Josip Plemelj
Josip Plemelj, leading Slovene mathematician, born: Thursday December 11, 1873 in the village Grad on Bled (Grad na Bledu), Slovenia, died: Monday May 22, 1967, in Ljubljana, Slovenia.
His father Urban, who was a joiner and a carver and as the same time also a crofter, died when Josip was one year old. His mother Marija, nee Mrak, hardly maintained her family. Only by a lucky chance she was able to send her son to school in Ljubljana. Plemelj attended high school from 1886 to 1894. After leaving examination he went to Dunaj (= Wienna) in 1894 where he had inscribed to Faculty of arts and
studied mathematics, physics and astronomy. His professors in Wienna were Gustav Ritter von Escherich for mathematical analysis, Leopold Bernhard Gegenbauer and Franz Mertens for arithmetic and algebra, Edmund Weiss for astronomy, Stefan's student Ludwig Boltzman for physics and the others.
File:Pl05.gif Josip Plemelj as a student in Wienna 1896
On May 1898 Plemelj presented his doctoral thesis under Escherich's advisory entitled O linearnih homogenih diferencialnih enačbah z enolično periodičnimi
koeficienti (Über lineare homogene
Differentialgleichungen mit eindeutigen periodischen Koeffizienten, About linear homogeneous differencial equations with uniform periodical coefficients). He proceeded with his study in Berlin (1899/1900) under German mathematicians Ferdinand Georg Frobenius and Lazarus Immanuel Fuchs and in Göttingen (1900/1901)
under Felix Christian Klein and David Hilbert.
On April 1902 he became a private senior lecturer at the University in Wienna.
1906 he was appointed for assistant at Technical university in Wienna. 1907 he became associate professor
and 1908 full professor for mathematics at the University in Černovice (= Chernivtsi, Russian = Chernovtsy
= Černovcy, русскый
= Черновцы,
Romanian = Cernauti, Czernowitz),
Ukrajina (= Ukraine). From 1912 to 1913 he was dean of this faculty. Goverment had forsibly ejected him as political unreliable to Moravska, Češka
(= Bohemia) in 1917. After the first World War he took part as a member of the university commission under Provincial goverment for Slovenia in Ljubljana in founding of first Slovene university. He was elected as a first Chancellor for a new university. The same year he was appointed professor for mathematics at Faculty of arts. First at Faculty of Arts and after the second World War at Faculty for Natural Science and Technology (FNT) he lectured mathematics for 40 years. He retired in 1957.
File:Plemelj josip 1952 Jakac.jpg Josip Plemelj on drawing of Slovene painter Božidar Jakac 1952
File:Plemelj josip 1963 90 let.jpg Josip Plemelj celebrating his 90th anniversary 1963
Plemelj had shown his great gift for mathematics already in elementary school. He mastered the whole high school subject already in the beginning of the fourth class. So he was able to help candidates for graduation with
exercises. At that time he discovered alone series for sin x and cos x. Actually he found a series for cyclometric function arc cos x and after that he just inverted this series and then guessed principle for coefficients. Yet he did not have a proof for that.
Plemelj had great joy for a difficult constructional tasks from geometry. From his high school days originates an elementary problem - his later construction of regular sevenfold polygon subscibed to circle otherwise exactly and not approximately with simple solution as an angle trisection which was yet not known in those days and which necessarily leads to the old Indian or Babylonian approximate construction. He started to occupy himself with
mathematics in fourth and fifth class of high school. Beside mathematics he interested also for natural science and specially for astronomy. He studied celestial mechanics already at high school. He liked observing the stars. His eyesight was so sharp he could see planet Venus as well in the daytime.
Let us hear about his early days in school in his own words: "It was the april 1891 in fifth class. The class had two rows of desks with crossing in between and I was sitting on the most side inside chair very rear. I think there were only two desks after me. Professor Borštner did not lectured. He had only given a lection from the book for the next lesson. He called to the blackboard two pupils and there he was discussing the subject and futhermore with this he included the whole class for cooperation. He used to have such a habit greatfully to give geometrical constructional tasks which he dictated from some
collection he brought with. Once he gave amongst the other a task: Draught a triangle if one side, its altitude and a difference of two angles along it are given. Classmates had appealed to me before the lesson if the task was a little bit hard. They could not solve this task after several lessons and he had asked them
the other. Borštner used to come from before the master's desk and he stopped ahead of me, sat toward me in the desk and hence he examined. After sometime he had said we should solve that task. Perhaps he was suspicious about that we had not yet solved it so he turned to me and asked me if I had tried this construction. I had said to him I could not find any path for the solution.Then he said he would show it in the next lesson. This had plucked up my courage to inspect it again. I had found a solution with subsidiary points, lines and so forth which seems to be inaccessible for human mind if the way which had inevitably led me to my aim is hidden from. Next lesson professor Borštner had sat toward me in the desk again. After customary examination of my classmates he said: ""Well, let us work out that construction task."" I whispered him I had succeeded till then and he said: ""So, let me show how had you done this."" He thought I would show him written on pa
said: ""Well, all-right."" He had steped aside and we went before the blackboard. I drew a triangle DABC with an ordinary analysis: Given side AB = c,its altitude vc and the difference 0 < a - b < p.
Let us draw a perpendicular AA'
from
A to
side AB
and let
AA' = 2 vc
. Let us bind
A'
with
C
in the way to be A'C = AC =
b and draw
A'B = m
. Along the side
A'C
let us gather out along
A' an
angle a
and on the left side of a triangle
A'B' = c
. On this way rised triangle is
D
A'B'C ~
D
ABC
. By
B
is an angle
Đ
A'B'C =
b
and an angle
Đ
A'CB' =
g
.
A triangle DBCB' is isosceles and Đ BCB' = 2 a , so we have Đ BB'C = p/2 - a. Now it is the angle Đ A'B'B = p/2 - a + b
and we can construct over the side BA' a circumferential angle of a certain circle. We get a point B' at once because it is A'B' = c.
Because a triangle D
BCB'
is isosceles a point C
lies on a symmetral of the side
BB'
where it intersects a parallel with
AB
in a given altitude
vc
. With this the triangle
D
ABC
is drawn. Professor Borštner was gazing when he saw this curious solution
and he held of his head:
""Aber um Hergottswillen, das ist doch harsträubend, das ist doch doch menschenunmöglich auf so einen Einfall zu kommen; sagen Sie mir doch, was hat Sie zu dieser Idee geführt!"" I said to him I had not
guessed this strange solution but I had asked myself about a trigonometric determination
of a triangle because I could not find the solution in the other way.
Geometric interpretation of this solution had led me up to this pure
geometrically understandable construction. We did not spoke anymore about this
with professor Borštner, but he had after that shown another easier
solution, which I could imitate from my own construction and which I had not
perceived because I had precisely traced way.
Trigonometric solution is easy: with altitude
vc
from point
C
to side AB
it breaks in two parts
vc cot
a
and
vc cot
b
.
Then we have:
vc (cot
a
+ cot
b
) = c
or
vc sin (
a + b
) = c sin
a
sin
b
.
We can then write:
2
vc sin (
a
+ b
) = c (cos ( a - b
) - cos (
a + b
) ).
Because it is a
+ b = p - g this equation is:
2vc sin g - c cos g = c cos (a - b).
From this equation we have to obtain an angle g.
The easiest way is if we introduce a certain subsidiary angle m. Namely we raise:
2 vc = m cos m,
c = m sin m.
Both equations give us for m
an uniform certain acute angle and for m a certain positive length. Equation for g is then:
m sin ( g - m ) = c cos (a - b).
We can consider this equation as a theorem of the sine for a certain triangle in which c and m are its sides and their opposite angles are g
- m and
p/2 ± (
a - b ) respectively ( lightgreen triangle on the
picture below ). In this quoted construction this triangle
is DBA'B', where BA' = m and A'B' = c, the angle Đ BB'A' = p/2
- a +b and the angle Đ A'BB' =
g - m, as we can easily see. In my construction this requested triangle is drafted twice. I saw later on we can interpret above equation with a triangle which has
a side AB already.
This leads us to very beautiful and short construction. Requested triangle is DK'AB.
Straight line CK = a
is symmetricaly displaced in CK' = a and at the same time is AK =
AK' = m. I had spoken as a professor in Černovice with two of my students about this elementary geometrical problem and I said that my high scool teacher had
dictated this task from a certain collection. They brought me a collection indeed where there was the exact picture from Borštner's construction. Unfortunately I had not written down a title of that book which was with no doubt professor Borštner's collection. Our teacher's library at classical gymnasium in Ljubljana does not have this book at present. But I got from there a Wiegand's book entitled Geometrijske naloge za višje gimnazije (Geometrische Aufgaben für Obergymnasien, Geometric exercises for upper gymnasiums), which does not have this exercise. I found in it a task: draught a triangle if we know one length of an angle symmetral from one point, perpendicular on this symmetral from the other point and an angle by the third
different solutions. The last has annotation: at the night of the 1st january 1940 after the New Year's Eve 1939".
His main research fields were (linear) differencial equations, integral equations, potencial
theory (of harmonic functions), theory of analitical functions and function theory. When he was studying at Göttingen, Holmgren had reported about theory which was developed by Erik Ivar Fredholm for linear integral equations of the 1st and the 2nd degree. Mathematicians from Göttingen began to work on this new research field under Hilbert's guidance. Plemelj was among the first who had done beginning work and he had achieved fine results. He had used integral equations in a potencial theory successfully.
His most important work in a potencial theory is a book entitled Raziskave v teoriji potenciala (Potentialtheoretische Untersuchungen, Researches in a potencial theory), " Preisschriften der fürstl. Jablonowskischen Gesselschaft in Leipzig", (Leipzig 1911, pp XIX+100) which 1911 received award from Scientific society of prince Jablonowski in Leipzig in the amount of 1500 marks and Richard Lieben award from University of Wienna in the amount of 2000 crowns. Argument for this was that this work was the most outstanding on the field of pure and applied mathematics which had been written
by any kind of 'Austrian' mathematician in the last three years. His most important work in general is with no doubt his original, marvellous and simple solution of the Riemann - problem
File:Svnm0001.gif
about the existence of differencial equation with given monodromic group. He published his solution 1908 in treatise entitled Riemannovi zbiri funkcij z dano monodromijsko grupo (Riemannsche Funktionenscharen mit gegebener Monodromiegruppe, Riemannian assemblages of functions with a given monodromic group), " Monatshefte für Mathematik und Physik" 19, W 1908, 211-246. In solving of the Riemann - problem Plemelj used equations about boundary values of holomorphic functions which he had discovered a short time before and which are now called after him Plemelj formulae or sometimes Sohotski
- Plemelj formulae after Russian mathematician Julijan (Yulian,Юлиан) Karl
(Карл) Vasiljevič (Vasilievich,Василйевич) Sohotski (Sokhotski,Сохотски) (1842 - 1927):
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File:Svnm002.GIF
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From his methods on solving the Riemann - problem had developed the theory of singular integral equations (MSC (2000) 45-Exx) which was entertained above all by the Russian school at the head of Nikolaj (Nikolay, Николай)
Ivanovič (Ivanovich, Иванович) Mushelišvili
(Muskhelishvili,Мусхелищвили)
(1891-1976). Also important are Plemelj's contributions to the theory of analitic functions in solving the problem of uniformization of algebraic functions, contributions on formulation of the theorem of analitic extension of designs and treatises in algebra and in number theory.
1912 he published very simple proof for the Fermat's last theorem for exponent n=5, which was first given almost simultaneously by Peter Gustav Lejeune Dirichlet 1828 and Adrien Marie Legendre 1830. Their proofs were pretty hard. Plemelj showed this proof as a nice example for usage of ring we get if we extend set of rational numbers with √ 5. He untied also Galois proof that modular equation can not be lowered.
His arrival in Ljubljana 1919 was very important for development of mathematics in Slovenia. As a good teacher he had raised several generations of mathematicians and engineers. His most famous student is Ivan Vidav. After the 2nd World War
Slovenska akademija znanosti in umetnosti (Slovenian Academy of Sciences and Arts) (SAZU)
had published his three year's course of lectures for students of mathematics:
Teorija analitičnih funkcij
(The theory of analitic functions),
(SAZU, Ljubljana 1953, pp XVI+516),
Diferencialne in integralske enačbe.
Teorija in uporaba (Differential and
integral equations. The theory and the application),
Plemelj found a formula for a sum of normal derivatives of one layered potencial in the internal or external region. He was pleased also with algebra and number theory, but he had published only few
contributions from these fields - for example a book entitled
Algebra in teorija števil
(Algebra and the number theory)
(SAZU, Ljubljana 1962, pp XIV+278) which was published abroad
as his last work Problemi v smislu
Riemanna in Kleina (Problems in
the Sense of Riemann and Klein) (edition and
translation by J. R. M. Radok, "Interscience Tract in Pure and Applied
Mathematics", No. 16, Interscience Publishers: John Wiley, New York, London,
Sydney 1964, pp VII+175). This work deals with questions which were of his most
interests and examinations. His bibliography includes 33 units, from
which 30 are scientific treatises and had been published among the others in a
magazines such as: "Monatshefte für Mathematik und Physik",
"Sitzungsberichte der kaiserlichen Akademie der Wissenschaften"
in Wien, "Jahresbericht der deutschen Mathematikervereinigung",
"Gesellschaft deutscher Naturorscher und Ärzte" in Verhandlungen,
"Bulletin des Sciences Mathematiques", "
Obzornik za matematiko in fiziko"
and "Publications mathematiques de
l'Universite de Belgrade". When French mathematician E. Picard denoted Plemelj's works as "deux excellents memoires", Plemelj became known in mathematical world.
Plemelj was regular member of SAZU since its foundation in 1938, corresponding
member of JAZU (Yugoslav Academy of Sciences and Arts) in Zagreb, Croatia since
1923, corresponding member of SANU (Serbian Academy of Sciences and Arts) in
Belgrade since 1930 (1931). 1954 he received the highest award for research in
Slovenia the Prešeren award. The same year he was elected for corresponding
member of Bavarian Academy of Sciences in Munich.
1963 for his 90th anniversary University of Ljubljana granted him title of the
honorary doctor. Plemelj was first teacher of mathematics at Slovene university
and 1949 became first honorary member of ZDMFAJ Yugoslav Union of societies of
mathematicians, physicists and astronomers). He left his villa in Bled to DMFA where today is his memorial room.
Plemelj did not prepared extra for lectures. He did not have any notes. He used to say that he had thought over the lecture subject on the way from his home in Gradišče to the University. His lectures were excellent. Students had impression that he was creating teaching material and that they were present in the formation of something new. He was writting formulae on the table beautifuly although they were composited from Greek, Latin or Gothic letters. He requested the same from students. They had to write distinct.
File:Plemelj josip predavanje t4.JPG Josip Plemelj during his lecture on algebra
Plemelj had very refined ear for language and he had made solid base for the
development of the Slovene mathematical terminology. He had accustomed students
for a fine language and above all for a clear and logical phraseology. For
example he was angry if they used the word
'rabiti'
('to use')
instead of the word
'potrebovati'
('to need').
For this reason he said:
"The engineer who does not know
mathematics never needs it. But if he knows it, he uses it
frequently". Happy were they who had an
opportunity to listen him during his lectures... His work lives on. And when all
mathematical problems will be solved it shall still live...
