Vojtěch Jarník


Vojtěch Jarník Czech pronunciation: ˈvojcɛx ˈjarɲiːk; December 22, 1897 – September 22, 1970 was a Czech mathematician He worked in number theory, mathematical analysis, and graph algorithms, and has been called "probably the first Czechoslovak mathematician whose scientific works received wide and lasting international response"1

Contents

  • 1 Education and career
  • 2 Contributions
    • 21 Number theory
    • 22 Mathematical analysis
    • 23 Combinatorial optimization
  • 3 Legacy
  • 4 Selected publications
  • 5 References
  • 6 Additional reading

Education and careeredit

Jarník was the son of Jan Urban Jarník cs, a professor of Romance language philology at Charles University;2 his older brother, Hertvík Jarník, also became a professor of linguistics3 Despite this background, Jarník learned no Latin at his gymnasium the CK 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 later3

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 Lerch3 In 1921 he completed a doctoral degree RNDr at Charles University with a dissertation on Bessel functions supervised by Petr,3 and returned to Charles University as an assistant to Petr314

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 19295 On his first return to Charles University he defended his habilitation,1 and on his return from the second visit, he was given a chair in mathematics as an extraordinary professor14 He was promoted to full professor in 1935 and later served as Dean of Sciences 1947–1948 and Vice-Rector 1950–19531 He retired in 196814

Jarník supervised the dissertations of 16 doctoral students Notable among these are Miroslav Katětov, a chess master who became rector of Charles University, Jaroslav Kurzweil, known for the Henstock–Kurzweil integral, and Slovak mathematician Tibor Šalát36

Contributionsedit

Although Jarník's 1921 dissertation1 and some of his later publications were 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 numbers4 He also made pioneering, but long-neglected, contributions to combinatorial optimization7

Number theoryedit

One of his theorems 1926, related to the Gauss circle problem, is that any convex curve with length L contains at most

3 2 π 3 L 2 / 3 + O L 1 / 3 L^+OL^

points of the integer lattice Neither the exponent of L nor the leading constant of this bound can be improved, as there exist convex curves with this many grid points89

Another theorem of Jarník in this area shows that, for any closed convex curve in the plane, the difference between the area it encloses and the number of integer points it encloses is at most its length10

In Diophantine approximation, Jarník proved 1928–1929 that the badly approximable real numbers the ones with bounded terms in their continued fractions have Hausdorff dimension one He also considered the numbers x for which there exist infinitely many good rational approximations p/q, with

| x − p q | < 1 q k \right|<

for a given exponent k > 2, and proved 1929 that these have Hausdorff dimension 2/k The second of these results was later rediscovered by Besicovitch11 Besicovitch used different methods than Jarník to prove it, and the result has come to be known as the Jarník–Besicovitch theorem12

Mathematical analysisedit

Jarník's work in real analysis was sparked by finding, in the unpublished works of Bernard Bolzano, a definition of a continuous function that was nowhere differentiable Bolzano's 1830 discovery predated the 1872 publication of the Weierstrass function, 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 of a closed interval does not have bounded variation 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 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 derivatives13

Combinatorial optimizationedit

In computer science and combinatorial optimization, Jarník is known for an algorithm for constructing minimum spanning trees that he published in 1930, in response to the publication of Borůvka's algorithm by another Czech mathematician, Otakar Borůvka14 Jarník's algorithm was later rediscovered in the late 1950s by Robert C Prim and Edsger W Dijkstra It is also known as Prim's algorithm or the Prim–Dikstra algorithm15

He also published a second, related, paper with Miloš Kössler cs 1934 on the Euclidean Steiner tree problem 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 researchers7

Legacyedit

Jarník was a member of the Czech Academy of Sciences and Arts, from 1934 as an extraordinary member and from 1946 as a regular member1 In 1952 he became one of the founding members of Czechoslovak Academy of Sciences14 He was also awarded the Czechoslovak State Prize in 19521

Jarníkova Street, the Jarníkova bus stop, and a commemorative sign honoring Jarník

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 C Strouhal17

A conference was held in Prague, in March 1998, to honor the centennial of his birth1

Selected publicationsedit

Jarník published 90 papers in mathematics,18 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, Zbl 50018902  A function with unbounded variation in all intervals has a dense set of points where a Dini derivative is infinite13
  • Jarník, Vojtěch 1926, "Über die Gitterpunkte auf konvexen Kurven" On the grid points on convex curves, Mathematische Zeitschrift in German, 24 1: 500–518, doi:101007/BF01216795, MR 1544776  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, Warszawa, 36: 91–106, Zbl 55071801  The badly-approximable numbers have Hausdorff dimension one11
  • Jarník, Vojtĕch 1929, "Diophantische Approximationen und Hausdorffsches Maß" Diophantine approximation and the Hausdorff measure, Matematicheskii Sbornik in German, 36: 371–382, Zbl 55071901  The well-approximable numbers have Hausdorff dimension less than one11
  • 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 trees7
  • Jarník, Vojtěch 1933, "Über die Differenzierbarkeit stetiger Funktionen" On the differentiability of continuous functions, Fundamenta Mathematicae in German, 21: 48–58, Zbl 000740102  Generic functions have infinite Dini derivatives at almost all points13
  • 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: 223–235, Zbl 000913106  The first serious treatment of the Steiner tree problem7

He was also the author of ten textbooks in Czech, on integral calculus, differential equations, and mathematical analysis18 These books "became classics for several generations of students"19

Referencesedit

  1. ^ a b c d e f g h i j k Netuka, Ivan 1998, "In memoriam Prof Vojtěch Jarník 22 12 1897 – 22 9 1970" PDF, News and Notes, Mathematica Bohemica, 123 2: 219–221 
  2. ^ Durnová 2004, p 168
  3. ^ a b c d e f Veselý, Jiří 1999, "Pedagogical activities of Vojtěch Jarník", in Novák, Břetislav, Life and work of Vojtěch Jarník, Prague: Union of Czech mathematicians and physicists, pp 83–94, ISBN 80-7196-156-6 
  4. ^ a b c d e O'Connor, John J; Robertson, Edmund F, "Vojtěch Jarník", MacTutor History of Mathematics archive, University of St Andrews 
  5. ^ Netuka 1998 and Veselý 1999; however, O'Connor and Robertson give his return dates as 1924 and 1928
  6. ^ Vojtěch Jarník at the Mathematics Genealogy Project,
  7. ^ a b c d Korte, Bernhard; Nešetřil, Jaroslav 2001, "Vojtěch Jarník's work in combinatorial optimization", Discrete Mathematics, 235 1–3: 1–17, doi:101016/S0012-365X0000256-9, MR 1829832 
  8. ^ Bordellès, Olivier 2012, "547 Counting integer points on smooth curves", Arithmetic Tales, Springer, p 290, ISBN 9781447140962 
  9. ^ Huxley, M N 1996, "22 Jarník's polygon", Area, Lattice Points, and Exponential Sums, London Mathematical Society Monographs, 13, Clarendon Press, pp 31–33, ISBN 9780191590320 
  10. ^ Redmond, Don 1996, Number Theory: An Introduction to Pure and Applied Mathematics, CRC Press, p 561, ISBN 9780824796969 
  11. ^ a b c Dodson, M M 1999, "Some recent extensions of Jarník's work in Diophantine approximation", in Novák, Břetislav, Life and work of Vojtěch Jarník, Prague: Union of Czech mathematicians and physicists, pp 23–36, ISBN 80-7196-156-6 
  12. ^ Beresnevich, Victor; Ramírez, Felipe; Velani, Sanju 2016, "Metric Diophantine approximation: Aspects of recent work", in Badziahin, Dzmitry; Gorodnik, Alexander; Peyerimhoff, Norbert, Dynamics and Analytic Number Theory: Proceedings of the Durham Easter School 2014, London Mathematical Society Lecture Note Series, 437, Cambridge University Press, pp 1–95, arXiv:160101948, doi:101017/9781316402696002  See Theorem 133 the Jarník–Besicovitch theorem, p 23, and the discussion following the theorem
  13. ^ a b c Preiss, David 1999, "The work of Professor Jarník in real analysis", in Novák, Břetislav, Life and work of Vojtěch Jarník, Prague: Union of Czech mathematicians and physicists, pp 55–66, ISBN 80-7196-156-6 
  14. ^ Durnová, Helena 2004, "A history of discrete optimization", in Fuchs, Eduard, Mathematics Throughout the Ages, Vol II, Prague: Výzkumné centrum pro dějiny vědy, pp 51–184, 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"
  15. ^ Sedgewick, Robert; Wayne, Kevin 2011, Algorithms 4th ed, Addison-Wesley Professional, p 628, ISBN 9780132762564 
  16. ^ Vojtěch Jarník International Mathematical Competition, retrieved February 16, 2017 
  17. ^ Miller, Jeff, Images of Mathematicians on Postage Stamps, retrieved 2017-02-17 
  18. ^ a b Novák, Břetislav, ed 1999, "Bibliography of scientific works of V Jarník", Life and work of Vojtěch Jarník, Prague: Union of Czech mathematicians and physicists, pp 133–142, ISBN 80-7196-156-6 
  19. ^ Vojtěch Jarník, Czech Digital Mathematics Library, 2010, retrieved 2017-02-17 

Additional readingedit

  • Novák, Břetislav, ed 1999, Life and work of Vojtěch Jarník, Prague: Union of Czech mathematicians and physicists, ISBN 80-7196-156-6 
  • Vojtěch Jarník digital archive, Czech Digital Mathematics Library 


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