Deane Montgomery was a mathematician specializing in topology, one of the contributors to the final resolution of Hilbert's fifth problem in the 1950s. He served as President of the American Mathematical Society from 1961 to 1962. Born in the small town of Weaver, Minnesota, he received his B. S. from Hamline University in St. Paul, MN and his Masters and Ph. D. from the University of Iowa in 1933. In 1941 Montgomery was awarded a Guggenheim Fellowship. In 1988, he was awarded the American Mathematical Society Leroy P. Steele Prize for Lifetime Achievement, he was a member of the United States National Academy of Sciences and of the American Academy of Arts and Sciences. With Leo Zippin: "A theorem on Lie groups". Bull. Amer. Math. Soc. 48: 448–452. 1942. Doi:10.1090/s0002-9904-1942-07699-3. MR 0006545. "Measure preserving homeomorphisms at fixed points". Bull. Amer. Math. Soc. 51: 949–953. 1945. Doi:10.1090/s0002-9904-1945-08477-8. MR 0013905. With Leo Zippin: "Two-dimensional subgroups". Proc. Amer. Math. Soc. 2: 822–838.
1951. Doi:10.1090/s0002-9939-1951-0047669-0. MR 0047669. With Leo Zippin: "Small subgroups of finite-dimensional groups". Proc Natl Acad Sci U S A. 38: 440–442. 1952. Doi:10.1073/pnas.38.5.440. PMC 1063582. PMID 16589121. Deane Montgomery and Leo Zippin, Topological Transformation Groups, Interscience Publishers, 1955. With Hans Samelson and C. T. Yang: "Groups on En with -dimensional orbits". Proc. Amer. Math. Soc. 7: 719–728. 1956. Doi:10.1090/s0002-9939-1956-0078643-0. MR 0078643. With C. T. Yang: "Orbits of highest dimension". Trans. Amer. Math. Soc. 87: 284–293. 1958. Doi:10.1090/s0002-9947-1958-0100272-7. MR 0100272. Deane Montgomery at the Mathematics Genealogy Project Interview with Montgomery about his experience at Princeton A biography of Montgomery A Tribute to Deane Montgomery, by Ronald Fintushel
Félix Édouard Justin Émile Borel was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Borel was born in Saint-Affrique, the son of a Protestant pastor, he studied at the Collège Sainte-Barbe and Lycée Louis-le-Grand before applying to both the École normale supérieure and the École Polytechnique. Although he qualified in the first position for both, he chose to attend the former institution in 1889; that year he won an annual national mathematics competition. After graduating in 1892, he placed first in the agrégation, a competitive civil service examination leading to the position of professeur agrégé, his thesis, published in 1893, was titled Sur quelques points de la théorie des fonctions. That year, Borel started a four-year stint as a lecturer at the University of Lille, during which time he published 22 research papers, he returned to the École normale in 1897, was appointed to the chair of theory of function, which he held until 1941.
In 1901, Borel married the daughter of colleague Paul Émile Appel. Émile Borel died in Paris on 3 February 1956. Along with René-Louis Baire and Henri Lebesgue, Émile Borel was among the pioneers of measure theory and its application to probability theory; the concept of a Borel set is named in his honor. One of his books on probability introduced the amusing thought experiment that entered popular culture under the name infinite monkey theorem or the like, he published a series of papers that first defined games of strategy. With the development of statistical hypothesis testing in the early 1900s various tests for randomness were proposed. Sometimes these were claimed to have some kind of general significance, but they were just viewed as simple practical methods. In 1909, Borel formulated the notion that numbers picked randomly on the basis of their value are always normal, with explicit constructions in terms of digits, it is quite straightforward to get numbers that are normal. In 1913 and 1914 he bridged the gap between hyperbolic geometry and special relativity with expository work.
For instance, his book Introduction Geometrique à quelques Théories Physiques described hyperbolic rotations as transformations that leave a hyperbola stable just as a circle around a rotational center is stable. In 1928 he co-founded Institut Henri Poincaré in Paris. In the 1920s, 1930s, 1940s, he was active in politics. In 1922, he founded Paris Institute of the oldest French school for statistics. From 1924 to 1936, he was a member of the French National Assembly. In 1925, he was Minister of Marine in the cabinet of fellow mathematician Paul Painlevé. During the Second World War, he was a member of the French Resistance. Besides the Centre Émile Borel at the Institut Henri Poincaré in Paris and a crater on the Moon, the following mathematical notions are named after him: Borel described a poker model which he coins La Relance in his 1938 book Applications de la théorie des probabilités aux Jeux de Hasard. Borel was awarded the Resistance Medal in 1950. "La science est-elle responsable de la crise mondiale?", Scientia: rivista internazionale di sintesi scientifica, 51, 1932, pp. 99–106.
"La science dans une société socialiste", Scientia: rivista internazionale di sintesi scientifica, 31, 1922, pp. 223–228. "Le continu mathématique et le continu physique", Rivista di scienza, 6, 1909, pp. 21–35. 8. Michel Pinault, Emile Borel, une carrière intellectuelle sous la 3ème République, Paris, L'Harmattan, 2017. Voir: michel-pinault.over-blog.com Quotations related to Émile Borel at Wikiquote French Wikisource has original text related to this article: Auteur:Émile Borel Media related to Émile Borel at Wikimedia Commons Wikilivres has original media or text related to this article: Émile Borel Works by or about Émile Borel at Internet Archive O'Connor, John J.. Author profile in the database zbMATH
The United Kingdom the United Kingdom of Great Britain and Northern Ireland, sometimes referred to as Britain, is a sovereign country located off the north-western coast of the European mainland. The United Kingdom includes the island of Great Britain, the north-eastern part of the island of Ireland, many smaller islands. Northern Ireland is the only part of the United Kingdom that shares a land border with another sovereign state, the Republic of Ireland. Apart from this land border, the United Kingdom is surrounded by the Atlantic Ocean, with the North Sea to the east, the English Channel to the south and the Celtic Sea to the south-west, giving it the 12th-longest coastline in the world; the Irish Sea lies between Great Ireland. With an area of 242,500 square kilometres, the United Kingdom is the 78th-largest sovereign state in the world, it is the 22nd-most populous country, with an estimated 66.0 million inhabitants in 2017. The UK is constitutional monarchy; the current monarch is Queen Elizabeth II, who has reigned since 1952, making her the longest-serving current head of state.
The United Kingdom's capital and largest city is London, a global city and financial centre with an urban area population of 10.3 million. Other major urban areas in the UK include Greater Manchester, the West Midlands and West Yorkshire conurbations, Greater Glasgow and the Liverpool Built-up Area; the United Kingdom consists of four constituent countries: England, Scotland and Northern Ireland. Their capitals are London, Edinburgh and Belfast, respectively. Apart from England, the countries have their own devolved governments, each with varying powers, but such power is delegated by the Parliament of the United Kingdom, which may enact laws unilaterally altering or abolishing devolution; the nearby Isle of Man, Bailiwick of Guernsey and Bailiwick of Jersey are not part of the UK, being Crown dependencies with the British Government responsible for defence and international representation. The medieval conquest and subsequent annexation of Wales by the Kingdom of England, followed by the union between England and Scotland in 1707 to form the Kingdom of Great Britain, the union in 1801 of Great Britain with the Kingdom of Ireland created the United Kingdom of Great Britain and Ireland.
Five-sixths of Ireland seceded from the UK in 1922, leaving the present formulation of the United Kingdom of Great Britain and Northern Ireland. There are fourteen British Overseas Territories, the remnants of the British Empire which, at its height in the 1920s, encompassed a quarter of the world's land mass and was the largest empire in history. British influence can be observed in the language and political systems of many of its former colonies; the United Kingdom is a developed country and has the world's fifth-largest economy by nominal GDP and ninth-largest economy by purchasing power parity. It has a high-income economy and has a high Human Development Index rating, ranking 14th in the world, it was the world's first industrialised country and the world's foremost power during the 19th and early 20th centuries. The UK remains a great power, with considerable economic, military and political influence internationally, it is sixth in military expenditure in the world. It has been a permanent member of the United Nations Security Council since its first session in 1946.
It has been a leading member state of the European Union and its predecessor, the European Economic Community, since 1973. The United Kingdom is a member of the Commonwealth of Nations, the Council of Europe, the G7, the G20, NATO, the Organisation for Economic Co-operation and Development and the World Trade Organization; the 1707 Acts of Union declared that the kingdoms of England and Scotland were "United into One Kingdom by the Name of Great Britain". The term "United Kingdom" has been used as a description for the former kingdom of Great Britain, although its official name from 1707 to 1800 was "Great Britain"; the Acts of Union 1800 united the kingdom of Great Britain and the kingdom of Ireland in 1801, forming the United Kingdom of Great Britain and Ireland. Following the partition of Ireland and the independence of the Irish Free State in 1922, which left Northern Ireland as the only part of the island of Ireland within the United Kingdom, the name was changed to the "United Kingdom of Great Britain and Northern Ireland".
Although the United Kingdom is a sovereign country, Scotland and Northern Ireland are widely referred to as countries. The UK Prime Minister's website has used the phrase "countries within a country" to describe the United Kingdom; some statistical summaries, such as those for the twelve NUTS 1 regions of the United Kingdom refer to Scotland and Northern Ireland as "regions". Northern Ireland is referred to as a "province". With regard to Northern Ireland, the descriptive name used "can be controversial, with the choice revealing one's political preferences"; the term "Great Britain" conventionally refers to the island of Great Britain, or politically to England and Wales in combination. However, it is sometimes used as a loose synonym for the United Kingdom as a whole; the term "Britain" is used both as a synonym for Great Britain, as a synonym for the United Kingdom. Usage is mixed, with the BBC preferring to use Britain as shorthand only for Great Britain and the UK Government, while accepting that both terms refer to the United K
Switzerland the Swiss Confederation, is a country situated in western and southern Europe. It consists of 26 cantons, the city of Bern is the seat of the federal authorities; the sovereign state is a federal republic bordered by Italy to the south, France to the west, Germany to the north, Austria and Liechtenstein to the east. Switzerland is a landlocked country geographically divided between the Alps, the Swiss Plateau and the Jura, spanning a total area of 41,285 km2. While the Alps occupy the greater part of the territory, the Swiss population of 8.5 million people is concentrated on the plateau, where the largest cities are to be found: among them are the two global cities and economic centres Zürich and Geneva. The establishment of the Old Swiss Confederacy dates to the late medieval period, resulting from a series of military successes against Austria and Burgundy. Swiss independence from the Holy Roman Empire was formally recognized in the Peace of Westphalia in 1648; the country has a history of armed neutrality going back to the Reformation.
It pursues an active foreign policy and is involved in peace-building processes around the world. In addition to being the birthplace of the Red Cross, Switzerland is home to numerous international organisations, including the second largest UN office. On the European level, it is a founding member of the European Free Trade Association, but notably not part of the European Union, the European Economic Area or the Eurozone. However, it participates in the Schengen Area and the European Single Market through bilateral treaties. Spanning the intersection of Germanic and Romance Europe, Switzerland comprises four main linguistic and cultural regions: German, French and Romansh. Although the majority of the population are German-speaking, Swiss national identity is rooted in a common historical background, shared values such as federalism and direct democracy, Alpine symbolism. Due to its linguistic diversity, Switzerland is known by a variety of native names: Schweiz. On coins and stamps, the Latin name – shortened to "Helvetia" – is used instead of the four national languages.
Switzerland is one of the most developed countries in the world, with the highest nominal wealth per adult and the eighth-highest per capita gross domestic product according to the IMF. Switzerland ranks at or near the top globally in several metrics of national performance, including government transparency, civil liberties, quality of life, economic competitiveness and human development. Zürich and Basel have all three been ranked among the top ten cities in the world in terms of quality of life, with the first ranked second globally, according to Mercer in 2018; the English name Switzerland is a compound containing Switzer, an obsolete term for the Swiss, in use during the 16th to 19th centuries. The English adjective Swiss is a loan from French Suisse in use since the 16th century; the name Switzer is from the Alemannic Schwiizer, in origin an inhabitant of Schwyz and its associated territory, one of the Waldstätten cantons which formed the nucleus of the Old Swiss Confederacy. The Swiss began to adopt the name for themselves after the Swabian War of 1499, used alongside the term for "Confederates", used since the 14th century.
The data code for Switzerland, CH, is derived from Latin Confoederatio Helvetica. The toponym Schwyz itself was first attested in 972, as Old High German Suittes perhaps related to swedan ‘to burn’, referring to the area of forest, burned and cleared to build; the name was extended to the area dominated by the canton, after the Swabian War of 1499 came to be used for the entire Confederation. The Swiss German name of the country, Schwiiz, is homophonous to that of the canton and the settlement, but distinguished by the use of the definite article; the Latin name Confoederatio Helvetica was neologized and introduced after the formation of the federal state in 1848, harking back to the Napoleonic Helvetic Republic, appearing on coins from 1879, inscribed on the Federal Palace in 1902 and after 1948 used in the official seal.. Helvetica is derived from the Helvetii, a Gaulish tribe living on the Swiss plateau before the Roman era. Helvetia appears as a national personification of the Swiss confederacy in the 17th century with a 1672 play by Johann Caspar Weissenbach.
Switzerland has existed as a state in its present form since the adoption of the Swiss Federal Constitution in 1848. The precursors of Switzerland established a protective alliance at the end of the 13th century, forming a loose confederation of states which persisted for centuries; the oldest traces of hominid existence in Switzerland date back about 150,000 years. The oldest known farming settlements in Switzerland, which were found at Gächlingen, have been dated to around 5300 BC; the earliest known cultural tribes of the area were members of the Hallstatt and La Tène cultures, named after the archaeological site of La Tène on the north side of Lake Neuchâtel. La Tène culture developed and flourished during the late Iron Age from around 450 BC under some influence from the Gree
Kazimierz Kuratowski was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Kazimierz Kuratowski was born in Warsaw, Vistula Land, on 2 February 1896, into an assimilated Jewish family, he was a son of Marek Kuratow, a barrister, Róża Karzewska. He completed a Warsaw secondary school, named after general Paweł Chrzanowski. In 1913, he enrolled in an engineering course at the University of Glasgow in Scotland, in part because he did not wish to study in Russian, he completed only one year of study when the outbreak of World War I precluded any further enrollment. In 1915, Russian forces withdrew from Warsaw and Warsaw University was reopened with Polish as the language of instruction. Kuratowski restarted his university education there the same year, this time in mathematics, he obtained his Ph. D. in 1921, in newly independent Poland. In autumn 1921 Kuratowski was awarded the Ph. D. degree for his groundbreaking work. His thesis statement consisted of two parts.
One was devoted to an axiomatic construction of topology via the closure axioms. This first part has been cited in hundreds of scientific articles; the second part of Kuratowski's thesis was devoted to continua irreducible between two points. This was the subject of a French doctoral thesis written by Zygmunt Janiszewski. Since Janiszewski was deceased, Kuratowski's supervisor was Stefan Mazurkiewicz. Kuratowski's thesis solved certain problems in set theory raised by a Belgian mathematician, Charles-Jean Étienne Gustave Nicolas, Baron de la Vallée Poussin. Two years in 1923, Kuratowski was appointed deputy professor of mathematics at Warsaw University, he was appointed a full professor of mathematics at Lwów Polytechnic in Lwów, in 1927. He was the head of the Mathematics department there until 1933. Kuratowski was dean of the department twice. In 1929, Kuratowski became a member of the Warsaw Scientific Society While Kuratowski associated with many of the scholars of the Lwów School of Mathematics, such as Stefan Banach and Stanislaw Ulam, the circle of mathematicians based around the Scottish Café he kept close connections with Warsaw.
Kuratowski left Lwów for Warsaw in 1934, before the famous Scottish Book was begun, hence did not contribute any problems to it. He did however, collaborate with Banach in solving important problems in measure theory. In 1934 he was appointed the professor at Warsaw University. A year Kuratowski was nominated as the head of Mathematics Department there. From 1936 to 1939 he was secretary of the Mathematics Committee in The Council of Science and Applied Sciences. During World War II, he gave lectures at the underground university in Warsaw, since higher education for Poles was forbidden under German occupation. In February 1945, Kuratowski started to lecture at the reopened Warsaw University. In 1945, he became a member of the Polish Academy of Learning, in 1946 he was appointed vice-president of the Mathematics department at Warsaw University, from 1949 he was chosen to be the vice-president of Warsaw Scientific Society. In 1952 he became a member of the Polish Academy of Sciences, of which he was the vice-president from 1957 to 1968.
After World War II, Kuratowski was involved in the rebuilding of scientific life in Poland. He helped to establish the State Mathematical Institute, incorporated into the Polish Academy of Sciences in 1952. From 1948 until 1967 Kuratowski was director of the Institute of Mathematics of the Polish Academy of Sciences, was a long-time chairman of the Polish and International Mathematics Societies, he was president of the Scientific Council of the State Institute of Mathematics. From 1948 to 1980 he was the head of the topology section. One of his students was Andrzej Mostowski. Kazimierz Kuratowski was one of a celebrated group of Polish mathematicians who would meet at Lwów's Scottish Café, he was a member of the Warsaw Scientific Society. What is more, he was chief editor in "Fundamenta Mathematicae", a series of publications in "Polish Mathematical Society Annals". Furthermore, Kuratowski worked as an editor in the Polish Academy of Sciences Bulletin, he was one of the writers of the Mathematical monographs, which were created in cooperation with the Institute of Mathematics of the Polish Academy of Sciences.
High quality research monographs of the representatives of Warsaw's and Lwów’s School of Mathematics, which concerned all areas of pure and applied mathematics, were published in these volumes. Kazimierz Kuratowski was an active member of many scientific societies and foreign scientific academies, including the Royal Society of Edinburgh, Germany, Hungary and the Union of Soviet Socialist Republics. In 1981, IMPAN, the Polish Mathematical Society, Kuratowski's daughter Zofia Kuratowska established a prize in his name for achievements in mathematics to people under the age of 30 years; the prize is considered the most prestigious of awards for young Polish mathematicians. Kuratowski’s research focused on abstract topological and metric structures, he implemented the closure axioms. This was fundamental for the development of topological space theory and irreducible continuum theory between two points; the most valuable results, which were obtained by Kazimierz K
Erich Kamke was a German mathematician, who specialized in the theory of differential equations. His book on set theory became a standard introduction to the field. Kamke was born in West Prussia, German Empire. After attending school in Stettin, Kamke studied mathematics and physics from 1909 at the University of Giessen and the University of Göttingen, he was a volunteer in the signals force in World War I. In 1919, he married Dora Heimowitch, the daughter of a Jewish businessman, he earned his doctorate in 1919 at the University of Göttingen under Edmund Landau with thesis Verallgemeinerungen des Waring-Hilbertschen Satzes. While teaching between 1920 and 1926, Kamke earned his habilitation at the University of Münster in 1922, he was appointed as a professor at the University of Tübingen in 1926. Because of his marriage and his opposition to National Socialism, he was denounced by fellow mathematician Erich Schönhardt and forced to retire in 1937. Following World War II, he was reappointed as a professor at the University of Tübingen, was instrumental in the organisation of a mathematical congress in Tübingen in autumn 1946, the first scientific congress in Germany after the war.
In 1948, he re-established the German Mathematical Society, was the chairman until 1952, when he became vice-president of the International Mathematical Union, which he remained until 1954. He died in Rottenburg am Neckar from a heart attack. Das Lebesguesche Integral. Eine Einführung in die neuere Theorie der reellen Funktionen, B. G. Teubner, Leipzig 1925 Mengenlehre, Sammlung Göschen/Walter de Gruyter, Berlin 1928 Differentialgleichungen reeller Funktionen, Akademische Verlagsgesellschaft, Leipzig 1930. Auflage 1962 in zwei Bänden: Band 1: Gewöhnliche Differentialgleichungen Band 2: Partielle Differentialgleichungen Einführung in die Wahrscheinlichkeitstheorie, S. Hirzel, Leipzig 1932 Differentialgleichungen. Lösungsmethoden und Lösungen I. Gewöhnliche Differentialgleichungen, Leipzig 1942 Differentialgleichungen. Lösungsmethoden und Lösungen II. Partielle Differentialgleichungen 1. Ordnung für eine gesuchte Funktion, Leipzig 1944 Das Lebesgue-Stieltjes-Integral, B. G. Teubner, Leipzig 1956
Germany the Federal Republic of Germany, is a country in Central and Western Europe, lying between the Baltic and North Seas to the north, the Alps to the south. It borders Denmark to the north and the Czech Republic to the east and Switzerland to the south, France to the southwest, Luxembourg and the Netherlands to the west. Germany includes 16 constituent states, covers an area of 357,386 square kilometres, has a temperate seasonal climate. With 83 million inhabitants, it is the second most populous state of Europe after Russia, the most populous state lying in Europe, as well as the most populous member state of the European Union. Germany is a decentralized country, its capital and largest metropolis is Berlin, while Frankfurt serves as its financial capital and has the country's busiest airport. Germany's largest urban area is the Ruhr, with its main centres of Essen; the country's other major cities are Hamburg, Cologne, Stuttgart, Düsseldorf, Dresden, Bremen and Nuremberg. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity.
A region named Germania was documented before 100 AD. During the Migration Period, the Germanic tribes expanded southward. Beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation. After the collapse of the Holy Roman Empire, the German Confederation was formed in 1815; the German revolutions of 1848–49 resulted in the Frankfurt Parliament establishing major democratic rights. In 1871, Germany became a nation state when most of the German states unified into the Prussian-dominated German Empire. After World War I and the revolution of 1918–19, the Empire was replaced by the parliamentary Weimar Republic; the Nazi seizure of power in 1933 led to the establishment of a dictatorship, the annexation of Austria, World War II, the Holocaust. After the end of World War II in Europe and a period of Allied occupation, Austria was re-established as an independent country and two new German states were founded: West Germany, formed from the American and French occupation zones, East Germany, formed from the Soviet occupation zone.
Following the Revolutions of 1989 that ended communist rule in Central and Eastern Europe, the country was reunified on 3 October 1990. Today, the sovereign state of Germany is a federal parliamentary republic led by a chancellor, it is a great power with a strong economy. As a global leader in several industrial and technological sectors, it is both the world's third-largest exporter and importer of goods; as a developed country with a high standard of living, it upholds a social security and universal health care system, environmental protection, a tuition-free university education. The Federal Republic of Germany was a founding member of the European Economic Community in 1957 and the European Union in 1993, it is part of the Schengen Area and became a co-founder of the Eurozone in 1999. Germany is a member of the United Nations, NATO, the G7, the G20, the OECD. Known for its rich cultural history, Germany has been continuously the home of influential and successful artists, musicians, film people, entrepreneurs, scientists and inventors.
Germany has a large number of World Heritage sites and is among the top tourism destinations in the world. The English word Germany derives from the Latin Germania, which came into use after Julius Caesar adopted it for the peoples east of the Rhine; the German term Deutschland diutisciu land is derived from deutsch, descended from Old High German diutisc "popular" used to distinguish the language of the common people from Latin and its Romance descendants. This in turn descends from Proto-Germanic *þiudiskaz "popular", derived from *þeudō, descended from Proto-Indo-European *tewtéh₂- "people", from which the word Teutons originates; the discovery of the Mauer 1 mandible shows that ancient humans were present in Germany at least 600,000 years ago. The oldest complete hunting weapons found anywhere in the world were discovered in a coal mine in Schöningen between 1994 and 1998 where eight 380,000-year-old wooden javelins of 1.82 to 2.25 m length were unearthed. The Neander Valley was the location where the first non-modern human fossil was discovered.
The Neanderthal 1 fossils are known to be 40,000 years old. Evidence of modern humans dated, has been found in caves in the Swabian Jura near Ulm; the finds included 42,000-year-old bird bone and mammoth ivory flutes which are the oldest musical instruments found, the 40,000-year-old Ice Age Lion Man, the oldest uncontested figurative art discovered, the 35,000-year-old Venus of Hohle Fels, the oldest uncontested human figurative art discovered. The Nebra sky disk is a bronze artefact created during the European Bronze Age attributed to a site near Nebra, Saxony-Anhalt, it is part of UNESCO's Memory of the World Programme. The Germanic tribes are thought to date from the Pre-Roman Iron Age. From southern Scandinavia and north Germany, they expanded south and west from the 1st century BC, coming into contact with the Celtic tribes of Gaul as well