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Subshift Of Finite Type
In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by a finite-state machine. The most widely studied shift spaces are the subshifts of finite type. Motivating examples A (one-sided) shift of finite type is the set of all sequences, infinite on one end only, that can be made up of the letters A, B, C, like AAA\cdots, ABAB\cdots, \dots. A (two-sided) shift of finite type is similar, but consists of sequences that are infinite on both ends. A subshift can be defined by a directed graph on these letters, such as the graph A \to B \to C \to A. It consists of sequences whose transitions between consecutive letters are only those allowed by the graph. For this example, the subshift consists of only three one-sided sequences: ABCABC\cdots, BCABCA\cdots, CABCAB\cdots. Similarly, the two-sided subshift described by th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bi-infinite Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toeplitz Systems
Toeplitz or Töplitz may refer to: Places * Töplitz, the German name of Toplița, a city in Romania * Toplița, Hunedoara, a commune in Romania * Teplice (archaic German: ''Töplitz''), Czech Republic People * Jerzy Toeplitz (1909–1995), co-founder of the Polish Film School * Kasper T. Toeplitz (born 1960), Polish-French composer * Otto Toeplitz (1881–1940), German Jewish mathematician See also * Dolenjske Toplice, a settlement in southeastern Slovenia * Toeplitz matrix, a structured matrix with equal values along diagonals * Toeplitz operator, the compression of a multiplication operator on the circle to the Hardy space * Toeplitz algebra, the C*-algebra generated by the unilateral shift on the Hilbert space * Toeplitz Hash Algorithm, used in many network interface controllers * Hellinger–Toeplitz theorem, an everywhere defined symmetric operator on a Hilbert space is bounded * Silverman–Toeplitz theorem, characterizing matrix summability methods which are re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturmian System
In mathematics, a Sturmian word (Sturmian sequence or billiard sequence), named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters. Such a sequence can be generated by considering a game of English billiards on a square table. The struck ball will successively hit the vertical and horizontal edges labelled 0 and 1 generating a sequence of letters. This sequence is a Sturmian word. Definition Sturmian sequences can be defined strictly in terms of their combinatoric properties or geometrically as cutting sequences for lines of irrational slope or codings for irrational rotations. They are traditionally taken to be infinite sequences on the alphabet of the two symbols 0 and 1. Combinatorial definitions Sequences of low complexity For an infinite sequence of symbols ''w'', let ''σ''(''n'') be the complexity function of ''w''; i.e., ''σ''(''n'') = the number of distinct contiguous subwords (factors) in ''w' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixing (mathematics)
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: e.g. mixing paint, mixing drinks, industrial mixing. The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including ''strong mixing'', ''weak mixing'' and ''topological mixing'', with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" condition than ergodicity). Informal explanation The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, dri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chacon System , a work by Johann Sebastian Bach
{{Disambiguation, geo ...
Chacon may refer to: * Chacón, a list of people with the surname Chacón or Chacon * Captain Trudy Chacon, a fictional character in the 2009 film ''Avatar'' * Chacon, New Mexico, United States, a town * Chacon Creek, a small stream in Texas, United States * ''Chacon'' (1912), a wooden fishing vessel in Alaska * ''Chacon'' (1918), a ship lost at sea in 1937 See also * Chaconne, a type of musical composition * ''Chaconne'' (ballet), a 1976 ballet * "The Chaconne", the last movement of Partita for Violin No. 2 (Bach) The Partita in D minor for solo violin, Bach-Werke-Verzeichnis, BWV 1004, by Johann Sebastian Bach, was written between 1717 and 1720. It is a part of his compositional cycle called Sonatas and Partitas for Solo Violin (Bach), Sonatas and Partita ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thue–Morse Sequence
In mathematics, the Thue–Morse or Prouhet–Thue–Morse sequence is the binary sequence (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. It is sometimes called the fair share sequence because of its applications to fair division or parity sequence. The first few steps of this procedure yield the strings 0, 01, 0110, 01101001, 0110100110010110, and so on, which are the prefixes of the Thue–Morse sequence. The full sequence begins: :01101001100101101001011001101001.... The sequence is named after Axel Thue, Marston Morse and (in its extended form) Eugène Prouhet. Definition There are several equivalent ways of defining the Thue–Morse sequence. Direct definition To compute the ''n''th element ''tn'', write the number ''n'' in binary. If the number of ones in this binary expansion is odd then ''tn'' = 1, if even then ''tn'' = 0. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Entropy
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most non-dissipative systems) as well as systems in thermodynamic equilibrium. Definition A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system :(X, \mathcal, \mu, T) with the following structure: *X is a set, *\mathcal B is a σ-algebra over X, *\mu:\mathcal\rightarrow ,1/math> is a probability measure, so that \mu (X) = 1, and \mu(\varnothing) = 0, * T:X \rightarrow X is a measurable transformation which preserves the measure \mu, i.e., \forall A\in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Manifold
In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only Connected space, connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RP''n'' is a closed ''n''-dimensional manifold. The complex projective space CP''n'' is a closed 2''n''-dimensional manifold. A Real line, line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Properties Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups. If M is a closed connected n-manifold, the n-th homology group H_(M;\mathbb) is \mathbb or 0 depending on whether M is Orientability, orientable or not. Moreover, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definition Given two differentiable manifolds M and N, a Differentiable manifold#Differentiability of mappings between manifolds, continuously differentiable map f \colon M \rightarrow N is a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. Two C^r-differentiable manifolds are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homoclinic Connection
{{unreferenced, date=December 2010 In dynamical systems, a branch of mathematics, a ''homoclinic connection'' is a structure formed by the stable manifold and unstable manifold of a fixed point. Definition for maps Let f:M\to M be a map defined on a manifold M, with a fixed point p. Let W^s(f,p) and W^u(f,p) be the stable manifold and the unstable manifold of the fixed point p, respectively. Let V be a connected invariant manifold such that : V\subseteq W^s(f,p)\cap W^u(f,p) Then V is called a homoclinic connection. Heteroclinic connection It is a similar notion, but it refers to two fixed points, p and q. The condition satisfied by V is replaced with: :V\subseteq W^s(f,p)\cap W^u(f,q) This notion is not symmetric with respect to p and q. Homoclinic and heteroclinic intersections When the invariant manifolds W^s(f,p) and W^u(f,q), possibly with p=q, intersect but there is no homoclinic/heteroclinic connection, a different structure is formed by the two manifolds, som ... [...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] |
