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Invariant Sigma-algebra
In mathematics, especially in probability theory and ergodic theory, the invariant sigma-algebra is a sigma-algebra formed by sets which are invariant set, invariant under a group action or dynamical system. It can be interpreted as of being "indifferent" to the dynamics. The invariant sigma-algebra appears in the study of ergodic systems, as well as in theorems of probability theory such as de Finetti's theorem and the Hewitt-Savage zero-one law, Hewitt-Savage law. Definition Strictly invariant sets Let (X,\mathcal) be a measurable space, and let T:(X,\mathcal)\to(X,\mathcal) be a measurable function. A measurable subset S\in \mathcal is called invariant set, invariant if and only if T^(S)=S. Equivalently, if for every x\in X, we have that x\in S if and only if T(x)\in S. More generally, let M be a group (mathematics), group or a monoid, let \alpha:M\times X\to X be a monoid action, and denote the action of m\in M on X by \alpha_m:X\to X. A subset S\subseteq X is \alpha-i ... [...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] |
Markov Kernel
In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space. Formal definition Let (X,\mathcal A) and (Y,\mathcal B) be measurable spaces. A ''Markov kernel'' with source (X,\mathcal A) and target (Y,\mathcal B), sometimes written as \kappa:(X,\mathcal)\to(Y,\mathcal), is a function \kappa : \mathcal B \times X \to ,1/math> with the following properties: # For every (fixed) B_0 \in \mathcal B, the map x \mapsto \kappa(B_0, x) is \mathcal A- measurable # For every (fixed) x_0 \in X, the map B \mapsto \kappa(B, x_0) is a probability measure on (Y, \mathcal B) In other words it associates to each point x \in X a probability measure \kappa(dy, x): B \mapsto \kappa(B, x) on (Y,\mathcal B) such that, for every measurable set B\in\mathcal B, the map x\mapsto \kappa(B, x) is measur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tail Event
The tail is the elongated section at the rear end of a bilaterian animal's body; in general, the term refers to a distinct, flexible appendage extending backwards from the midline of the torso. In vertebrate animals that evolved to lose their tails (e.g. frogs and hominid primates), the coccyx is the homologous vestigial of the tail. While tails are primarily considered a feature of vertebrates, some invertebrates such as scorpions and springtails, as well as snails and slugs, have tail-like appendages that are also referred to as tails. Tail-shaped objects are sometimes referred to as "caudate" (e.g. caudate lobe, caudate nucleus), and the body part associated with or proximal to the tail are given the adjective " caudal" (which is considered a more precise anatomical terminology). Function Animal tails are used in a variety of ways. They provide a source of thrust for aquatic locomotion for fish, cetaceans and crocodilians and other forms of marine life. Terrestrial specie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shift Space
Shift may refer to: Art, entertainment, and media Gaming * ''Shift'' (series), a 2008 online video game series by Armor Games * '' Need for Speed: Shift'', a 2009 racing video game ** '' Shift 2: Unleashed'', its 2011 sequel Literature * ''Shift'' (novel), a 2010 alternative history book by Tim Kring and Dale Peck * ''Shift'' (novella), a 2013 science fiction book, part two of the Silo trilogy by Hugh Howey * Shift the Ape, a character in ''The Chronicles of Narnia'' novel series * Shift (DC Comics), a DC Comics character who is a fragment of Metamorpho * Shift (Marvel Comics), a Marvel Comics Marvel Comics is a New York City–based comic book publishing, publisher, a property of the Walt Disney Company since December 31, 2009, and a subsidiary of Disney Publishing Worldwide since March 2023. Marvel was founded in 1939 by Martin G ... character who is a clone of Miles Morales Music * ''Shift'' (Nasum album), 2004 * Shift (The Living End album) * Shift (music) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Finetti Theorem
In probability theory, de Finetti's theorem states that exchangeable observations are conditionally independent relative to some latent variable. An epistemic probability distribution could then be assigned to this variable. It is named in honor of Bruno de Finetti, and one of its uses is in providing a pragmatic approach to de Finetti's well-known dictum "Probability does not exist". For the special case of an exchangeable sequence of Bernoulli random variables it states that such a sequence is a "mixture" of sequences of independent and identically distributed (i.i.d.) Bernoulli random variables. A sequence of random variables is called exchangeable if the joint distribution of the sequence is unchanged by any permutation of a finite set of indices. In general, while the variables of the exchangeable sequence are not ''themselves'' independent, only exchangeable, there is an ''underlying'' family of i.i.d. random variables. That is, there are underlying, generally unobserv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Measure
In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces, except that there can be many natural choices for the product measure. Let (X_1, \Sigma_1) and (X_2, \Sigma_2) be two measurable spaces, that is, \Sigma_1 and \Sigma_2 are sigma algebras on X_1 and X_2 respectively, and let \mu_1 and \mu_2 be measures on these spaces. Denote by \Sigma_1 \otimes \Sigma_2 the sigma algebra on the Cartesian product X_1 \times X_2 generated by subsets of the form B_1 \times B_2, where B_1 \in \Sigma_1 and B_2 \in \Sigma_2: \Sigma_1 \otimes \Sigma_2 = \sigma\left( \lbrace B_1 \times B_2 \mid B_1 \in \Sigma_1, B_2 \in \Sigma_2 \rbrace \right) This sigma algebra is called the ''tensor-product σ-algebra'' on the product space. A ''product measure'' \mu_1 \times \mu_2 (also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint (mutually exclusive) events by the measure should be the sum of the probabilities of the events; for example, the value assigned to the outcome "1 or 2" in a throw of a dice should be the sum of the values assigned to the outcomes "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a set function \mu to be a probability measure on a σ-algebra are that: * \mu must return results in the unit interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function (mathematics), function in which * the Domain of a function, domain is the set of possible Outcome (probability), outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the Range of a function, range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the Real number, real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutations
In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is &nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Sigma-algebra
Product may refer to: Business * Product (business), an item that can be offered to a market to satisfy the desire or need of a customer. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Product (mathematics) Algebra * Direct product Set theory * Cartesian product of sets Group theory * Direct product of groups * Semidirect product * Product of group subsets * Wreath product * Free product * Zappa–Szép product (or knit product), a generalization of the direct and semidirect products Ring theory * Product of rings * Ideal operations, for product of ideals Linear algebra * Scalar multiplication * Matrix multiplication * Inner product, on an inner product space * Exterior product or wedge product * Multiplication of vectors: ** Dot product ** Cross product ** Seven-dimensional cross product ** Triple product, in vector calculus * Tensor product Topology * Product topology Algebraic topology * Cap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Set-theoretic definition A rigorous definition of the Cartesian product re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
