Some Notes on Mathematics in Latvia Through the Centuries
Daina Taimina, Cornell University
with writing input by Ingrida Henina, University of Latvia
( please send comments to email@example.com )
( this material translated by Bohdan Zograf in Belorussian )
This paper was written in 1997 as a compilation of all materials available for me at the time and being away from Latvia. To write a full account would be too big task for me alone and having no support.I hope that Latvian mathematicians one day will take this paper as a basis and build up an extensive webpage. Please accept my apologies for information which is not included in this paper - it is by no means done intentionally. For more complete information on mathematics in Latvia please visit the webpage of Latvian Mathematical Society .
Unfortunately none of the people to inhabit the land of present-day Latvia in the ninth millennium B.C. left their memoirs. New Indo-European tribes, living by stock raising and farming, appeared here in the second millennium B.C. They were the ancestors of the Baltic tribes - the Letts (the Kursi, Zemgali, Latgali) and the Lithuanians. The Latvian nationality subsequently came into being, as a result of the convergence of these tribes, sharing similar languages, cultures and economic ties. Historians have found the Balts first were mentioned in the book of the ancient historian Herodotus - in his Book IV of Historiae he described the nation Neuru which was later identified with ancient Balts. The peoples of the Mediterranean were interested in the Baltics mostly because of the amber found there. We can find the Baltic peoples mentioned also in works of Claudius Ptolemaeus (around 150 A.D.). In one of the IX century chronicles there was the first mention of a Latvian tribe - the Chori, but still in Western Europe there was not much known about the nations living on the coast of the Baltic Sea. One of the significant sources about the ancient Balts is the Icelandic historical Sagas (Islendingasògur and Konungasògur). [Radins, 1996]
The very first written documents about the history of Latvia (written in Latvia in the German language) are Henricus Lettus' Chronicles of Livonia. Henricus was a German monk who came with crusaders in the 12th century to bring Christianity to the Baltic lands. In the Chronicles there is nothing about the mathematical knowledge of the ancient Latvians. All we can tell now about their using numbers and calculations is what we find from the oral tradition in ancient Latvian folk songs (dainas) and folk tales, and from archaeological sources. In 1914 Professor J. Zalktis counted which were the most used numbers in ancient Latvian folk songs. The most often used number was 3 (like it is also in many folk traditions), and following 3 in order of frequency 2, 9, 5, 6, 1, 100, 200, 3 times 9 and only then comes the Biblical number 7, which was expected to be used more often. In the Chronicles there was not mentioned that the first school in Riga was founded in 1211, although Chronicles were describing time period from 1186 till 1227. About counting in that time one can judge from amounts of money mentioned there.
There is an evidence of geometry in ancient Latvian signs. Circle, square and regular triangle were well known geometric shapes. For example, being Northern nation and mostly peasants Latvians always had a very special relationship with Sun. There were used several geometric shapes for Sun. Simplest one was a circle. But usually the Sun sign was based on an octagon or a square divided in 9 parts. Among those ancient signs we can find also a sign of infinity -- two intertwining spirals.
During the Middle Ages, schools in Latvia mostly were just elementary schools to prepare children from wealthy German families to be able to continue their education in Germany. Only by the end of 16th century did there appear the first Rechnenschule -- schools for local Latvian people to prepare them to work in commerce. By 1600, in Riga (the capital of Latvia) there were already 3 schools where mathematics was one of the main subjects.
The very first notes which document the appearance of scientific ideas in Latvia we can find in letters of Copernicus to his contemporaries in Riga - the archbishop of Riga Janis VII Blankenfeld was Copernicus study mate in University in Bologna and possibly was participating on defending Copernicus doctoral thesis. In 1597 in a letter to Vicekancler of Sweden the famous Danish astronomer Tycho Brahe asked about permission to build an astronomical observatory on Doles Sala (an island in the Daugava River near Riga), but it was never built. A significant event in the Sixteenth Century was the opening of the first public library (Bibliotheca Rigensis) in Riga in 1524. But the attitude toward science and new ideas in feudal Riga was negative. In 1544, a famous German traveller S. Minster in his "Cosmography" wrote: "Es seind allein die kauffleut und reichen bey jnen in grossen achtung aber die gelerten do nicthtsz (only merchants and the rich do they honour, but scientists not at all)". [Stradins, 1982]
The first papers with scientific content were published in Latvia already in 1632, but unfortunately there was nothing about mathematics. The first scientific institution in Latvia (Academia Petrina) was founded in 1775 in Jelgava. There was an idea to organize a university there but at that time the Polish Kingdom had great influence and they did not allow a university to open for religious reasons. The very first professor of mathematics in Academia Petrina, which was an academic gymnasium, was Wilhelm Beitler (1745-1811) who graduated from the Law School, Tubingen University, and was interested in mathematics and physics. He lectured there for 36 years and in 1778 published a paper "New analysis of cubic equations" which we can count as the first scientific paper in mathematics published in Latvia. He also organized the first astronomical observatory in Latvia. Wilhelm Beitler was also a foreign member of Academy of Sciences of St. Petersburg. At that time all scientific papers in Latvia were published in German.
In 1813, Magnus Georg Pauker (1787-1855) became a mathematics professor in Academia Petrina. He graduated from the Dorpat University (now Tartu University in Estonia) and first worked as an engineer on building the very first optic telegraph line in Russia from St. Petersburg to Carskoje Selo. In 1811, he returned to the university as a lecturer and in 1813 awarded his Ph.D. for a thesis in solid physics. In 1815, he started to organize the first scientific society in Latvia -- Kurland Society of Literature and Arts. Pauker was in charge of the Section of Mathematics, Astronomy and Geodesy. During 1819-1822 Pauker published in two volumes the scientific papers of members of society. Among them was also a paper by Pauker himself about the construction of the regular 257-sided polygon. The existence of a construction of such a polygon had been proved before by Carl Friedrich Gauss. Pauker quoted Gauss' letter of January 2, 1820, where Gauss sent to Pauker some of his own calculations done in 1796. Gauss was elected as an honorary member of Kurland Society of Literature and Arts and in his letter from among all his titles he chose this one to sign the letter. But even the authority of Gauss did not protect Pauker from fights with his colleagues in the Society who blamed him for being wasteful with society money publishing such unnecessary calculations. Pauker resigned as secretary of Society and returned to this post again only after 25 years when he was already retired from teaching. This gap in his administrative duties was fruitful for his scientific work. Soon he became noticed as a specialist in data processing theory and mathematical statistics. He was one of the very first who recognized how valuable is the method of least squares for working with large experimental data bases. [Meders, 1928]
Until the beginning of the 19th century schools were taught only in German. The very first arithmetic textbook in the Latvian language was written by a German priest, Christopher Harder, and published in 1806. Its title was "A little counting book for the sake of all unlettered people who value wisdom and lucidity of mind and for Latvians to increase their happiness, wisdom, understanding and benefit". For comparison, the first textbook of geometry in Latvian appeared in 1862. [Taimina, 1990]
In 1796, Latvia became as a part of the Russian Empire but the authorities in Latvia remained German. Latvians themselves were still allowed only to be peasants or servants. The 1840's and 1850's Latvia saw widespread peasant revolts, and many people changed their religious from Lutheran to Orthodox to protest against German priests. During these years there started a movement of "New Latvians", to give more education to Latvians, and confirm that to be a Latvian did not have to mean to be only a peasant or servant. Educational reforms were widespread and by the end of the 19th Century, there was already 95% literacy in both sexes between ages 10 to 19. [Taimina, 1990]
The first ethnic Latvian who studied mathematics in a university was Karlis Viljams (1777-1847), the son of a bondsman. He taught himself to read in German by reading a parallel (Latvian and German) Bible -- all mathematics texts at that time were in German. Impressed by his talent, Baron Vrangel decided to give Viljams his freedom and let him go to study mathematics. In 1808, the Rector of Dorpat University (now Tartu University in Estonia) wrote a letter to the Czar of Russia asking for a stipend for Viljams. There is not a lot known about his further life because Latvians after graduating from a foreign university were not allowed to come back to Latvia -- they could not get jobs there. If we are looking at what courses the very first students took in the university we can see that most of them took at least one mathematics course. They felt intuitively that mathematical knowledge would help them. Of the 22 Latvians who studied mathematics in Tartu University before 1890, 8 graduated and only 2 of them returned back to Latvia but were not working in mathematics. [Rabinovics, 1961]
One of the very first Latvians who studied mathematics in the Tartu University, 1847-1852, was Karlis Petersons (1828-1881). Later he was a private tutor of mathematics in Moscow and together with 6 others (who all were professors in Moscow University) was a founder of the Moscow Mathematical Society in September 1864. Petersons made significant contributions in differential geometry. In 1853, he wrote a paper "Curving surfaces" where he derived the, so called, Petersons-Codazzi equations. In 1868, Petersons found the curvature of minimal surfaces and wrote some more papers on differential geometry. He also wrote papers on partial differential equations. Karlis Petersons is recognized as a founder of Moscow school of differential geometry. But this recognition was not during his life time. In 1891 a German geometer A. Foss published a paper in which he was discussing a new notion in differential geometry called P-surfaces. Foss mentioned that he learned about these surfaces from a little brochure "Ueber Curven ad Flachen. Deutsch bearbeitet vom Author Karlis Petersons. Erste Lieferung. Moskau and Leipzig, 1868", but he knew nothing about the author. This remark took attention of another German geometer P. Stackel who was also interested in the history of mathematics and he decided to search for information about the mathematician who was using differential geometry notions discovered by German mathematicians significantly later. P. Stackel had a good friend, professor A. Kneser, who was at the time teaching in Tartu University and he asked his friend to look for what he could find in archives about Karlis Petersons. Kneser learned that one of Petersons' supervisors was Ferdinand Minding. He found in an archive Peterson's manuscript which was evaluated by F. Minding "Ausgezeichnet!". It appeared to be Peterson's doctoral thesis which showed that in 1853 Peterson's used the same formula as O. Bonnet. But Bonnet's formulas were published in 1867. In his work Petersons used the same method as was used independently by Italian geometers Mainardi, whose results were first published in 1857, and Codazzi, who published his results 10 years later. [Phillips, 1979; Kolmogorov, 1996]
Karlis Petersons worked in Moscow as a teacher of mathematics. His main work was in differential geometry and he obtained an honorary doctorate from the University of Odessa in 1879 for work on partial differential equations. Largely because he was not teaching at a university his results were not well known, but they did influence D. Egorov in Moscow and Petersons gained an international reputation only when Darboux and Bianchi used his results. A class of surfaces is named after him. [Rossinskii, 1949, 1952,Youskevitch, Grigorian]
Petersons' most important paper was "On the ratios and relationships between curved surfaces" (1866), devoted to deformation of surfaces, which laid the foundation for a series of papers on the problem of bending on a principal basis (i.e., preserving the conjugacy of a certain net on the surface), the first example of which for deformation of surfaces of revolution was found by Minding in 1838. Petersons' paper "On curves on surfaces" (1867) and the book "Uber Curven und Flächen" (1868) were devoted to differential geometry. Some of the results of these works of Petersons' were later duplicated by G. Darboux and other foreign geometers; but, after E. Cosserat's translations of Peterson's main 1866-1867 works were published in Toulouse in 1905, his work achieved general recognition. [Stackel, 1901; Depman, 1952; Rabinovics, 1966; Youschkevitch, 1968; Gray, 1980]
Among seven founders of Moscow Mathematical Society was also August Davidov (1823-1885) who was born in Libav (now Liepaja, Latvia) and graduated from high school in Kuldiga (Latvia). August Davidov obtained a doctorate in 1841 from Moscow University and taught there for 35 years various courses in mathematics and mechanics. For 12 years he was a head of the Faculty of Physics and Mathematics there. He was the first president of Moscow mathematics Society (1866-1885). Davidov was the first to give a general analytic method for determining the position of equilibrium of a floating body. He also worked on partial differential equations, elliptic functions and the application of probability to statistics. [Grigorian]
Looking through biographies of mathematicians we can find that in 1861 in Riga was born Fedor Molin (1861-1941). Fedor Molin wrote a thesis under Klein's supervision after attending lectures by Klein and Carl Neumann. He was professor in Tomsk for most of his career. He worked in the theory of algebras and the theory of group representations. Molin's connection with Riga was not long. After graduating Riga gymnasium in 1879, next year he entered university in Dorpat (now Tartu University, Estonia) and to support his studies his family moved to Dorpat. Molin was a talented student with broad range of interests. He loved to learn languages. He knew German, Estonian, French, Swedish already before he entered gymnasium where he learned Greek, Hebrew, Latin, English, Italian. Later he learned Spanish, Portugese, Dutch, Norwegian. [Kanunov, 1983]. But as we see from this list of languages he had no need to learn Latvian even living in Latvia for 18 years. This shows insignificance of Latvian language in educated circles.
In 1862, the Riga Politechnikum was founded as an higher educational institution for preparing engineers. Working there as mathematics professors were H. Weidemann, G. Kieseritzky, G. Bungner, and starting from 1895, Piers Bohl (1865-1921), the most outstanding mathematician in Latvia in the first half of 20th century.
Piers Bohl was born in the little town of Valka, on the Latvian-Estonian border, in a family of Jewish merchants. His first education was from private tutoring, but later he attended the city school. In 1878, he entered the classical German gymnasium in Vilandi (Estonia). At that time his parents were not wealthy because we can find name of Piers Bohl among the students who came from poor families and got stipends if they had good success in their studies. In the gymnasium, mathematics was taught 4-5 hours per week. It was taught by Hugo Weidemann (1854-1887) who graduated from Tartu (Dorpat) University. Weidemann used his rights as teacher to make his own curriculum and he was teaching his students about different functions. After graduating from the gymnasium, Piers Bohl entered Tartu University in the Faculty of Physics and Mathematics. Students at that time could freely choose subjects they will be studying. Attending classes was not obligatory. The only requirement for graduation was to pass three parts of the graduation examinations. When a student decided that he was ready for passing one of the examinations he reported to dean in writing and then received permission to sit for the examination. Students also had to defend a thesis. If they successfully did that they were awarded a degree, "mathematics candidate" or "astronomy candidate", depending on the subject of thesis they had written. If a student could not defend his thesis, after graduation from the university he got a "real student" certificate. Candidate degree in mathematics was equivalent master's degree in US. Master's degree in 1893 was equvalent to adoctorte in US. There was (and it still exist in Latvia) next academic degree then called doctorate that could be gained only after candidate had done an outstanding work in his chosen field, and this degree allowed the holder to be called a professor. Piers Bohl passed the first third of the graduation exam in December 1885, the next he passed in the beginning of following year. In the same year, 1886, he participated in the annual student's scientific paper competition and received a gold medal for his work "Invariants of linear differential equations and their applications". The last third of the graduation exam Piers Bohl passed in August 1887 and got a diploma of "mathematics candidate". After two weeks he passed an exam on didactics of mathematics and wrote a paper "The value of gymnasium education". For that he got a senior teachers diploma. He started his teaching career as a private teacher in Levi Estate (Estonia) but then taught in the teacher's seminar in Irlava (Latvia). There he wrote his first scientific publications, "Molecule attraction law" and "About some generalization of Kepler's 3rd law". In 1889, his name appeared again in the list of Tartu University students -- he started to work on his master thesis. His dissertation was "Development of single variable functions with multivariable trigonometric series proportional to one variable". Although it was defended in January, 1893, other mathematicians started to pay attention to Bohl's ideas only 10 years later when the French astronomer E. Esclangon in 1903 independently discovered the same thing and suggested to use a convenient term for it -- quasi periodic (almost periodic) functions. The notion of quasi-periodic functions was generalized still further by Harald Bohr when he introduced almost periodic functions.
Piers Bohl got his doctorate in November of 1900 defending the thesis "On Some Differential Equations of a General Character Applicable in Mechanics". In his doctoral dissertation, Bohl, following Henri Poincare and A. Kneser, presented a new development of topological methods for systems of first order differential equations. To investigate the existence and properties of the integral solutions of such systems he applied a series of theorems he had developed and proved about points that remain fixed under continuous mappings of n-dimensional sets of points. L. Brouwer's famous theorem (1910) about the existence of a fixed point for any continuous mapping of a sphere onto itself can be easily obtained as a consequence of one of the propositions completely demonstrated in Bohl's "Uber die Bewegung eines mechanischen Systems in der Nahe einer Gleichgewichtslage" (1904). [Reizins, 1973, 1974, 1977; Myshkis, 1974,Youskevitch]
Piers Bohl started to teach in the Riga Politechnikum in 1895 as chair of the Department of Mathematics. In 1896, there came a new directive that all subjects must be taught in Russian (before that it was German). Bohl could do that because he should have experienced already before teaching in Russian in Estonia where Russian language became the state language a little earlier than in Latvia. (Estonia and Latvia both at that time were just provinces of Russia). Bohl's hobby was chess. He actively participated in the Latvian chess-players team. His style of playing interested such an outstanding chess player at that time as E. Lasker. One of the chess openings discovered by Piers Bohl is known in the literature as "the Riga version of the Spanish game". [Rabinovics, 1956; Myshkis, 1965]
The higher mathematics textbook written by Piers Bohl included analytic geometry and calculus and was used for 20 years. Bohl wrote several papers in the function theory and the theory of ordinary differential equations. Bohl studied questions regarding whether the fractional parts of certain functions give a uniform distribution. His work in this area was carried forward independently by Weyl and Sierpinski. There are many seemingly simple questions in this area which still seem to be open. For example it still seems unknown whether the fractional parts of (3/2)n form a uniform distribution on (0,1) or even if there is some finite subinterval of (0,1) which is avoided by the sequence.
Bohl had no family and no close friends. He lived only for science and he was indifferent to glory. When he found some new result in mathematics he said he could not believe that nobody had noticed it before.
During the World War I, the Riga Politechnikum was evacuated to Moscow. Piers Bohl also lived and worked there. Later, in 1919, he returned back to Riga where he started to lecture as a professor in the University of Latvia which was founded in September 1919. Two years later Piers Bohl had a cerebral hemorrhage and died. [Kneser, 1925; Myshkis, 1955; Gaiduks, 1982; Kul'vetsas, 1986]
The necessity to establish in Riga a higher educational institution to prepare specialists for engineering and commerce like ETH in Zurich was discussed by the Riga Stock Committee already in 1857. On May 16, 1861, Russian Czar Alexander II signed the regulations of the Riga Politechnikum. First classes started on October 2, 1862. Riga Politechnikum was a private educational institution, classes there were held in German up to 1896 when all classes in the future had to be in Russian. In 1863, there was started the Departments of Engineering, Chemistry, Agriculture; in 1864, the Department of Mechanics; in 1868, the Department of Commerce; and, in 1869, the Department of Architecture. It was the very first higher educational institution in Latvia. All the first and second year students took higher mathematics: analytic geometry and calculus. Riga Politechnikum had a good reputation for the high level of studies there. At the start of World War I, Riga Politechnikum was evacuated to Moscow and later to Ivanovo (Russia) where, using the Politechnikum's laboratory equipment and library as its base, the Innovations Politechnikum Institution was established. It never came back to Riga. In Riga, German authorities allowed the Baltische Technische Hochschule to open, but it was there only from October 1918 up to January 1919 when Soviet authorities named it the Latvia High School. In May, it got back the name, Baltische Technische Hochschule, but this institution did not start the next semester. On September 28, 1919, the University of Latvia open in a building of the Riga Politechnikum which had been empty after war and there were some professors who had returned or had stayed in Riga during the war and who started to lecture at the university. [Henina, 1991]
Let us return back to Riga Politechnikum and look at what the curriculum in mathematics had been there. The very first mathematics curriculum was prepared by the physics professor Ernst Nauk (1819-1875) who came from Germany. Mathematics courses were divided into two levels depending upon specialties of the students. For students who specialized in mechanics, geodesy, engineering, and architecture a higher level of mathematics was taught than for other students. In the first two years of study only theoretical mathematics was taught. The students' course load in the first year was 36 hours per week of which 8 hours was mathematics; and in the second year 4 hours (out of 32 hours) of instruction per week was in mathematics. Besides calculus and analytic geometry there were courses on descriptive geometry, projective geometry, spherical geometry, location geometry, mathematical physics, and mathematical geography. Starting from 1877-79 there were changes in the curriculum. It became obligatory to pass an oral exam in mathematics before defending the diploma thesis. The first mathematics professors in Riga Politechnikum were Gustav Schmidt (1826-1883), who came from Austria, and Gustav Cefuss, who earlier was a privatdozent at Heidelberg University (Germany). Before he arrived he wanted to know whether it will be possible to read the Crelle Mathematical Journal in a library in Riga. Both of them did not find Riga attractive enough to stay and after a year they left. In 1864, Gustav Kieseritzky (1830-1896) arrived in Riga. He was a graduate from Tartu University and became a mathematics professor in Riga Politechnikum for the next 32 years. He was a director of the Politechnikum during 1875-1885. Descriptive geometry was taught by two professors, well known at that time in Europe, Anton Schell (1836-1909) and, when he left for Vienna, Alexander Beck (1847-1926) who came from Zurich (Switzerland). Both of them had their main publications on the subjects of geodesy and astronomy. A. Beck was on the board of the Politechnikum and was very careful about choosing the candidates for professorships there. Later, geometry was taught by Karl Kupffher (1872-1935) who graduated from Tartu University as a mathematics and botany candidate. There has already been mentioned the mathematics professor Piers Bohl. A. Beck also invited as a mathematics professor, Alfreds Meders (1873-1944), whose scientific interests were differential geometry and mathematical analysis. Alfreds Meders graduated from Tartu University's Faculty of Physics and Mathematics in 1895 and received his M.A. from the University of St. Petersburg in 1906. He taught at the University of Latvia until 1939. Alfreds Meders was the first one in Latvia who wrote a paper about the history of mathematics "Direct and Indirect connections between Gauss and the University of Tartu". [Meders, 1928]
Mathematics at the University of Latvia in 1920's was mainly taught for students of the Faculty of Mathematics and Natural Sciences. Students there could major in mathematics, physics, geophysics, astronomy, biology, and geography. There were general courses in mathematics such as analytic geometry and calculus; but for students majoring in mathematics obligatory were also differential geometry, descriptive geometry, differential geometry, complex analysis, number theory, probability, and honors algebra. They had to take physics and mechanics and later spherical trigonometry and astronomy. Occasionally, there were instructions in some other mathematics subjects which students could chose. For students who wanted to get a teacher's certificate obligatory were several courses in pedagogics and didactics of mathematics but they still had to pass the certification exam at the Ministry of Education. Almost all students majoring in mathematics became teachers. To graduate from the university students had to pass six exams during a month (three in physics and mechanics, differential equations - oral and written, and complex analysis). To get an graduate degree two years after graduating the university students had to defend thesis and then they got a degree cand. math. which later was substituted with a master degree. In the late 1930's there were more new mathematics courses developed, in 1938/39 physical education and foreign language studies were included in general curricula. In 1940, after the Soviet occupation, students could not anymore chose the sequence of courses -- there was a strict sequence of courses established, and the length of studies were extended to 5 years.
The first dean of the Faculty of Mathematics and Natural Sciences was professor Edgars Lejnieks. E. Lejnieks was born in Riga in 1889. His interest in mathematics started already in school when his first paper about harmonic series was published in Odessa in a journal "Bulletin of experimental physics and elementary mathematics" (1907). In 1906 there were published solutions of 42 problems signed by the student, Lejnieks. In 1907 he went to Moscow to study mathematics at Moscow University. He was actively participating in editing papers in several journals and also he was an author of problems published in these journals. He was especially interested in modern elementary geometry and around that time there appeared several papers about triangle geometry written by Lejnieks. He also was doing research in number theory and algebra. [Lejnieks, 1911] After graduating from the university he started his teaching carrier in Moscow's St. Maria Women's Institute and also in the Moscow School for Painters, Sculptors, and Architects. Among his students was the Russian poet Vladimir Majakovskij. In 1912 he became a vice editor of a journal "Matematiceskoje Obrazovanije" which was published until 1917. After receiving his Master's degree in 1914 Lejnieks had a chance to go to Gottingen University for research purposes. There he attended lectures of D. Hilbert and E. Landau. But after the beginning of World War I he came back to Moscow where he continued his lecturing in mathematics at several institutions. In 1919, Lejnieks returned to Riga and taught at the University of Latvia until 1934. He was in charge of developing the mathematics program in the university and also he was lecturing in many courses himself. In addition, he was a founder of the University of Latvia Library. He was one of the first Latvian mathematicians who established international contacts with mathematicians from other countries -- he was a participant in the International Congresses of Mathematicians in 1928 (Bologna) and 1932 (Zurich), and the First Congress of Soviet Mathematicians in Kharkov, 1930. The scientific interests of Lejnieks were mostly in triangle geometry and number theory. [Gaiduks, 1962; Hovanskij, 1968]
Being always very busy (Lejnieks taught 15 hours a week while the usual full time course load was 6 hours per week), he never wrote his lecture notes in number theory and algebra -- these were published later by his students, Ernests Fogels and Arvids Lusis. Lejnieks' lecture notes in triangle geometry were published only in 1993. His health was not strong enough to handle all his duties and he retired from the university in 1934 because of eye illness. He died in 1937 at the age of 48. [Rabinovics, 1961]
We already mentioned that one of the first professors of mathematics in the University of Latvia was A. Meders. Alfreds Arnolds Adolfs Meders was born October 1, 1873, in Riga into a Baltic German family. His father was a high school mathematics teacher. In 1890 Meders graduated from high school and studied mathematics at Dorpat (Tartu) University where he graduated in 1895 with a higher degree. He had close scientific connections with Prof. A. Kneser who was a leading mathematician at Dorpat University at that time and who also had graduated from the university. From 1897 until 1918 Meders was teaching in the Riga Politechnical Institut, first as an assistant of K. Kupfer and later as a docent and adjunct professor together with P. Bohl. In 1906 he got his masters degree from St. Petersburg University. He was a professor at the University of Latvia from 1919 until 1939. Before 1927 he was lecturing in Russian but later his classes were held only in his native German. Mostly, this change was because scientific terminology in higher mathematics had not yet been developed and he wanted to escape from language mistakes which would make fun for his students. In 1938 he was awarded an honorary doctorate from the University of Latvia.
A. Meders' scientific work was mostly devoted to differential geometry (various singularities of spacial curves) and calculus. His papers were published mostly in German scientific journals but he had publications also in Latvia. Meders' scientific interests were not only devoted to mathematics. He was actively involved in Society of Natural Scientists in Riga (Naturforscher Verein zu Riga) where he often gave presentations not only about mathematical issues but also about astronomy, biology (especially birds), and meteorology. [Meders, 1896, 1899, 1906, 1910, 1911]
Meders started the fall semester in 1939 but he was on a list of people who were required to repatriate to Germany and he had to leave. It was emotionally very hard for him to leave Riga, his family home. He died in Poznan (Poland) in 1944.
Meders was a professor to all distinguished Latvian mathematicians who graduated from the University of Latvia in 1920's and 1930's: these included A. Lusis, E. Leimanis, A. Putnis, E. Fogels, E. Grinbergs, G. Engelis, N. Brazma, S. Mihelovics, and others. [Engelis, 1994; Mihelovics, 1994]
Among students of Professor Meders likely was also Lipman Bers (1914-1993), who became a President of the American Mathematical Society (1975-1976). Bers was born in Riga on May 22, 1914, but spent his first 4 years in St. Petersburg, Russia. In 1919 his family returned to Latvia which by then was an independent country. Both of his parents were involved in the developing school system in Latvia. Latvian language for the first time became a state language and it was finally possible for Latvians to get education in their own language. But it was also possible for other nationalities living in Latvia to get at least an elementary education in their native languages. Bers father became the director of a Yiddish gymnasium which was one of the few public Yiddish gymnasiums in the world. [Albers, 1990]. Lipman Bers graduated from this gymnasium and applied to the University of Zurich. But he stayed there only for a term because he could not get money from home as a result of the economic turmoil starting in Europe (most countries introduced rules against sending out currency). He came back to Riga and entered the University of Latvia. But there he took only few courses and passed few exams -- mostly he was involved in politics, according to his own words [Albers, 1990]. We have not yet found records in the archives of the University of Latvia describing which courses Bers had taken. In 1934, Lipman Bers had to flee Latvia because of his involvement in various underground activities. He never returned to Latvia. For further information on Bers, see [Albers, 1990; Notices, 1995].
The first mathematician who graduated from the University of Latvia and later spent all his working life lecturing at the university was Professor Arvids Lusis. He was born in 1900 into a family of peasants and got his first education in a village school. He graduated from a gymnasium in 1919 and in the same year started his studies at the University of Latvia. His parents could not support his studies financially and therefore he was also attending teacher preparation courses to be able to get a teacher's certificate and to teach in school while he was studying at the university. He was teaching mathematics, cosmography and physics at the Teachers Training Institute in Jelgava in 1923 and taught there for next 11 years. Here we should explain that the first time Latvia got its independence was after World War I in 1918. And this was the first time when students could get an education in their native language. Therefore a lot of new teachers were needed. The budget of the new republic was not large but anyway 17.5% of it was dedicated to education.
In 1924 Lusis graduated from the university and was invited to continue his research work in mathematics. During summer semesters of 1926 and 1927 he was visiting the Mathematics Institute of the Leipzig's University (Germany) where he was working with professors L. Lichtenstein and O. Helder. In 1928 he published a paper about permutable functions and Volterra integral equations and he became a privatdozent of the University of Latvia. [Hammerstein, 1932] During the years 1928-1935 he lectured in different courses in theoretical mechanics and applied mathematics. In 1938 he got his Ph.D. for a thesis about problems of permutable function theory. Also during World War II he continued his research on differential and integral equations. Since 1940 he was a professor in the University of Latvia until he died in 1969. He participated in the ICM in 1936 (Oslo) and 1966 (Moscow) and in many other scientific conferences. He was a member of Mathematical Society of France and during the 1950s he wrote for reference journal "Matematika". But mostly he is remembered as a teacher of many Latvian mathematicians who started their research work in the second half of the 20th century. His students remember him as an excellent lecturer with very clear presentation style. [Reizins, 1970] Professor A.Lusis was one of the first mathematicians who wrote about the history of mathematics in Latvia. [Lusis, 1948; 1950; 1958; 1966; Detlovs, 1968]
Between 1919 and 1939 twelve actively working Latvian mathematicians published 57 papers and also 4 new textbooks in mathematics for the university. Mathematicians were working individually -- everybody had his own scientific interests: A. Lusis was interested in integral equations, A. Putnis worked on partial differential equations, K. Zalts studied nomography, E. Fogels was interested in number theory, E. Grinbergs first interests were geometry but later he became interested in graph theory, and N. Brauers-Brazma worked on problems in theory of functions. [Reizins, 1975]
Janis Tomsons graduated from University of Latvia and started to work there in 1929. He spent all his life in teaching. In 1930 two more assistants started to work in Mathematics Seminar - E. Mednis (we have no other information) and Eizens Leimanis, who became a privatdozent in 1935.
Eizens Leimanis was born on April 10, 1905 on Vecbaizas estate near Valmiera into a farmer's family. First he received home schooling; but in 1911, when his parents moved to Riga, Leimanis went to elementary school and later graduated from Riga's First Gymnasium in 1924. That same year he started his studies at the University of Latvia from which he obtained a master's degree in 1929. He was an assistant in the descriptive geometry department, but in 1930/31 he also was teaching mathematics at the Riga School of Commerce. In 1931, he went to Leipzig University where he took some pure mathematics classes and later he went to Copenhagen Astronomical Observatory to continue his studies in celestial mechanics. In 1935 he received a doctorate for a thesis in algebraic geometry and became a privatdozent of the Department of Pure Mathematics in the University of Latvia. From November 1935 until July 1936 he was in Paris at the H. Poincare Institute doing research in the area of differential equations and celestial mechanics. In 1937 he became a dozent in Department of Theoretical Astronomy and Analytical Mechanics at the University of Latvia. During his time in Paris Leimanis also took classes at the University of Paris and the College of France, he also participated in a seminar led by J. Hadamard. In 1936 he participated in the International Congress of Mathematicians in Oslo. Leimanis continued to lecture at the University of Latvia during World War II until he was forced to go to Danzig as a refugee in 1944. Later he was lecturing at the Baltics University in Hamburg and Pineberg, Germany. In 1949 he emigrated to Canada. He was a professor at Columbia University in Vancouver. His scientific interests in mathematics were mostly in applied mathematics but also he had many publications in the history of mathematics, in philosophy, and in religion. The whole list of his publication contains about 110 titles. He died in 1992. [Leimanis, 1940; 1943; 1946; 1958; 1991]
In the 1930's Latvian mathematicians were interested in astronomy because that was a time of intensive work at the Astronomical Observatory of the University of Latvia.
Alfreds Putnis was born on March 18, 1907 in Riga. He received his education in Moscow and in Aluksne and Riga in Latvia, where he graduated from a gymnasium in 1923. That same year he entered a military school from which he graduated in 1926. He was an officer in the Latvian Army until 1937. He entered the University of Latvia as a student in 1928 and graduated from the university in 1933 with cand. math. degree and started his teaching in the Faculty of Natural Sciences. He was elected as a dozent in 1936. In 1935 he spent a semester at Geneva University, Switzerland, and in summer 1938 he did research in theoretical aerodynamics in the University of Paris with Professor Peress. Putnis died in 1940. [Putnis, 1935a, 1935b, 1936, 1938]
Karlis Zalts was born in Latvia on March 10, 1885. In 1904, after graduating from the Real Gymnasium in Jelgava, he entered the Kiev Polytechnic Institute (Ukraine) from which he graduated in 1912 as an engineer. He started his teaching in Kiev and then returned to Latvia in 1921. During the period 1921-1938 he was teaching classes in mathematics for engineering students at the University of Latvia. He had several publications about counting machines, indexes in statistics, and nomography. In 1928 he started his studies again - this time becoming a student in the Faculty of Mathematics and Natural Sciences. In 1937 he got his Masters degree for his thesis on problems in nomography. He was a dozent in the faculty until fall 1944. In February 1944 he defended his thesis on the geometry of deformations using vectors.
The life of Zalts is one of the tragic examples of the Latvian intelligentsia which was heavily affected by WWII. In 1944 Zalts was taken to Dresden by the Nazis where he was employed in a military plant. He continued researches in geometrical optics because the plant in which he was employed was involved with optical equipment. On May 1, 1945, this plant was taken by the Red Army and until September 1, 1945, Zalts was used as an interpreter for the Red Army. Later, he was an interpreter and scientific consultant for the Special Constructor Bureau in Moscow. He was allowed to return back to Riga on March 16, 1946. At first, he was working in the library but in September he was allowed to start teaching at the University of Latvia in the Faculty of Engineering Sciences. In 1946, the Latvian Academy of Sciences was organized. First, it was planned that Zalts would be included as a member of one of the institutes of the academy, but then his vita was found to be suspicious. Zalts had many publications during the 1920's and 1930's not only in mathematics but also in folklore, education, and philosophy, and he was also actively involved in the writing of the Latvian Encyclopedia. In the Soviet time all those sources were on a list of literature not accessible to the general public, except with special permission. Because of this, Zalts became an "unwelcome person" which meant that he could not publish the results of his research, attend scientific conferences, nor be promoted, even though he was allowed to continue to lecture at the University of Latvia. He died in 1953.
During the World War II, the University of Latvia continued to operate but the three occupations (Soviet, 1940-41; German, 1941-44; Soviet, 1944-1991) greatly affected scientific activities. Many of the intelligentsia in Latvia were sent to Siberia in 1940-41 and 1944-49, or sent to German concentration camps, or conscripted in either the German or Soviet Armies, or were killed, or went to the West at the end of World War II during a brief gap between the German and Soviet occupations.
In October 1944 after Soviet Army defeated Nazi forces in battles for Riga, the university continued to work under the Soviet regime. There were only 30 students left on a Faculty of Physics and Mathematics and 9 instructors (N. Brazma, who was a dean of the Faculty of Physics and Mathematics, E. Arins, A. Erglis, E. Fogels, A. Grava, A. Lusis, Z. Plume, J. Rats, and J. Tomsons). There were more mathematicians left in Riga at that time but not all of them would be allowed to teach in the university because of their "suspicious past". At that time many excellent mathematicians became school teachers which had a great influence on the level of general mathematical education. [Andzans, 1995]
Before World War II main research work in mathematics was done in the University of Latvia. Some work was also done in the Baltic-German Institute but we do not know about it very much. Mostly because the archives of this institute were taken away by Germans during their repatriation in 1939, and people lost contacts with their colleagues over the years of political turbulence. After World War II there were several new higher education institutions organized in Latvia were significant research work in mathematics was done: Academy of Agriculture, Riga Pedagogical Institute, Riga's Institute of Engineers of the Civil Air Fleet, and Pedagogical Institutes in Liepaja and Daugavpils. In 1946 the Latvian Academy of Sciences was founded. Research work in mathematics in academy was organized in the Institute of Physics and Mathematics which was reorganized as Institute of Physics in 1950. [Detlovs, 1968]
The first director of this institute was mathematician N. Brazma. Nikolajs Brazma (Brauers) was born in 1913 in a middle class family. While still attending the gymnasium he attended piano classes in the Academy of Music but decided not to become a professional musician. In 1931 he became a student of the Faculty of Mathematics and Natural Sciences of the University of Latvia from which he graduated with honors in 1936. After graduating from the University of Latvia he started his research work. In 1939 he was in Denmark where he worked with professor M. Bohr in the theory of quasi-periodic functions. He started to teach in the University of Latvia in 1938 until 1957. From 1944 until 1950 he was a Chair of Department of General Mathematics. In 1946 he defended his thesis and got his degree, candidate of sciences (Ph.D.), on investigations about unique solutions of hyperbolic differential equations. He was working together with professor A. Myshkis who worked in Riga during 1947-1953 and who influenced the interest of Latvian mathematicians in the theory of differential equations. N. Brazma became a docent in 1956. In 1957 he for a while was teaching in the Academy of Agriculture, but since 1958 he was working in Riga Polytechnic Institute (now Riga Technical University) developing new courses in mathematics there. He died in 1966. [Brazma, 1951; 1955; 1964; 1968]
The first chair of the Department of Mathematics in the newly organized Institute of Physics and Mathematics was Professor Arvids Lusis. One of the first mathematicians who also started to work in mathematical researches in the newly established institute was Ernests Fogels. Fogels was born in 1910 in Nigrande, Latvia. His parents were poor farmers. He attended the Second Gymnasium in Riga. At a mathematical competition he was rewarded with a book on number theory. This was an extra stimulus to arouse his interest in mathematics. In 1928 he became a student of the Faculty of Mathematics and Natural Sciences of the University of Latvia. As he also had a talent for painting, he was also attending the Academy of Fine Arts. In the meantime he worked as a clerk and later as a school teacher of mathematics to earn money for his tuition. He graduated from the university in 1933, and in 1935 he started to lecture at the university, mainly in algebra and number theory. At the end of 1938 he went on probation course to Cambridge University, England. His original plan to work under the supervision of G.H. Hardy failed, but he was willingly accepted by A.E. Ingham who proposed to improve the estimate of the difference between two consecutive primes. [Fogels, 1938a; 1938b] The beginning of the World War II in 1939 interrupted this research. In 1940 E. Fogels was appointed as an associate professor at the University of Latvia. In 1947 he defended his thesis "On mean values of arithmetical functions", the main part of which was written in early forties, and got a degree of candidat nauk. In 1947 he also became a research fellow at the Institute of Physics and Mathematics of the Latvian Academy of Science (LAS). In 1950 after mathematics was abandoned in this institute E. Fogels stated to lecture at Riga Pedagogical Institute where during the next 8 years he was lecturing in almost all mathematics courses and wrote about 30 sets of lecture notes for his students. This left practically no time for his research. In 1958 the Pedagogical Institute was closed. Because of that closure and his poor health until 1961 E. Fogels held no official position. From 1961-1966 he was a research fellow at the Radio Astrophysical Observatory of the LAS. This period was fruitful for his scientific activities. He obtained rather strong results on the density of zeros of different zeta-functions, on the distribution of primes in arithmetical progressions, on various algebraic fields, and on binary and ternary quadratic forms. [Fogels, 1963; 1964] E. Fogels made reports at seminars in Moscow and Leningrad (Russia). Yu.V. Linnik and others suggested that he should prepare a doctoral thesis on the basis of his most important results and special permission was even given by the Higher Certification Commission. However, E. Fogels did not want to go through all bureaucracy. For some time E. Fogels was a reviewer for review journals. He was extremely careful and verified each formula, which took him too much time. Therefore he soon gave up this work, remaining only on the editorial board of "Acta Arithmetica" from 1967 until his death in 1985. [Kubilius, 1991]
In 1962 again there was serious research work in mathematics started in a section of mathematical physics of the Institute of Physics of the LAS. Leader of those researches was professor L. Reizins who later organized the Laboratory of Mathematics and was one of the leading mathematicians in Latvia.
Linards Reizins was born on January 14, 1924, in Riga, Latvia, to a family of teachers. After finishing Elementary School No. 14 of Riga, he went to the Second Gymnasium of Riga. His studies were interrupted by the Second World War. When Nazis army entered Riga, L. Reizins was at a sports camp in Burtnieki and the roads to Riga were blocked. With a group of young men he went first to Valmiera and then to Estonia. Near Paide they were scattered and L. Reizins was arrested by the German soldiers and taken as a prisoner to Riga. He was released in spring 1942 and managed to pass his high-school final exams. He tried to find a job that would help avoid recruitment in the German army. In the summer he worked in the countryside. Then he took a preparatory course as a physical education teacher, and as well as he worked for the "Telefunken" company. He spent the last months of war in hiding. After World War II, L. Reizins began studying mathematics at the University of Latvia. He participated in the students' research society, the youth organization, and the sports club. In 1948 he graduated cum laude from the University and continued research as a staff member in the Department of Mathematical Analysis, and soon took up a postgraduate course with Professor A. Lusis, specializing in the field of differential equations. L. Reizins continued work on quasihomogeneous differential equations which he began as part of his theses. He generalized the concepts of exceptional direction and normal domain in order to analyze the structure of trajectories in the neighborhood of an isolated stationary point in three-dimensional space. L. Reizins formulated and solved some discrimination problems. His first publication, "The behavior of the integral curves of a system of three differential equations in the neighborhood of a singular point", appeared in the Proceedings of the Latvian Academy of Sciences in 1951. The American Mathematical Society took interest in it, and published it in 1955 in the American Mathematical Translations. This was a rare case, when a university graduation paper aroused the interest of A.M.S. mathematicians. Reizins had a successful career; however, he lost his job during a soviet political campaign in 1949, when he had to discontinue also his postgraduate studies. Until 1959 he worked at the Riga Secondary School No. 7 where he taught mathematics, was the headmaster's assistant and lectured in a course for teachers; and, naturally, continued his research. L. Reizins generalized the concept of exceptional direction, formulated and solved discrimination problems in n-dimensional space. In 1959, L. Reizins successfully defended his first thesis, "The behavior of trajectories in the neighborhood of a stationary point in the three-dimensional space", at the Tartu State University. Professor of Moscow University V. V. Nemytskii in his reference to this theses gave particular prominence to two theorems, deeming them as the "first theorems of topological equivalence in the neighborhood of a stationary point of higher order in n-dimensional space." In these theorems, L. Reizins found sufficient conditions under which a perturbed homogeneous dynamical system in three-dimensional space is locally almost equivalent to its transaction. This high appraisal by V. V. Nemytskii in many respects determined L. Reizins' further investigations. In 1957 he was a junior research fellow at the Astronomy Department of the Latvian Academy of Sciences; in 1958, he was promoted to Scientific Secretary, and in 1961 he became a senior research fellow. In 1963 he became head of the Department of Mathematics at the Institute of Physics of the Latvian Academy of Sciences. In 1969, this Institute included a research group led by Eduards Riekstins (a specialist in asymptotic expansion theory), together with a Computing Center. L. Reizins wanted to create an Institute of Mathematics affiliated to the Academy of Sciences. L. Reizins worked at the Institute of Physics until his death and contributed much through his intensive research. The main problem of the qualitative theory of differential equations is the classification of differential equations according to significant properties of solutions. Such a classification allows the change of complicated systems of differential equations into simpler systems. In a sufficiently small neighborhood of the invariant set, sufficient classification is achieved by applying the concept of dynamical (topological) equivalence. In his doctoral theses, L. Reizins started to investigate conditions under which two systems of differential equations are equivalent. In 1962 he generalized the Hartman-Grobman theorem in the case of an elementary cycle. For this purpose he introduced the pseudo- local coordinates, and hence the problem of topological structure in the neighborhood of the cycle was reduced to the investigation of semiperiodical system in the neighborhood of origin. He proved that if the cycle was elementary, then the corresponding system was equivalent to its linear part. Further, L. Reizins obtained, independently of V. M. Alekseev, a formula for the relationship between the solutions of full and of truncated systems of differential equations. Hence, he proved that there exists a dynamical equivalence between dichotomic differential systems in the neighborhood of nonelementary stationary points and in the neighborhoods of nonelementary cycles. [Reizins, 1971]. In 1971, L. Reizins received his Dr. Sc. degree in the Belorussian State University. For solving discrimination problems, L. Reizins used Lyapunov functions [Reizins, 1986]. Professor A. D. Myshkis drew Professor L. Reizins attention to Pfaffs' equations which were then an incompletely studied problem. L. Reizins, together with his first post-graduated student, Inta Karkliòa, used the concept of topological equivalence for the classification of such equations. During his last years, L. Reizins again took up Pfaff's equations. He introduced for the study of Pfaff`s equations the concept of noncontinuable solution in a multi-sheeted space, as well as limit set and prolongation of orbits. Meanwhile, he was developing material for a new monograph on Pfaffian equations. He was an outstanding authority in his field. He was the scientific advisor to a number of young mathematicians who later become successful researchers and professors. L. Reizins reviewed for the journals "Mathematical Reviews", "Zentralblatt fur Mathematik" and "Russian Mathematical Reviews". He is also an author of many articles in various encyclopedias and was a member of several academic boards dealing with mathematical problems. Starting with 1958, L. Reizins resumed his position as lecturer at the University of Latvia. In 1969 he became associate professor and in 1979 full professor. He lectured on the qualitative theory of differential equations and guided students' work on their graduation papers. Part of his lectures formed a textbook [Reizins, 1977]. He was actively involved in research and in developing methodological topics. In 1980 he was the chairman of a large conference of university teachers in Riga. He published 140 articles and abstracts, of which 38 are connected with the history of mathematics, and 9 are expository popular science articles reflecting astronomy problems published in Latvia in the late 1950's and early 1960's. [Lusis, 1966; Detlovs, 1968; Reizins, 1970; 1973; 1975; 1977; Myshkis, 1974; Kaòevskij, 1978] Of his work on the history of research, one should first mention his work on the study of Piers Bohl's (1865-1921) heritage. In 1965, during the Bohl Readings in Riga, dedicated to his centenary, a decision was adopted to publish the complete works of P. Bohl in Russian. They were published in 1974 under L. Reizins editorship [Reizins, 1974]. When editing the above work, a lot of mathematical problems emerged while discussing P. Bohl's research in comparison with different theorems of various authors. [Reinfelds, 1994]
In 1961 there was an Institute of Electronics and Computers organized in Latvian Academy of Sciences where many mathematicians moved to work from the Institute of Physics, but there was no special mathematics department. In 1959 at the University of Latvia (the first in three Baltic States) a Computing Center of research character was established which in 1994 became the Institute of Mathematics and Computer Science. The first director and organizer of it was professor E. Arins. He with E. Grinbergs and J. Daube were the three who started applications of mathematics and computer science in Latvia. They all three met in 1930 starting their studies at the University of Latvia in the Faculty of Mathematics and Natural Sciences. But before that they came from different places. Janis Daube was born in 1910 near Krustpils, Latvia. After graduating from high school he worked as a clerk for a year and then studied at the same time while keeping a full time clerks job with a health insurance company. He graduated from the University of Latvia in 1939. During the 1940's he worked as a constructor in the Laboratory of Measuring Appliances and Radio. In 1949 he started to work in the Institute of Physics of Academy of Sciences. Under his leadership a computer net in Latvia was developed. Since 1961 he was working in the Computer Center of the University of Latvia. He died in 1982. [Dambitis, 1996]
Emanuel Grinbergs was born in 1911 in St. Petersburg. His father was a Latvian who was a bishop in Russian Lutheran Church. When his father died the family returned to Riga in 1923. In 1927 he, as a winner of high school student mathematics competition, went to Lille, France, where he studied in the licee. Later, he studied mathematics in the University of Latvia which he graduated cum laude in 1934. In 1935 and 1936 he was awarded a K. Morbergs Stipend to continue his studies in France at the Ecole Normale in Paris. He attended classes of famous mathematicians who at that time were organizing the Bourbaki group. His first publication was in geometry [Grinbergs, 1936]. He also participated in International Congress of Mathematicians in Oslo, 1936. He was elected as privatdocent in the University of Latvia in 1937 and starting January, 1938 he lectured in different geometry courses. In 1943 he defended his doctoral thesis but in 1944 he was called up in the Latvian Legion and for that he was kept until 1946 in a filtration camp in Kutaisi, Georgia. There he was used as a specialist for doing computations connected with building problems. After returning to Latvia he was not allowed to continue his docentship and his doctoral degree was denied by Soviet authorities. So he started to work as an ordinary worker in radio manufacture. Soon his great mathematical abilities were showing up in thinking about different theoretical problems of radio manufacturing. He developed original theory for analysis and synthesis of electrical filters, using function approximation theory and generalized Euler symbolics which made electric circuits easy to describe mathematically. His work was later adopted throughout the Soviet Union. In 1954/55 he was allowed again to lecture on the Faculty of Physics and Mathematics of the University of Latvia. In 1956 he was invited to work at the Institute of Physics of the Latvian Academy of Sciences. In spring 1960 he defended another thesis and received a degree of "candidat fiziko-matematiceskih nauk". The same year he started to work at the Computer Center of the University of Latvia were he could actively do research in different areas. He was leading researches in mathematical models of electrical chains which in the 1970's became mathematical methods of electronic schemes. E. Grinbergs was leading a group which made in 1962 a computer model for the hull of a ship which allowed computers to help decide how to cut the steel sheets during ship building. This method was introduced in all ship building companies all over USSR. Working on these problems the theory of splines were developed but its theoretical part was not published at the time, it was left only in technical descriptions. While lecturing on probability theory Grinbergs became interested in Markov processes, and he with some co-authors developed methods which could be used in telephone wiring. E. Grinbergs also is known for his work in graph theory. There is a well known formula in graph theory called Grinbergs formula, sometimes mistakenly referred to as a formula of Russian mathematician [Stewart, 1992]. Grinbergs was a shy man who left much of his work unpublished. He left some 10,000 hand-written notes when he died in 1982, many of them containing new and interesting results. [Riekstins, 1993]
Eizens Arins was born in 1911 in Krasnojarsk, Russia, where his worker father was in exile. His mother died there in 1918 and, in 1920, father and son returned to Latvia. In 1929 he graduated from high school in Daugavpils, Latvia, and next year he started his studies at the University of Latvia. Because he had to support himself, he worked as a clerk for the insurance company. Arins graduated from the university in September of 1941 and continued to work as a mathematician in the insurance company. In December of 1944 he was invited to lecture in the Faculty of Physics and Mathematics of the University of Latvia. His university diploma later was not recognized by Soviet authorities, so he had to graduate from the faculty again which he did with honors in 1946. While lecturing in university he was also doing researches in the Institute of Physics of the Academy of Sciences of Latvia in 1946-1951 and 1956-1960. In 1954 Arins defended his thesis and received a degree of kandidat fiziko-matematicheskih nauk. He became a docent in 1955, but he did not become a full professor until 1973. He was the first director of the Computer Center of the University of Latvia until 1978. He had great organizational talent and under his leadership and collaborating with E. Grinbergs and J. Daube the Computer Center became not only the leading center for researches in the computer science but also in theoretical mathematics. He died in 1987. [Dambitis, 1996]
E. Arins and J. Daube convinced young Aivars Lorencs that he can do researches in a new field of mathematics at that time - theoretical computer science. Despite his blindness (he lost his eyesight as a boy playing with a grenade after World War II) he became a professor of the University of Latvia and is still doing active research work in mathematics. He got his candidate of sciences degree in 1964 and doctor of sciences degree in mathematical logic and algorithm theory in 1979. He was collaborating with a well known Russian mathematician Professor Andrey Markov. After that he continued his research on problems of structural synthesis of probabilistic automatas. [Lorencs, 1968; 1974; 1975]. Recently, Professor Lorencs is interested in problems of cryptology.
After the World War II active research work in mathematics started also in the Faculty of Physics and Mathematics of the University of Latvia. It was mentioned already in a biography of Professor Lusis about the interest in the theory of integral equations. The other scientific school was led by Professor Eduards Riekstins who was interested in asymptotic methods in the theory of functions.
Eduards Riekstins was born in 1919 as the eighth child in the family of a boatman and dressmaker on Doles Sala near Rîga. He graduated from Riga's First Gymnasium in 1937 and started to work as a clerk to raise money for his studies. In 1938 he became a student of the Faculty of Mathematical and Natural Sciences of the University of Latvia. As an excellent student he was allowed to make use of the President Foundation to cover tuition. In 1941 he received a grant from the German government. In 1943 Riekstins graduated cum laude from the university and attempted to find a job in the war industry, which would help to avoid being drafted into the Nazi army. He was an accountant at the Kopperscmidt Plant for less than a year. In spite of poor vision, he was drafted by the German army on August 4, 1944. He served some time along the route to Danzig and later as a stableman in Italy. He was wounded there and underwent surgery after which he did not return to German army but hid with Italian farmers and guerrillas. When the war ended, he returned home through all of Europe becoming a prisoner of the Americans and the Russians. He arrived back in Riga in August, 1945. The same year he got a position as a lecturer in the Faculty of Physics and Mathematics of the University of Latvia. In 1952 he received his degree for the thesis "The decomposition method for generalized system of telegraph equations". In 1955 he was promoted to docent and in 1953 he became a Chair of the Department of General Mathematics until 1970. He was actively involved in both research and teaching. He developed new courses, wrote several textbooks for high school and university students and actively participated in organizing mathematical competitions for school students. His textbooks are still being used by university students. In order to become actively involved in the research on the theory of asymptotic expansions of functions, Prof. Riekstins in 1969 joined the Laboratory of Mathematics at the Institute of Physics of the LAS. In 1970 he completely left the university, although he was occasionally lecturing in the mid 1970's. He acted as reviewer for the journals Matematika (Russian) since 1953, Mathematical Reviews since 1964, and Zentrablat fur Mathematik since 1974. He wrote 5 monographs [Riekstins 1974, 1977, 1981, 1986, 1991]. Such monographs, devoted to the theory of the functions appeared for the first time, and have been greatly appreciated by the mathematical community in both journal reviews and letters addressed to the author. For this research as a top mathematician in Latvia he received in 1991 the Keldish Award of the Latvian Academy of Sciences. The number of his publications totals more than 120. More than 30 of them are in the history of mathematics. Although seriously ill, Professor E. Riekstins continued to work on his next monograph about Lommel functions. This work was interrupted by his death in 1992. Selected bibliography of E. Riekstins published works (91 titles) can be found in [Cirulis, 1994].
Another outstanding Latvian mathematician who actively was involved in teaching and research in the Faculty of Physics and Mathematics of the University of Latvia was Dr. H.C. Georgs Engelis who worked in the field of the theory of functions and differential equations.
Georgs Engelis was born in 1917 in Petrograd, Russia, where his family was living during the World War I. His father was a school teacher and in 1918 the family returned to Latvia. In 1931 he became a student in Liepaja Gymnasium where he was originally interested in biology but was greatly influenced by his excellent mathematics teacher P. Kalnins and decided to devote his life to mathematics. The Engelis family was poor and so to support his student who graduated with honors from the gymnasium Kalnins left in his last will some money for Engelis' university stipend. Engelis became a student of the University of Latvia in 1936 and after graduating from the university he became there an assistant and was instructing classes in theoretical mechanics, calculus, differential equations, number theory. In 1942 he got his master's degree. In the summer of 1944 he was working at his parents farmland and when Red Army entered Riga he decided for safety reasons not to return back to the university. He became a school teacher of physics and mathematics in Liepaja, Latvia. In 1946 he returned back to Riga and was a research fellow at the Latvian Academy of Sciences. He was one the very first Latvian mathematicians after World War II who passed exams required for those who wanted to defend a thesis for obtaining a scientific degree. In 1947 he went to Leningrad, Russia, to make a presentation in a seminar led by professor L.V. Kantorovitch about generalizing Green functions. But his scientific career was interrupted for political reasons. His biography was not acceptable because he was studying in the university during the independent republic and also because he was working there during the German occupation. G. Engelis decided not to waste time on fighting to get permission to defend his thesis. He did not leave research but mostly he was concentrating on teaching. During his 45 years in the university he developed 18 mathematics courses, wrote 8 textbooks for students and did translations of Russian textbooks into Latvian. He was also actively involved working with school students -- in organizing mathematics competitions and lecturing for teachers and students in different workshops. He had about 20 scientific publications. [Engelis, 1964; 1974; Sujetin, 1988]. Engelis also presented his work at the ICM in 1966 in Moscow. In 1974 he finished a draft of his thesis "On n-dimensional analogs of classical orthogonal polynomials". His thesis were defended in 1976 at the Moscow Institute of Electronic Mechanical Engineering. He became a docent in 1980. He retired in 1986 and was still doing research and was working on calculus textbook when in 1997 he was accidentally killed by a street car. [Henina, 1997]
In the 1960's and 1970's mathematical researches in Latvia were characterized by the scientific school which were taking shape at that time. Best known are researches in function theory, especially connected with finding new effective methods in asymptotic development of functions (E. Riekstins, T. Cirulis, G. Engelis, M. Belovs). The first researches in functional analysis in Latvia were done by S. Krachkovskij who was interested in Fredholm type functional equations. Later he with M. Goldman did researches in spectral theory of limited linear operators. These researches further were developed by J. Engelsons, I. Karklins, U. Raitums. N. Brazma started researches in mathematical physics and partial differential equations. Later he was joined by A. Myshkis who encouraged the researches of L. Reizins on differential equations.
Researches in mathematical logic started with V. Detlovs who in early 1950's proved the equivalence of algorithmic functions (as defined by A.A. Markov using the notion of NORMAL ALGORITHMS) and partial recursive functions, and the equivalence of total algorithmic functions and total recursive functions. In the 1960's J. Barzdins was investigating universality problems in the theory of growing automata, estimated the complexity of symmetry recognition by Turing machines, and explored the behavior of different type of automata depending on its topology. [Barzdins, 1964; 1965; Kolmogorov, 1967; Trakhtenbrot, 1973] Janis Barzdins is born in 1937, Latvia. In 1959 he graduated from the University of Latvia. He was a graduate student of Professor Trakhtenbrot in Novosibirsk where he defended his Ph.D. thesis in 1965. He got his Doctor of Science degree (Mathematics) in 1976 from the Institute of Mathematics (Novosibirsk) of the Soviet Academy of Sciences. From 1965 he worked in the Computer Center of the University of Latvia. Since 1997 he is Director of the Institute of Mathematics and Computer Science of the University of Latvia. Since 1985 J. Barzdins is Professor at the University of Latvia and since 1992 Head of Division of Computer Science of the Faculty of Physics and Mathematics. Since 1992 he is a Full Member of the Latvian Academy of Sciences of Latvia. Currently he is interested in inductive synthesis and learning theory, languages and tools. [Barzdins, 1991; 1996; 1997]
The theory of probabilistic automata also interested Professor R. Freivalds. Rusins-Martins Freivalds was born in Latvia in 1942. He graduated from the University of Latvia in 1965. He received his Ph.D. (1971) from the Institute of Mathematics, Novosibirsk, and DSc (1976) from Moscow University. Since 1992 he is Dr. habil. math. and a member of the Latvian Academy of Sciences (1992). He is also a member of the European Association for Theoretical Computer Science (1979). Since 1970 he has been working at the Institute of Mathematics and Computer Science of the University of Latvia. He has given invited talks on his research in complexity of computation and inductive inference at conferences in USA, Japan, and others, and served in program committees of international conferences. Computer education is another interest of his, this has resulted in educational TV shows, textbooks (including one published in a million copies), 162 published papers and 3 books. [Ambainis, 1998; 1999a; 1999b; Apsitis, 1999; Freivalds, 1998]
In the beginning of the 1990's, along with the reforms in whole country to build an independent and democratic republic, reforms also took place in science and education. In the University of Latvia intensive work was done in order to make their programs closer to the standards of Western universities. Before 1991 higher educational institutions in Latvia were organized in the same way as it was all over the Soviet Union -- studies were 5 years and finished with the university diploma. Now bachelors and master programs are separated. Many institutes of the Academy of Sciences were included in the University of Latvia to stimulate scientific researches among professors of the university and also to involve researches into the teaching process.
One of the first such institutes was the Institute of Mathematics of the Academy of Sciences and the University of Latvia. It was organized on a basis of the two mathematics research laboratories of the Institute of Physics of the LAS and professors from the Departments of Differential Equations and General Mathematics of the Faculty of Physics and Mathematics. The first director and initiator of the institute was professor Andris Buikis who with professor Harijs Kalis are the leaders in the researches in mathematical modeling in Latvia. A. Buikis is also interested in Filtration Theory, Mathematical Problems for Layered Media ,Numerical Methods for Partial Differential Equations, Biophotons, and the Philosophy of Science. He obtained his degree Dr. habil. math. (Doctor of Science in former USSR) from the Kazan University, Russia in 1988. [Buikis, 1995]
Professor Harijs Kalis received his degree of Dr. habil. phys. (Doctor of Science in former USSR) from St. Petersburg Technical University in 1991 for thesis "The Development and Application of Special Computational Methods for Calculation of Flow for Viscous Incompressible Electrically-Conducting Fluid with Large Parameters" and his degree Dr. habil. math. from the University of Latvia in 1993 for thesis "The Development and Application of Special Numerical Methods for Solving Problems for Mathematical Physics, Hydrodynamics and Magnetohydrodynamics". His research concerns are in special finite difference approximations and numerical solutions of systems of differential equations of mathematical physics and fluid mechanics with large parameters at first order derivatives or small parameters at second order derivatives. The so-called uniform numerical (special) methods of E. Doolan, J. Miller, W.H. Schilder, D. Allen, R. Southwell and A. Il`yn are being developed for application to nonlinear problems of mathematical physics and MHD for viscous incompressible fluid, which cannot be solved by classical methods. This provides the basis for the development of special monotonous vector difference schemes with a perturbation coefficient of the function matrix. Since 1964, he has published a monograph on the numerical methods of solutions of differential equations ; 136 original scientific papers, proceedings and abstracts on the numerical simulation of MHD flow for viscous incompressible fluid in the electromagnetic field, about the mathematical modeling in aluminum reduction cell, on the numerical solution of the heat transfer equations in multi-layer medium, the special finite-difference approximations of the equations of mathematical physics; and 22 textbooks or booklets for students about numerical methods of differential equations in mathematical physics. [Kalis, 1995]
Mathematicians of the institute are continuing traditions of research in theory of differential equations started by L. Reizins and E. Riekstins. Now the leading mathematician in this direction is Professor Andrejs Reinfelds. His main interests are Qualitative Theory of Differential Equations, Dynamical and Semidynamical Systems, Difference Equations, Impulse Differential Equations, Application of Differential Equations, and Acturial Mathematics. He received his scientific degree Dr. habil. math. for the thesis "Reduction Principle of Differential Equations" from the University of Latvia in 1988. [Reinfelds, 1996a; 1996b; 1996c; 1997a, 1997b, 1997c]
Researches in the theory of asymptotic developing of the functions started by Professor E. Riekstins now are led by Professor Teodors Cirulis [Reinfelds, 1999].
In 1997 another department of the Faculty of Physics and Mathematics joined the Mathematics Institute. This was Department of Mathematical Analysis where active researches in theoretical topology are done. This direction is led by Professor Aleksandrs Sostaks who is now interested in fuzzy sets and fuzzy structures [Sostaks, 1996]. [Reinfelds, 1999]
In 1994 the Computer Center of the University of Latvia was reorganized as the Institute of Mathematics and Informatics with the main research directions: differential equations, ordinary and partial, mathematical modeling of physical processes, and theory of optimal control for partial differential equations. In the field of ordinary differential equations the investigations are concentrated around the existence of solutions for boundary value problems for equations and systems of ordinary differential equations of the second and higher order, including a priori estimates, and uniqueness and multiplicity of solutions (Yu. Klokov, A. Lepin, F. Sadirbaev). A special class of exact solutions for the nonlinear Klein-Gordon equation has been constructed, and geometrical properties of these solutions are described (V.Gudkov). In the field of mathematical modeling the main efforts are devoted to numerical methods. At present half-implicit difference schemes have been elaborated for the Fokker-Planck type equations describing the electron transport in semiconductors, as well as for the equations describing semiconductor fluid dynamics in semihydrodynamic approximation, dynamics of liquid and gaseous flows in porous medium with account for the capillary pressure, and the penetration and condensation of metal vapor in ceramics (J.Kaupuzs, J.Rimshans). Mathematical simulation of electrodiffusion in electrolytes has been done (B.Martuzans, Yu.Skril). In the field of theory of optimal control problems for partial differential equations investigations mainly concern various extensions (relaxation, convexication, G-closure) of original problems. Effective descriptions for successful extensions for optimal material layout problems in the case of a single equation are obtained (U. Raitums). [Henrad, 1998; Klokov, 1998a; Klokov, 1998b; Lepin, 1997; Raitums, 1989; 1990; 1997; 1999; Sadirbaev, 1996; Zaytsev, 1999].
Mathematical problems of theoretical computer science are developed further in system modeling and design, telecommunications software, simulation of discrete event systems, inductive synthesis and computational learning theory, computational complexity, graph drawing, and natural language processing. A research direction where results of mathematicians of the institute are internationally recognized is "Inductive Synthesis and Computational Learning Theory". In the 1970's important results concerning inductive synthesis theoretical limits were obtained (J. Barzdins, R. Freivalds, K. Podnieks). In the 1980's a new model of inductive synthesis (so called "dots expressions") was developed and corresponding inductive inference rules formalized (J. Barzdins, A. Brazma). At the beginning of the 1990's a new efficient inductive synthesis algorithm based on term rewriting systems was found (G. Barzdins). In recent years a new approach to practical inductive synthesis based on hypothesis space restriction by means of attribute grammars has been developed (U. Sarkans, J. Barzdins). Another research direction in recent years has been the application of inductive synthesis to biocomputing and genome informatics (A. Brazma). In cooperation with Helsinki and Bergen universities and European Institute of Bioinformatics a new algorithm has constructed for discovery of statistically important patterns in biosequences. The research on recursion-theoretical level was also continued (R. Freivalds, K. Apsitis, J. Viksna). Transformations preserving synthesizability are shown to be a powerful tool to characterize the synthesis types. Synthesis of real-valued functions is shown to differ very much from synthesis of recursive functions. A research program in quantum computation has been started. It has been proved that quantum finite automata can be of exponentially smaller size compared with equivalent deterministic or even randomized finite automata. Together with European partners the 5-th Framework project "Quantum Algorithms and Information processing" has been won by R. Freivalds.
Another priority area for the Institute is graph theory and graph drawing. Very efficient incremental layout algorithms for interactive design of graph-like diagrams are developed (P. Kikusts et al.). These algorithms have won the first and third prizes in the Graph-Drawing Contest '95 in Passau (Germany) and the first and third prizes in the Graph-Drawing Contest '99 in Prague (Czech Republic).
Mathematicians are doing researches not only in institutes connected with the University of Latvia. Professor J. Carkovs holds a position in Riga Technical University. Jevgenijs Carkovs was born on December 8, 1935, in Rostov-on-Don, Russia. He graduated from Chernovtsy State University, Ukraine, in 1959. He got his degree candidate of sciences from the University of Latvia in 1966, but his degree of doctor of sciences from the Mathematical Institute of Ukraine in 1983. He became a head of the laboratory in the Computer Center of the University of Latvia in 1972, but in 1979 he became a Professor of the Riga Technical University. He has been a Visiting Scholar and Visiting Professor at Science & Technology University of Hong Kong (Hong Kong, 1995, 1996); Michigan State University (East Lansing, MI, USA, 1996); California Institute of Technology (Los Angeles, CA, USA, 1996); Linkoeping University (Sweden, 1994, 1995); Bremen University (Germany, 1993); Nish University and Belgrade University (Yugoslavia, 1989). His scientific interests are Probability Theory, Mathematical Statistics, Stochastic Differential Equations, Markov Dynamical Systems, Stochastic Analysis of Securities, Dynamical Systems. [Carkovs, 1986; 1995; 1997]
In 1993, the Latvian Mathematical Society was founded. Originally there were only 66 members in it but this number is increasing. We can say that the Latvian Mathematical Society consists of two thirds of all sufficiently actively working Latvian mathematicians. Latvian Mathematical Society was accepted as a member of the European Mathematical Society on July, 1996. Since January 1996, Latvia is also a member of the International Mathematical Union. As a new society of mathematicians Latvian Mathematical Society is happy to collaborate with mathematical societies in other countries -- Estonia, Slovakia, Catalonia, Poland, Serbia, Germany. Especially Latvian Mathematical Society appreciates support by American Mathematical Society.
Latvian Mathematical Society was present at numerous significant mathematical conferences, workshops and congresses in the persons of its members. Let us give just some of the examples of these activities. In 1993 A. Sostaks participated in the Congress of the Mathematicians of Southern Africa. R. Freivalds participated several times in international conferences on Algorithmic Learning Theory. In particular, in October 1996 in Sidney, Australia, he gave a talk presenting joint results with his student A. Ambainis about transformations preserving learnability. In 1994 professor R. Freivalds was a member of the program committee of the 13th World Computer Congress held in Hamburg, Germany. In September 1998 professor R. Freivalds served as a Chair of the Program Committee at the conference on Randomized Algorithms in Brno, Czech Republic. Professors A. Buikis and H. Kalis regularly participate in Conferences of the European Consortium for Mathematics in Industry. U. Raitums and J. Vucans participated in the conference of Modeling and Optimization of distributed parameter systems, organized under the auspices of the International Federation for Information Processing in Warsaw, Poland, 1995. In July 1996 A. Reinfelds took part in the 2nd International Congress of Analysis in Athens, Greece. Eight mathematicians from Latvia (A. Aboltins, S. Asmuss, J. Carkovs, U. Raitums, A. Reinfelds, I. Strazdins, D. Taimina, J. Vucans) participated at the International Congress of Mathematicians in Zurich, 1994, and 6 mathematicians (S. Asmuss, J. Carkovs, U. Raitums, F. Sadyrbayev, A. Sostaks, I. Strazdins) at the next congress in Berlin, 1998. In 1996 J. Carkovs participated in the XVII International Conference on Probabilistic Mechanics and Structural Reliability (Worchester, U.S.A.) and in May 1997 he took part in the First Pan-China Conference on Differential Equations in Kunming, China. D. Taimina represented Latvia at 3 invitation only ICMI Studies: Teaching Geometry for 21st Century (Catania, Italy, 1995), the Role of History of Mathematics in Teaching and Learning Mathematics (Luminy, France, 1998), and Teaching Mathematics at the University Level (Singapore, 1998). In July 1998 A. Andzans and A. Cibulis were invited to participate in the 3rd WFNMC Congress (mathematics competitions) in China at which A. Andzans was a recipient of a Paul Erdos Award for 1998 to honor his "significant contribution to the enrichment of mathematics learning in Latvia". A. Andzans has also participated in two International Congresses of Mathematics Education (Quebec, Canada, 1992; Seville, Spain, 1996). [Sostaks, 1998] Daina Taimina presented a paper at the Joint Meetings of the American Mathematical Society and MAA (San Antonio, TX, USA, 1999, Washington, DC, 2001).
The last several examples show that great attention has been devoted in Latvia to the quality of mathematics education in all levels. As it was already discussed before, after World War II many outstanding mathematicians became mathematics teachers and gave great contribution to the teaching of a young generation of mathematicians. We have to mention here the famous "teacher of math teachers" in Latvia, Professor Janis Mencis; and two mathematics teachers with 55 years of teaching experience, A. Grava and A. Sika. A student of A. Grava, Agnis Andzans, is now a professor at the University of Latvia and is also a leader of the Corresponding School of Mathematics for school students grades 5-12. The Corresponding School of Mathematics was organized in 1969 and since then it became also the center for organizing mathematics competitions at all levels in Latvia. The Latvian team has successfully participated in International Mathematics Olympiads. Agnis Andzans who got his scientific degree in Theoretical Computer science but decided to devote his efforts to mathematics education. He is also an author of many school textbooks in mathematics and is a leader of the school informatization project in Latvia.
We understand that there is a lot of the great work done by mathematicians in Latvia during past decade left out in this limited story about mathematics in Latvia over centuries. Now when we can take more and more advantage of modern technologies it is possible to get more updated and full information on websites about mathematics in Latvia such as:
Latvian Mathematicians: http://www.lza.lv/MATHEM.htm
Faculty of Physics and Mathematics of the University of Latvia: http://www.lu.lv/e_strukt/fakult/phmath/info/
Institute of Mathematics of the University of Latvia and Latvian Academy of Sciences: http://www.lza.lv/inst/in07.htm
Specialized Institute in Mathematical Statistics of Riga Technical University: http://www.itl.rtu.lv/iti/
Institute of Mathematics and Computer Science: http://www.latnet.lv/LU/MII/
Latvian Education Informatization System: http://www.liis.lv/english/main.htm
[Albers, 1990] More Mathematical People, ed. Donald J. Albers, Gerald L. Alexanderson, Constance Reid, HBJ, Boston, 1990, pp. 1-21.
[Ambainis, 1998] A. Ambainis, R. Freivalds, 1-way quantum finite automata: strengths, weaknesses and generalizations, Proceedings of the 39th Symposium on Foundations of Computer Science, Palo Alto, California, November 1998, p.332-341.
[Ambainis, 1999a] A. Ambainis, R. Bonner, R. Freivalds, M. Golovkins, M. Karpinski, Quantum finite multitape automata, Lecture Notes in Computer Science, 1999, v.1725, p.336-344.
[Ambainis, 1999b] A. Ambainis, R. Bonner, R. Freivalds, A. Kikusts, Probabilities to accept languages by quantum finite automata, Lecture Notes in Computer Science, 1999, v.1627, p.174-183.
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