Jarnik was appointed to a chair of mathematics at the Charles University of Prague in 1928. He held this post until he retired in 1968 having taught at the University for a total of 47 years.
The main topic of Jarnik'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 ∈ Z contained in a circle centre O, radius r. There exists a constant C and a number k with
| R(r) - πr2| < Crk.Let d be the minimal value of k. Gauss proved in 1837 that d ≤ 1. Sierpinski improved the inequality to d ≤ 2/3 in 1904. Landau also made important contributions and in 1915 Hardy and Landau proved that d > 1/2. In 1923 it was proved that d < 2/3.
Jarnik 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. Jarnik showed that for certain closed curves the error term does have d = 2/3. He studied the problem for the particular case of the ellipsoid in a series of papers.
Another area of number theory which interested Jarnik 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 Jarnik'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.
Jarnik's character is described in :-
Jarnik 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.As well as being an editor of Acta Arithmetica from the beginning of the journal, Jarnik 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.
Article by: J J O'Connor and E F Robertson
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