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Markov Shift
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. Definition Let V be a finite set of n symbols (alphabet). Let ''X'' denote the set V^\mathbb of all bi-infinite sequences of elements of ''V'' together with the shift operator ''T''. We endow ''V'' with the discrete topology and ''X'' with the product topology. A symbolic flow or subshift is a closed ''T''-invariant subset ''Y'' of ''X'' Xie (1996) p.21 and the associated language ''L''''Y'' is the set of finite subsequences of ''Y''.Xie (1996) p.22 Now let A be an n\times n adjacency matrix with entries in . Using these elements we construct a directed graph ''G''=(''V'',''E'') with ''V'' the set of vertices and ''E'' the set of edges containing the directed ed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Language
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expressions engines, which are augmented with features that allow recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognized by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician Stephen Cole Kleene). In the Chomsky hierarchy, regular languages are the languages generated by Type-3 grammars. Formal definition The collection of regular languages over an alphabet Σ is defined recursively as follows: * The empty language Ø is a regular language. * For each ''a'' ∈ Σ (''a'' belongs to Σ), the singleton language is a regular language. * If ''A'' is a regular language, ''A''* ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deterministic Automaton
In computer science, a deterministic automaton is a concept of automata theory in which the outcome of a transition from one state to another is determined by the input. A common deterministic automaton is a deterministic finite automaton (DFA) which is a finite state machine where for each pair of state and input symbol there is one and only one transition to a next state. DFAs recognize the set of regular languages and no other languages. A standard way to build a deterministic finite automaton from a nondeterministic finite automaton is the powerset construction In the theory of computation and automata theory, the powerset construction or subset construction is a standard method for converting a nondeterministic finite automaton (NFA) into a deterministic finite automaton (DFA) which recognizes the sa .... References Automata (computation) {{Comp-sci-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automata Theory
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word ''automata'' comes from the Greek word αὐτόματος, which means "self-acting, self-willed, self-moving". An automaton (automata in plural) is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically. An automaton with a finite number of states is called a Finite Automaton (FA) or Finite-State Machine (FSM). The figure on the right illustrates a finite-state machine, which is a well-known type of automaton. This automaton consists of states (represented in the figure by circles) and transitions (represented by arrows). As the automaton sees a symbol of input, it makes a transition (or jump) to another state, according to its transition function, which takes the previous state and current input symbol as its arguments. Automata theo ... [...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 In mathematics, the Silverman–Toeplitz theorem, first pro ... [...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'' o ... [...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: mixing paint, mixing drinks, industrial mixing, ''etc''. 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" notion than ergodicity). Informal explanation The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, drinks, ... [...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 Partitas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thue–Morse Sequence
In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. The first few steps of this procedure yield the strings 0 then 01, 0110, 01101001, 0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. The full sequence begins: :01101001100101101001011001101001.... The sequence is named after Axel Thue and Marston Morse. 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. For this reason John H. Conway ''et al''. called numbers ''n'' satisfying ''tn'' = 1 ''odious'' (for ''odd'') numbers and numbers for which ''tn''&n ... [...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 \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Manifold
In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. A 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. Open manifolds For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact. Abuse of language Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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