Vojtěch Jarník | |
|---|---|
 | |
| Born | (1897-12-22)22 December 1897 |
| Died | 22 September 1970(1970-09-22) (aged 72) |
| Known for | |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Charles University |
| Doctoral advisor | Karel Petr |
| Other academic advisors | Edmund Landau |
| Doctoral students | |
Vojtěch Jarník (Czech pronunciation:[ˈvojcɛxˈjarɲiːk]; 22 December 1897 – 22 September 1970) was aCzechmathematician. He worked for many years as a professor and administrator atCharles University, and helped found theCzechoslovak Academy of Sciences. He is the namesake ofJarník's algorithm forminimum spanning trees.
Jarník worked innumber theory,mathematical analysis, andgraph algorithms. He has been called "probably the first Czechoslovak mathematician whose scientific works received wide and lasting international response".[1] As well as developing Jarník's algorithm, he found tight bounds on the number ofinteger lattice points onconvex curves, studied the relationship between theHausdorff dimension of sets of real numbers and how well they can beapproximated by rational numbers, and investigated the properties ofnowhere-differentiable functions.
Jarník was born on 22 December 1897. He was the son ofJan Urban Jarník [cs], a professor ofRomance languagephilology atCharles University,[2] and his older brother, Hertvík Jarník, also became a professor of linguistics.[3] 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.[3]
He studied mathematics and physics at Charles University from 1915 to 1919, withKarel Petr as a mentor. After completing his studies, he became an assistant to Jan Vojtěch at theBrno University of Technology, where he also metMathias Lerch.[3] In 1921 he completed a doctoral degree (RNDr.) at Charles University with a dissertation onBessel 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 withEdmund 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 hishabilitation, 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 areMiroslav Katětov, achess master who became rector of Charles University,Jaroslav Kurzweil, known for theHenstock–Kurzweil integral, Czech number theoristBohuslav Diviš, and Slovak mathematicianTibor Šalát.[3][6]
He died on 22 September 1970, at the age of 72.[1]
Although Jarník's 1921 dissertation,[1] like some of his later publications, was inmathematical analysis, his main area of work was innumber theory. He studied theGauss circle problem and proved a number of results onDiophantine approximation,integer lattice point problems, and thegeometry of numbers.[4] He also made pioneering, but long-neglected, contributions tocombinatorial optimization.[7]
TheGauss circle problem asks for the number of points of theinteger lattice enclosed by a givencircle.One of Jarník's theorems (1926), related to this problem, is that any closed strictlyconvex curve with lengthL passes throughat most
points of the integer lattice. The in this formula is an instance ofBig O notation. Neither the exponent ofL 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, theabsolute 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 inDiophantine approximation, the study of the approximation ofreal numbers byrational numbers.He proved (1928–1929) that the badly approximable real numbers (the ones with bounded terms in theircontinued fractions) haveHausdorff 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 numbersxfor which there exist infinitely many good rational approximationsp/q, with
for a given exponentk > 2, and proved (1929) that these have the smaller Hausdorff dimension2/k. The second of these results was later rediscovered byBesicovitch.[11] Besicovitch used different methods than Jarník to prove it, and the result has come to be known as the Jarník–Besicovitch theorem.[12]
Jarník's work inreal analysis was sparked by finding, in the unpublished works ofBernard Bolzano, a definition of acontinuous function that was nowheredifferentiable. Bolzano's 1830 discovery predated the 1872 publication of theWeierstrass 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 areal-valued function of aclosed interval does not havebounded variation in any subinterval, then there is a dense subset of its domain on which at least one of itsDini 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 byStefan Banach andStefan Mazurkiewicz thatgeneric functions (that is, the members of aresidual 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.[13]
Incomputer science andcombinatorial optimization, Jarník is known for analgorithm for constructingminimum spanning trees that hepublished in 1930, in response to the publication ofBorůvka's algorithm by another Czech mathematician,Otakar Borůvka.[14]Jarník's algorithm builds a tree from a single starting vertex of a givenweighted 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 byRobert C. Prim andEdsger W. Dijkstra. It is also known as Prim's algorithm or the Prim–Dijkstra algorithm.[15]
He also published a second, related, paper withMiloš Kössler [cs] (1934) on the EuclideanSteiner tree problem. In this problem, one must again form a tree connecting a given set of points, with edge costs given by theEuclidean 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 byGauss), and it already contains "virtually all general properties of Steiner trees" later attributed to other researchers.[7]
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 ofCzechoslovak 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 inOstrava, is named in his honor,[16] as is Jarníkova Street in theChodov district ofPrague. A series ofpostage stamps published by Czechoslovakia in 1987 to honor the 125th anniversary of theUnion of Czechoslovak mathematicians and physicists included one stamp featuring Jarník together withJoseph Petzval andVincenc Strouhal.[17]
A conference was held in Prague, in March 1998, to honor the centennial of his birth.[1]
Since 2002, ceremonialJarník's lecture is held every year atFaculty of Mathematics and Physics, Charles University, in a lecture hall named after him.[18]
Jarník published 90 papers in mathematics,[19] including:
He was also the author of ten textbooks in Czech, onintegral calculus,differential equations, andmathematical analysis.[19] These books "became classics for several generations of students".[20]
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