Vojtěch Jarník
Vojtěch Jarník | |
---|---|
Born | |
Died | 22 September 1970 | (aged 72)
Nationality | Czechoslovakia |
Known for |
|
Scientific career | |
Fields | Mathematics |
Institutions | Charles University |
Doctoral advisor | Karel Petr |
Other academic advisors | Edmund Landau |
Doctoral students |
Vojtěch Jarník (Czech pronunciation:
Jarník worked in
Education and career
Jarník was born on 22 December 1897. He was the son of
He studied mathematics and physics at Charles University from 1915 to 1919, with Karel Petr 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.[3] In 1921 he completed a doctoral degree (RNDr.) at Charles University with a dissertation on Bessel functions supervised by Petr,[3] then returned to Charles University as Petr's assistant.[3][1][4]
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.[5] On his first return to Charles University he defended his habilitation, 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.[1][4]
Jarník supervised the dissertations of 16 doctoral students. Notable among these are
He died on 22 September 1970, at the age of 72.[1]
Contributions
Although Jarník's 1921 dissertation,[1] like some of his later publications, was in mathematical analysis, his main area of work was in number theory. He studied the Gauss circle problem and proved a number of results on Diophantine approximation, lattice point problems, and the geometry of numbers.[4] He also made pioneering, but long-neglected, contributions to combinatorial optimization.[7]
Number theory
The Gauss circle problem asks for the number of points of the integer lattice enclosed by a given circle. One of Jarník's theorems (1926), related to this problem, is that any closed strictly convex curve with length L passes through at most
points of the integer lattice. The in this formula is an instance of Big O notation. Neither the exponent of L nor the leading constant of this bound can be improved, as there exist convex curves with this many grid points.[8][9]
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 between the area it encloses and the number of integer points it encloses is at most its length.[10]
Jarník also published several results in Diophantine approximation, the study of the approximation of real numbers by rational numbers. He proved (1928–1929) that the badly approximable real numbers (the ones with bounded terms in their continued fractions) have Hausdorff dimension 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 x for which there exist infinitely many good rational approximations p/q, with
for a given exponent k > 2, and proved (
Mathematical analysis
Jarník's work in
Combinatorial optimization
In
He also published a second, related, paper with Miloš Kössler (1934) on the Euclidean Steiner tree problem. In this problem, one must again form a tree connecting a given set of points, with edge costs given by the Euclidean distance. 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), and it already contains "virtually all general properties of Steiner trees" later attributed to other researchers.[7]
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.[1] In 1952 he became one of the founding members of Czechoslovak Academy of Sciences.[1][4] He was also awarded the Czechoslovak State Prize in 1952.[1]
The Vojtěch Jarník International Mathematical Competition, held each year since 1991 in Ostrava, is named in his honor,[16] as is Jarníkova Street in the Chodov district of Prague. A series of postage stamps 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 and Vincenc Strouhal.[17]
A conference was held in Prague, in March 1998, to honor the centennial of his birth.[1]
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.[18]
Selected publications
Jarník published 90 papers in mathematics,[19] including:
- Jarník, Vojtěch (1923), "O číslech derivovaných funkcí jedné reálné proměnné" [On derivative numbers of functions of a real variable], Časopis Pro Pěstování Matematiky a Fysiky (in Czech), 53: 98–101, JFM 50.0189.02. A function with unbounded variation in all intervals has a dense set of points where a Dini derivative is infinite.[13]
- Jarník, Vojtěch (1926), "Über die Gitterpunkte auf konvexen Kurven" [On the grid points on convex curves], S2CID 117747514. Tight bounds on the number of integer points on a convex curve, as a function of its length.
- Jarník, Vojtĕch (1928–1929), "Zur metrischen Theorie der diophantischen Approximationen" [On the metric theory of Diophantine approximations], Prace Matematyczno-Fizyczne (in German), 36, Warszawa: 91–106, JFM 55.0718.01. The badly-approximable numbers have Hausdorff dimension one.[11]
- Jarník, Vojtĕch (1929), "Diophantische Approximationen und Hausdorffsches Maß" [Diophantine approximation and the Hausdorff measure], JFM 55.0719.01. The well-approximable numbers have Hausdorff dimension less than one.[11]
- Jarník, Vojtěch (1930), "O jistém problému minimálním. (Z dopisu panu O. Borůvkovi)" [About a certain minimal problem (from a letter to O. Borůvka)], Práce Moravské Přírodovědecké Společnosti (in Czech), 6: 57–63. The original reference for Jarnik's algorithm for minimum spanning trees.[7]
- Jarník, Vojtěch (1933), "Über die Differenzierbarkeit stetiger Funktionen" [On the differentiability of continuous functions], Fundamenta Mathematicae (in German), 21: 48–58, Zbl 0007.40102. Generic functions have infinite Dini derivatives at almost all points.[13]
- Jarník, Vojtěch; Kössler, Miloš (1934), "O minimálních grafech, obsahujících n daných bodů" [On minimal graphs containing n given points], Časopis pro Pěstování Matematiky a Fysiky (in Czech), 63 (8): 223–235,
He was also the author of ten textbooks in Czech, on
References
- ^ ..
- ^ Durnová (2004), p. 168.
- ^ ISBN 80-7196-156-6.
- ^ a b c d O'Connor, John J.; Robertson, Edmund F., "Vojtěch Jarník", MacTutor History of Mathematics Archive, University of St Andrews
- ^ Netuka (1998) and Veselý (1999); however, O'Connor and Robertson give his return dates as 1924 and 1928.
- ^ Vojtěch Jarník at the Mathematics Genealogy Project,
- ^ MR 1829832.
- ISBN 9781447140962.
- ISBN 9780191590320.
- ISBN 9780824796969.
- ^ ISBN 80-7196-156-6.
- S2CID 119304793. See Theorem 1.33 (the Jarník–Besicovitch theorem), p. 23, and the discussion following the theorem.
- ^ ISBN 80-7196-156-6.
- ISBN 9788072850464. See in particular page 127: "Soon after Borůvka's published his solution, another Czech mathematician, Vojtěch Jarník, reacted by publishing his own solution," and page 133: "Jarník’s article on this topic is an extract from a letter to O. Borůvka".
- ISBN 9780132762564..
- ^ "Vojtěch Jarník International Mathematical Competition". Retrieved 16 February 2017.
- ^ Miller, Jeff. "Images of Mathematicians on Postage Stamps". Retrieved 2017-02-17..
- ^ Ceremonial Lectures, mff.cuni.cz
- ^ ISBN 80-7196-156-6.
- ^ Vojtěch Jarník, Czech Digital Mathematics Library, 2010, retrieved 2017-02-17.
Further reading
- Novák, Břetislav, ed. (1999), Life and work of Vojtěch Jarník, Prague: ISBN 80-7196-156-6.
- Vojtěch Jarník digital archive, Czech Digital Mathematics Library
External links
- Media related to Vojtěch Jarník at Wikimedia Commons