# Alexandru Froda

Alexandru Froda
Born July 16, 1894
Bucharest, Romania
Died October 7, 1973 (aged 79)
Bucharest, Romania
Residence  Romania
Nationality  Romania
Fields Mathematician
Institutions University of Bucharest
Alma mater University of Bucharest
University of Paris
Known for Froda's theorem

Alexandru Froda (July 16, 1894, Bucharest, Romania – October 7, 1973, Bucharest, Romania) was a well-known Romanian mathematician with important contributions in the field of mathematical analysis, algebra, number theory and rational mechanics. In his 1929 thesis he proved what is now known as Froda's theorem.[1]

## Life

Alexandru Froda was born in Bucharest in 1894. In 1927 he graduated from the University of Sciences (now the Faculty of Mathematics from the University of Bucharest). He received his Ph.D. from the University of Paris and from University of Bucharest in 1929. He was elected president of the Romanian Mathematical Society in 1946. In 1948 he became professor at the Faculty of Mathematics and Physics at the University of Bucharest.

## Work

Froda's major contribution was in the field of mathematical analysis. His first important result[1] was concerned with the set of discontinuities of a real-valued function of a real variable. In this theorem Froda proves that the set of simple discontinuities of a real-valued function of a real variable is at most countable.

In a paper[2] from 1936 he proved a necessary and sufficient condition for a function to be measurable.

In the theory of algebraic equations Froda proved[3] a method of solving algebraic equations having complex coefficients.

In 1929 Dimitrie Pompeiu conjectured that any continuous function of two real variables defined on the entire plane is constant if the integral over any circle in the plane is constant. In the same year[4] Froda proved that, in the case that the conjecture is true, the condition that the function is defined on the whole plane is indispensable. Later it was shown that the conjecture is not true in general.

In 1907 D. Pompeiu constructed an example of a continuous function with a nonzero derivative which has a zero in every interval. Using this result Froda finds a new way of looking at an older problem[5] posed by Mikhail Lavrentyev in 1925, namely whether there is a function of two real variables such that the ordinary differential equation ${\displaystyle dy=f(x,y)dx}$ has at least two solutions passing through every point in the plane.

In the theory of numbers, beside rational triangles[6] he also proved several conditions[7][8][9][10][11] for a real number, which is the limit of a rational convergent sequence, to be irrational, extending a previous result of Viggo Brun from 1910.[12]

In 1937 Froda independently noticed and proved the case ${\displaystyle n=1}$ of the Borsuk-Ulam theorem.

Froda's theorem

## References

1. ^ a b Alexandru Froda, Sur la distribution des proprietes de voisinage des functions de variables reelles, These, Harmann, Paris, 3 December 1929
2. ^ A. Froda, Propriétés caractérisant la mesurabilité des fonctions multiformes et uniformes des variables réelles, Comptes Rendus de l'Académie des Sciences, Paris, 1936, t.203, p.1313
3. ^ A. Froda, Résolution générale des équations algébriques, Comptes Bendus de l'Academie de Sciences, Paris, 1929, t.189, p.523
4. ^ A. Froda, Sur la propiete de D. Pompeiu, concernant les integrales des fonctions a deux variables reelles, Bulletin de la Soc. Roumaine des Sciences, Bucharest, 1935, t.35, p.111-115
5. ^ A. Froda, Ecuatii diferential Lavrentiev si functii Pompeiu, Bul. Stiint. Acad. RPR, nr. 4, 1952
6. ^ A. Froda, Triunghiuri Rationale, Com. Acad. RPR, nr.12, 1955
7. ^ A. Froda, Criteres parametriques d'irrationallite, Mathematica Scandinavica, Kovenhava, Vol. 13, 1963
8. ^ A. Froda, Sur l'irrationalite des nombres reels, definis comme limite, Revue Roumanie de mathematique oures et appliquees, Bucharest, vol.9, facs.7, 1964
9. ^ A. Froda, Extension effective de la condition d'irrationalite de Vigg Bran,Revue Roumaine de nathematique pures et appliquees, Bucharest, vol.10, no.7, 1965, p.923-929
10. ^ A. Froda, Sur le familles de criteres d'irrationalite, Math. Z., 1965, 89, p.126-136
11. ^ A. Froda, Nouveaux criteres parametriques d'irrationalite, C. d'Acad. des Sciences, Paris, t.261, p.338-349
12. ^ Viggo Brun, Ein Satz uber Irrationalitat, Aktiv fur Mathemaik, 09 Naturvidensgab, Kristiania, vol.31, H3, 1910.