Alexandru Froda
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Alexandru Froda (July 16, 1894 – October 7, 1973) was a
Romania Romania ( ; ro, România ) is a country located at the crossroads of Central Europe, Central, Eastern Europe, Eastern, and Southeast Europe, Southeastern Europe. It borders Bulgaria to the south, Ukraine to the north, Hungary to the west, S ...
n
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
with contributions in the field of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
,
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
and rational mechanics. In his 1929 thesis he proved what is now known as
Froda's theorem In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and th ...
.


Life

Alexandru Froda was born in
Bucharest Bucharest ( , ; ro, București ) is the capital and largest city of Romania, as well as its cultural, industrial, and financial centre. It is located in the southeast of the country, on the banks of the Dâmbovița River, less than north of ...
in 1894. In 1927 he graduated from the University of Sciences (now the Faculty of Mathematics of the
University of Bucharest The University of Bucharest ( ro, Universitatea din București), commonly known after its abbreviation UB in Romania, is a public university founded in its current form on by a decree of Prince Alexandru Ioan Cuza to convert the former Princel ...
). He received his
Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...
from the
University of Paris , image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin) , mottoeng = Here and a ...
in 1929 under the direction of
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Math ...
. 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 Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
. His first important result 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 from 1936 he proved a necessary and sufficient condition for a function to be
measurable In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
. In the theory of
algebraic equations In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
, Froda proved a method of solving algebraic equations having complex coefficients. In 1929
Dimitrie Pompeiu Dimitrie D. Pompeiu (; – 8 October 1954) was a Romanian mathematician, professor at the University of Bucharest, titular member of the Romanian Academy, and President of the Chamber of Deputies. Biography He was born in 1873 in Broscăuți, ...
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 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 posed by
Mikhail Lavrentyev Mikhail Alekseevich Lavrentyev (or Lavrentiev, russian: Михаи́л Алексе́евич Лавре́нтьев) (November 19, 1900 – October 15, 1980) was a Soviet Union, Soviet mathematician and hydrodynamics, hydrodynamicist. Early years ...
in 1925, namely whether there is a function of two real variables such that the
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
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 he also proved several conditions for a real number, which is the limit of a
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
convergent
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
, to be
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
, extending a previous result of
Viggo Brun Viggo Brun (13 October 1885 – 15 August 1978) was a Norwegian professor, mathematician and number theorist. Contributions In 1915, he introduced a new method, based on Legendre's version of the sieve of Eratosthenes, now known as the ' ...
from 1910.Viggo Brun, ''Ein Satz uber Irrationalitat'', Aktiv fur Mathematik, 09 Naturvidensgab, Kristiania, vol. 31, H3, 1910. In 1937 Froda independently noticed and proved the case n=1 of the
Borsuk–Ulam theorem In mathematics, the Borsuk–Ulam theorem states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are ...
. He died in Bucharest in 1973.


See also

*
Froda's theorem In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and th ...


References

{{DEFAULTSORT:Froda, Alexandru 20th-century Romanian mathematicians Scientists from Bucharest Academic staff of the University of Bucharest 1894 births 1969 deaths University of Bucharest alumni University of Paris alumni Romanian expatriates in France