# Vojtěch Jarník

### Quick Info

Born
22 December 1897
Prague, Bohemia (now Czech Republic)
Died
22 September 1970
Prague, Czechoslovakia (now Czech Republic)

Summary
Vojtěch Jarník was a Czech mathematician who worked in number theory, mathematical analysis, and graph algorithms.

### Biography

Vojtěch Jarník's father was Jan Urban Jarník, the professor of Romanic philology at the Charles University of Prague. Jan Urban Jarník (1848-1923) had graduated from the University in Vienna where he had studied Czech, German, French, Italian, English, Sanskrit and comparative grammar of Indo-European languages. From 1882 he taught as a professor at the Czech University in Prague, where he founded a seminar on Romance philology. Jan Urban Jarník married Enrikou Eysselt-Klimpély (born 1856) in 1876. The first of their children was Hertvík Jarník (1877-1938) who followed his father in becoming a professor of Romanic philology in Brno. Vojtěch was the youngest of his parents' six children.

Jarník attended the First Czech Real Gymnasium, similar to the German Realschule, in Ječná Street in Prague. Jirí Vesely writes [12]:-
We do not know whether Jarník's choice of school was connected with the wish to master modern languages or whether the choice was influenced just by the site of the school, which was located quite close to the place where the Jarníks were living at that time.
Whatever his reason for attending this school the education he received there meant that he had problems with entering a university. He graduated from the Real Gymnasium on 7 July 1915, but without having studied Latin. Latin was necessary for admission as a student at the Charles University in Prague where he wished to go to study mathematics and physics. However, he was admitted in 1915 as an extraordinary student, only becoming a properly registered student three semesters later after passing a Latin examination. He received an exemption from military service on 17 January 1917.

At the Charles University, Jarník attended mathematics lectures delivered by Karel Petr (1868-1950), Bohuslav Hostinsky (1884-1951), Karel Rychlík (1885-1968), Jan Sobotka (1862- 1931), Bohumil Bydzovsky (1880-1969) and Václav Láska (1862-1943). He was most influenced by Karel Petr who had moved from Brno to Prague in 1902. Jarník also attended physics lectures by Bohumil Kucera (1874-1921), Václav Posejpal (1874-1935), Vladimír Václav Heinrich (1884-1965) and František Závisška (1879-1945). His university education was broad for, in addition to mathematics and physics, he took courses on philosophy, psychology, chemistry, Czech literature, and German literature. Jarník graduated from the Charles University in 1919.

After graduating, Jarník was appointed as an assistant to Jan Vojtěch (1879-1953) at the Technical University of Brno. Jan Vojtěch, after teaching in secondary schools in Prague, Olomouc, and Brno had taught at the Technical University of Brno from 1916, being appointed as an extraordinary professor on 25 February 1918. His main mathematical interests were on the theory of transformations, the theory of plane curves of the sixth degree and projective geometry. In Brno, Jarník met Matyáš Lerch who influenced his mathematical development. While carrying out his duties in Brno, Jarník continued to work on his doctoral thesis and he submitted On the roots of Bessel functions (Czech) to the Charles University in Prague and, after defending his thesis, was awarded his doctorate in 1921.

After the award of his doctorate, he returned to Prague where he was appointed as an assistant to Karel Petr. As Petr's assistant he worked on analysis and number theory. In particular he studied the number theory works of Edmund Landau. At this time Petr was writing his book Pocet diferenciální (cást analytická) and Jarník helped with proof-reading but also made improvements to the text which Petr acknowledged in the published work. During this period Jarník published On Bolzano's function (Czech) (1922) in which he [4]:-
... proved among other that Bolzano's function is in fact the oldest example of a continuous nowhere differentiable function.
In 1923 he went to the University of Göttingen to work with Edmund Landau. Returning to his post in Prague in February 1925, later that year, in December, he moved from his assistant position to that of docent after submitting his habilitation dissertation O mřížových bodech v rovině . He began lecturing, his first lecture course in 1925-26 was 'Lebesgue Integration', and, in the following year, he gave lectures as part of a course on 'Discussions on newer directions in mathematics'. He was again to visit Göttingen in session 1927/28 when he again worked with Edmund Landau. One year after his return to Prague he was promoted to Extraordinary Professor. In 1935 he was appointed to a chair of mathematics at the Charles University of Prague. He held this post until he retired in 1968 having taught at the University for a total of 47 years.

The main topic of Jarník's research was number theory. One of the problems which he worked on extensively was related to the Gauss circle problem. Let $R(r)$ denote the number of points $(m, n)$ with $m, n \in \mathbb{Z}$ contained in a circle centre O, radius $r$. There exists a constant $C$ and a number $k$ with
$| R(r) - \pir^{2}| < Cr^{k}.$
Let $d$ be the minimal value of $k$. Gauss proved in 1837 that $d ≤ 1$. Sierpinski improved the inequality to $d ≤ \large\frac{2}{3}\normalsize$ in 1904. Landau also made important contributions and in 1915 Hardy and Landau proved that $d > \large\frac{1}{2}\normalsize$. In 1923 it was proved that $d < \large\frac{2}{3}\normalsize$.

Jarník and Landau studied the same problem for curves and surfaces other than circles. Here one is interested in the difference between the number of lattice points within the closed surface and the volume enclosed by the surface. Jarník showed that for certain closed curves the error term does have $d = \large\frac{2}{3}\normalsize$. He studied the problem for the particular case of the ellipsoid in a series of papers.

Another area of number theory which interested Jarník was Diophantine approximation. He wrote papers on this topic spanning the years 1928 to 1969. During the decade 1939-49 he wrote a series of papers dealing with the geometry of numbers, in particular dealing with Minkowski's inequality for convex bodies.

Around 60 of Jarník's 90 papers were written on number theory. Many of the others were written on functions of a real variable, particularly during the years 1933-36, where he studied Dini derivatives and approximate derivatives of continuous functions. He also wrote on rearrangement of infinite series, trigonometric series and other areas of analysis. Two of his lesser known works are discussed in [3]. These are O jistém problému minimálním (1930) in which he presented a simpler proof of the Minimal Spanning Tree Problem first solved by O Boruvka in 1926. The second paper, written jointly with M Kössler, is O minimálních grafech obsahujících n danych bodu (1934) [3]:-
... the first two sections ... are devoted to general properties of "Steiner trees." It appears that virtually all general properties of Steiner trees are explicitly stated in [the paper]. Even today they are attributed to others and even today one can find in [this paper] arguments superior to those in common use ...
There was already tension around the Charles University from the beginning of 1938, with everyone worrying about German aggression. In September 1938 the Munich agreement was signed by Germany, France, the United Kingdom and Italy, allowing Germany to annex German speaking parts of Czechoslovakia which were then named the Sudetenland. After this, although the Charles University continued to operate, both staff and students were very nervous. What they all feared happened in March 1939 when German troops took over the whole of Czechoslovakia. On 14 March, with Prague covered in snow, Jarník burnt all his correspondence. On the following day Nazi troops took over Prague and closed the Charles University. Only in May 1945, after the liberation of Prague, did the university again begin operating. At this time Jarník told all his colleagues:-
Forget about everything, get down to Mathematics again.
Jarník then had the extremely arduous task of trying to reform and rebuild higher education in the Charles University and in the country as a whole. He was Dean of the Faculty of Science (1947-48), Vice-Dean of the Faculty of Science (1948-49), and Vice-Rector of the Charles University (1950-53). He was a founder member of the Czechoslovak Academy of Sciences in 1952 and served as chairman of its Mathematical-Physical Section from 1952 to 1955. He was chairman of the Academy's Mathematics Board during 1964-66. The many administrative positions he held made finding time for teaching and research difficult but, of the two, it was teaching that took priority.

Let us now look at some comments from Jarník's students, the first from before the Nazi takeover of Prague, the second soon after teaching began following the end of World War II. Štefan Schwartz entered Charles University in 1932 and began attending Jarník's lectures. He writes [10]:-
Jarník knew exactly what we had learnt at secondary school. Therefore he started with inequalities and the absolute value. (Not, of course, such neck-breaking examples which are now forced into secondary-school subject matter by immature reformers.) His lectures were transparent, delivered in a cultivated language. Jarník did not lecture "slowly", but no rush was ever felt. We felt that everything was well thought out and organized beforehand. I cannot remember whether he did ever consult his notes. In the first seminar he "apologized" several times that he would teach the theory of real numbers only in the second year. His lectures were accompanied with practical exercises named 'Elementary Problems of Higher Analysis', which he conducted himself. The lectures and practical exercises were attended by 15-20 students, who did not miss a single lecture; some 5-10 other students did not attend the lectures so regularly. Jarník was then 35 years old. He was so "tactful" that he tried not to call us to the blackboard to solve problems: mostly we volunteered. He was extremely patient. At the oral examination after the first term I learnt that Jarník knew my name and also how I performed in the seminar of Prof Bydžovsky. In this way I discovered that even the teachers "were gossiping".
Tibor Šalát entered the Faculty of Science of Charles University in the autumn of 1946 and, shortly after this, he attended lectures by Jarník [9]:-
His looks were the first thing that impressed me. He reminded me more of an elegant Frenchman than a Czech teacher, and many of my colleagues agreed with me. But it was his lectures, well thought out and adapted for beginning mathematicians, which left in me an unforgettable impression. Their influence accompanied me all my life as a University teacher. His famous tactfulness manifested itself at seminars (which he led himself) among other ways by his reluctance to force students to go to the blackboard to solve examples, even if it was their duty to do so. Those who willingly volunteered were rewarded by having to answer only theoretical questions in the examination, not also having to solve numerical examples. Let me note that at that time only students on scholarships had to pass these so called "partial" examinations. Nonetheless, all students preparing for teacher's career had to pass the so called state colloquia preceding the state examinations.
Again in [7] Bretislav Novák writes of Jarník's teaching and his care of young mathematicians:-
An outsider could get the impression that Professor Jarník has few students, that he did not establish his own school as other mathematicians of his stature usually did. However, the fact is that practically all today's Czech mathematicians can be considered directly or indirectly Jarník's students, and that everybody in this country who is engaged in Mathematics has been, to a greater or smaller extent, influenced by Jarník's personality. Professor Jarník was aware of the fact that, for the sake of our Mathematics, it was of much greater importance to pass to a large number of young people an affection for Mathematics and to give them firm foundations both in knowledge and methods of scientific work than to educate a few narrow specialists in his own field. In the same way, he preferred filling in the gaps in our mathematical literature to writing himself ten or twenty research papers.
Perhaps one of the best indications of Jarník's dedication to training young mathematicians is the fact that he supervised the doctoral dissertations of 37 students.

Jarník's character is described in [6]:-
Jarník was an outstanding teacher who was able to transmit his enthusiasm for mathematics to his students. ... his profound humane erudition, his tact and his pure human character resulted in an admiration and deep respect from all who have known him personally.
Another description of his character is given in [2]:-
Academician Jarník is a modest man, sometimes too modest, who always and under all circumstances behaves with the utmost tact. Never and to no one does he show his professional superiority. He is patient, sometimes even too patient. In situations when much younger and much less experienced colleagues lose their temper, Jarník keeps his head. In addition to high intelligence and capability to assess all matters in a wider context, this also requires lots of trained personal discipline and willpower which not everyone possesses. For these rare qualities, Jarník is not only respected and appreciated by all his friends but also they make him really very popular. He has broad knowledge of culture, a passion for music and has been regular visitor to Prague concert halls for several decades. He plays tennis well, is a fair skier, and loves Nature and hiking.
In [12] Štefan Schwartz wrote:-
Whenever it was possible I never missed an opportunity to talk with him, to ask him for his opinion. It was always an enjoyable event for me. Even now, being myself well over seventy, I see in front of me a man with deep humane feelings, a man of clear character, such as I have rarely met throughout my life.
As well as being an editor of Acta Arithmetica from the founding of the journal, Jarník was active in organising university education and scientific research throughout Czechoslovakia. He was honoured by many scientific societies, in particular being elected to the Czechoslovak Academy of Sciences. He was awarded a State Prize in 1952.

As we noted above, Jarník retired from the Charles University in 1968. Sadly he only had a short retirement before his death at the age of 72. On 16 March 1998 the Faculty of Mathematics and Physics of Charles University, the Union of Czech Mathematicians and Physicists, and the Mathematical Institute of the Academy of Sciences of the Czech Republic jointly organised an international conference to celebrate the centenary of the birth of Vojtech Jarník. The book [1], from which we have referenced papers [3], [4], [8], [9], [10], and [12], records some of the contributions to this conference. He has also been honoured by having the Jarník International Mathematical Competition named for him, see [14].

### References (show)

1. B Novák (ed), Life and work of Vojtěch Jarník (Society of Czech Mathematicians and Physicists, Prague, 1999).
2. V Knichal and Š Schwarz, Academician Vojtech Jarník Sexagenarian, Casopis pest. mat. 8 (1957), 463-492.
3. B Korte and J Nešetřil, Vojtěch Jarník's work in combinatorial optimization, in B Novák (ed), Life and work of Vojtěch Jarník (Society of Czech Mathematicians and Physicists, Prague, 1999), 37-54.
4. I Netuka, Life and work of Professor Vojtěch Jarník (22. 12. 1897 - 22. 9. 1970), in B Novák (ed), Life and work of Vojtěch Jarník (Society of Czech Mathematicians and Physicists, Prague, 1999), 11-16.
5. I Netuka, In memoriam Prof Vojtěch Jarník (22. 12. 1897 - 22. 9. 1970), News and Notes, Mathematica Bohemica 123 (2)(1998), 219-221.
6. B Novak and St Schwarz, Vojtěch Jarník (22.12.1897 - 22.9.1970), Acta Arithmetica 20 (1972), 107-115.
7. B Novak, In Memoriam Prof Vojtech Jarník, Pokroky Mat. Fyz. Astronom. 16 (1971), 1-5.
8. D Preiss, The work of Professor Jarník in real analysis, in B Novák (ed), Life and work of Vojtěch Jarník (Society of Czech Mathematicians and Physicists, Prague, 1999), 55-66.
9. T Šalát, My recollections of Professor V Jarník, in B Novák (ed), Life and work of Vojtěch Jarník (Society of Czech Mathematicians and Physicists, Prague, 1999), 111-112.
10. Š Schwartz, Recalling academician Vojtěch Jarník, in B Novák (ed), Life and work of Vojtěch Jarník (Society of Czech Mathematicians and Physicists, Prague, 1999), 95-102.
11. Š Schwartz, Niekolko vzpomienok na akademika Vojtecha Jarníka, Pokroky Mat. Fyz. Astronom. 35 (1990), 340-345.
12. J Vesely, Pedagogical activities of Vojtěch Jarník, in B Novák (ed), Life and work of Vojtěch Jarník (Society of Czech Mathematicians and Physicists, Prague, 1999), 83-94.
13. Vojtěch Jarník, Czech Digital Mathematics Library (2010). https://dml.cz/jarnik-en
14. Vojtěch Jarník International Mathematical Competition. http://vjimc.osu.cz