B.N.
Prasad, was born on the 12th of January, 1899, in a respectable
middle class family of MohammadabadGohna in the Azamgarh District
of Uttar Pradesh (India). He was the youngest son of his parents.
His father, the late Sri Ram Lal, was a practical man of foresight
who tried to give the best possible education to his sons. Young
Badri Nath Prasad was full of promise in his boyhood. He had a
remarkably strong memory. I remember his recalling to me once
as to how in his school days he performed the amazing feats of
committing to memory long verses by reading them only once or
twice. His mother and his elder brother seem to have exercised
stronger influence in shaping his personality than his father.
In reminiscent moods B.N. Prasad used to talk with refreshing
enthusiasm of the great qualities of head and heart possessed
by his mother. He used to recall with a feeling of reverence that
on every occasion he paid his obeisance to his mother. She gave
him the blessing: " Be Victorious".
He
received his early education through the medium of the Urdu language
but later changed to Hindi, with the result that he had an almost
equal command over both of these languages. He enjoyed reading
both Urdu and Hindi poetry. His elder brother was a college student
at Allahabad and B.N. Prasad also came to that town and joined
the local C.A.V. High School, which has now grown into an Intermediate
College. He passed the Matriculation Examination of the Calcutta
University from the V.M.H.E. School of Swan in the present state
of Bihar and the Intermediate (Science) and B.Sc. (Hons.) examination
from the Patna College, Patna, which was then affiliated to the
Calcutta University. He was a regular and hardworking student
right from the beginning and had a uniformly brilliant academic
career throughout. As a schoolboy Prasad used to take part in
games as well and played hockey particularly well. He passed the
M.Sc. examination in Mathematics of the Banaras Hindu University
in the year 1921, obtaining a first class and securing the first
position in order of merit among all the successful candidates
at the Master's Degree Examination of that year in different subjects.
As a student of the M.Sc. (Previous) class during the academic
year 191920, Prasad came in contact with the late Professor Ganesh
Prasad, who recognised his talents and inspired him to do creative
research work in Mathematics. Those were the days when almost
all the bright Indian students aspired to join the civil services.
B.N. Prasad was actually offered a lucrative post by the then
provincial government on the basis of his brilliant performance
at the M.Sc. examination. This was not a small attraction, particularly
when members of his family strongly desired that he should become
an officer in the civil services, but his love for Mathematics
and his devotion to his teacher Prof. Ganesh Prasad, who strongly
advised him to take to the noble profession of teaching and research,
were so great that he politely declined the invitation of the
Government and decided to pursue his study of Mathematics. It
must be said to the credit of the late Dr. Ganesh Prasad that
he aroused a real interest in the minds of his students for doing
creative work in Mathematics and succeeded in training a number
of bright young men as researchers.
In
July 1921, B.N. Prasad joined the Department of Mathematics of
the Banaras Hindu University as a research scholar working under
the supervision of Prof. Ganesh Prasad. He was appointed on the
8th of July 1922, as an Assistant Professor of Mathematics at
the Banaras Hindu University. As is usually the lot of the young
University lecturers, in India at least, B.N. Prasad, had to shoulder
a heavy teaching load. But inspite of this, he continued his studies
of advanced topics in Mathematics with unabated zeal and published
during 192223 two original research papers on the properties
of nondifferentiable functions.
As
he was preparing to get settled at Banaras, with regard to his
research activities, his teacher Dr. Ganesh Prasad resigned his
Professorship at Banaras in the year 1923 owing to some serious
differences on questions of administrative policy with the university
authorities and joined the Calcutta University as the Hardinge
Professor of Higher Mathematics. Shortly after Dr. Gnash Prasad's
departure from Banaras, some of the other teachers also left the
department of Mathematics to join other universities and young
B.N. Prasad had to shoulder almost the entire load of postgraduate
teaching during that year. He had to teach topics as widely different
as the Theory of Functions and Mechanics, but he did his job with
creditable success. It appears that B.N. Prasad did not feel very
happy to stay at Banaras after the departure of Dr. Ganesh Prasad
and so he too left Banaras and joined the University of Allahabad
as a Lecturer in Mathematics on the 17th of July, 1924, During
the brief period of two years that he taught at Banaras, he became
quite popular as a teacher and attracted a number of good students
who were so devoted to him that some of them (M. L. Misra being
one of them) migrated with him to the University of Allahabad.
He
lectured at Allahabad to both under graduate and post graduate
classes on different topics of pure and applied mathematics. His
involvement in the teaching programme retarded the progress of
his research to some extent. However, the few years spent at Allahabad
before his departure to England in 1929, proved to be fruitful
in that he made an intensive study of the work done on the theory
of trigonometric series by W.H. Young, G.H. Hardy, J.E. Littlewood,
E.C. Titchmarsh, H. Lebesgue, A. Denjoy, and other continental
mathematicians and was fascinated by their profound researches.
Obtaining study leave from the University of Allahabad, he went
to the United Kingdom in the year 1929. He stayed at Edinburgh
for a few months and did some work on the theory of Fourier series,
but later proceeded to Liverpool to work with Prof. E.C. Titchmarsh.
He completed his thesis in a short period of 1½ years and
obtained the Ph.D. degree of the University of Liverpool in 1931.
Professor Titchmarsh was highly impressed by the work of B.N.
Prasad and writing about him he says, "I found him an extremely
industrious and intelligent worker who had plenty of ideas of
his own and merely asked for my advice and direction".
After the completion of his Ph.D. thesis at Liverpool, Prasad
went to Paris, the great center for mathematical research and
learning, and worked there with the celebrated French mathematician
Arnaud Denjoy. His wellknown thesis entitled, "Contribution
á l'étude de la séries conjuguée d'une
série de Fourier" prepared in less than a year under
the guidance of Prof. Denjoy, earned him the degree of 'Docteur
és Science' (State D.Sc.) of the University of Paris with
mention 'très honorable' The 'Suotenance' (The French world
Soutenace, which literally means defence, is uded to describe
the ceremony of defending the thesis by a candidate in public
before the Jury of Examiners.) of his thesis for the 'doctorat
d'Etat'took place on the 4th of June 1932, before a Jury of which
Prof. Emil Borel was the President and Prof. A. Denjoy and Prof.
G. Valiron were the two examiners. His examiners were greatly
impressed by the quality of his thesis and Prof. Borel while announcing
the result of the "Soutenance" paid compliments to Prasad
by saying: 'Nous ne pouvons que nous nous réjouir de voir
I'un de plue distingués parmi les jeunes mathematiciens
hindous se rattacher par ses travaux à I'école mathematique
francaise' (English Translation: 'We can only rejoice to see one
of the most distinguished of the young Indian mathematicians associating
himself, through his work, with the French School of Mathematics'.)
(quoted from LE TEMPS of Paris dated June 5, 1932 p.6). Bearing
testimony to the merit of Prasad's thesis Prof. Borel wrote: "Le
jury a été trés favourablement impressionné
par ses qualitiés dont il fait preuve en traitant un subject
difficile et en offerant des resultats importants et nouveaux
dans une question déja travaillée par de nombreaux
mathématiciens. II a fait preuve egalement, dans la soutenance,
qualitiés de claritié et de precision qui montrent
que il sera un excellent professur". (English Translation:
"The jury have been very favourably impressed by his qualities
of which he has given proof in treating a difficult subject and
in obtaining new and important results in a field in which a number
of mathematicians have already worked. During the Soutenance of
his thesis he has equally well given proof of possessing qualities
of clarity which show that he will be an excellent professor".).
B.N. Prasad was perhaps the first Indian to be awarded the State
D.Sc. Degree in mathematics of the University of Paris and it
was indeed a creditable achievement. During his stay in Europe,
Prasad visited various centers of mathematical research in England,
Scotland, Ireland, France, Germany and Italy and came in contact
with mathematicians like Hadamard, Hilbert, Whittaker, Hardy,
Lebesgue and others, in addition to those with whom he worked.
He returned to India in July 1932, to resume his duties at the
University of Allahabad as a Lecturer in Mathematics.
B.N.
Prasad returned to India with great academic achievements to his
credit, but the University of Allahabad failed to give due recognition
to his merits and scholarship. He remained a Lecturer for fourteen
long years after his return from France and was made a Reader
only on April 8, 1946, when the then ViceChancellor, Dr. Amaranatha
Jha, being conscious of the injustice done to Dr. Prasad, created
a new post of Reader in the Mathematics Department. It was an
irony of fate that a man who eminently deserved to hold a chair
of Mathematics in any university in India remained only a Lecturer
during the most creative period of his life.
On
his return from Europe, Prasad had to face many difficulties.
There were financial problems as also domestic ones. His hopes
and claims for receiving a promotion in the university remained
unfulfilled, but his spirit never got subdued. He had a keen desire
to create an active centre of research in Analysis at Allahabad.
He encouraged and inspired bright students to pursue higher studies
and research. Those were the days of depression when it was difficult
to get employment and money. Scholarships were too few. Most of
the bright students were opting for the administrative services.
It was very difficult to pursuade them to take to research as
the prospects of good employment for a researcher in Mathematics
were not very encouraging. B.N. Prasad's own example was there
before them. Inspite of all these difficulties, he succeeded in
training a few research workers during the years preceding the
second world war. It was his indomitable courage and invincible
determination that helped him in keeping the torch of research
burning at Allahabad and it is gratifying to recall that his efforts
bore rich fruits during the forties when a number of bright young
researchers gathered round him and Allahabad thrived as a leading
centre of research in Analysis in India. He succeeded in building
at Allahabad a strong School of Research on Summability theory,
which flourished and developed considerably during his life time
and which is very appropriately associated with his name. It is
perhaps true to say that B.N. prasad trained a larger number of
researchers than any other individual mathematician in India.
Some of his research students serving the cause of mathematical
education and research in India are: M.L. Misra at Sagar ( retired
recently as Professor of Mathematics), P.L. Bhatnagar at Bangalore,
U.N. Singh at Baroda, J.A. Siddiqi and S.M. Mazhar at Aligarh,
T. Pati at Jabalpur, S.R. Sinha, D.P. Gupta, S.N. Bhatt, T. Singh
and N.D. Mehrotra at Allahabad, Mrs. Pramila Srivastava and L.M.
Tripathi at Varanasi and Mrs. Sulaxana Kumari Gupta at Delhi.
Dr.
Prasad was elected President of the Section of Mathematics and
Statistics of the 32nd session of the Indian Science Congress
held at Nagpur in January 1945. His Presidential Address delivered
on this occasion, gave an excellent survey of the work done, upto
the end of 1944, concerning the summability problems ( of various
kinds) of a Fourier series and its Conjugate series. Written in
an attractive style and marked for its clarity of expression and
thoroughness of trcatment, this address gives an admirable account
of the vast progress made upto that time in respect of researches
in the theory of summability of a Fourier series. He gave several
other addresses of this type later in his life including his last
address which he delivered as the General President of the Indian
Science Congress at Chandigarh on January 3, 1966, a forthnight
before his death, and in which he gave a survey of the recent
researches concerning the absolute summability of infinite series
B.N.
Prasad had willingly chosen to stay at Allahabad all these years
even though he received offers of higher posts from several other
universities of India. However he yielded once to outside pressure,
when he accepted the invitation of the Government of Bihar to
work as Professor of Mathematics and Chairman of the Mathematics
Department at the Science College of Patna. Soon after joining
this post in the month of March 1949, he realized that the atmosphere
there was not congenial to his academic pursuits. Consequently
he resigned his professorship at Patna and came back to his old
post at Allahabad in January 1951. He never entertained the idea
of leaving Allahabad again thereafter.
B.N.
Prasad was married on the 29th May, 1923, to Smt. Lakshmi Devi,
a very accomplished lady, highly cultured and kindhearted. The
Prasads had three children: one son Prakash Chandra and two daughters,
Indu Prabha and Arun Prabha. They lived happily till the time
of Mrs. Prasad's death. She passed away on the 15th of September
1954, at Patna, after a prolonged illness. This tragedy gave a
stunning blow to Dr. Prasad. It created a void in his life which
tormented his mind constantly, but he concealed his grief well
within himself, remaining outwardly calm and composed. Only those
who were in very close contact with him, could perceive how lonely
he felt within himself after the death of his wife. This was,
perhaps the main reason for his keeping himself extremely busy
and accepting many outstation engagements.
In October, 1954, Prasad went to Montevideo (Uruguay, South America)
to attend the Eighth General Conference of the UNESCO as a member
of the Government of India's delegation which was headed by the
then VicePresident of India, Dr. S. Radhakrishnan. This trip
to South America gave him an opportunity to visit various universities
in the U.S.A., where he was invited to give research lectures.
He revisited Paris after an interval of about 22 years and renewed
his contacts with mathematician friends there.
After
gaining independence, India embarked on a big programme of planned
economy. One of the first steps taken in planning scientific education
on modern lines was to build a chain of national laboratories.
However, no national institutes for higher studies and research
in Mathematics were created besides the Tata Institute of Fundamental
Research, which was founded by a private trust at Bombay and has
subsequently been financed almost entirely by the Central Government.
People felt that it was desirable to have a few more institutes
for promoting quality research in Mathematics. B.N. Prasad had
slightly different ideas in this regard. He was convinced that
the universities were the fountainheads of higher learning and
research and that they should never be allowed to starve in respect
of facilities. By offering better conditions of service, higher
scales of pay and better facilites to work, the national laboratories
and institutes would attract talents from the universities causing
a serious drain on their resources and, as an inevitable consequence,
damaging the quality of research and teaching in the universities.
He, therefore, desired that good University Departments should
be given facilities to function as centres of Advanced Research
and learning and advocated a healthy coordination between research
and teaching. He wanted India to have institutions like the "École
Normale Superieur of Paris. He expressed his ideas on these questions
very clearly and emphatically in his General Presidential Address,
delivered at the 53rd session of the Indian Science Congress,
at Chandigarh on January 3, 1966.
B.N.
Prasad was also not quite happy with the working and the affairs
of the Indian Mathematical Society. During his tenure of office
as the President of the Society in the year 1961, he remedied
many of its organizational defects. He believed that the duties
of a mathematical society should not only be limited to organizing
annual conferences and publishing journals, but they should also
devise and successfully execute constructive programmes for promoting
quality research in Mathematics. With this end in view, he founded
the Allahabad Mathematical Society at Allahabad in December 1958.
He ardently desired this Society to function as an effective forum
for promoting the cause of advanced studies and research in Mathematics
and, as the President of this Society, he constantly endeavoured
to achieve this objective. The Allahabad Mathematical Society
started the publication of a research journal called the Indian
Journal of Mathematics. This Journal attained an international
status very soon. Prasad completely identified himself with the
welfare of the Society (even the office of the Society was accommodated
in his own house at Allahabad), and had a very bright vision regarding
its future growth. It is, indeed, unfortunate for the society
that he is no more to guide its destiny. He has left behind many
ideas and plans unfulfilled; however, he had the satisfaction
of putting the Society on a firm footing and of giving it the
initial inspiration and impetus.
B.N.
Prasad acted as officiating Professor and Head of the Department
of Mathematics in the University of Allahabad for about two years
beginning from November 1, 1958. He was appointed Professor of
Mathematics and Head of the Mathematics Department on a permanent
basis with effect from August 16, 1960, just about five months
before his retirement from the service of the University. During
the brief period that he was the Head of the Department, he succeeded
in infusing fresh vigour into the academic life of the Department.
He retired from the active service of the University of Allahabad
on the 11th of January 1961, after rendering to the university,
for over 36 years, a service of rare distinction of which any
mar can justly be proud. On the day of his retirement he was given
a memorable and touching farewell by his pupils and friends. That
was, in a sense, a measure of the high esteem in which he was
held by them.
B.N.
Prasad received many honours. He was one of the earliest Fellows
of the National Institute of Sciences of India (an F.N.I.), was
the first mathematician to be elected General President of the
Indian Science Congress, was the President and a Fellow of the
National Academy of Sciences of India, and was the President of
the Vijnan Parishad. The President of India conferred upon him
the title of "Padma Bhushan" in the year 1963, in recognition
of his meritorious services to the cause of Science and Education,
and a year later the President again honoured him by nominating
him to be a Member of the Rajya Sabha, the upper house of the
Indian Parliament.
The
guiding principle of B.N. Prasad's life was to seek perfection.
He sought perfection in everything he did; in writing a research
paper or a book, in giving a lecture to his class, in building
a house, in doing gardening, in every situation whatsoever, it
was this cardinal principle which consciously and unconsciously
guided his thoughts and actions. He never did anything halfheartedly.
He put his heart and soul in every undertaking of his, big or
small, and never left things to chance. He was a man of refined
taste and highly aesthetic outlook which permeated his whole personality.
He would express his annoyance unhesitatingly to any one (even
to his wife and children, to his friends and pupils) who failed
to come upto his standards of perfection, and was very outspoken
and frank in expressing his views and feelings.
That
he was an exacting taskmaster many know, but very few know how
sincerely he strove for the welfare of his pupils. He would thoughtfully
and enthusiastically plan for their wellbeing and goad them to
fulfil those plans just as he would constantly urge and inspire
them to do creative research work in Mathematics. It is perhaps
his pupils who miss him most.
The name B.N. Prasad has come to be associated with an extensive
crosssecton of Analysis, including numerous subjects such as
the convergence and summability problems of a Fourier series,
its conjugate series, the derived series of a Fourier series and
its cojugate series; the theory and applications of absolute summability
of infinite series in general, summability factors; Fourier integrals;
radial variation of analytic functions, nonsummability in the
sense of Abel, the theory of generalized derivatives, strong summabilityespecially
with Riesz's typical means, the Gibbs phenomenon and the generalized
jump of a function, the multiplication of Dirichlet series and
the second theorem of consistency. The numerous doctoral theses
prepared from time to time under his supervision and the large
number of research papers published in the important research
journals of the world in the wake of his pioneering contributions
on these subjects, bear eloquent testimony to the intrinsic value
and power of Prasad's outstanding work.
Work
done in collaboration with his pupils:
Izumi and Kawata ( 1938 ) and Cheng (1947) extended Prasad's classical
result on A summability factors of Fourier series; Pati (1954)
obtained generalizations which contained as special cases both
of these results. In 1957, Prasad and Bhatt gave further extensions
and a number of allied results of various kinds in the same direction.
Prasad and Siddiqi (1949) extended the scope of summability of
the derived Fourier series by applying Norlund means instead of
Cesaro means which had been used by previous authors. Later (1950)
they obtained a very general result concerning the Norlund summability
of the rth derived Fourier series, generalizing previous results
by Zygmund, Astrachan and Wang.
Prasad and U.N. Singh worked on the strong summabiliity of order
unity of the derived series of a Fourier series at a point, the
generating function being assumed to be of bounded variation in
the fundamental interval. Their result was a source of inspiration
for some of the subsequent researches on the subject.
After the classical work of Hardy and Riesz, Hirst, Kuttner and
Chandrasekharan on the 'second theorem of consistency', Prasad
and Pati made a study of the 'second theorem' as well and developed
his idea of the unification of the two theorems of consistency
of Riesz means. They examined the question of relative inclusion
between Riesz methods in which 'type' and 'order' were both changed.
In a later paper, Prasad and Pati studied the multiplication of
absolutely summable Dirichlet series.
While
Prasad and Pramila Srivastava (1960) have studied the strong Riesz
summability of Dirichlet series, Prasad and D.P. Gupta (1965)
have studied the convergence of ultraspherical lacunary series.
All
these papers were written in the forties, fifties and sixties
and one can discern the stamp of the master on their style. But
this is not all. During this period more than 120 valuable research
papers have been produced by the research students of B.N. Prasad
and they have been printed in almost all the important scientific
and mathematical research journals being published from different
parts of the world. 
1. 

'On
nondifferentiable functions which have progressive or regressive
derivatives for certain values of the variable', Proc. Benaras
Math. Soc., III (1921), 1.4. 
2. 

'The
absolute summability (A) of Fourier series', Proc. Edinburgh
Math. Soc. (2), 2 (1930), 129134. 
3. 

'A
theorem on the Cesaro summability of the Allied series of
a Fourier series', Jour. London Math. Soc. 6 (1931), 274278. 
4. 

'Sur
la convergence de la série conjuguée d'une série
de Fourier', Comptes Rendus de l'Académie des sciences
de Paris, 193 (1931), 11591162. 
5. 

'Sur
la summabilté de la série conjuguée dune
série de Fourier', Comptes Rendus de l Academie des
Sciences de Paris, 193 (1931), 13861387. 
6. 

'Contribution
a létude de la série conjuguée d'une
série de Fourier', Liovelles Jour., 9(1932), 153205. 
7. 

'On
the summability (C,1) of the conjugate series of a Fourier
Series', Annali di Matematica Pura ed Applicata (4), XI (1932),
207214. 
8. 

'A
new theorem on the representation of a function by Fourier's
single integral', Jour. London Math. Soc., 7(1932), 3638 
9. 

'Nonsummability
of the conjugate series of a Fourier series', Annals of Math.
(2) 33 (1932), 771772. 
10. 

'Direct
proof of Young's theorem for the convergence of the conjugate
series of a Fourier series', Bull. Calcutta Math. Soc. 24
(1932), 135142. 
11. 

'Remarks
on infinite derivatives', Jahresbericht der deutschen Mathematiker
Vereinigung, 31 (1982), 174175. 
12. 

'On
the summability of Fourier series and the bounded variation
of power series', Proc. London Math. Soc. (2), 35(1933), 407424. 
13. 

'Note
on the summability of the conjugate series of a Fourier series',
Tôhoku Math. Jour., 36 (1933), 223224. 
14. 

'On
the summability of Fourier series by arithmetic means', Proc.
U.P. Acad. Sc., 4 (1934), 3946. 
15. 

'Note
on the convergence of the conjugate series of a Fourier series',
Proc. U.P. Acad. Sc., 4 (1934), 125128. 
16. 

'On
the summability (C, 1) of Fourier series', Math. Zeit., 40(1935),
496502. 
17. 

'Notes
on a theorem of Hardy and Littlewood concerning the Cesáro
summability of the allied series of a Fourier series', Abstracts,
Indian Science Congress, 1943. 
18. 

'On
the Norlund summability of derived Fourier series', Proc,
Nat. Inst. Sc. (India), 16 (1950), 7182 (jointly with J.A.
Siddiqi). 
19. 

'On
the Norlund summability of rth derived Fourier series', Jour.
Indian Math. Soc., 14 (1951), 159170 (jointly with J.A. Siddiqi). 
20. 

'On
the strong summability of the derived Fourier series and its
conjugate series', Math. Zeit., 56 (1951), 280288 (jointly
with U.N. Singh). 
21. 

'On
the theorems of consistency for Riesz summability', Abstracts,
Proc. International Congress of Mathematicians, 1954, 461462. 
22. 

'The
summability factors of a Fourier series', Duke Math. Jour.,
24 (1957), 103117 (jointly with S.N. Bhatt). 
23. 

'On
the second theorem of consistency in the theory of absolute
Riesz summability', Trans. American Math. Soc., 85 (1957),
122123 (jointly with T. Pati). 
24. 

'On
the Gibbs phenomenon for Nörlund means', Indian Jour.
Math., 1(1958), 2128 (jointly with J.A. Siddiqi). 
25. 

The
second theorem of consistency for Riesz summability",
Abstracts, Proc. Inter. Cong. Mathematicians, Edinburgh 1958. 
26. 

'Recent
researches on the second theorem of consistency', Bull. Calcutta
Math. Soc., Jubilee Volume (1958), 225233. 
27. 

'The
second theorem of consistency in the theory of absolute Reisz
summability', Math. Ann., 140 (1960), 187197 (jointly with
T. Pati). 
28. 

'On
the multiplication of absolutely summable Dirichlet series',
Jour. Indian Math. Soc., Golden Jubilee Volume (1960), 421431
(jointly with T. Pati). 
29. 

'On
strong Riesz summability of Dirichlet series', Proc. Nat Inst.
Sc. (India), (Supplement II), 26(1960) 180209 (jointly with
Pramila Srivastava). 
30. 

'On
convergence of ultrapherical lacunary series', Annali di Matematica
Pura ed Applicata, 67 (1965), 394404 (jointly with D.P. Gupta). 

List
of presidential Addresses delivered by Prof. B.N. Prasad 
1. 

'The
summability of a Fourier series and its conjugate series',
Presidential Address, Section of Mathematis and Statistics,
32nd Session of the Indian Science Congress, Nagpur, 1945. 
2. 

'The
absolute summability of Fourier series', Presidential Address,
Section of Physical Sciences, National Academy of Sciences
of India, 29th Annual Session, Gorakhpur, 1960. 
3. 

'Mathematics
in India (Part A: General) and 'Certain aspects of the researches
on Dirichlet series' (Part B: Technical), Presidential Address,
27th Conference of the Indian Mathematical Society, Ahmedabad,
1961. 
4. 

'Hindi
madhyam, Ganitiya adhyayana aur anusandhana' (Hindi medium,
mathematical studies and research), Presidential Address,
Vijnan Parishad, Delhi, Sept. 1963. 
5. 

'Science
in India' (Part A: General) and 'Recent researches in the
absolute summability of infinite series and their applications'
(Part B: Technical), General Presidential Address, 53rd Session
of the Indian Science Congress, Chandigarh, 1966. 



