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Dynkin Diagram G2b
Dynkin (Russian: Дынкин) is a Russian masculine surname, its feminine counterpart is Dynkina. It may refer to the following notable people: * Aleksandr Dynkin (born 1948), 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 finite-dimensional highest-weight representations of a compact simple Lie algebra \mathfrak g relates their trace forms via \frac= \frac. In the particular case where \lambda is the highest root, so that ... {{surname Russian-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Dynkin
Alexander A. Dynkin (Russian: Александр Александрович Дынкин; born 30 June 1948) 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, ''Institute of World Economy and International Relations'' (IMEMO) (Russian Academy of Sciences, 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 Sciences, Russian Academy of Science * Economic adviser to Prime Minister of Russia, 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 As ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugene Dynkin
Eugene Borisovich Dynkin (; 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 as Dynkin's Jewish origin; the special efforts of Andrey Kolmogoro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin Diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), 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 graph, directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected graph, undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter–Dynkin Diagram
In geometry, a Harold Scott MacDonald Coxeter, Coxeter–Eugene Dynkin, Dynkin diagram (or Coxeter diagram, Coxeter graph) is a Graph (discrete mathematics), graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group. A class of closely related objects is the Dynkin diagrams, which differ from Coxeter diagrams in two respects: firstly, branches labeled "" or greater are Directed graph, directed, while Coxeter diagrams are Undirected graph, undirected; secondly, Dynkin diagrams must satisfy an additional (Crystallographic restriction theorem, crystallographic) restriction, namely that the only allowed branch labels are and Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras. Description A Coxeter group is a group that admits a presentation: \langle r_0,r_1,\dots,r_n \mid (r_i r_j)^ = 1 \rangle where the are integers that ... [...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 \cdots 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 Dyn ... [...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 function applied to a Feller process 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 a Feller process with infinitesimal generator A. For a point x in the state-space of X, let \mathbf P^x denote the law of X given initial datum X_0=x, and let \mathbf E^x denote expectation with respect to \mathbf P^x. Then for any function f in the domain of A, and any stopping time \tau with \mathbf Etau+\infty, Dynkin's formula holds: : \mathbf^ (X_)= f(x) + \mathbf^ \left \int_^ A f (X_) \, \mathrm s \right Example: Itô diffusions Let X be the \mathbf R^n-valued Itô diffusion solving the stochastic differential equation :\mathrm X_ = b(X_) \, \mathrm t + \sigma (X_) \, \mathrm B_. The ... [...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 sets on ,1 If T\colon X\to Y, and (Y,) is a measurable space, then :\sigma(T)\ \stackrel\ \ is the smallest \sigma-algebra on X such that T is \sigma(T) / -measurable. Statement of the lemma Let T\col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin Index
In mathematics, the Dynkin index I() of finite-dimensional highest-weight representations of a compact simple Lie algebra \mathfrak g relates their trace forms via \frac= \frac. In the particular case where \lambda is the highest root, so that V_\lambda is the adjoint representation, the Dynkin index I(\lambda) is equal to the dual Coxeter number. 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. See also * Killing form In mathematics, the Killing form, na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
