include, among others, a statement of the Pythagoras theorem, approximation of the number `pi', approximation of square root of number two upto five decimal places, etc. Undoubtedly, two of the most striking contributions of ancient Indian mathematics are the decimal place value system, which seems to have been discovered as early as in the Harappan period, and the number zero, `Sunya', which could be considered as a profound gift of India to the mankindcirculation in the domains of mathematics and philosophy. To put this in the words attributed to Laplace: \The ingenious method of expressing every possible number using a set of ten symbols (each having a place value and an absolute value) emerged in India. Its simplicity lies in the wayfor it facilitated calculation and placed arithmetic fore- mostMember's amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond two of the greatest men of antiquity-Archimedes and Apollonius"; and the next quote from Bourbaki [7], p. 46: not * The text of the Presidential Address (general) delivered at the 82nd Annual Conference of the Indian Mathematical Society held at the University of Kalyani, Kalyani-741 235, Nadia, West Bengal, India during December 27 - 30, 2016. c Indian Mathematical Society, 2017 . 1 2 D. V. PAI \.......It must be noted moreover that the conception of zero as a number (and not as a simple symbol of separation) and its introduction into calculations also count amongst the original contributions of the Hindus." Indeed, it is often said that the early seeds, sown in India, led through the efforts of the Arabs, to the revival of mathematics in Europe, the Middle East and even China. Chronologically, next comes mathematics of Jains, the so-called Jaina mathematics, whose characteristic was fascination for large numbers and attempts to understand various types of infinities and infinitecimals. Buddhists also under- stood infinite and indeterminate numbers. The apex of mathematical achievements of ancient India occured during the so-called classical period of Indian mathe- matics, which saw the legendary contributions of the mathematician-astronomers: Aryabhatta (476-550), Varahamira (505-587), Brahmagupta (598-670), Bhaskara (600-680) and Bhaskaracharya (1114-1185). The schools established by some of them are well acknowledged universally. Much later, during the 16th century came the flourishing school of Kerala, whose prominant mathematicians were Madhava, Parameshvara, Nilkantha, Jyestadevan and Achyutan. It has been recognized since the early forties that this school has anticipated by more than 200 years (albeit, with less of rigour than their western counterparts), a number of mathematical results in analysis involving infinite series (such as the Gregory-Newton series for the inverse tangent, etc.) and calculus, whichcopy- were later invented by Newton and Leibniz in the 18th century.
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