Nikolai Nikolaevich Luzin at the crossroads of the dramatic events of the European history of the first half of the 20th century

Nikolai Nikolaevich Luzin’s life (1883–1950) and work of this outstanding Russian mathematician, member of the USSR Academy of Sciences and foreign member of the Polish Academy of Arts and Sciences, coincides with a very difficult period in Russian history: two World Wars, the 1917 revolution in Russia, the coming to power of the Bolsheviks, the civil war of 1917–1922, and finally, the construction of a new type of state, the Union of Soviet Socialist Republics. This included collectivization in the agriculture and industrialization of the industry, accompanied by the mass terror that without exception affected all the strata of the Soviet society. Against the background of these dramatic events took place the proces of formation and flourishing of Luzin the scientist, the creator of one of the leading mathematical schools of the 20th century, the Moscow school of function theory, which became one of the cornerstones in the foundation of the Soviet mathematical school. Luzin’s work could be divided into two periods: the first one comprises the problems regarding the metric theory of functions, culminating in his famous dissertation Integral and Trigonometric Series (1915), and the second one that is mainly devoted to the development of problems arising from the theory of analytic sets. The underlying idea of Luzin’s research was the problem of the structure of the arithmetic continuum, which became the super task of his work. The destiny favored the master: the complex turns of history in which he was involved did not prevent, and sometimes even favored the successful development of his research. And even the catastrophe that broke out over him in 1936 – “the case of Academician Luzin” – ended successfully for him.Nikołaj Nikołajewicz Łuzin na skrzyżowaniu dramatycznych wydarzeń w historii Europy pierwszej połowy XX wieku Życie Mikołaja Nikołajewicza Łuzina (1883–1950) i twórczość wybitnego rosyjskiego matematyka, członka Akademii Nauk ZSRR i zagranicznego członka Polskiej Akademii Umiejętności, przypadają na bardzo trudny okres w historii Rosji: dwie wojny światowe, rewolucja 1917 w Rosji, dojście do władzy bolszewików, wojna domowa 1917–1922, wreszcie budowa nowego typu państwa – Związku Socjalistycznych Republik Radzieckich, obejmująca kolektywizację w rolnictwie i industrializację przemysłu, czemu towarzyszył masowy terror, który bez wyjątku dotknął wszystkie warstwy społeczeństwa radzieckiego. Na tle tych dramatycznych wydarzeń przebiegał proces powstawania i rozkwitu naukowca Łuzina, twórcy jednej z głównych szkół matematycznych XX wieku – moskiewskiej szkoły teorii funkcji, która stała się jednym z kamieni węgielnych radzieckiej szkoły matematycznej. Twórczość Łuzina można podzielić na dwa okresy: pierwszy obejmuje zagadnienia dotyczące metrycznej teorii funkcji, których kulminacją jest jego słynna rozprawa Całka i szeregi trygonometryczne (1915), a drugi, poświęcony głównie rozwojowi problemów wynikających z teorii zbiorów analitycznych. Ideą leżącą u podstaw badań Łuzina był problem struktury kontinuum arytmetycznego, który stał się nadrzędnym zadaniem jego pracy. Przeznaczenie sprzyjało mistrzowi: złożone zwroty historii, w które był wplątany, nie przeszkadzały, a czasem nawet sprzyjały pomyślnemu rozwojowi jego badań. I nawet katastrofa, która wybuchła w 1936 roku – „przypadek akademika Łuzina” – zakończyła się dla niego pomyślnie

Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous

Let A stand for the class of all almost continuous functions from R to R and let A(A) be the smallest cardinality of a family F ⊆ R R for which there is no g: R → R with the property that f + g ∈ A for all f ∈ F. We define cardinal number A(D) for the class D of all real functions with the Darboux property similarly. It is known, that c \u3c A(A) ≤ 2 c [10]. We will generalize this result by showing that the cofinality of A(A) is greater that c. Moreover, we will show that it is pretty much all that can be said about A(A) in ZFC, by showing that A(A) can be equal to any regular cardinal between c + and 2c and that it can be equal to 2c independently of the cofinality of 2c . This solves a problem of T. Natkaniec [10, Problem 6.1, p. 495]. We will also show that A(D) = A(A) and give a combinatorial characterization of this number. This solves another problem of Natkaniec. (Private communication.

The Russian Federation and the conflicts in former Yugoslavia, 1992-1995.

The thesis examines the evolution of Russian policy towards the Yugoslav conflicts from the start of 1992, when the Russian Federation became an independent state, to the Dayton Accords that ended the Bosnian conflict in December 1995. In Part I, I discuss rival international relations theories in the post-Cold War world and apply them to the debate over foreign policy in Russia and Russian perceptions of the Yugoslav conflicts. Part II examines the evolution of Russian policy towards the Yugoslav conflicts until the end of 1993. January to autumn 1992 was the 'liberal internationalist' phase of Russian policy, when the government promoted co-operation with the West in order to achieve a settlement of the Yugoslav conflicts, and a domestic backlash put pressure on the government to adjust its approach. A transitional phase followed, from autumn 1992 to the end of 1993, during which the government developed a more assertive great power policy based on relative domestic consensus. Part III shows this neo-realist policy in action. Russian policy makers used the Sarajevo crisis of February 1994 to demonstrate Russia's great power status. They also sought to prevent developments considered to be harmful to Russia's national interests, in particular military action by NATO against the Bosnian Serbs. For a period, other powers recognised that Russian opinions must be taken into account. But in summer 1995, Western policy makers ignored Russian objections and Russia played a secondary role in achieving a peace settlement. Russian policy makers attempted to use the Yugoslav conflict to demonstrate Russia's great power status and its independence from the West, but Russia lacked the power and influence for the policy to be effective. Russian policy contributed to the failure of the 'international community' to achieve a just settlement in Bosnia-Herzegovina, and added to the divisions developing between Russia and the West