Alexandru Froda
Encyclopedia
Alexandru Froda 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
.
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.
. 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 the theory of algebraic equations Froda proved 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 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 posed by Mikhail Lavrentyev
in 1925, namely if there is a function of two real variables such that the ordinary differential equation
has at last 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
convergent sequence
, to be irrational, extending a previous result of Viggo Brun
from 1910.
In 1937 Froda independently noticed and proved the case
of the Borsuk-Ulam theorem.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
with important contributions in the field of mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and rational mechanics. In his 1929 thesis he proved what is now known as Froda's theorem
Froda's theorem
In mathematics, Froda's theorem, named after Alexandru Froda, describes the set of discontinuities of a real-valued function of a real variable. Usually, this theorem appears in literature without Froda's name being mentioned. However, this result was first proven by A...
.
Life
Alexandru Froda was born in BucharestBucharest
Bucharest is the capital municipality, cultural, industrial, and financial centre of Romania. It is the largest city in Romania, located in the southeast of the country, at , and lies on the banks of the Dâmbovița River....
in 1894. In 1927 he graduated from the University of Sciences (now the Faculty of Mathematics from the University of Bucharest
University of Bucharest
The University of Bucharest , in Romania, is a university founded in 1864 by decree of Prince Alexander John Cuza to convert the former Saint Sava Academy into the current University of Bucharest.-Presentation:...
). He received his Ph.D. from the University of Paris
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
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 analysisMathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
. 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 the theory of algebraic equations Froda proved a method of solving algebraic equations having complex coefficients.
In 1929 Dimitrie Pompeiu
Dimitrie Pompeiu
-Biography:After studying in Dorohoi and Bucharest, he went to France, where he studied mathematics at the University of Paris . He obtained a Ph.D. degree in mathematics in 1905 with a thesis, On the continuity of complex variable functions, written under the direction of Henri Poincaré...
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 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 posed by Mikhail Lavrentyev
Mikhail Lavrentyev
Mikhail Alekseevich Lavrentyev or Lavrentiev was an outstanding Soviet mathematician and hydrodynamicist.-Biography:...
in 1925, namely if there is a function of two real variables such that the ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
has at last 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
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
convergent sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
, to be irrational, extending a previous result of Viggo Brun
Viggo Brun
Viggo Brun was a Norwegian mathematician.He studied at the University of Oslo and began research at the University of Göttingen in 1910. In 1923, Brun became a professor at the Technical University in Trondheim and in 1946 a professor at the University of Oslo...
from 1910.
In 1937 Froda independently noticed and proved the case
of the Borsuk-Ulam theorem.