Alexandru Froda

Alexandru Froda
Born	(1894-07-16)July 16, 1894 Bucharest, Kingdom of Romania
Died	October 7, 1973(1973-10-07) (aged 79) Bucharest, Socialist Republic of Romania
Nationality	Romania
Alma mater	University of Bucharest University of Paris
Known for	Froda's theorem
Scientific career
Fields	Mathematician
Institutions	University of Bucharest
Thesis	Sur la distribution des propriétés de voisinage des functions de variables réelles  (1929)
Doctoral advisor	Émile Borel

Alexandru Froda (July 16, 1894 – October 7, 1973) was a Romanian mathematician with contributions in the field of mathematical analysis, algebra, number theory and rational mechanics. In his 1929 thesis he provided the namesake proof of an often unnamed theorem now sometimes called Froda's theorem.

Life

Alexandru Froda was born in Bucharest in 1894. In 1927 he graduated from the University of Sciences (now the Faculty of Mathematics of the University of Bucharest). He received his Ph.D. from the University of Paris in 1929 under the direction of Émile Borel. ^{[1] [2]}

Froda was elected president of the Romanian Mathematical Society in 1946. In 1948 he became professor in the Faculty of Mathematics and Physics of the University of Bucharest.

Work

Froda's major contribution was in the field of mathematical analysis. His first important result ^[3] 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 ^[4] from 1936 he proved a necessary and sufficient condition for a function to be measurable. In the theory of algebraic equations, Froda proved ^[5] 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 ^[6] 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 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 ^[7] 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 ^[8] he also proved several conditions ^{[9] [10] [11] [12] [13]} 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. ^[14]

In 1937 Froda independently noticed and proved the case ${\displaystyle n=1}$ of the Borsuk–Ulam theorem. He died in Bucharest in 1973.

See also

• Froda's theorem

References

1. Alexandru Froda at the Mathematics Genealogy Project
2. Alexandre Froda. "Sur la distribution des propriétés de voisinages des fonctions de variables réelles" (PDF). Sudoc.
3. Froda, Alexandre (3 December 1929). Sur la distribution des propriétés de voisinage des functions de variables réelles (PDF) (Thesis). Paris: Hermann. JFM   55.0742.02.
4. Froda, A. (1936). "Propriétés caractérisant la mesurabilité des fonctions multiformes et uniformes des variables réelles". Comptes rendus de l'Académie des Sciences de Paris . 203: 1313–1315. Zbl   0015.39905.
5. Froda, A. (1929). "Résolution générale des équations algébriques". Comptes rendus de l'Académie des Sciences de Paris . 189: 523–525.
6. A. Froda, Sur la proprieté de D. Pompeiu, concernant les integrales des fonctions a deux variables réelles, Bulletin de la Société Roumaine des Sciences, Bucharest, 1935, vol. 35, 111-115. Zbl   0008.34502
7. A. Froda, Ecuații diferentiale Lavrentiev și funcții Pompeiu, Buletin științific – Academia Republicii Populare Române, nr. 4, 1952, 801-814. Zbl   0082.29803
8. A. Froda, Triunghiuri Raționale, Comunicări Academia Republicii Populare Române, nr. 12, 1955
9. A. Froda, Critères paramétriques d'irrationallité, Mathematica Scandinavica, Kovenhava, vol. 13, 1963
10. A. Froda, Sur l'irrationalite des nombres reels, definis comme limite, Revue Roumanie de mathématique pures et appliquées, Bucharest, vol.9, facs.7, 1964
11. A. Froda, Extension effective de la condition d'irrationalité de Vigg Bran, Revue Roumaine de mathématique pures et appliquées, Bucharest, vol. 10, no. 7, 1965, 923-929
12. A. Froda, Sur le familles de critères d'irrationalité, Mathematische Zeitschrift, 1965, 89, 126–136
13. A. Froda, Nouveaux critères parametriques d'irrationalité, Comptes rendus de l'Académie des Sciences de Paris, vol. 261, 338–349
14. Viggo Brun, Ein Satz uber Irrationalitat, Aktiv fur Mathematik, 09 Naturvidensgab, Kristiania, vol. 31, H3, 1910.