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Math. Student 2021-Part-1 ISSN: 0025-5742 THE MATHEMATICS STUDENT Volume 90, Nos. 1-2, January - June (2021) (Issued: April, 2021) Editor-in-Chief M. M. SHIKARE EDITORS 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 Clare D Cruz L. Sunil Chandran J. R. Patadia C. S. Aravinda A. S. Vasudeva Murthy Indranil Biswas Timothy Huber T. S. S. R. K. Rao Atul Dixit PUBLISHED BY THE INDIAN MATHEMATICAL SOCIETY www.indianmathsociety.org.in THE MATHEMATICS STUDENT Edited by M. M. SHIKARE 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. research papers, 2. 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. 3. general survey articles, popular articles, expository papers and Book-Reviews. 4. problems and solutions of the problems, 5. new, clever proofs of theorems that graduate / undergraduate students might see in their course work, 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 A LTEX and .pdf file including figures and tables to the Editor M. M. Shikare on E-mail: [email protected] along with a Declaration form downloadable from our website. Manuscripts (including bibliographies, tables, etc.) should be typed double spaced on A A4 size paper with 1 inch (2.5 cm.) margins on all sides with font size 11 pt. in LTEX. 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 A format. In LTEX the following preamble be used as is required by the Press: \ documentclass[11 pt,a4paper,twoside,reqno]{amsart} \ usepackage {amsfonts, amssymb, amscd, amsmath, enumerate, verbatim, calc} \ renewcommand{\ baselinestretch}{1.2} \ textwidth=12.5 cm \ textheight=20 cm \ topmargin=0.5 cm \ oddsidemargin=1 cm \ evensidemargin=1 cm \ pagestyle{plain} The details are available on Society’s website: www.indianmathsociety.org.in Authors of articles / research papers printed in the the Mathematics Student as well as in the Journal shall be entitled to receive a soft copy (PDF file) of the paper published. There are no page charges for publication of articles in the journal. All business correspondence should be addressed to S. K. Nimbhorkar, Treasurer, Indian Mathematical Society, C/O Dr. Mrs. Prachi Kulkarni, Ankur Hospital, Tilaknagar, Aurangabad 431 001 (MS), India. E-mail: [email protected] or [email protected] In case of any query, one may contact the Editor through the e-mail. Copyright of the published articles lies with the Indian Mathematical Society. ISSN: 0025-5742 THE MATHEMATICS STUDENT Volume 90, Nos. 1-2, January - June (2021) (Issued: April, 2021) Editor-in-Chief M. M. SHIKARE EDITORS 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 Clare D Cruz L. Sunil Chandran J. R. Patadia C. S. Aravinda A. S. Vasudeva Murthy Indranil Biswas Timothy Huber T. S. S. R. K. Rao Atul Dixit PUBLISHED BY THE INDIAN MATHEMATICAL SOCIETY www.indianmathsociety.org.in ISSN: 0025-5742 ii c THE INDIAN MATHEMATICAL SOCIETY, 2021. This volume or any part thereof should not be reproduced in any form without the written permission of the publisher. This volume is not to be sold outside the 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 circulation. Published by Prof. Satya Deo for the Indian Mathematical Society, type set by M. M. Shikare, “Krushnakali”, Survey No. 73/6/1, Gulmohar Colony, Jagtap Patil Estate, Pimple Gurav, Pune 411061 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. 90, Nos. 1-2, January-June (2021) CONTENTS 1. B. Sury View of mathematics by our society 1– 8 and what our role could be 2. B. Sury Linear algebraic groups over 9–31 global and local fields-some themes 3. Dinesh Khurana Some glimpses into noncommutative 33–44 ring theory 4. Sanoli Gun Large Fourier coefficients of modular forms 45–61 5. Mythily An invitation to differential equations 63–75 Ramaswamy 6. Khushbu Das Recursive subsequence of various 77– 84 Devbhadra Shah Fibonacci-type sequences 7. Rakesh Barai On decomposition of a rational prime 85–102 in a cubic field 8. A. D. Godase Hyperbolic k-Fibonacci and k-Lucas 103–116 quaternions 9. D. Barman Unique common fixed point results 117–132 K. Sarkar and of integral type contraction condition K. Tiwary in 2-Banach spaces 10. S. B. Patil and Initial value problem for nth order 133–141 D. S. Palimkar random differential inclusion 11. Amrik Singh Extensions of Ramanujan’s three series 143–153 1 Nimbran for π and related series 12. Arpita Ghosh and A contractible graph with eleven vertices 155–159 Surojit Ghosh and no gluable edge 13. N. A. Malik and A new approach to good matrices 161–175 H. Mahato 14. B. G. Khedkar Complex inversion formula for 177–182 S. B. Gaikwad modified - Stieltjes transform iv Problems and Solutions 15. - Problem Section 183– 192 ******* The Mathematics Student ISSN: 0025-5742 Vol. 90, Nos. 1-2, January-June (2021), 1–8 VIEW OF MATHEMATICS BY OUR SOCIETY AND WHAT OUR ROLE COULD BE* B SURY It is a great honor to have been elected to the office of the President of the Indian Mathematical Society. The genesis of the Indian Mathematical Soci- ety has the following interesting history. On April 4, 1907, V. Ramaswamy Aiyar and 20 founding members founded the ‘Analytic club’ on the lines of the Edinburgh Mathematical society. The idea was put forth in a letter in December 1906; note that Einstein’s theory of relativity and Ramanujan’s works came to the world’s notice in 1906 - a golden year indeed. The IMS is the oldest mathematical society of our country, and has had such luminaries as Vijayaraghavan, Vaidyanathaswamy, Hansraj Gupta and P L Bhatnagar among its Past Presidents. I am also grateful for the opportunity to ad- dress this august gathering. Even though the pandemic has not allowed us to meet in person, it could be a blessing in disguise as perhaps more people can virtually participate. I would like to share my thought about the following three aspects of Math- ematics: (i) How Indian society views mathematics, and what we should or could do to effectively create awareness among the public to the importance of this discipline; (ii) Ways and means of motivating students who are not particularly inter- ested in the subject due to the way it is being taught; and (iii) (This has to do with the mathematical community) What might the future scenario of research in mathematics look like. ∗ This article is based on the text of the Presidential Address (General) given by Prof. B. Sury at the 86th Annual Conference of the IMS-An International meet held at Vellore Inst. of Tech., Vellore (T. N.) using online mode during December 17-20, 2020. © Indian Mathematical Society, 2021 . 1 2 B SURY Before talking about these three points, let us recognize that mathematics is a social enterprise that is expected to further human goals and aspira- tions. A mathematical conference is not only about discussions that are mathematical, but the purpose is also to socialize, reinvigorate friendships, and engage in informal conversations. The point is to remind ourselves that it is about being part of a community that is united by its love for, and belief in, the importance of mathematics. Irrespective of whether our interest is in teaching, or research or reaching out to society, all of us are essential to the continued health of our discipline. In that aspect, an online meeting falls woefully short. The belief is that a good education in math- ematics benefits the individual as well as society, but the benefits should include empowerment of the individual, development of analytical/logical thinking and also gains that are practical such as the support of scientific and business aspirations. Aspect I : Our society’s view of mathematics The view of the mathematical layman towards practising mathematicians like us is based mainly on our holding positions in institutions rather than actual appreciation - this is understandable. It is akin to what many math- ematicians think of Ramanujan. Their appreciation is mostly based on reading encomiums about Ramanujan by some top mathematicians, cou- pled with personal aspects as to how he faced hardships, poverty etc. An appreciation of his original ideas and an understanding of what Ramanujan actually did and how fertile his mind was, is only by a small minority. Many people think of mathematicians as people who can do fast calcula- tions and always deal with numbers. My favorite reply to this often is that the only numbers I may come across in my research are page numbers of manuscripts! I am sure that most of us face the following experience. In any general get-together, when people hear about your profession, they say, “Oh, I was so bad at mathematics! I just managed to scrape through my exams." We are led to think that they perceive us as higher entities of some sort.
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