Vojtěch Jarník (; 22 December 1897 – 22 September 1970) was a
Czech
Czech may refer to:
* Anything from or related to the Czech Republic, a country in Europe
** Czech language
** Czechs, the people of the area
** Czech culture
** Czech cuisine
* One of three mythical brothers, Lech, Czech, and Rus
*Czech (surnam ...
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, mathematical structure, structure, space, Mathematica ...
. He worked for many years as a professor and administrator at
Charles University
Charles University (CUNI; , UK; ; ), or historically as the University of Prague (), is the largest university in the Czech Republic. It is one of the List of oldest universities in continuous operation, oldest universities in the world in conti ...
, and helped found the
Czechoslovak Academy of Sciences. He is the namesake of
Jarník's algorithm for
minimum spanning tree
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. ...
s.
Jarník worked in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
,
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 ( ...
, and
graph algorithms. He has been called "probably the first Czechoslovak mathematician whose scientific works received wide and lasting international response".
As well as developing Jarník's algorithm, he found tight bounds on the number of
lattice points on
convex curves, studied the relationship between the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...
of sets of real numbers and how well they can be
approximated by rational numbers, and investigated the properties of
nowhere-differentiable functions.
Education and career
Jarník was born on 22 December 1897. He was the son of , a professor of
Romance language
The Romance languages, also known as the Latin or Neo-Latin languages, are the languages that are Language family, directly descended from Vulgar Latin. They are the only extant subgroup of the Italic languages, Italic branch of the Indo-E ...
philology
Philology () is the study of language in Oral tradition, oral and writing, written historical sources. It is the intersection of textual criticism, literary criticism, history, and linguistics with strong ties to etymology. Philology is also de ...
at
Charles University
Charles University (CUNI; , UK; ; ), or historically as the University of Prague (), is the largest university in the Czech Republic. It is one of the List of oldest universities in continuous operation, oldest universities in the world in conti ...
, and his older brother, Hertvík Jarník, also became a professor of linguistics.
[.] Despite this background, Jarník learned no Latin at his gymnasium (the C.K. české vyšší reálné gymnasium, Ječná, Prague), so when he entered Charles University in 1915 he had to do so as an extraordinary student until he could pass a Latin examination three semesters later.
He studied mathematics and physics at Charles University from 1915 to 1919, with
Karel Petr
Karel Petr (; 14 June 1868, Zbyslav, Austria-Hungary – 14 February 1950, Prague, Czechoslovakia) was a mathematician from Bohemia in Austria-Hungary and later Czechoslovakia
Czechoslovakia ( ; Czech language, Czech and , ''Česko-Slov ...
as a mentor. After completing his studies, he became an assistant to Jan Vojtěch at the
Brno University of Technology, where he also met
Mathias Lerch.
In 1921 he completed a doctoral degree (RNDr.) at Charles University with a dissertation on
Bessel function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
s supervised by Petr,
then returned to Charles University as Petr's assistant.
While keeping his position at Charles University, he studied with
Edmund Landau at the University of Göttingen from 1923 to 1925 and again from 1927 to 1929. On his first return to Charles University he defended his
habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in Germany, France, Italy, Poland and some other European and non-English-speaking countries. The candidate fulfills a university's set criteria of excelle ...
, and on his return from the second visit, he was given a chair in mathematics as an extraordinary professor. He was promoted to full professor in 1935 and later served as Dean of Sciences (1947–1948) and Vice-Rector (1950–1953). He retired in 1968.
Jarník supervised the dissertations of 16 doctoral students. Notable among these are
Miroslav Katětov, a
chess master
A chess title is a title regulated by a chess governing body and bestowed upon players based on their performance and rank. Such titles are usually granted for life. The international chess governing body FIDE grants several titles, the most pres ...
who became rector of Charles University,
Jaroslav Kurzweil, known for the
Henstock–Kurzweil integral, Czech number theorist
Bohuslav Diviš, and Slovak mathematician
Tibor Šalát.
He died on 22 September 1970, at the age of 72.
Contributions
Although Jarník's 1921 dissertation,
like some of his later publications, was in
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 ( ...
, his main area of work was in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
. He studied the
Gauss circle problem and proved a number of results on
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated ...
,
lattice point problems, and the
geometry of numbers
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice (group), lattice in \mathbb R^n, and the study of these lattices provides fundam ...
.
He also made pioneering, but long-neglected, contributions to
combinatorial optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combina ...
.
Number theory

The
Gauss circle problem asks for the number of points of the
integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice (group), lattice in the Euclidean space whose lattice points are tuple, -tuples of integers. The two-dimensional integer lattice is also called the s ...
enclosed by a given
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
.
One of Jarník's theorems (
1926
In Turkey, the year technically contained only 352 days. As Friday, December 18, 1926 ''(Julian Calendar)'' was followed by Saturday, January 1, 1927 '' (Gregorian Calendar)''. 13 days were dropped to make the switch. Turkey thus became the ...
), related to this problem, is that any closed strictly
convex curve with length passes through
at most
:
points of the integer lattice. The
in this formula is an instance of
Big O notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
. Neither the exponent of nor the leading constant of this bound can be improved, as there exist convex curves with this many grid points.
Another theorem of Jarník in this area shows that, for any closed convex curve in the plane with a well-defined length, the
absolute difference
The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y, and is a special case of the Lp distance fo ...
between the area it encloses and the number of integer points it encloses is at most its length.
Jarník also published several results in
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated ...
, the study of the approximation of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s by
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s.
He proved (
1928–1929) that the badly approximable real numbers (the ones with bounded terms in their
continued fraction
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
s) have
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...
one. This is the same dimension as the set of all real numbers, intuitively suggesting that the set of badly approximable numbers is large. He also considered the numbers
for which there exist infinitely many good rational approximations , with
:
for a given exponent , and proved (
1929) that these have the smaller Hausdorff dimension . The second of these results was later rediscovered by
Besicovitch.
[.] Besicovitch used different methods than Jarník to prove it, and the result has come to be known as the Jarník–Besicovitch theorem.
Mathematical analysis
Jarník's work in
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
was sparked by finding, in the unpublished works of
Bernard Bolzano
Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal ...
, a definition of a
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
that was nowhere
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
. Bolzano's 1830 discovery predated the 1872 publication of the
Weierstrass function
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiab ...
, previously considered to be the first example of such a function. Based on his study of Bolzano's function, Jarník was led to a more general theorem: If a
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real ...
of a
closed interval
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
does not have
bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
in any subinterval, then there is a dense subset of its domain on which at least one of its
Dini derivatives is infinite. This applies in particular to the nowhere-differentiable functions, as they must have unbounded variation in all intervals. Later, after learning of a result by
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
and
Stefan Mazurkiewicz that
generic functions (that is, the members of a
residual set of functions) are nowhere differentiable, Jarník proved that at almost all points, all four Dini derivatives of such a function are infinite. Much of his later work in this area concerned extensions of these results to approximate derivatives.
[.]
Combinatorial optimization

In
computer science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
and
combinatorial optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combina ...
, Jarník is known for an
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
for constructing
minimum spanning tree
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. ...
s that he
published in 1930, in response to the publication of
Borůvka's algorithm by another Czech mathematician,
Otakar Borůvka
Otakar Borůvka (10 May 1899 – 22 July 1995) was a Czech mathematician. He is best known for his work in graph theory..
Education and career
Borůvka was born in Uherský Ostroh, a town in Moravia, Austria-Hungary (today in the Czech Republic) ...
.
Jarník's algorithm builds a tree from a single starting vertex of a given
weighted graph by repeatedly adding the cheapest connection to any other vertex, until all vertices have been connected.
The same algorithm was later rediscovered in the late 1950s by
Robert C. Prim and
Edsger W. Dijkstra
Edsger Wybe Dijkstra ( ; ; 11 May 1930 – 6 August 2002) was a Dutch computer scientist, programmer, software engineer, mathematician, and science essayist.
Born in Rotterdam in the Netherlands, Dijkstra studied mathematics and physics and the ...
. It is also known as Prim's algorithm or the Prim–Dijkstra algorithm.
He also published a second, related, paper with (
1934
Events
January–February
* January 1 – The International Telecommunication Union, a specialist agency of the League of Nations, is established.
* January 15 – The 8.0 1934 Nepal–Bihar earthquake, Nepal–Bihar earthquake strik ...
) on the Euclidean
Steiner tree problem
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a ...
. In this problem, one must again form a tree connecting a given set of points, with edge costs given by the
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
. However, additional points that are not part of the input may be added to make the overall tree shorter. This paper is the first serious treatment of the general Steiner tree problem (although it appears earlier in a letter by
Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
), and it already contains "virtually all general properties of Steiner trees" later attributed to other researchers.
Recognition and legacy
Jarník was a member of the Czech Academy of Sciences and Arts, from 1934 as an extraordinary member and from 1946 as a regular member.
In 1952 he became one of the founding members of
Czechoslovak Academy of Sciences.
He was also awarded the Czechoslovak State Prize in 1952.
The Vojtěch Jarník International Mathematical Competition, held each year since 1991 in
Ostrava
Ostrava (; ; ) is a city in the north-east of the Czech Republic and the capital of the Moravian-Silesian Region. It has about 283,000 inhabitants. It lies from the border with Poland, at the confluences of four rivers: Oder, Opava (river), Opa ...
, is named in his honor, as is Jarníkova Street in the
Chodov district of
Prague
Prague ( ; ) is the capital and List of cities and towns in the Czech Republic, largest city of the Czech Republic and the historical capital of Bohemia. Prague, located on the Vltava River, has a population of about 1.4 million, while its P ...
. A series of
postage stamp
A postage stamp is a small piece of paper issued by a post office, postal administration, or other authorized vendors to customers who pay postage (the cost involved in moving, insuring, or registering mail). Then the stamp is affixed to the f ...
s published by Czechoslovakia in 1987 to honor the 125th anniversary of the
Union of Czechoslovak mathematicians and physicists included one stamp featuring Jarník together with
Joseph Petzval
Joseph Petzval (6 January 1807 – 17 September 1891) was a mathematician, inventor, and physicist best known for his work in optics. He was born in the town of Szepesbéla in the Kingdom of Hungary (in German: Zipser Bela, now Spišská Belá in ...
and
Vincenc Strouhal.
A conference was held in Prague, in March 1998, to honor the centennial of his birth.
[.]
Since 2002, ceremonial ''Jarník's lecture'' is held every year at
Faculty of Mathematics and Physics, Charles University, in a lecture hall named after him.
Selected publications
Jarník published 90 papers in mathematics,
[.] including:
*. A function with unbounded variation in all intervals has a dense set of points where a Dini derivative is infinite.
*. Tight bounds on the number of integer points on a convex curve, as a function of its length.
*. The badly-approximable numbers have Hausdorff dimension one.
*. The well-approximable numbers have Hausdorff dimension less than one.
*. The original reference for
Jarnik's algorithm for minimum spanning trees.
*. Generic functions have infinite Dini derivatives at almost all points.
*. The first serious treatment of the
Steiner tree problem
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a ...
.
He was also the author of ten textbooks in Czech, on
integral calculus
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Int ...
,
differential equations, and
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 ( ...
.
These books "became classics for several generations of students".
[.]
References
Further reading
*.
*
External links
*
{{DEFAULTSORT:Jarnik, Vojtech
1897 births
1970 deaths
Mathematicians from Prague
Czechoslovak mathematicians
Academic staff of Charles University
Charles University alumni