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Dynkin Diagram A2
Dynkin (Russian: Дынкин) is a Russian masculine surname, its feminine counterpart is Dynkina. It may refer to the following notable people: * Aleksandr Dynkin, Russian economist * Eugene Dynkin (1924–2014), Soviet and American mathematician known for ** Dynkin diagram ** Coxeter–Dynkin diagram ** Dynkin system ** Dynkin's formula ** Doob–Dynkin lemma ** Dynkin index In mathematics, the Dynkin index I() of a finite-dimensional highest-weight representation of a compact simple Lie algebra \mathfrak g with highest weight \lambda is defined by \text_= 2I(\lambda) \text_, where V_0 is the 'defining representat ... {{surname Russian-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Dynkin
Alexander A. Dynkin (Russian: Александр Александрович Дынкин; born 30 June 1946) is a Russian economist whose research interests and publications have been in growth, forecasting, international comparisons, technological innovation and energy studies. He is the President of the ''Institute of World Economy and International Relations'' (IMEMO) (Russian Academy of Science) http://www.imemo.ru/en/struct/director.php Institute of World Economy and International Relations: Brief biography of Alexander A. Dynkin (Accessed Jan 2011) Notability * Elected for life as a full member of the Russian Academy of Science * Economic adviser to Prime-Minister of Russia (1998-1999) * 1986 Order of the ''Sign of Worship'' * 2006 ''Friendship Order'' * Keynote Speech at UNIDO's Proceedings of the Industrial Development, Forum and Associated Round Tables, Vienna en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_typ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugene Dynkin
Eugene Borisovich Dynkin (russian: link=no, Евгений Борисович Дынкин; 11 May 1924 – 14 November 2014) was a USSR, Soviet and American mathematician. He made contributions to the fields of probability and algebra, especially Semisimple Lie group, 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 Saint Petersburg, 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 Master of Science, 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 pr ... [...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 semisimple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter–Dynkin Diagram
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge), that is, the amount by which the angle between the reflective planes can be multiplied to get 180 degrees. An unlabeled branch implicitly represents order-3 (60 degrees), and each pair of nodes that is not connected by a branch at all (such as non-adjacent nodes) represents a pair of mirrors at order-2 (90 degrees). Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" ... [...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 Dynkin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin's Formula
In mathematics — specifically, in stochastic analysis — Dynkin's formula is a theorem giving the expected value of any suitably smooth statistic of an Itō diffusion at a stopping time. It may be seen as a stochastic generalization of the (second) fundamental theorem of calculus. It is named after the Russian mathematician Eugene Dynkin. Statement of the theorem Let ''X'' be the R''n''-valued Itō diffusion solving the stochastic differential equation :\mathrm X_ = b(X_) \, \mathrm t + \sigma (X_) \, \mathrm B_. For a point ''x'' ∈ R''n'', let P''x'' denote the law of ''X'' given initial datum ''X''0 = ''x'', and let E''x'' denote expectation with respect to P''x''. Let ''A'' be the infinitesimal generator of ''X'', defined by its action on compactly-supported ''C''2 (twice differentiable with continuous second derivative) functions ''f'' : R''n'' → R as :A f (x) = \lim_ \frac or, equivalently, :A f (x) = \sum_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doob–Dynkin Lemma
In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin (also known as the factorization lemma), characterizes the situation when one random variable is a function of another by the inclusion of the \sigma-algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the \sigma-algebra generated by the other. The lemma plays an important role in the conditional expectation in probability theory, where it allows replacement of the conditioning on a random variable by conditioning on the \sigma-algebra that is generated by the random variable. Notations and introductory remarks In the lemma below, \mathcal ,1/math> is the \sigma-algebra of Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin Index
In mathematics, the Dynkin index I() of a finite-dimensional highest-weight representation of a compact simple Lie algebra \mathfrak g with highest weight \lambda is defined by \text_= 2I(\lambda) \text_, where V_0 is the 'defining representation', which is most often taken to be the fundamental representation if the Lie algebra under consideration is a matrix Lie algebra. The notation \text_V is the trace form on the representation \rho: \mathfrak \rightarrow \text(V). By Schur's lemma, since the trace forms are all invariant forms, they are related by constants, so the index is well-defined. Since the trace forms are bilinear forms, we can take traces to obtain :I(\lambda)=\frac(\lambda, \lambda +2\rho) where the Weyl vector :\rho=\frac\sum_ \alpha is equal to half of the sum of all the positive roots of \mathfrak g. The expression (\lambda, \lambda +2\rho) is the value of the quadratic Casimir in the representation V_\lambda. The index I(\lambda) is always a positiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
