Contributions of Nepali mathematicians in modern period: An overview

The mathematical sense in human beings developed the mathematical genesis that paved the way for development and human civilizations. The paper aims at exploring the contributions of Nepali mathematicians since 1951AD to 2015 AD. Despite the fact that writing about the contributions of mathematicians as a part of the history of mathematics which is an arduous task, the paper seeks to explore and analyze Nepali mathematicians’ contributions, that identifies them at the national and international mathematics arena.

Introduction

Mathematics plays an important role in the progress and prosperity of the different disciplines, such as, language, culture, statistics, literature, infrastructure, liberty, tourism, music, commerce, agriculture, education, politics, and economics. To know the development process of mathematics, it is necessary to understand its history. So, we need to seek our history ourselves [1] . The analytical study of Nepali mathematician's contributions was started by Naya Raj Pant, Shankar Raj Pant, Ram Man Shreshtha, Eka Ratna Acharya, Kanhaiya Jha, Neelam Subedi and Dipak Basyal and so on. It is evolving as an interesting field to new researchers in mathematics in the present time.In this area,the first doctoral work was completed by Eka Ratna Acharya entitled "Naya Raj Pantaka Ganitiya Kritiharu ko Adhyayan" [5] from Nepal Sanskrit University, Dang. This is the first academic endeavor to recognize the contributions of Nepali mathematicians. Furthermore, he completed his postdoctoral study entitled "Development of Numerals Systems in Nepal" from University of Crete, Greece [10] . After that, it has gained momentum as an emerging field in Nepali context. In a sense, this article is a byproduct of this process, and it tries to add recent activities of Nepali mathematician's works in mathematics development on the basis of the sources. Likewise, it is crucial to understand the history of mathematics for as understanding of mathematics. It is mathematics that reflects some of the noblest thoughts to various generations for ages. The world's mathematical system has highly influenced Nepali mathematical system [20] . The past is a treasure trove of information. Men's achievements are documented in history. Historical research use scientific methods as an alternative to recounting and interpreting historical events [12] . Mathematical culture, biographical information, and contributions of Nepali mathematicians would help to develop the mathematical program that aims to meet the challenges and needs of the changing societies. The expression 'contributions of Nepali mathematicians' means the work of Nepali academicians that are exposed to mathematical worlds as information [19] . The study of historical contributions [9] of Nepali mathematicians is a way of motivating learners that shows the significant area of mathematics which helps to understand and follow mathematical concepts and processes. so, the development of mathematical science and technology can be the barometer of human development in every aspect. The information supports to make awareness and seems the paradigm shift in the field of mathematics [5] . Due to the cultural, geographical, and religious attachments, the Nepali education system was highly reflective of South-East Asia's education. Indeed, In the ancient period "Gurukula" was the place where history of mathematics was taught [27] . The Gurukula education system continued for many years, and there were no formal schools as we see in today world. In this system, basic mathematics courses were offered in the name of Jyoutisha Shasta [29] ; teachers' teachings Jyoutisha were popular mathematics books written in Sanskrit like Siddhanta Siromany of Bhaskaracharya (II) [11] . In this context, this paper aims at outlining remarkable contributions of Nepali mathematicians in the modern period in Nepal from 1951 to 2015. Mathematics education is deeply rooted in the context of Nepali education system from the very beginning. Therefore, the mathematics that developed prior to the written recorded history is termed as antiquity of mathematics. In Nepal'shigh Himalayan region, many mathematical symbols in the forms of rock art or formation of rooms to administrator with decorated utensils, and cave like Zhong Kiore Cave of at Mustang [7] are found in the antiquity. Similarly, researchers also studied Cretan hieroglyphs as the elementary phase of development of numerals these are sources of Nepali mathematician's contribution in the sector of mathematics. So, this paper is mostly based on Eka Ratna Acharya's documentation regarding Nepali mathematician's works. The essential idea that links classical and modern mathematics is based on some thought and concepts of the mathematics [6] . Here, my concern is to expose the brief information of the contributions of Nepali mathematicians either they introduced its practice or developed it. These mathematicians have been contributing regularly to the development of Nepali mathematics in such fields as astrological mathematics, history of mathematics, Nepali history, Nepali and Sanskrit literature.

Objectives

The primary objective of this paper is to expose Nepali mathematicians' contribution to the development of mathematics in the national and international arena. It has the following specific objectives: • To identify the eminent Nepali mathematicians, form Modern Period and to elucidate their contributions in global context. • To identify the relevance of their contributions.

Methodology

To fulfill the objective of this research, it adopts the descriptive explanatory approach. It intervenes the historical research paradigms and follows the systematics documents analysis procedure to come into the conclusion. Therefore, the past events have been analyzed in the historical context. Its secondary sources are located in personal collections. Also, exact materials may be documented as well. To accomplish the proposed work, I have consulted several old documents that were written by different writers in different periods of related field.

Some Prominent Nepali Mathematicians in the Modern Period

Narayan Bahadur Manandhar was the first person to pass M.A. in mathematics. A short introduction of Narayan Bahadur Manandhar is found in "Hridaya," the manuscript of Nepal Bhasha council. As per the book, Narayan Bahadur Manandhar was born in 1026 Nepal Sambat (1906) and died in 1095 N.S. (1974). His permanent address is Gaur, Rautahat of Narayani Zone. Furthermore, his qualification was a double M.A. in Sanskrit and mathematics [5] . He was the first person to pass M.A. in mathematics. In the beginning, Manandhar taught at Durbar High school and later taught at Tri-Chandra College. Moreover, his grandson said that the Rana Prime minister Juddha Shamsher had offered him the post of Nayab Subba (Assistant Executive), that he rejected. Then Juddha Shamsher ordered his people to arrest and imprison him. Later Manandhar desired to continue teaching, and he was appointed as a teacher at Durbar High School. Thus, he was one of the most successful teachers of mathematics [9] . Khadka Man Malla Singh had important contribution in the Nepali history of mathematics. He had translated the books of mathematics form English into Nepali. But he did not merely translatethem; rather he also inserted his views and mathematical contents from his own side in those books. He has abundantly mentioned Arithmetical words like ancient measurement, weight, time, etc. [5] . The books written by Malla have proved that mathematics is a scientific language too. He had published many mathematics books from in 1941. His sincerity in studying his books assisted in understanding mathematics deeply. The book called Nepali Arithmetic is a translation of "Junior Arithmetic." The book was published four times per thousand pieces every year. It consists different mathematical units related to weight, measurement, time, counting, etc. Likewise, Malla had written the book mathematic part-I in 2003. Furthermore, Jay Deva Bhattarai has mentioned this fact in his book. Named "Saahityakar Parichayara Abhibhyakti" [5,31] . Indra Nath Aryal (5 th Feb, 1921-27 th May, 2020), was born at Pyukhatole, Kathmandu. Aryal is seeing as a perfect mathematician. He completed his M.A. from Poona University in 1959. Aryal taught at Patan High School from 1948 to 1963. When he returned from the U.S.A., he started teaching Mathematics at the Central Department of Education at Tribhuvan University [4] . He always took mathematics as a practical subject. Indeed, he always focuses on the practical and applicable parts of mathematics. At the same time, he has applied mathematics with several other contexts. He has been advocating that mathematics can be explained and used as a language. A very few mathematicians have made such efforts to popularize mathematics as an international language. Generally, mathematics is seen integrated with the setting as it is the product of nature and society. Aryal's main contexts of interest is practical mathematics, explanation, and expansion of mathematics. Aryal had made an extensive contribution to the PRIME (Program for Revision and Improvement in mathematics Education), which was set up in 1968 [4] . Indra Nath Aryal has explained mathematics as actual science in Nepali context. Most of these treaties explain how mathematics describes the life and the universe. He has given several examples of experimental mathematics in Ganita 1, Ganita 2, and Ganita 3. He has talked about various aspects of mathematics linking with his motherland Nepal. He has used mathematical instruments in the physical division of Nepal. He has depicted Nepal as a land of four castes and thirty-six sub-castes. Describing Nepal from a mathematical perspective, he has made the study of mathematics more interesting and fun generating. His contributions have been recognizedin the identification the actual form of mathematic as depicted in Ganita 1, Ganita 2, and Ganita 3. His honor as mathematician lies in the fact that he has given his new and distinct dimensions [4] . He made a notable contribution to mathematics teaching during his active teaching period. He is best known for his expository work to link mathematics attributions with "NEPAL."Aryal won awards from RONAST and the Mathematics Education Council (MEC) for his contribution to mathematics teaching [13] . He is famous as a living legend in Nepali mathematics. He can classically express mathematics with a practical base. Indeed, he retired from T.U. in 1983. Even after his retirement, he is continuously making significant contributions in Mathematics [4] . Dhupa Ratna Bajracharya (19 th January, 1930-15 th Nov, 2020) was referred to by the moniker DR. Bajracharya. In 1953, he graduated with an M.Sc. in mathematics from the University of Allahabad, India. Bajrachary was the first president of the Nepal Mathematical Society (1979-1987) and the head of Tribhuvan University's Central Department of Mathematics (1973-1991). Parental instructions for the development of mathematics and mathematics education have been presented by him [3] . He wrote a number of textbooks. One of his most well-known textbooks, Basic Mathematics, is currently available at the + 2 level and is intended for intermediate students. In the course of his career, he held positions as chairman of the mathematics instruction committee in 1973 while on a deputation from the Nepali government, professor-in-charge at Patan College from August 8, 1954, to August 6, 1962, and professor of mathematics at Tri-Chandra College from 1964 until his retirement in 1990. On September 19, 2014, in honor of 56th Math Day, he received recognition from the Department for his significant contributions to the field of mathematics education [3] . He wrote higher secondary level Basic Mathematics part I and II. Besides these publications, he presented many papers in the workshop, papers related to teaching, and lectures on Nepal's development of mathematics. Council awards him for mathematics Education in 2006 and the Royal Nepal Academic of Science and Technology in 1998. He has spent more than 81 years as an active and energetic person in educational and mathematical activities and encourages Youngers in mathematics. Indeed Prof. Bajracharya is as our pride in mathematics and society [3,26] . Rameshswor Prasad Manandhar was a well-known mathematician in Nepal. He was born on 2 nd Feb. 1936 at Pakanajol, Kathmandu Nepal. He passed MSc. degree in mathematics from TU. He completed his PhD degree in 1968 on the topic "Study Meijer of Bessel Transform" under the guidance of Prof. K. M. Saxena from Ranchi University, in India. However, he discussed the convergence and uniform convergence of the transform. He defined as,

He clarified the recognized relationships between the convergence properties of the corresponding transform and the order properties of the determining function [22] . After completing his study from Ranchi University, he joined Institute of Engineering. From 28/8/1969 to 18/1/1971, he worked as visiting lecturer in mathematics at Engineering Institute. Dr. Manandhar is well equipped in his subjectteaching mathematics. Thus, students were pleased to attend his classes for his nice and effective method of teaching. He appointed as a Lecture in 1961, and promoted to Reader in 1977 and to Professor in 1985. He had more than 35 years' experience of teaching mathematics from school level to postgraduate and research level. Moreover, hehas expressed a new inversion operator [23] , for the Meijer Bessel Transforms (Manandhar, 1976). He was the former Head of Department, Central Department of Mathematics, TU, Kirtipur from 1991 to 1998 and former president of Nepal Mathematical Society from 1993 to 2005. He retired from TU Central Department of Mathematics in 1999 [9] . Manandhar has been active as supervisor of many PhD research scholars. He has successfully supervised PhD of many PhD research scholars in mathematics like Bhanu Chandra Bajracharya entitled 'Extension of Some Convolution Transform' in 1980. Similarly, Manandhar supervised Hari Bhakta Shrestha's PhD entitled 'Study of Integral Transform' in 1980 for the first time from Nepali mathematicians, Tribhuvan University. The Institute of Science and Technology thank fully acknowledges the contribution made by Manandhar [21] . He has published several research papers in different International journals and has participated international seminars, workshops and conferences [24,31] . Gauree Shankar Shrestha was born in 1933 on 13 th January at Khachhen Tole, ward no. 16 of Lalitpur Municipality. His teacher Indra Nath Aryal was a well-known mathematician in Nepal. His dedication and valuable contributions in the Nepali mathematical domain are highly appreciable. He got M.Sc. degree in mathematics in 1957 from the University of Patna . Ram Man Shrestha (1940-2020) was an Emeritus, Professor of mathematics at the Central Department of Mathematics at Tribhuvan University. Indeed, he was born on 5 May 1938 at old Baneshwor, Kathmandu Metropolis City. He passed M. Sc. from Tribhuvan University in 1961 and Ph.D. in 1965 from Banaras Vishow Hindu University in the title "Study of Generalized Transform" under the subject of Integral Transform. He was the first PhD degree holder in mathematics of Nepal [31,34] . In 1995, Shrestha supervised Shanti Bajracharya's Ph.D. from Tribhuvan University. As an eminent intellectual, he had many contributions to the development of mathematics in the country. He hadgood knowledge of the field of mathematics. He published two dozen of research works in various National and International and he had expressed the views on the development of Lichchhavi and other numeral systems in different civilizations and communities. According to Eka Ratna Acharya, Shreshtha had given new and systematic pattern of Lichchhavi numeral systems [8,32] . He was an active academician at NAST, served as the Member of Parliament and served as the Rector of Tribhuvan University. He was also a Senior Associate at International Center for Theoretical Physics (ICTP) and a former Humboldt Fellow [16] . He was the first to get fellowship from ICTP, Italy. He wrote, edited, and published many mathematics books, reference materials and dozens of research articles for school and higher levels. He was as an evaluator of different mathematical international researches. He was awarded by the Nepal Mathematical Society, Central Department of Mathematics, and many other organizations [8,18] . A mathematical paper titled "Mathematics Education for the 21st Century New Nepal" [33] , written by Ram Man Shrestha, was released by the Nepal Academy of Science and Technology [NAST] against this backdrop. It serves as a foundational element for the examination of Nepali mathematics through cross-cultural comparisons. Additionally, it marks a turning point for mathematical education and research in Nepal. Following these two books, other articles-including biographies of mathematicians-are published both domestically and abroad. The historical development of mathematics is based on numerals, manuscripts, inscriptions, and modern concepts. He appeared enthusiastic about mathematics in Nepal. Moreover, he has classified the time of the historical development of mathematics into three periods, the early ages (pre-classical period-before 500B. C.E), the Middle Ages (classical periodc, 500 B.C.E-c. 1600 C.E), and the modern Ages (modern period-c.1600 C.E.-onwards). In this work, the conservation areas, libraries and ancient Nepali mathematicians' contributions are not well explained. However, in many of the seminars, workshops, and mathematical programs, he has given PowerPoint presentations and oral notes on the historical development of mathematics in Nepal. Shrestha has expressed the applications of mathematics to social sciences. In the 2009 Establishment of Nepal Mathematics Centre, Ram Man Shrestha was the first founder president [9,33] . He has published several research papers and general articles in national and international journals and newspapers. Among those are some fruitful research articles for researchers and the development of mathematics in Nepal. He has published mathematics text books for different levels including Modern Intermediate Trigonometry which was revised by Satya Narayan Raj Bhandari, Analytic Geometry, 1968, Undergraduate Algebra with Mohan Bir Singh in 1977, Basic Mathematics with Dhupa Ratna Bajracharya in 1979. Mathematics for Grade IV and its Teacher's Guide with Satya Narayan Raj Bhandari and others in 1972. He wrote school level text books and teacher's guide with Satya Narayan Raj Bhandari in 1972. His books have been published by Ratna Pustak Bhandar, Curriculum Development Centre TU, Janak Shiksha Samagri Kendra [31] . He was thefounder secretary of Nepal Mathematical Society. He was the member of Indian Mathematical Society, and American Physical Society. He worked as a teacher of mathematics at primary, lower and upper secondary (1953 -1961) and Intermediate, Bachelor and Master Levels (1962 -1966 -1968). He participated (and on many occasions organized) various seminars and workshops in mathematics. He had been research guidance for many Post-Graduate and Doctorate Candidates. He died in 2020. He has published an important article regarding to The Lichchhavi Numeral and Changu Narayan Manu Script [32,35] . Shankar Raj Pant (1940) is a well-known mathematician in Nepal. Pant was born on October 08, 1940, in Chiti, Lampung, and Gandaki Zone. He was an Emeritus Professor of Tribhuvan University from 1997.He did his master's degree in mathematics from the Central Department of Mathematics, Kirtipur, in 1962. He completed his doctorate in mathematics from Ranchi State Government University, Bihar, India, in 1967 under Prof. K. M. Saxena's guidance. The title of his Ph.D. thesis is "Generalized Laplace Transform and it is Cesaro Mean". His area of specialization is Generalized Laplace Transform and Integral Transform [15,17] . He has included research work a brief survey of various generalizations of the Laplace integral in his PhD. He defined this as, F(s) =∫ 𝑒 -𝑠𝑡 ∞ °∅(𝑡) 𝑑𝑡.Here, f(s) is called the generating function and ∅(𝑡) the determining function. In short, we write as L {∅ } = f(s). Pant has studied in his thesis the following generalization ,𝑓(𝑠) = ∫ 𝑒 -𝜆𝑠𝑡 (𝑠𝑡) -𝑚-1/2 𝑀 𝑘,𝑚 (𝑠𝑡) 𝜙(𝑡) 𝑑𝑡 (𝜆 > It is obtained by putting 𝜆 = 1/2 in (i) and reduces to the Laplace transform for k= m + ½ [28,31] . He has taught mathematics in different campuses as a Professor, Central Department of Mathematics, T.U. 1990, Reader, Central Department of Mathematics, T.U. (1977-1990), Lecturer, Trichandra College, (1967-1969). He headed the department from 2002 to 2005. He was the Nepal Mathematical Society (NMS)'s secretary in 1988 and then was elected the NMS president in 2003. He has played important role in NMS and is the key person to bring society to this stage. During his tenure, he sped up different activities of NMS. NMS observed and celebrated Mathematics Annual Day every year since his leadership. He initiated NMS Newsletter, which is being published every year. He also contributed the Nepali mathematical department of mathematics [8,15] . Indeed, he has given the exposure of the Nepal Mathematical Society among the mathematics professors, teachers, students and general sympathizers of mathematics and the society takes speed in publication of papers and newsletters in the research-based approach. Moreover, he has five-decade long experience of teaching mathematics. However, he has played key role as a supervisor in research. He has co-supervised five students. He has been engaged in the Central Department of Mathematics, TU. He also became the HOD of the department. He has pointed out an importance of spirituality in mathematics. He is a founder member of Nepal Mathematical Society. He has contributed a lot to publish proceeding National conference on mathematics and Newsletter. Similarly, he is very popular as a man of honor and a scientist. He has also played an instrumental role as a supervisor in many international journals and articles [8,32] . Furthermore, he is a lifelong member of Nepal Mathematical Society. Besides, he is a member of Indian Society for History of Mathematics (ISHM). He is a visiting professor for Japan. He is an advisor of many academic and social organizations. He has written dozens of textbooks of mathematics for school and higher secondary levels. Along with this he has edited hundreds of course books, references, research articles and so on. Thus, he is one of the brilliant stars in the area of mathematics (Guru Felicitation, 2016). He has published biographical sketches of some Nepali mathematicians like Prof. Gobinda Dev Pant, Prof. Mitra Nath Devkota, Prof. Naya Raj Pant and Prof. Keshav Dev Bhattarai. This is Pant's remarkable contribution in Nepali mathematics. He has published more than a dozen of research papers in reputed national and international journals [14,15] . He has received many awards and honors. He won Colombo plan Fellowship in 1905, India. (Fellowship and grant). He is awarded by Mahendra Vidya Bhusan and Gold Medal in 1969, Education Award: National Education Committee in 1983, Guru Felicitation in 2011 Guru Felicitation in 2016andNepal Mathematical Society Felicitated Honor.Also he is felicitated by Universal Peace Federation Interrelation and International Federation for World Peace as the 'Ambassador for Peace' [15,30] . In the scientific journal Scientific World, he wrote a well-known historical paper titled Mathematics in Nepal: A historical examination. This study represents a significant milestone in the historical development of mathematics education in Nepal and may be the first to provide a brief history of the subject [29] . This study is considered a landmark in the field of research on the history of mathematics in the setting of Nepal.

Conclusions

On the basis of the reviews and the reflections of abovementioned documents, one can say that Nepali mathematicians play a stimulating role in the blend of the development of mathematics nationally and internationally which is the base of globalization of mathematics. The first person to graduate from Calcutta University with a master's degree in mathematics major was Narayan Bahadur Manandhar. Khadka Man Malla Singh has had important contribution to the development of Nepali history of mathematics. He has translated the books of mathematics into Nepali form English. Mathematics is practically expressed in a classical manner by Indra Nath Aryal. He made a significant contribution to the 1968-founded PRIME (Program for Revision and Improvement in Mathematics Education). In addition, Dhupa Ratna Bajracharya served as the Nepal Mathematical Society's first president from 1979 to 1987. He was previously the head of Tribhuvan University's Central Department of Mathematics. He gave parents advice on how to foster their children's mathematical development and education. Rameshswor Prakash Manandhar has expressed a new inversion operator for the Meijer Bessel Transforms. Indeed, Gauree Shankar Shrestha's dedication and Valuable Contribution to the Nepal Mathematical domain is highly appreciable. Similarly, Ram Man Shreshtha was as the announcer of the search of mathematics in real practice. He played key role in blending information and communication technology for globalization of mathematics. Thus, A mathematical report written by Ram Man Shrestha for mathematics education in the twenty-first-century New Nepal was released by the Nepal Academy of Science and Technology (NAST). It is regarded as a turning point in Nepali mathematics teaching and research. Numerous studies, including biographies of mathematicians, are published both nationally and internationally following the publication of these two volumes. In Nepal, Shankar Raj Pant wrote a well-known historical paper titled Mathematics that is regarded as a journal in the scientific community. This paper serves as a foundational resource for researching the historical evolution of mathematics and mathematics education in the Nepali setting. It may be the first to provide a brief history of mathematics in Nepal.

Acknowledgement

AcknowledgementsI like to express the sincere professor thanks to ancestors of Mathematicians specially professors Shankar Raj Pant, Santosh Man Maske, Dinesh Raj Pant and Chet Raj Bhatta that encouraged the history of Mathematics directly and Indirectly.

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