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Member's Copy - Not for Circulation the MATHEMATICS STUDENT Edited by M ISSN: 0025-5742 THE MATHEMATICS STUDENT Volume 88, Nos. 1-2, January-June (2019) (Issued: May, 2019) 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 Subhash J: Bhatt L: Sunil Chandran J: R: Patadia C: S: Aravinda A: S: Vasudeva Murthy Indranil Biswas Timothy Huber T: S: S: R: K: Rao Clare D0Cruz Atul Dixit PUBLISHED BY THE INDIAN MATHEMATICAL SOCIETY www.indianmathsociety.org.in (Dedicated to Prof. N. K. Thakare in honour of his long and dedicated services to the Indian Mathematical Society) Member's copy - not for circulation 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, 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 LATEX 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 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. In LATEX the following preamble be used as is required by the Press: n documentclass[11 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 n pagestylefplaing 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 with watermarked \Author's copy") of 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 towards LATEX typesetting. 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, the Editor may be contacted. Copyright of the published articles lies with the Indian Mathematical Society. Member's copy - not for circulation ISSN: 0025-5742 THE MATHEMATICS STUDENT Volume 88, Nos. 1-2, January-June (2019) (Issued: May, 2019) 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 Subhash J: Bhatt L: Sunil Chandran J: R: Patadia C: S: Aravinda A: S: Vasudeva Murthy Indranil Biswas Timothy Huber T: S: S: R: K: Rao Clare D0Cruz Atul Dixit PUBLISHED BY THE INDIAN MATHEMATICAL SOCIETY www.indianmathsociety.org.in (Dedicated to Prof. N. K. Thakare in honour of his long and dedicated services to the Indian Mathematical Society) Member's copy - not for circulation ISSN: 0025-5742 ii c THE INDIAN MATHEMATICAL SOCIETY, 2019. 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 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 Satya Deo for the Indian Mathematical Society, type set by J. R. Patadia 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. Member's copy - not for circulation The Mathematics Student ISSN: 0025-5742 Vol. 88, Nos. 1-2, January-June, (2019) CONTENTS A: Articles on Professor N. K. Thakare 1. S. K. Nimbhorkar Professor N. K. Thakare: A brief mathemat- 01{15 M. P. Wasadikar ical profile and M. M. Pawar 2. Satya Deo Professor N. K. Thakare: On his retirement from the Indian Mathematical Society 17{20 3. A. K. Agarwal Professor N. K. Thakare: some reminiscences 21{24 4. J. R. Patadia Professor N. K. Thakare: through my eyes 25{26 B: Articles based on Pres. address, ple- nary lecture and Award Lectures 5. Sudhir Ghorpade Presidential Address - General 27-35 6. Steven Dale Resolution of singularities and local uniform- 37-48 Cutkosky ization 7. Manoranjan Hydrodynamic instabilities in porous media 49-66 Mishra flows 8. S. S. Khare Trends in Bordism, G-Bordism theory and 67-90 fixed point set: a survey C: Expository articles, classroom notes, etc. 9. R. P. The space C[0; 1] can not be locally compact: 91-93 Pakshirajan a measure theoretic proof 10. Prasanna Kumar The Sendov conjecture for polynomials 95-112 11. M. Ram Murty A simple proof that ζ(2) = π2=6 113-115 D: Research papers 12. Amrik Singh Some series involving the central binomial 117-124 Nimbran and coefficient with sums containing π2=5 Paul Levrie 13. Amrik Singh Sums of series involving central binomial 125-135 Nimbran coefficients and harmonic numbers 14. Avijit Sarkar and On N(κ)-para conpact 3-manifolds with 137-145 Rajesh Mondal Ricci solitons 15. Abhijit A. J. and Number of non-primes in the set of units 147-152 A. Satyanarayana modulo n Reddy Member's copy - not for circulation iv 16. Md. Yasir Feroz Instability of Lagrange interpolation on 153-158 Khan, R. Biswas H1(Ω) \ C(Ω) and R. C. Sau 17. Shane Chern On the conjecture of George Beck. II 159-164 18. S. K. Roushon A sufficient condition for a 2-dimensional 165-171 orbifold to be good 19. R. P. Pakshirajan On the law of the iterated logarithm for the 173-177 iterated brownian sequence E: Problems and Solutions 19. - Problem Section 179-188 ******* Member's copy - not for circulation The Mathematics Student ISSN: 0025-5742 Vol. 88, Nos. 1{2, January-June (2019), 01{15 PROFESSOR N. K. THAKARE: A BRIEF MATHEMATICAL PROFILE∗ SHRIRAM K. NIMBHORKAR, MEENAKSHI P. WASADIKAR AND MADHUKAR M. PAWAR Professor Nimba Krishna Thakare was born on April 22, 1938 at the village Morane, Dist. Dhule in Maharashtra State, India. After completing his school edu- cation in Dhule, he completed graduation from Fergusson College, Pune and M. Sc. (Mathematics) from the Royal Institute of Science (Govt. Institute of Science), Bom- bay in 1962. After M. Sc. he served in some col- leges under Bombay university, including, its P. G. Centre at Panaji, Goa. He com- pleted Ph.D. (Mathematics) in 1972 under the guidance of Late Prof. B. R. Bhonsle, in Special Functions. He was the first researcher to receive the Ph. D. degree in Mathematics from Shivaji University, Kolhapur. Days at Shivaji University (1969{1977): Prof. Thakare joined Shivaji University in 1969. During his tenure at Shivaji University he introduced several new courses for M. Sc. students such as Banach Algebra, Distribution Theory, Lattice Theory, Topological Vector Spaces and Lie theoretic approach to Special Function Theory. In fact, he started researches in several areas such as orthogonal polynomials, bi-orthogonal polynomials, Geometry of Banach spaces, Operator theory and Lattice theory. His general approach for research was to tackle the open problems posed by eminent mathematicians in that particular area. ∗ This article is written on the occasion of retirement of Prof. N. K. Thakare as the General Secretary of the Indian Mathematical Society. 1 Member's copy - not for circulation 2 S. K. NIMBHORKAR, M. P. WASADIKAR AND M. M. PAWAR R. Askey, [J. Math. Anal. Appli., 24(1968), 677{685] queried: \ Whether dual series equations involving Jacobi polynomials of differ- ent indices could be solved. " N. K. Thakare [Zentl. Aung. Math. Mech. 54(1974), 283{284] explic- itly solved the dual series equations involving Jacobi polynomials of differ- ent indices by modifying Noble's multiplying factor technique. K. R. Patil and N. K. Thakare [J. Math. Phy., 18(1977), 1724{1726] gave solutions to the dual series equations involving Laguerre - Konhauser polynomials of the first kind and of different indices. N. K. Thakare and B. K. Karande obtained several q-identities and gave solutions to many problems posed by L. Carlitz. Prof. Thakare's substantial contribution is the unification of all the classical orthogonal polynomials; namely the Jacobi polynomials, the La- guerre polynomials and the Hermite polynomials.
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