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Mirimanoff
Dmitry Semionovitch Mirimanoff (; 13 September 1861, Pereslavl-Zalessky, Russia – 5 January 1945, Geneva, Switzerland) was a member of the Moscow Mathematical Society in 1897. And later became a doctor of mathematical sciences in 1900, in Geneva, and taught at the universities of Geneva and Lausanne. Mirimanoff made notable contributions to axiomatic set theory and to number theory (relating specifically to Fermat's Last Theorem, on which he corresponded with Albert Einstein before the First World WarJean A. Mirimanoff. Private correspondence with Anton Lokhmotov. (2009)). In 1917, he introduced, though not as explicitly as John von Neumann later, the cumulative hierarchy of sets and the notion of von Neumann ordinals; although he introduced a notion of regular (and well-founded set) he did not consider regularity as an axiom, but also explored what is now called non-well-founded set theory and had an emergent idea of what is now called bisimulation. Life Dmitry Semio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every Empty set, non-empty Set (mathematics), set ''A'' contains an element that is Disjoint sets, disjoint from ''A''. In first-order logic, the axiom reads: \forall x\,(x \neq \varnothing \rightarrow (\exists y \in x) (y \cap x = \varnothing)). The axiom of regularity together with the axiom of pairing implies that Russell paradox, no set is an element of itself, and that there is no infinite sequence (a_n) such that a_ is an element of a_i for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains. The axiom was originally formulated by von Neumann; it was adopted in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-well-founded Set Theory
Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation. The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as an axiom. Although a number of axiomatic systems of non-well-founded sets were proposed afterwards, they did not find much in the way of applications until the book Non-Well-Founded Sets by Peter Aczel introduces hyperset theory in 1988. The theory of non-well-founded sets has been applied in the logical modelling of non-terminating computational processes in computer science ( process algebra and final semantics), linguistics and natural language semantics (situation theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pereslavl-Zalessky
Pereslavl-Zalessky (, ), formerly known as Pereyaslavl-Zalessky, or simply Pereyaslavl, is a town in Yaroslavl Oblast, Russia, located on the main Moscow–Yaroslavl road and on the southeastern shore of Lake Pleshcheyevo at the mouth of the Trubezh River. Population: History It was founded in 1152 by George I of Vladimir as a projected capital of Zalesye (). The inhabitants of the nearby town of Kleshchin were relocated to the new town. Between 1175 and 1302, Pereslavl was the seat of a principality; in 1302, it was inherited by the prince of Moscow following the childless death of Dmitry of Pereslavl's son Ivan. Pereslavl-Zalessky was devastated numerous times by the Mongols between the mid-13th century and the early 15th century. In 1611–1612, it suffered from the Polish invasion. In 1688–1693, Peter the Great built his famous "fun flotilla" on Lake Pleshcheyevo for his own amusement, including the so-called Peter's little boat (''botik''), which could be con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Switzerland
Switzerland, officially the Swiss Confederation, is a landlocked country located in west-central Europe. It is bordered by Italy to the south, France to the west, Germany to the north, and Austria and Liechtenstein to the east. Switzerland is geographically divided among the Swiss Plateau, the Swiss Alps, Alps and the Jura Mountains, Jura; the Alps occupy the greater part of the territory, whereas most of the country's Demographics of Switzerland, 9 million people are concentrated on the plateau, which hosts List of cities in Switzerland, its largest cities and economic centres, including Zurich, Geneva, and Lausanne. Switzerland is a federal republic composed of Cantons of Switzerland, 26 cantons, with federal authorities based in Bern. It has four main linguistic and cultural regions: German, French, Italian and Romansh language, Romansh. Although most Swiss are German-speaking, national identity is fairly cohesive, being rooted in a common historical background, shared ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bisimulation
In theoretical computer science a bisimulation is a binary relation between state transition systems, associating systems that behave in the same way in that one system simulates the other and vice versa. Intuitively two systems are bisimilar if they, assuming we view them as playing a ''game'' according to some rules, match each other's moves. In this sense, each of the systems cannot be distinguished from the other by an observer. Formal definition Given a labeled state transition system , where is a set of states, \Lambda is a set of labels and → is a set of labelled transitions (i.e., a subset of S \times \Lambda \times S), a bisimulation is a binary relation R \subseteq S \times S, such that both and its converse R^T are simulations. From this follows that the symmetric closure of a bisimulation is a bisimulation, and that each symmetric simulation is a bisimulation. Thus some authors define bisimulation as a symmetric simulation. Equivalently, is a bisimulatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist and mathematician. He is a professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer science. Knuth has been called the "father of the analysis of algorithms". Knuth is the author of the multi-volume work '' The Art of Computer Programming''. He contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process, he also popularized the asymptotic notation. In addition to fundamental contributions in several branches of theoretical computer science, Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces. As a writer and scholar, Knuth created the WEB and CWEB computer programming systems des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Désiré André
Désiré André (André Antoine Désiré) (March 29, 1840, Lyon – September 12, 1917, Paris) was a French mathematician, best known for his work on Catalan numbers and alternating permutations. Biography He is the son of Auguste Antoine Désiré André, shoemaker in Lyon, and his wife Antoinette Magdalene Jar. He entered the École Normale Supérieure in 1860 and passed the Agrégation in Mathematics in 1863. He defended his doctoral thesis on 25 March 1877. He was a student of Charles Hermite (1822–1901) and Joseph Bertrand (1822–1900). Starting as a teacher at the Lycée de Troyes, he went on to Collège Sainte-Barbe, then to the University of Burgundy, University of Dijon and finally became professor of mathematics at Collège Stanislas de Paris from 1885 to 1900. He was a laureate of the Ministère de l'Instruction Publique, member of the Circolo Matematico di Palermo and the Commission Internationale Permanente de Bibliographie Mathématique. He was made a Knight ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand's Ballot Theorem
In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives ''p'' votes and candidate B receives ''q'' votes with ''p'' > ''q'', what is the probability that A will be strictly ahead of B throughout the count under the assumption that votes are counted in a randomly picked order?" The answer is :\frac. The result was first published by W. A. Whitworth in 1878, but is named after Joseph Louis François Bertrand who rediscovered it in 1887. In Bertrand's original paper, he sketches a proof based on a general formula for the number of favourable sequences using a recursion relation. He remarks that it seems probable that such a simple result could be proved by a more direct method. Such a proof was given by Désiré André, based on the observation that the unfavourable sequences can be divided into two equally probable cases, one of which (the case where B receives the first vote) is easily computed; he proves the equality b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Swiss Citizen
The primary law governing nationality of Switzerland is the Federal Act on Swiss Citizenship, which came into force on 1 January 2018. Switzerland is a member state of the European Free Trade Association (EFTA) and the Schengen Area. All Swiss nationals have automatic and permanent permission to live and work in any European Union (EU) or EFTA country. Swiss nationals are citizens of their municipality of origin, their canton of origin, and the Confederation, in that order: a Swiss citizen is defined as someone who has the citizenship of a Swiss municipality (article 37 of the Swiss Federal Constitution). They are entered in the family register of their place of origin. The manner by which Swiss citizens acquire their place of origin differs depending on whether they acquired Swiss citizenship by filiation (jus sanguinis), ordinary naturalisation, or facilitated naturalisation. Marriage has in and of itself no effect on the places of origin of the spouses. Terminology The dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russian Revolution
The Russian Revolution was a period of Political revolution (Trotskyism), political and social revolution, social change in Russian Empire, Russia, starting in 1917. This period saw Russia Dissolution of the Russian Empire, abolish its monarchy and adopt a socialist form of government following two successive revolutions and Russian Civil War, a civil war. It can be seen as the precursor for Revolutions of 1917–1923, other revolutions that occurred in the aftermath of World War I, such as the German Revolution of 1918–1919. The Russian Revolution was a key events of the 20th century, key event of the 20th century. The Russian Revolution was inaugurated with the February Revolution in 1917, in the midst of World War I. With the German Empire inflicting defeats on the front, and increasing logistical problems causing shortages of bread and grain, the Russian Army was losing morale, with large scale mutiny looming. Officials were convinced that if Tsar Nicholas II abdicated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saint Petersburg
Saint Petersburg, formerly known as Petrograd and later Leningrad, is the List of cities and towns in Russia by population, second-largest city in Russia after Moscow. It is situated on the Neva, River Neva, at the head of the Gulf of Finland on the Baltic Sea. The city had a population of 5,601,911 residents as of 2021, with more than 6.4 million people living in the Saint Petersburg metropolitan area, metropolitan area. Saint Petersburg is the List of European cities by population within city limits, fourth-most populous city in Europe, the List of cities and towns around the Baltic Sea, most populous city on the Baltic Sea, and the world's List of northernmost items#Cities and settlements, northernmost city of more than 1 million residents. As the former capital of the Russian Empire, and a Ports of the Baltic Sea, historically strategic port, it is governed as a Federal cities of Russia, federal city. The city was founded by Tsar Peter the Great on 27 May 1703 on the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moscow
Moscow is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia, standing on the Moskva (river), Moskva River in Central Russia. It has a population estimated at over 13 million residents within the city limits, over 19.1 million residents in the urban area, and over 21.5 million residents in Moscow metropolitan area, its metropolitan area. The city covers an area of , while the urban area covers , and the metropolitan area covers over . Moscow is among the world's List of largest cities, largest cities, being the List of European cities by population within city limits, most populous city entirely in Europe, the largest List of urban areas in Europe, urban and List of metropolitan areas in Europe, metropolitan area in Europe, and the largest city by land area on the European continent. First documented in 1147, Moscow became the capital of the Grand Principality of Moscow, which led the unification of the Russian lan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
