**Tiberiu Popoviciu**was born in Arad which, at the time of his birth was in the autonomous region of Transylvania in Austro-Hungary. This remained the case until the end of World War I when, following the Treaty of Trianon, it became part of Romania. He attended primary school in Arad, moving on to the high school in Arad which is now the National College "Moise Nicoara". This school was founded in 1873 when Arad was part of Austro-Hungary and only received the name "Moise Nicoara" in 1919 when Transylvania became part of Romania. He showed remarkable ability in mathematics at this high school, and in 1923 he became the editor of

*Jurnal Matematic*, the school mathematical journal, a task he continued with for two years. The

*Gazeta Matematica*(the Mathematical Gazette) ran annual mathematics contests and, in 1924, Popoviciu won the national Rumania-wide contest. In the same year, he graduated from the high school and entered the Faculty of Sciences at the University of Bucharest.

Popoviciu attended a variety of courses in the Faculty of Sciences at the University of Bucharest, specialising in mathematics. His professors were some famous mathematicians of that time, such as: David Emmanuel, Gheorghe Țițeica, Dimitrie Pompeiu, and Anton Davidoglu. He graduated from the University of Bucharest in 1927 and, despite having to compete against strong competition, he was admitted to the École Normale Superieure in Paris later that year. Between 1927 and 1930, as well as courses at the École Normale Superieure, he also attended mathematics courses at the Sorbonne. During his time in Paris, he attended courses given by world-leading mathematicians such as: Émile Picard, Edouard Goursat, Jacques Hadamard, Elie Cartan, Paul Montel, Ernest Vessiot, Gaston Julia, and Jean Chazy.

In October 1928 Popoviciu was awarded his bachelor's degree in Mathematics and then began research for his Ph.D. thesis under the guidance of Paul Montel. On 12 June 1933 Tiberiu Popoviciu defended, with great distinction, his Ph.D. thesis *Sur quelques propriétes des fonctions d'une ou de deux variables réelles* . In this thesis he generalised the notion of convex functions, defining the convex functions of higher order. Over the following years, he published his thesis and a series of papers on this topic beginning with *Sur le prolongement des fonctions convexes d'ordre supérieur* (1934), *Sur l'approximation des fonctions convexes d'ordre supérieur* (1934) and *Notes sur les fontions convexes d'ordre supérieur* (1936). We give more information on this important research below.

After his return to Romania in 1933, Popoviciu began his university activities at Cluj where he was appointed first as librarian and then, after a year, as assistant lecturer in the Faculty of Sciences. From 1934 to 1936 he was an assistant to the professor of Algebra, Theodor Angheluță. On 1 April 1936 he left Cluj when he was appointed as a lecturer in the Faculty of Sciences at Cernăuți. This university had been founded in 1875 when this region was part of Austro-Hungary but it became Romanian in 1918 and new buildings had been built by the Romanian government. Originally teaching had been in German but, when Popoviciu taught there, the language of instruction was Romanian. On 4 June 1937 he was elected as a corresponding member of the Romanian Academy of Sciences. However, in 1940, during World War II, Cernăuți and the surrounding area was captured by the Soviet troops and it became part of Ukraine. The language of instruction became Ukrainian. At this time Popoviciu left Cernăuți and he was appointed as an assistant professor of Mathematical Analysis at the University of Bucharest. He held this position from 1940 to 1942 when he was appointed as Professor of Function Theory at the University of Iași. This university was founded in 1860 and today is known as the Alexandru Ioan Cuza University.

In 1946 Popoviciu returned to Cluj where he was appointed professor in the Chair of Algebra and Number Theory. At this time there were two universities in Cluj, the Romanian University of Cluj which had been the King Ferdinand I University, was renamed Babeș University (after the Romanian natural scientist Victor Babeș) while the Hungarian University of Cluj was named the Bolyai University (after the mathematician János Bolyai). The two universities combined to become the Babeș Bolyai University in 1959. After two years in the Chair of Algebra and Number Theory, Popoviciu was appointed to the Chair of Mathematical Analysis in 1948. He became the head of this department and, in the same year, was elected as a corresponding member of the Romanian Academy.

Popoviciu made important contributions in Mathematical Analysis, Approximation Theory, Convexity, Numerical Analysis, Functional Equations, Algebra and Number Theory. One of his most important scientific contributions, mentioned above, is the concept of convex functions of higher order (as a generalization of the notion of convex function) given in his Ph.D. thesis and then published in *Mathematica* in 1934. Most of the results concerning the theory of convex functions of higher order are contained in his famous book *Les fonctions convexes* , Actualites Scientifique et Industrielles, Paris, 1944. Edwin F Beckenbach (1906-1982) writes in a review:-

The importance of Popoviciu's work is stressed in [13] where the author indicates that with this work, begun in his thesis and continued in many papers and the bookIn the first chapter the author develops fundamental properties of functions of order n, in the second he continues the study of functions of order n and considers various inequalities, in the third he examines several generalizations of functions of order n, while he devotes the fourth and last chapter to the discussion of functions of two or more variables. Proofs are omitted deliberately, and very few applications are made.

*Les fonctions convexes*:-

Tiberiu Popoviciu is the founder of the Cluj School of Numerical Analysis. His work on this topic began in 1938 with the paper... he laid the foundations of a new chapter in modern mathematics. The road Tiberiu Popoviciu opened in the Constructive Theory of Functions and Functional Analysis allowed the establishment of linear functionals that intervene in the expression of the remainder in approximation operations. Obviously, all the results referring to the evaluation of the approximation order by positive and linear operators are included here, as are the convergence criteria for sequences of such operators, the study of remainders in linear methods of approximation, the comparative theory of different ways of organising computations in interpolation, the criteria whose certain satisfaction engenders representations of special forms, and many other results. Why is it important to highlight all these things? Because Tiberiu Popoviciu's very outlook on Computation Practice transformed what was referred as Numerical Calculation before into a new discipline, Numerical Analysis, in the modern sense of this concept. This point of view was understood and assumed by Tiberiu Popoviciu's students and underlies the foundation of what is now known as the Romanian School of Numerical Analysis and Approximation Theory.

*Sur les solutions bornées et les solutions mesurables de certaines écuations fontionnelles*. It was, however, only in the early 1950s that he began to publish a whole series of important papers on the topic. Because of his efforts, in 1957 the Institute for Scientific Computing was founded in Cluj. Elena Popoviciu writes [5]:-

Popoviciu was the director of this Institute which, in 1961, produced one of the first Romanian computers DACICC-1 (Dispozitiv Automat de Calcul al Institutului de Calcul din Cluj). Then, in 1969, also in Cluj, they produced DACICC-200 - one of the best performing Romanian computers of the Sixties. Florian Potra said in the interview [4]:-In1957, Tiberiu Popoviciu founded the first Institute for Scientific Computing, with a multi-disciplinary structure, in south-eastern Europe. The Institute of Numerical Analysis in Cluj, part of the Romanian Academy, succeeded in gathering around it a group of gifted mathematicians, under the guidance of Tiberiu Popoviciu. The department of computing machines of the Institute brought together a multi-disciplinary team(mathematicians, economist, physicist engineers)with the aim of building some very advanced computers.

In 1964 Popoviciu married the mathematician Elena Moldovan, the daughter of Ioan Moldovan and of his wife Rozalia. Elena Popoviciu, who has a biography in this archive, is the author of [5], [6], [7], [8], and [9].Students in the special mathematics high school class had access to the Institute for Scientific Computing(Institutul de Calcul)of the Romanian Academy. They had their own computers, built there, and we learned to do some machine language programming. That was in the1960s, and it was interesting. I kept up a long relationship with the director of that institute, the Academician Tiberiu Popoviciu. He was about60when I met him as a high school student. Later he was my Master's thesis advisor. Because of my admiration for Tiberiu Popoviciu, I combined some good classical analysis with some numerical analysis, and a little bit of functional analysis ...

Some other achievements of Tiberiu Popoviciu were: the reactivation, in 1958, of the journal *Mathematica *(*Cluj*), the founding, in 1972, of the journal *Revue d'Analyse Numerique et de Theorie de l'Approximation*, and the opening, in 1967, of a research seminar: 'The Itinerant Seminar on Functional Equations', later renamed as 'The Itinerant Seminar on Functional Equations, Approximation and Convexity'. Tiberiu Popoviciu was, in November 1948, elected a corresponding member and from 20 March 1963 a full member of the Romanian Academy. He was also for more than 30 years the president of the Cluj branch of the Romanian Mathematical Society.

Popoviciu published *Numerical Analysis. Introductory notions of approximate calculation* (Romanian) in 1975. It was the first book in an intended series on Computing Theory, Numerical Analysis and Information Theory. A review of the book states:-

However, his early death meant that he was unable to complete his intended task. He was a very active, creative and prolific mathematician until his unexpected death in 1975, after just half year from the moment of the abolition of his Institute of Scientific Computing by the Communist regime. After his death, the Institute of Scientific Computing was transferred from the Romanian Academy to the National Education Minister and the number of researchers was reduced from 48 to 6.This is planned to be the first volume in a series on numerical analysis. It is introductory, elementary and very thorough: the material covered is confined to the study of decimal representations of real numbers, approximations to real numbers, error bounds in general, the approximate execution of the fundamental operations, and the numerical calculation of the Lagrange interpolating polynomial.

We note that the Tiberiu Popoviciu High School of Computer Science was founded in 1971 and named for Popoviciu as the founder of computer science in Romania. The Institute for Scientific Computing of the Romanian Academy is now named the "Tiberiu Popoviciu" Institute for Scientific Computing.

Let us end this biography by giving a quote by Tiberiu Popoviciu:-

Algebra is a science of the equalities while Analysis is one of the inequalities.

**Article by:** *J J O'Connor* and *E F Robertson*