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Proceedings of Telangana Academy of Sciences TAS Special Issue Vol.1, Issue 1, 2020 Mathematical Sciences Frontiers in Mathematics Guest Editor Prof. B. Sri Padmavati School of Mathematics and Statistics University of Hyderabad Hyderabad - 500046, TS, India TAS Telangana Academy of Sciences Executive Council - 2018-2020 Editorial Board Prof. K. Narasimha Reddy President Editor of Publications (Former V.C., M. G Univ., Nalgonda) Dr. S. M. Reddy H.No: 12-13-1126/B-305 Prof. P. VenugopalReddy Vice-President Kalakriti Tarnaka - 500 017, (Dept. of Physics, OU, Hyderabad) [email protected]., Prof.Ch. SanjeevaReddy Vice-President Dr. G.V. Madhava Sharma (Dept. of Chemistry, K.U. Warangal) Chief Scientist, CSIR-IICT, Hyderabad-500 007 [email protected] Dr. S. Chandra Sekhar Hon. Secretary (Director, CSIR-IICT, Hyderabad) Dr.G. Bhanuprakash Reddy Scientist-F, NIN, Hyderabad - 500007 Dr. Y. Purushotham Hon. Treasurer [email protected] (C-MET, Hyderabad) Dr. M. Sujatha Prof. S.M. Reddy Editor of Publications Pr. Scientist, Oil seeds Research, (Dept. of Botany, K.U. Warangal) Hyderabad - 5000030 [email protected] Executive Council Prof D. Prasada Rao Chairman, Indo-US Super Specialty Hospital, Prof. G. Bagyanarayana Hyderabad [email protected] (Former VC, Palamuru University) Dr. G. Bhanuprakash Reddy Prof. V. Malla Reddy H.No.-1-9-30(C-77), National Institute of Nutrition (NIN), Hyderabad Ravindranagar Colony, Road No - 1, Hyderabad - 500 007 Dr. V. Chakravarthi [email protected] (University of Hyderabad, Hyderabad) Dr. G. Madhusudhana Reddy Dr. G. Madhusudhana Reddy Scientist G, DMRL, Hyderabad-500 058 (Outstanding Scientist & Director, DMRL, Hyderabad) gmreddy [email protected] Dr. G. Sundararajan Dr. G Parthasarathy (IIT-Madras, Hyderabad) Chief Scientist, NGRI, Hyderabad - 500 007 [email protected] Dr. G. Parthasarathy (NGRI, Hyderabad) Prof. M. Ghanashyam Krishna School of Physics, UOH, Hyderabad - 500 046 Dr. K. Jeevan Rao [email protected] (PJTSAU, Hyderabad) Prof Bir Bahadur Prof. K. Janardhan Reddy The Odyssey 4a, 10-2-289/46, (Dept. of Botany, OU, Hyderabad) Shanthinagar, Masab Tank, Hyderabad - 500 028 [email protected] Dr.R.B.N. Prasad Ex-Officio Members Chief Scientist, IIct Hyderabad - 500007 [email protected] Dr. Ch. Mohan Rao (Former Director, CSIR-CCMB, Hyderabad) Dr. B. Sri Padmavati Professor, Principal Secretary School of Mathematic & Statistics, University of (Higher Education Dept. Govt. of Telangana) Hydebad, Gachinowli, Hyderabad - 500046. Email: [email protected] Telangana Academy of Sciences Regional Coordinators Hyderabad & Ranga Reddy Karimnagar & Adilabad Nizamabad & Medak Prof. K. Janardhan Reddy Prof. M.Vithal Dr. Y. Purushotham H.No.11-14- 234/B3, Road No.11, Dept. of Chemistry, Scientist 'D' C- MET, Hyderabad-051 Haripriya Colony, Saroornagar, Hyderabad-500 035. Osmania University, Hyderabad-500 007 [email protected] [email protected] [email protected] Warangal & Khammam Nalgonda & Mahboobnagar Prof. S. M. Reddy Dr. G. Bhanuprakash Reddy H.No: 12-13-1126/B-305 Kalakruti, Tarnaka - 500017, Scientist-F, NIN, Hyderabad - 500007 [email protected] geereddy @yahoo.com FOREWORD The Telangana Academy of Sciences (TAS) has been engaged in the Science popularization activities including Publications, Journals and Special Issues on different occasions. The recent publication activities include “Diabesity: The Unacknowledged Indian Knowledge”, “Jivanayanam lo Rasayanalu “ and “21st Century Noble Awards in Chemistry”. As a part of its continued efforts in this direction, a Special Issue of the Proceedings of the Telangana Academy of Sciences entitled ''Frontiers In Mathematics'' has been brought out now with Prof. B. Sri Padmavati, School of Mathematics and Statistics, University of Hyderabad as a Guest Editor. This issue contains different research topics of Mathematics with leading authors drawn from different national and international Centers of Excellence including California Institute of Technology, Emory University of USA, besides premier institutions like TIFR, Chennai Mathematical Institute, University of Hyderabad, IIT Delhi- Kanpur- Palakkad-Kharagpur-Hyderabad , University of Mumbai, Indian Statistical Institute, Kolkata. I hope this Proceedings will serve as a pace setter for detailed investigations in Mathematics and Applied Mathematics and be beneficial for the young researchers in the field and also for aspiring mathematicians. I express my gratitude to Prof. B. Sri Padmavati, Guest Editor of the special issue and to Prof. S.M. Reddy, Editor of Publications for bringing out such an excellent special issue entitled “Frontiers in Mathematics”. S. Chandrasekhar Hon. Secretary, TAS From the Guest Editor It gives me immense pleasure to bring out this special issue of the Proceedings of Telangana Academy of Sciences entitled “Frontiers in Mathematics” as a Guest Editor. This issue includes articles by some very eminent and senior mathematicians and also by some promising young mathematicians. This issue contains and assortment of topics in mathematics and applied mathematics that will appeal to serious researchers in quest of original results, to aspiring mathematicians trying to get a flavour of some research topics in mathematics and also to those who just enjoy and love reading about mathematics and mathematicians. I thank all the authors who contributed their original results, all those who contributed expository articles and also those who wrote some other very interesting, insightful and illuminating articles for this special issue. I also thank all the reviewers who spared their precious time to send their valuable comments and suggestions to the authors. Prof. B. Sri Padmavati Guest Editor From the Editor of Publications Telangana Academy of Sciences, the erstwhile AP Akademi of Sciences, is one of the oldest of the state Academies of the country, and has been serving the cause of science in particular and society at large in a befitting manner. It has been publishing scientific documents of international, national and regional relevance since its inception in 1963. It has been publishing the official journal of the academy covering the areas of Life Sciences, Chemical, Physical, Engineering and Earth Sciences. In view of its long standing contributions, we are making earnest efforts to make it vibrant which can bring name and fame to the Telangana Sciences in particular and to India at large. India has made great strides in different fields of sciences and made remarkable efforts to realize different revolutions such as green, white, blue, yellow and rainbow which made India one among different countries in the world to become self-sufficient in agricultural products. Even though our R&D contributions in the world stand at 3.2%, no institute stands in the list of top 200 research organizations in the world, even though the contributions of some individuals are of high standard. There is a need for concerted efforts in basic research which can be translated into world class technologies. The Academy has initiated printing of special issues and books on popular sciences in which established researchers were requested to contribute for this purpose. We are now ready with some special issues including the present one on ‘Mathematics‘. I thank Prof. B. Sri Padmavati, Guest Editor, and all the contributors for their valuable articles. We also look forward to the whole hearted co-operation from the Fellows of TAS through their contributions of high quality scientific and research articles for publication in these Proceedings. S.M. Reddy Editor of Publications, TAS Contents 1 Some Highlights of Indian Contributions to Mathematics in the 20th Century. M.S.Raghunathan, CEBS, Mumbai. 1 2 Diophantine Equations : An Introduction. Dinakar Ramakrishnan, California Institute of Technology, U.S.A. 12 3 Fermat's Last Theorem in a Polynomial Ring. Rajat Tandon, Hyderabad. 22 4 The Folding Mathematics. Archana S. Morye, University of Hyderabad. 26 5 Enumeration of Groups in Varieties of A-groups: A Survey. Geetha Venkataraman, Ambedkar University Delhi. 38 6 Local Global Principle for Quadratic Forms. V.Suresh, Emory University, U.S.A. 46 7 On the Non-Vanishing of the Fourier Coefficients of Primitive forms. Tarun Dalal and Narasimha Kumar, IIT Hyderabad. 52 8 A Partial Survey on Some Characteristic p Invariants. V. Trivedi, Tata Institute of Fundamental Research, Mumbai. 65 9 Symbolic Powers, Set-Theoretic Complete Intersection and Certain Invariants. Clare D'Cruz, Chennai Mathematical Institute. 75 10 Differential Equations and Monodromy. T.Venkataramana, Tata Institute of Fundamental Research, Mumbai. 85 11 Zabreiko's Lemma: Unified Treatment of Four Fundamental Theorems in Functional Analysis. S. Kumaresan, Indian Institute of Technology Kanpur. 103 12 Gromov's Theory of h-Principle. Mahuya Datta, ISI Kolkata. 113 13 Extendable Continuous Self Maps and Self Homeomorphisms on Subsets of !2. P. Chiranjeevi, University of Hyderabad. 124 14 Gap Formula for Symmetric Operators. S. H. Kulkarni, IIT Palakkad and G. Ramesh, IIT Hyderabad. 129 15 Fixed Point Theorems and Applications to Fluid Flow Problems. G P Raja Sekhar and Meraj Alam, IIT Kharagpur. 134 16 Geometrical Acoustic Waves in van der Waals Gases. K Ambika, Nirma University, Ahmedabad and R Radha, University of Hyderabad. 147 17 A Survey of Age-Structured Population Models in Population Dynamics. Joydev Halder and Suman Kumar Tumuluri, University of Hyderabad. 156 18 On a Question
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