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Eugene Dynkin
Eugene Borisovich Dynkin (russian: link=no, Евгений Борисович Дынкин; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician. He made contributions to the fields of probability and algebra, especially semisimple Lie groups, Lie algebras, and Markov processes. The Dynkin diagram, the Dynkin system, and Dynkin's lemma are named after him. Biography Dynkin was born into a Jewish family, living in Leningrad until 1935, when his family was exiled to Kazakhstan. Two years later, when Dynkin was 13, his father disappeared in the Gulag. Moscow University At the age of 16, in 1940, Dynkin was admitted to Moscow University. He avoided military service in World War II because of his poor eyesight, and received his MS in 1945 and his PhD in 1948. He became an assistant professor at Moscow, but was not awarded a "chair" until 1954 because of his political undesirability. His academic progress was made difficult due to his father's fate, as well a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saint Petersburg
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), is the List of cities and towns in Russia by population, second-largest city in Russia. It is situated on the Neva River, at the head of the Gulf of Finland on the Baltic Sea, with a population of roughly 5.4 million residents. Saint Petersburg is the List of European cities by population within city limits, fourth-most populous city in Europe after Istanbul, Moscow and London, 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 Russia's Imperial capital, and a Ports of the Baltic Sea, historically strategic port, it is governed as a Federal cities of Russia, federal city. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
USSR
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen national republics; in practice, both its government and its economy were highly centralized until its final years. It was a one-party state governed by the Communist Party of the Soviet Union, with the city of Moscow serving as its capital as well as that of its largest and most populous republic: the Russian SFSR. Other major cities included Leningrad (Russian SFSR), Kiev ( Ukrainian SSR), Minsk ( Byelorussian SSR), Tashkent ( Uzbek SSR), Alma-Ata ( Kazakh SSR), and Novosibirsk (Russian SFSR). It was the largest country in the world, covering over and spanning eleven time zones. The country's roots lay in the October Revolution of 1917, when the Bolsheviks, under the leadership of Vladimir Lenin, overthrew the Russian Provisional G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great powers—forming two opposing military alliances: the Allies of World War II, Allies and the Axis powers. World War II was a total war that directly involved more than 100 million Military personnel, personnel from more than 30 countries. The major participants in the war threw their entire economic, industrial, and scientific capabilities behind the war effort, blurring the distinction between civilian and military resources. Air warfare of World War II, Aircraft played a major role in the conflict, enabling the strategic bombing of population centres and deploying the Atomic bombings of Hiroshima and Nagasaki, only two nuclear weapons ever used in war. World War II was by far the List of wars by death toll, deadliest conflict in hu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gulag
The Gulag, an acronym for , , "chief administration of the camps". The original name given to the system of camps controlled by the State Political Directorate, GPU was the Main Administration of Corrective Labor Camps (, )., name=, group= was the government agency in charge of the Soviet Union, Soviet network of Correctional labour camp, forced labour camps which were set up by order of Vladimir Lenin, reaching its peak during Joseph Stalin's rule from the 1930s to the early 1950s. English-language speakers also use the word ''gulag'' in reference to each of the forced-labor camps that existed in the Soviet Union, including the camps that existed in the History of the Soviet Union (1927–1953), post-Lenin era. The Gulag is recognized as a major instrument of political repression in the Soviet Union. The camps housed a wide range of convicts, from petty criminals to political prisoners, a large number of whom were convicted by simplified procedures, such as NKVD troikas or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kazakhstan
Kazakhstan, officially the Republic of Kazakhstan, is a transcontinental country located mainly in Central Asia and partly in Eastern Europe. It borders Russia Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eigh ... to Kazakhstan–Russia border, the north and west, China to China–Kazakhstan border, the east, Kyrgyzstan to Kazakhstan–Kyrgyzstan border, the southeast, Uzbekistan to Kazakhstan–Uzbekistan border, the south, and Turkmenistan to Kazakhstan–Turkmenistan border, the southwest, with a coastline along the Caspian Sea. Its capital is Astana, known as Nur-Sultan from 2019 to 2022. Almaty, Kazakhstan's largest city, was the country's capital until 1997. Kazakhstan is the world's largest landlocked country, the largest and northernmost Muslim world, Muslim-majority cou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin System
A Dynkin system, named after Eugene Dynkin is a collection of subsets of another universal set \Omega satisfying a set of axioms weaker than those of -algebra. Dynkin systems are sometimes referred to as -systems (Dynkin himself used this term) or d-system. These set families have applications in measure theory and probability. A major application of -systems is the - theorem, see below. Definition Let \Omega be a nonempty set, and let D be a collection of subsets of \Omega (that is, D is a subset of the power set of \Omega). Then D is a Dynkin system if # \Omega \in D, # D is closed under complements of subsets in supersets: if A, B \in D and A \subseteq B, then B \setminus A \in D, # D is closed under countable increasing unions: if A_1 \subseteq A_2 \subseteq A_3 \subseteq \ldots is an increasing sequenceA sequence of sets A_1, A_2, A_3, \ldots is called if A_n \subseteq A_ for all n \geq 1. of sets in D then \bigcup_^\infty A_n \in D. It is easy to check that any D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin Diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra. The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' Dynkin diagrams will be explicitly so named. Classification of semisi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings ( Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Lie Group
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form, κ(x,y) = tr(ad(''x'')ad(''y'')), is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals; * the radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple. Semisimple Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
