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Bratteli–Vershik Diagram
In mathematics, a Bratteli–Veršik diagram is an ordered, essentially simple Bratteli diagram (''V'', ''E'') with a homeomorphism on the set of all infinite paths called the Veršhik transformation. It is named after Ola Bratteli and Anatoly Vershik. Definition Let ''X'' = be the set of all paths in the essentially simple Bratteli diagram (''V'', ''E''). Let ''E''min be the set of all minimal edges in ''E'', similarly let ''E''max be the set of all maximal edges. Let ''y'' be the unique infinite path in ''E''max. (Diagrams which possess a unique infinite path are called "essentially simple".) The Veršhik transformation is a homeomorphism φ : ''X'' → ''X'' defined such that φ(''x'') is the unique minimal path if ''x'' = ''y''. Otherwise ''x'' = (''e''1, ''e''2,...) , ''e''''i'' ∈ ''E''''i'' where at least one ''e''''i'' ∉ ''E''max. Let ''k'' be the smallest such integer. Then φ(' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bratteli Diagram
In mathematics, a Bratteli diagram is a combinatorial structure: a Graph (discrete mathematics), graph composed of vertices labelled by positive integers ("level") and unoriented edges between vertices having levels differing by one. The notion was introduced by Ola Bratteli in 1972 in the theory of operator algebras to describe directed sequences of finite-dimensional algebras: it played an important role in Elliott's classification of approximately finite-dimensional C*-algebra, AF-algebras and the theory of subfactors. Subsequently Anatoly Vershik associated dynamical systems with infinite paths in such graphs. Definition A Bratteli diagram is given by the following objects: * A sequence of sets ''V''''n'' ('the vertices at level ''n'' ') labeled by positive integer set N. In some literature each element v of ''V''''n'' is accompanied by a positive integer ''b''''v'' > 0. * A sequence of sets ''E''''n'' ('the edges from level ''n'' to ''n'' + 1 ') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ola Bratteli
Ola Bratteli (24 October 1946 – 8 February 2015) was a Norwegian mathematician. He was a son of Trygve Bratteli and Randi Bratteli (née Larssen). He received a PhD degree in 1974. He was appointed as professor at the University of Trondheim in 1980 and at the University of Oslo in 1991. He was a member of the Norwegian Academy of Science and Letters. Selected works *with Derek W. Robinson: ''Operator Algebras and Quantum Statistical Mechanics'' (Springer-Verlag, 2 volumes, 1980) *''Derivations, Dissipations and Group Actions on C*-algebras'' (Springer-Verlag, 1986) *with Palle T. Jørgensen: ''Wavelets through a looking glass, the world of the spectrum'' (Birkhäuser, 2002) See also *Approximately finite-dimensional C*-algebra *Bratteli diagram *Bratteli–Vershik diagram In mathematics, a Bratteli–Veršik diagram is an ordered, essentially simple Bratteli diagram (''V'', ''E'') with a homeomorphism on the set of all infinite paths called the Veršhik transform ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anatoly Vershik
Anatoly Moiseevich Vershik (; 28 December 1933 – 14 February 2024) was a Soviet and Russian mathematician. He is most famous for his joint work with Sergei V. Kerov on representations of infinite symmetric groups and applications to the longest increasing subsequences. Biography Vershik studied at Leningrad State University (later renamed to Saint Petersburg State University), receiving his doctoral degree in 1974; his advisor was Vladimir Rokhlin. Vershik worked at the St. Petersburg Department of Steklov Institute of Mathematics and at Saint Petersburg State University. In 1998–2008, he was the president of the St. Petersburg Mathematical Society. In 2012, Vershik became a fellow of the American Mathematical Society. In 2015, he was elected a member of Academia Europaea. His doctoral students include Alexander Barvinok, Dmitri Burago, Anna Erschler, Sergey Fomin, Vadim Kaimanovich, Sergei Kerov, Alexander N. Livshits, Andrei Lodkin, Nikolai Mnev, and Natal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minor (graph Theory)
Minor may refer to: Common meanings * Minor (law), a person not under the age of certain legal activities. * Academic minor, a secondary field of study in undergraduate education Mathematics * Minor (graph theory), a relation of one graph to another * Minor (matroid theory), a relation of one matroid to another * Minor (linear algebra), the determinant of a square submatrix Music * Minor chord * Minor interval * Minor key * Minor scale People * Minor (given name), a masculine given name * Minor (surname), a surname Places in the United States * Minor, Alabama, a census-designated place * Minor, Virginia, an unincorporated community * Minor Creek (California) * Minor Creek (Missouri) * Minor Glacier, Wyoming Sports * Minor, a grade in Gaelic games; also, a person who qualifies to play in that grade * Minor league, a sports league not regarded as a premier league ** Minor League Baseball Minor League Baseball (MiLB) is a professional baseball organization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-quasi-ordering
In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set X is a quasi-ordering of X for which every infinite sequence of elements x_0, x_1, x_2, \ldots from X contains an increasing pair x_i \leq x_j with i x_2> \cdots) nor infinite sequences of ''pairwise incomparable'' elements. Hence a quasi-order (''X'', ≤) is wqo if and only if (''X'', <) is well-founded and has no infinite antichains. Ordinal type Let be well partially ordered. A (necessarily finite) sequence of elements of that contains no pair with is usually called a ''bad sequence''. The ''tree of bad sequences'' is the tree that contains a vertex for each bad ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definitions A binary relation \,\si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Relation
A symmetric relation is a type of binary relation. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: : \forall a, b \in X(a R b \Leftrightarrow b R a) , where the notation ''aRb'' means that . An example is the relation "is equal to", because if is true then is also true. If ''R''T represents the converse of ''R'', then ''R'' is symmetric if and only if . Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation. Examples In mathematics * "is equal to" ( equality) (whereas "is less than" is not symmetric) * "is comparable to", for elements of a partially ordered set * "... and ... are odd": :::::: Outside mathematics * "is married to" (in most legal systems) * "is a fully biological sibling of" * "is a homophone of" * "is a co-worker of" * "is a teammate of" Relationship to asymmetric and antisymmetric relations By definition, a nonempty relation cannot be both symmetric and asymmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Conjugacy
In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially. To illustrate this directly: suppose that f and g are iterated functions, and there exists a homeomorphism h such that :g = h^ \circ f \circ h, so that f and g are topologically conjugate. Then one must have :g^n = h^ \circ f^n \circ h, and so the iterated systems are topologically conjugate as well. Here, \circ denotes function composition. Definition f\colon X \to X, g\colon Y \to Y, and h\colon Y \to X are continuous functions on topological spaces, X and Y. f being topologically semiconjugate to g means, by definition, that h is a surjection such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Odometer
In mathematics, a Markov odometer is a certain type of topological dynamical system. It plays a fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem of H. Dye asserts that every ergodic nonsingular transformation is orbit-equivalent to a Markov odometer. The basic example of such system is the "nonsingular odometer", which is an additive topological group defined on the product space of discrete spaces, induced by addition defined as x \mapsto x+\underline, where \underline:=(1,0,0,\dots). This group can be endowed with the structure of a dynamical system; the result is a conservative dynamical system. The general form, which is called "Markov odometer", can be constructed through Bratteli–Vershik diagram to define ''Bratteli–Vershik compactum'' space together with a corresponding transformation. Nonsingular odometers Several kinds of non-singular odometers may be defined. These are sometimes referred to as adding machin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
