Introduction

Professor Abul Hasan Siddiqi is a leading spirit behind the satellite conference ICM-2010 on “Mathematics in Science and Technology” 15-17 August 2010 and the workshop on “Inverse Problems and wavelets with applications to real world systems” on 14 August 2010. He is the founder secretary of Indian Society of Industrial and Applied Mathematics and has done yeoman service in establishing and expanding its activities. In this article we describe about his academic collaboration in different countries of the world and the areas of mathematics he has endeavoured to promote along with resume of his academic contributions.

He has collaborated at different levels with researchers of various areas in countries such as Germany, USA, Canada, France, Italy, Belgium, Hungary, Turkey, Saudi Arabia, Uzbekistan, Algeria, Iran, Oman, and Kuwait. We would also like to mention his research contributions and something about his books and edited proceedings of International conferences held at different places. He has visited and served more than 20 countries between 1971 and 2010. He was an active faculty member of Aligarh Muslim University (AMU) holding various important positions including Chairman, Dean, Pro-Vice-Chancellor and Acting Vice-Chancellor. His research collaborators are holding important positions in different parts of the world. It is a rare honour for any individual that his name is recommended thrice during the period of nine years for the highest administrative position of A.M.U by the executive council.

Professor Siddiqi started his research career at Aligarh Muslim University, India; one of the oldest central university in India established in 1920. It may be recalled that well known mathematicians such as Andre Weil, D.D. Kusambhi, S.M. Shah, J.A. Siddiqi, U.N. Singh, T. Hussain, Q.I. Rehman, J.S. Rastogi were faculty members of this university. Faculty members and old boys of this university have always been in the forefront of national development and some of them have held very important positions for example, Dr. Zakir Hussain was the third President of India, the present Vice-President (Mr. Hamid Ansari) and UPSC Chairman (Prof. D.P. Agarwal) are old boys of AMU.

Academic collaboration

In the initial stage of his research work he benefited a lot from late Prof. Jamil A. Siddiqi, Prof. S.M. Mazhar (now in Kuwait University) and Prof. Q.I. Rehman (now in Montreal University). In June 1971 he was awarded German Academic Exchange programme fellowship to work with Prof. Horst Leptin at Heidelberg University (Angewandte Institute of Mathematik). While in Heidelberg he learnt contemporary trends in abstract Harmonic analysis. He also availed his visit to work with Dr. S. Gaehler of German Academy of Sciences, Berlin to learn 2-dimensional analogues of Banach and metric spaces known as 2-metric and 2-Banach spaces. After return from Germany, he delivered lectures on these themes in different conferences in India and Iran. Many Indian and Iranian researchers started working in the areas of 2-metric and 2-Banach spaces. He jointly with S. Gaehler and S.C. Gupta published a paper in Mathematische Nachrichten on “Contributions to Non-Archimedean Functional Analysis”, besides several papers joint with Prof. Esfahanizadeh, (Tabriz), Ahmad Khan, Ataullah Siddiqi, Mohiuddin (India), Mohammad Zadeh, Subhan Allahi (from Tabriz University). Prof. Siddiqi was visiting Professor of Tabriz University, Iran from Dec. 1972 to August 1975, supervised the work of M.S. Dissertation and initiated the research programme. He himself benefitted from the contact of distinguished mathematicians invited in the annual conferences of Iranian mathematical Society in 1973, 1974, 1975. Between 1975 and 1980 Professor Siddiqi took effective steps to propagate research activities in 2-Normed spaces, Non-Archimedean Functional Analysis and Walsh Fourier Analysis. To achieve this goal he invited Prof. Grand de Kimpe of Belgium to AMU who spent a month in India and delivered series of lectures.

In the area of Walsh-Fourier analysis he collaborated with a distinguished Professor of Electrical Engineering Prof. M.U. Siddiqi in A.M.U. who was subsequently HOD of Electrical Engineering of IIT Kanpur and now in Malaysia. He learnt from him application of Walsh-Fourier Analysis to Differential Equations and other engineering problems. During the above mentioned period he established an interdisciplinary research group and persuaded a mechanical engineering professor to guide a student of mathematics in fluid mechanics. Between 1980 and 1983 he worked as a Professor in the Constantine University of Algeria under Indo-Algerian agreement and was Incharge of the research activities of mathematics in the University Research Centre. Several faculty members and post graduate students worked with him such as Dr. Sherif, Prof. Abu Bakar Merouni (Subsequently he became Vice-Rector of Setif University), Dr Yasin Chweiter, Dr. Merhoun, Dr Mustafa. During his stay in Algeria he was motivated by French mathematicians Prof. L. Tartar, late Prof. (Ms.) Madame J Boujot of Orlean University and Prof. R. Lozi to work in the areas of Variational Inequalities, optimization in Hilbert Spaces and Optimal Control. He pursued these fields rigorously in subsequently years. After returning to India, he was sanctioned a major research project of CSIR on Variational Inequalities in which one of the author (Prof. Q.H. Ansari) worked as a research fellow. On the initiative of Prof. Siddiqi and late Prof. Madame J Boujot of Orlean University, an Indo-French workshop and conference was organized in May 1985 at TIFR centre, IISC Bangalore for about 3 weeks by NBHM. Several distinguished French researchers in PDE in general and Variational Inequality in particular delivered lectures and interacted with the participants. Many beneficiaries of this conference and workshop are well known mathematicians of the present day.

Between 1985 and 1998 he took several steps to establish research collaboration between AMU and other institutions such as Orlean Univeristy France, Kaiserslautern university Germany, McMaster University, Canada, Tenesse University, Knoxville, Concordia University, Canada, Laval university Canada, Universities in Rome and Pisa. He could establish effective contact with Prof. Helmut Neunzert group at Kaiserslautern Germany. In 1989 Prof. Neunzert and his colleague Prof. Brokate visited India several times and interacted with Indian researchers not from A.M.U. only but from Allahabad University, Guru Nanak Dev University, IIT Delhi, Chennai, Roorkee, Bombay, Kharagpur and IISC Bangalore etc. A fairly good number of Indians were trained at Neunzert’s group who are now occupying good positions in the different parts of the world and engaged in research of applied nature. One notable beneficiary is Dr. Akhtar Khan of Rochester Institute of Technology who was inspired by Prof. Siddiqi.

As pursuance of his objective he could arrange visits of Prof. U.Mosco (Rome & new USA), Prof. F.Genesee (Pisa), Prof. A.Bensonssan (Paris) Prof. R.Lozi (Nice) and Prof. R.T.Rockefellar (USA) to AMU besides visits of several young researchers of Europe and USA. Professor Siddiqi visited different universities is US and Canada such as South University of Florida, Duquesne University, University of Tennessee, McMaster University and Laval University in March – April 1991. Prof. Siddiqi had close contact with Prof. T.Hussain of McMaster University. Prof. Hussain visited several times to AMU between 1985 and 1996. Prof. Siddiqi was a guest of honor in the function arranged at the time of his retirement in May 1994. On this occasion an international conference was organized by the department of McMaster University which was attended by distinguished functional analysts of that period including Walter Rudin. The Proceeding has been published in Pitman Lecture Notes in Mathematics.

Between 1989 and 1999 Prof. Siddiqi visited Kaiserslautern practically every year and made several valuable research contribution in the form of research papers a book on “Topics in Industrial Mathematics-Case Studies” joint with H. Neunzert, Kluwer Academic Publisher,1997 and edited a volume jointly by Prof. Martin Brokate entitled “Functional Analysis with Applications” Pitman Lecture Notes series in mathematics, Longman/CRC 1998. On the initiative of Prof. Siddiqi and Prof. Neunzert a workshop on Industrial Mathematics sponsored by UGC was organized in Delhi in Feb 1999 in which thirty HOD’s of northern universities in India participated and benefited from the delibration of this workshop. Senior Indian Professors working in the applications of mathematics to real world problems also participated in this workshop. Neunzert and his research group along with Prof. Engle and Shashi Maier delivered lectures in this workshop.

Prof. Siddiqi had visited several institutions in Germany but his visit to Prof. M. Zowe at Erlangen proved very useful as he could collaborate with Prof. M. Kocvara. Prof. Siddiqi and one of the author (Prof. Manchanda) could write a paper in the area of financial mathematics entitled “Two-Step Algorithm for American Option Pricing” published IMA Journal of Business and Industry 2000. They also wrote a paper on Wavelet based solution of Black-Scholes Model of option pricing. Prof. Siddiqi and Prof. Kocvara edited the proceeding of ISIAM annual conference in GNDU, Amritsar Jan 2001 published by Kulwer Academic Publisher, 2002. The proceeding contains some valuable papers by P.L.Butzer, J.R.Higgins, G.Fairweather, N.I.Gold, P.Toint, P.N.Shivakumar et al.

Association with ICTP

Professor Siddiqi has continuously visited ICTP Trieste Italy, an UNESCO organization since 1987 in various capacity such as federation arrangement, visiting mathematician, short time visitor, regular associate, guest of the director, senior associate. He has been also appointed as the visiting scholar/consultant (office of external activities) to Istanbul, Turkey for the period 2007-2010. Visits to ICTP enabled Prof. Siddiqi to collaborate with scientists of the different developing countries of the world including India in a congenial atmosphere. During his visits he could work with Prof. Zafer Aslan of Turkey (Istanbul) and Prof. Rasulova Mukhayo of Uzbekistan (Tashkent) and has published several joint papers. He is co-chair of International workshop on wavelets organized on Wavelets in Istanbul since 2005. He has motivated many researchers in Turkey to work in the applications of wavelet to real world problems. Proceedings of these workshops are published every year. Currently he is consultant in a research project of Dr. Hulya Sevindir of Kocaeli University, Izmit, Turkey.

Stay of Prof. Siddiqi at KFUPM (Saudi Arabia)

Prof. Siddiqi was invited to give series of lectures at King Fahad University of Petroleum and Minerals (KFUPM), Dhahran, King Saud University, Riyadh and Umm-Al-Qura University, Mecca in May-June 1998. At the end of the visit the KFUPM offered him a post of professorship which he accepted and joined in the first week of the October of the same year. It may be noted that this university was established by the largest oil company of the world ARAMCO in 1963 adjacent to its premises in Dhahran (Saudi Arabia). It is considered as one of the best university of the developing country. Its campus facilities and library can be easily compared to a few best universities of the world. Prof Siddiqi stayed in KFUPM till 2007/2008. During this period of about 10 years he collaborated with Prof. K.M. Furati (Former HOD of the Department of Mathematical Sciences), Dr. Tawfiq Hattan (Present HOD of the Department of Mathematical Sciences), Prof M. El Gebeily, Prof Al-Homidan (Dean, Faculty of Science, KFUPM), Dr. M.A. Sohail, Dr. Fauzi khene, Dr. Aiman Mukheimer, Dr M. Feroz, Mr Samsuddin Khan, Mr S. Rehman. During his stay he wrote 44 research papers, 3 books and edited proceedings of 5 International conferences, all published by reputed publishers of the world. The proceedings were co-edited by Prof. I.S. Duff, O. Christensen, Dr. M. Kocvara, Prof. K.M. Furati and M.Z. Nashed et al. He was the lead technical editor of “The Theme issue on Wavelet and Fractal methods in Science and Engineering” of the Arabian Journal for Science and Engineering in two volumes, volume 28(1C) 2003 and volume 29(2C) 2004. In these two volumes there are 25 contributors from different parts of the world, notable among them are S. Dahlke, G. Teschke, V. Wickerhauser, O. Christensen, C. Canuto, G. Naldi, K.Urban, P. Venini, D. Saupe, Maria Rosaria Lancia, F. Nekka. Prof. G. Korvin was the technical editor and the guest technical editors were Prof. U. Mosco, S. Dahlke and W. Freeden. Prof. Siddiqi was appointed the Director of the Research lab in Wavelet & Variational Methods of the Department of Mathematical Sciences KFUPM. He was also consultant in a major research project of ARAMCO completed by the research institute of KFUPM. This was a big project in which good numbers of experts of different discipline were involved. Prof. Siddiqi has completed four research projects and one book writing project as the Principal investigator during his stay in KFUPM. On his initiative several new courses at the under graduate and post graduate level were introduced. On his initiative and an international symposium on Industrial Mathematics was held in February 2004 at KFUPM in which Dr. Noel Barton, Prof. Martin Brokate et al were invited speakers. Prof. H. Neunzert visited KFUPM twice on his initiative. Prof. H.D. Mittlmen of Arizona State University visited KFUPM for one month. Prof. A.H.Siddiqi was his visit coordinator. On his initiative Prof. K.R. Sreenivasan, Director, ICTP visited KFUPM and other institutions in Saudi Arabia.

After returning to India 2007/2008 Prof. Siddiqi joined the Sharda Group of Institutions and started collaborating with engineers in the areas of Wavelet, ANFIS and NeuroFuzzi method and their application s medical signal such as EEG, EOG, ECG and meteorological problems. He visited Sultan Qaboos University, Muscat, Oman, thrice in 2009 first as a speaker in modeling conference in February, as a consultant in a research project of Dr. M. Boulbrachene in May and as a visiting consultant from September 2009 to January 2010. He is actively collaborating with Dr. M. Boulbrachene in the area of Numerical simulation of variational inequalities, particularly parabolic. He was the main functionary in organizing an international conference on “Modeling of Engineering and Technological Problems” at BMAS, Agra in which more than 200 scientists and engineers participated including 25 well known scientists and engineers from abroad. The Proceedings of this conference edited jointly by Prof. Siddiqi, Prof. A.K.Gupta (Principal, BMAS) and Prof. M. Brokate (Technical University of Munich).

Areas of Mathematics pursued by Prof. Siddiqi and his co-workers.

1. Fourier and Dyadic Harmonic Analysis: Professor Siddiqi tried to popularize certain areas with which researchers in our country were not familiar. In sixties most of the researchers in northern India were engaged in the summability of Fourier series and special functions. Young Siddiqi tried to introduce to Indian researchers new areas such as classes of Fourier coefficients, approximation by Fourier series and orthonormal system other than trigonometric system such as Walsh and Haar systems. He pursued vigorously Walsh Fourier Analysis and collaborated with engineers and Hungarian mathematicians F.Schipp, F.Moricz and S.Fridli. Prof. Moricz visited him thrice and has written two valuable papers while Fridli visited him once and had hosted him in Hungary. He has written recently a joint paper establishing fairly general results in the area of approximation by Walsh polynomial. Prof. Schipp visited him twice in India, Prof. Sididiqi and his, co-workers learnt from him Haar Vilenkin system. Prof. William Wade, USA visited him in India and also invited him to visit Tennessee University. He has cited many places the book of Prof. Siddiqi on Walsh function published by AMU Press 1978 and several of his results. Some of the results obtained by Prof. Siddiqi in these areas are given in the next section.
2. Two Dimensional Analogues of Banach Spaces: The concept of two dimensional analogues of metric, normed and Banach spaces were introduced and studied by Sigfried Gaehla of the German Academy of Science, Berlin around 1963. Prof. Siddiqi introduced this topic to researchers in India, Iran and Algeria. He spent a month with Dr. Gaehler in Berlin. He wrote several papers on non-Archimedean (ultra) 2-matric and 2-normed spaces, quasinormed spaces, 2-semi inner product spaces, orthogonality in 2-normed spaces, ultra m-metric and non-Archimedean m-normed spaces. Some typical results obtained by him & his collaborators are given in the next section.
3. Non Archimedean functional analysis: Besides concepts of compact operators and fixed point in non-Archimedean functional analysis, Prof. Siddiqi introduced the concept of invariant means in non-archimedean Functional Analysis which is published in Springer Lecture Notes in Math. Vol. 399, Springer Berlin 1974.
4. Variational Inequalities: Variational Inequalities, introduced and studied by celebrated mathematicians like J.L.Lions, G.Stampacchia, G. Fichera in early sixties were not know in Indian subcontinent till 1985. Prof. Siddiqi popularized this subject in Indian subcontinent and he guided successfully 10 research scholars in this area, many of them are very active researchers including one of the present author. He has studied applications of variational inequalities in some problems of superconductivity and elasticity (rigid punch problem) and wrote joint papers with his co-workers. A resume of same of his research work in this area will be presented in the next section.
5. Wavelet Theory: According to Professor Siddiqi wavelets were introduced to him by Prof. H. Neunzert while driving from Kaiserslautern to Darmstadt in June 1990. He tried to introduce this subject to his Ph.D students in early nineties in particular, and other Indian researchers in general. One of the authors (P. Manchanda) came in contact with him in an International Conference at AMU organized by him in 1996. Prof. Siddiqi joined KFUPM in 1998 and devoted his full time to promote study of wavelets theory and its applications. Some of this efforts and results in this area will be mention in the next section. He completed 5 research projects related to this theme while serving KFUPM.
6. Fractals: He was motivated by a distinguished electrical engineering professor (Prof. M.N.Faruqi, Former Deputy Director IIT Kharagpur & Vice Chancellor AMU) in 1993 to take up the study of fractals and their applications in image processing. He guided research in this area and one of his Ph.D student (Mr. Aiman Mukhimar) worked in this area, got Ph.D degree and now HOD of applied sciences in the Prince Sultan University, Riyad, Saudi Arabia. One of the author of this article also published papers in collobration with him. Some of the results obtained by him will be presented in the latter part of this article.
7. Industrial and Financial Mathematics: Writings of Prof. H.Neunzert inspired him to take up the study of industrial and financial mathematics. He has made serious efforts to initiate teaching and research of industrial mathematics in India. He established the Indian society of Industrial and Applied Mathematics (SIAM) with the help of senior academicians of the country such as Prof. J.N.Kapoor, Prof. H.P.Dikshit, Prof. Izhar Hussain, Prof. U.P. Singh, Prof. D.K.Sinha, Prof. V.P.Saxena, Prof. G.C.Sharma, Prof. N.K.Gupta, Prof. Ishwar Bhola, Prof. Karmeshu, Prof. O.P.Bhutani, Prof. Rudraiah. He is credited for organizing Biennale Conferences of the society and international conferences since 1992. The proceedings of these conferences are published by reputed publishers. The society is publishing its journal of which Prof. Sididiqi is the editor-in-chief. It is printed and distributed jointly by Anamaya & Taylor & Francis. Researchers from all over the world are contributing to this journal.
8. Oil Exploration and Wavelets: Professor Siddiqi has worked on the role of wavelets to oil industry as a consultant in a research project of the largest oil company of the world – ARAMCO. He also completed two research projects on applications of meteorological data of the Kingdom of Saudi Arabia. His work is brief described in next section.
9. Wavelets Inverse Problems and Medical Signals: Several collaborators of Professor Siddiqi namely Prof. R.C.Singh, Prof. Hulya Selvinder, Prof. Zafer Aslan, Ms. Noor – E – Zahra, Mr. Vivek Singh Bhadouria and Ms. Shafali Pande are engaged in the study of this area. Some of their results will be presented during the workshop and the Satellite Conference (14-17 August 2010, India Habitat Centre, Lodhi Road, New Delhi). These researchers are also studying applications of wavelets methods combined with ANFIS, SVM and Neuro-Fuzzy methods to study behaviour of time series representing real systems.