ISSN: 0025-5742 THE MATHEMATICS STUDENT Volume 77, Numbers 1 - 4, (2008) copy- Edited by J. R. PATADIA circulation (Issued: March, 2009) Member'sfor not PUBLISHED BY THE INDIAN MATHEMATICAL SOCIETY (In the memory of late Professor R. P. Agarwal) THE MATHEMATICS STUDENT Edited by J. R. PATADIA In keeping with the current periodical policy, THE MATHEMATICS STUDENT (STUDENT) will seek to publish material of interest not just to mathematicians with specialized interest but to undergraduate and postgraduate students and teachers of mathematics in India. With this in view, it will publish material of the following type: 1. Expository articles and popular (not highly technical) articles. 2. Classroom notes (this can include hints on teaching certain topics or filling vital gaps in the usual treatments of topics to be found in text-books or how some execises can and should be made an integral part of teaching, etc.) 3. Problems and solutions, News and Views. 4. 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Authors of articles/research papers printed in the STUDENT are chargednot page charges at the rate of US dollars Thirty-five per page (Rs. 100 per page for authors in India). Business correspondence relating to the STUDENT should be addressed to Dr. S. K. Nimbhorkar, Treasurer, Indian Mathematical Society, Dept. of Mathematics, Dr. B. A. M. University, Aurangabad-434 004 (MS), India. ISSN: 0025-5742 THE MATHEMATICS STUDENT Volume 77, Numbers 1 - 4, (2008) copy- Edited by J. R. PATADIA circulation (Issued: March, 2009) Member'sfor not PUBLISHED BY THE INDIAN MATHEMATICAL SOCIETY (In the memory of late Professor R. P. Agarwal) c THE INDIAN MATHEMATICAL SOCIETY, 2009. This volume or any part there of 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 copy- Indian Mathematical Society. Member's copy is strictly for personal use. It is not intended for sale. circulation Member'sfor not Published by V. M. Shah for the Indian Mathematical Society, type set by Sunil B. Thakare, at the Department of Mathematics, University of Pune, Pune-411 007 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 Vol. 77, Nos. 1-4, (2008) CONTENTS M. A. PATHAN: OBITUARY - Professor R. P. Agarwal (1925 - 2008). 01 N. K. THAKARE: R. P. Agarwal (15th June, 1925-9th February, 2008) 09 - A Ratan (Jewel) in mathematics. A. K. AGARWAL: The importance of Mathematics. 13 A. K. AGARWAL: Ramanujan's contributions and some recent advances in the 19 theory of partitions. Abstracts of Papers submitted for presentation at the 74th Annual Conference of The 37 Indian Mathematical Society held at the University of Allahabad, Allahabad{211 002 during December 27{30, 2008. S. G. DANI: On some large subsets of Euclidean spaces involved in geometry, 75 Dynamics and Diophantine approximation. M. RAM MURTY: Transcendental Numbers and Zeta Functions. 87 G. S. SRIVASTAVA: Entire functions extensions of continuous functions - 103 a brief survey. S. ARUMUGAM: Coloring and Domination in Graphs. 119 SUKUMAR DAS ADHIKARI: The relation between two combinatorics 131 group invariants: history and ramifications. copy- S. GADGIL: Orders and surgery on manifolds. 145 G. S. SALUJA: Random fixed point of Three-step random iterative process with 161 errors for asymptotically quasi-nonexpansive random operator. MRIDULA GARG AND TRIPATI GUPTA: On certain new summation formulae for 177 Fox H-function. B. S. K. R. SOMAYAJULU AND KADAMBINI KAR: Some new properties of 183 prime powers. T. S. NANJUNDIAH: The nowhere differentiable continuous functions of 193 Weierstrass and Van der Waerden. circulation ARABINDI SATYA NARAYAN RAO: On the existence of a unique almost periodic 197 solution of a non-linear abstract differential equation. DRAGU ATANASIU AND DENNIS NEMZER: Extending the Laplace Transform. 203 RALPH CHILL: Three variations on Newtons Method. 213 AKHILESHMember's PRASAD: Hankelfor Transform on Gevrey space. 231 not iv S. K. JENA: Method of infinite ascent applied on mA6 + nB3 = C2. 239 V. A. HIREMATH AND S. S. GRAMOPADHYE: Dualization of Cyclic Purity. 247 RAMJI LAL: 74th IMS Conference A Brief Report. 257 IMS Sponsored Lectures. 265 ******* copy- circulation Member'sfor not The Mathematics Student Vol. 77, Nos. 1-4,(2008), 01-08 Ratan Prakash Agarwal (June 15, 1925 - Februarycopy- 9, 2008) OBITUARY Professor Ratan Prakash Agarwal (1925 - 2008) India has produced many mathematicians who worked in the fields of Number theory, Partition theory, Combinatorics and Special functions and made outstanding contributions. Brilliance with numbers is a curious thing. The number theorist of the 20th century,circulation S. Ramanujan is rightly called the gift of India. In the years past, people with a gift for numbers overcame vast odds to find an outlet for their genius. S. S. Pillai and T. Vijayragh- van, the great mathematical wizards of the 20th century are undoubtedly the gift of number theoristsfor of India to the world. India lost one of her brightestMember's jewels in the fields of Special Functions, Number theory and Par- tition theory with the passing away on February 9, 2008, of Professor R. P. Agarwal, formerly Professor and Head of the Department of Mathematics and Astronomy, University of Lucknow and Vice Chancellor of Universities of Rajasthannot and Lucknow. A true visionary who became an institution in his life time and whose work on mock theta and related functions, are com- parable to well known mathematicians of the world which is quite evident from the recognitions and appreciations he received from the world famous mathematicians. Between 1994 and 2003, Indian researchers accounted for 2.21 percent of c Indian Mathematical Society-2009. 1 2 world’s publication of research in mathematics. India had been the largest contributor in the field of special functions and number theory in mathe- matical publications output in the world. Indian researchers have done well in these specific areas and the major reason is because some ‘dedicated’ and ‘monastic’ mathematicians like Prof. R. P. Agarwal who worked with the raw material to produce very different results. One of Prof. Agarwal’s achievements is formation of a school of research in Ramanujan mathemat- ics, q-series, q-transforms, mock theta functions, combinatorics and con- tinued fractions. He was one of the most successful guides. In his school one can find research teams where juniors can freely contradict their seniors in ways that are essential to the scientific temper; dissent is, after all, the beginning of discovery in his opinion. Professor Ratan Prakash Agarwal was born on June 15, 1925 in Basti, U.P. He was Ph.D from Lucknow University(1950) and also from London University(1953) under the supervision of W.copy- N. Baily of Bedford College. He was on the faculty of Lucknow University from 1946 to 1985 save for a break of three years at University of Gorakhpur where he was Head of the Department of Mathematics (1963-66). He was a Fulbright Visiting Research Professor at West Virginia University, USA during 1967-68. He was invited as Visiting Professor by Pennsylvania State University, USA in 1979. On return from University of Gorakhpur to University of Lucknow, Prof Agarwal started collecting a band of enthusiasticcirculation workers around him. He encouraged the teachers in the universities of Gorakhpur and Lucknow to devote them to research. He organized study groups for teachers, seminars and workshops and participated vigorously in them. More than 23 persons including Prof. W. H. Abdi,for Prof Arun Verma, Prof. R. Y. Denis and Prof. M.Member's A. Khan obtained doctorate degrees working under him in different classifications of analysis. He was a rare supervisor who has not bothered to have a joint paper with his Ph.D and D. Sc. students. In course of a few years he published a number of papers which had unmistakable marks of originalitynot in them, and his great power as a mathematician began to be appreciated.