# Don Showcases the Power of Mathematics

A Professor of Mathematical Analysis, Department of Mathematics, College of Physical Sciences (COLPHYS) of the University, Professor James Oguntuase, has said that Mathematics helps in finding solutions to problems and in particular, Hardy Type Inequalities, which play a key role in this regard, because they are basically the celebration of averages that are very useful in all areas of human life.

Professor Oguntuase, who is a Fellow of the Nigerian Mathematical Society (FNMS), stated this while discussing some of his research breakthroughs in Mathematical Analysis, with special focus on the Theory of Inequalities. According to him, his research interests include Gronwall Type Inequalities, Hardy Type Inequalities, Integrodifferential Inequalities, Convex Functions and Inequalities on Time Scales.

The Don said his research on Gronwall Type Inequalities started in 1990 when he wrote his M.Sc Thesis on Gronwall-Bellman Type Inequalities. He noted that Gronwall Inequalities are used to establish bounds on differential and integral equations. Furthermore, they play a vital role in finding the existence, uniqueness and stability of solutions of differential and integral equations. He noted that he had written many highly-cited papers in books and reputable international journals in this area of research. Professor Oguntuase, who is the immediate past Chairman, Publications Committee of the University, further added that he had done a lot of research on Hardy Type Inequalities.

He added that this area of research started more than 100 years ago (In 1915, as G.H. Hardy needed an estimate for the arithmetic means in his 1920 classical paper). The Don informed that his late supervisor, Professor Christopher Imoru, introduced him into this area of research during his Ph.D work at the Obafemi Awolowo University, Ile-Ife, saying “I was inspired by the great work he has done in this area of research and also a lot of research materials he made available to me”. The Don further said that he was also motivated to work on Hardy Type Inequalities because of the rich research materials he received from Professor Lars-Erik Persson (Of Sweden) and Professor Alios Kufner (Of Czech Republic). “Indeed, the two of them made the drafts of their book on the subject-area available to me both in soft and hard copies. I was opportune to meet with the above eminent professors during the 2006/2007 Academic Session when I was in Lulea, Sweden, under the Swedish Institute Guest Fellowship for research. The meeting made us to do some collaborative research and also provided me with opportunity to serve as external examiners to two of their Licentiate students. One of the best research works that I did with Professor Lars-Erik Persson was ‘Refinements of Hardy Inequalities for Superquadratic and Subquadratic Functions’ and this work was published in the Journal of Mathematical Analysis and Applications. This work is the first known application of Hardy type Inequalities for Superquadratic and Subquadratic Functions. This paper is significant in that it was the paper in the literature to have a turning point of Hardy Type Inequalities at Point p=2, instead of the usual Turning Point p=1. Another significant results obtained (which would appear in 2017), is on a Limit Hardy Type Inequalities with three sharp constants at the left sides of the integrals. In fact, this is the only paper in the literature on Hardy Type Inequalities with this property”, he added.

Speaking further, Professor Oguntuase, who is a 2003 Young African Mathematician Award holder, said that he also worked on Integral Inequalities such as, Gronwall, Gronwall-Bellman-Bihari and integrodifferential inequalities. According to him, these Integral Inequalities were studied on the real line and in some cases, they are studied in Rn. Professor Oguntuase, an alumnus of the University of Ado-Ekiti, Ekiti State and the Obafemi Owolowo University, Ile-Ife, also stated that during his Master’s degree at OAU, Ile-Ife, he was motivated to work on the Theory of Loops. He said he was equally able to publish papers on the Theory of Loops because of the way Professor Adewale Solarin, another expert in the profession, handled the course called Non-associative Algebric Systems, during the M.Sc. programme.

Highlighting some of the achievements of his research, Professor Oguntuase stated that research in Mathematics was for solving problems. He said that he had worked in the area of analysis, which involved abstract-thinking where theorems, lemmas, corollaries, axioms, among others, were needed in establishing results, noting that this area of Mathematics is called Pure Mathematics and that the results were then used by the Applied Mathematicians, Engineers and Economists in finding solutions to their models and also to help them know if solution would exist or not.

The Professor of Mathematical Analysis explained further that his reason for focusing on Mathematical Analysis was that it could help to develop people’s mental reasoning and also allows for clear judgment in any situation. He observed that in Pure Mathematics, which involves a lot of abstractions, one could obtain results without actually knowing where it would be applied, noting that somebody could come up later and find the applications for the result.

According to Professor Oguntuase, Hardy type inequalities were very useful in studying and predicting stock exchange because of the variable limits in the Hardy Integral Operators, adding that “if you are an Engineer, you can go to the field and get your model, which is like a differential equation, but the Mathematician can tell you what you need in order to find the solution”. He added that “when an engineer has problems in building or construction, mathematicians would assist them in knowing what a particular beam can carry in order to reduce collapse of buildings or bridges”. He noted that it is usually required that Engineers should have good knowledge of Mathematics, adding that in Medicine; in the area of Mathematical Biology; which deals with sickle cell patients, they could develop modern solution that can help predict when the next crisis would occur, so that the crisis can be averted. He added that Mathematics is all-embracing, as it helps in national planning, budget planning, academic planning, even for wives at home and in every facet of life.

Speaking on the challenges being encountered, Professor Oguntuase lamented that in this part of the world, Mathematicians are not appreciated, noting that there was the problem of funding, alleging that he had never received any grant for research from FUNAAB. He added that he was fortunate to get grants from the Swedish government, where he got a one-year research scholarship and this had made him to record a major breakthrough in his research. He noted that while he was in Sweden, he was provided with everything he needed such as computers, printers, books, journals, constant electricity, workshops and conferences sponsorship, adding that he was also lucky to have won the 2002 Mathematics Research Fellowship of the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, which also contributed a lot to his career development.

Professor Oguntuase, who had been a Senior Associate of the Abdus Salam International Centre for Theoretical Physics (2010-2015), said that his research work improved during this period and would always be grateful to the Centre. He was also given the grant by the International Mathematical Union (IMU), to attend the World Mathematical Congress in India. The Congress was organised for Mathematicians every four years, adding that the last one was held in 2014, in South Korea while the next one would hold in Brazil, in 2018. Professor Oguntuase noted that he was lucky to have gotten most of his grants through external assistance, as he had always been invited, almost every year, to the Bouchet School at the University of Ghana, Legion, Ghana, since 2006. He revealed that another factor that helped him to overcome the numerous challenges facing Mathematicians was the intervention by the National Mathematical Centre, Abuja where he was invited, more than 12 times, during his postgraduate studies. According to him, such foundation postgraduate and research-oriented programmes actually prepared him for research, saying “the fact that resource persons usually come from Europe, United States of America, Australia, among others, encouraged us, coupled with the fact that transport, accommodation, feeding and stipends were always provided, helped us to concentrate in our studies”.

The Don, therefore, recommended that the repositioning of the National Mathematical Centre, Abuja to serve as a Centre of Excellence for Mathematics research in Nigeria, where Mathematicians, both young and old, would turn to for the latest work in their various fields of research, to find solution to the various problems in the society. He further observed that Mathematicians are endangered species, who must be encouraged. He stated that in the 1980s and 1990s, young Mathematicians were invited to the National Mathematical Centre, Abuja for four weeks to be trained in Foundation Postgraduate and Research Oriented Programmes, respectively; free-of-charge. I am a beneficiary of such programmes and many Professors today are products of such programmes. But today, things have changed as young Mathematicians are now required pay money to visit the Centre. He also suggested that, government should make a legislation, whereby Mathematics graduates, especially, First Class graduates, would be given employment in the University to teach Mathematics and they should start as Assistant Lecturers, adding that Mathematics teachers at all levels should be given an enhanced package, in order to encourage more people to study Mathematics.

Furthermore, he recommended that favouritism in the posting of teachers in secondary schools should be curbed, alleging that government officials often influence posting of teachers, so that their wives and relations would not be posted to rural areas, thereby, making the rural areas devoid of qualified teachers, particularly, those that are degree holders. He also called on the University to allocate special fund to the research works of Mathematicians, because the only laboratory they have, are conferences and workshops. The Don suggested that universities should give Mathematicians special allowances and also make the entry point for Master’s holder to be Lecturer II. Furthermore, “the University should, as a matter of duty, sponsor the Ph.D of young Mathematicians, adding that this would encourage them to be focused and stay in the University to make their own positive contributions to its development”, he stated.

Professor Oguntuase, an alumnus of the University, having bagged a Postgraduate Diploma in Computer Sciences, revealed that he had cooperated with more than 23 researchers from over 13 different countries, noting that a researcher should be able to spread out to other places in order to publish good papers in reputable journals.