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c_o1cifbyqottv
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Minuscule representation
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Summary
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Minuscule_representation
|
The minuscule representations are indexed by the weight lattice modulo the root lattice, or equivalently by irreducible representations of the center of the simply connected compact group. For the simple Lie algebras, the dimensions of the minuscule representations are given as follows.
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c_8rig8o5xai1c
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Minuscule representation
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Summary
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Minuscule_representation
|
An (n+1k) for 0 ≤ k ≤ n (exterior powers of vector representation). Quasi-minuscule: n2+2n (adjoint) Bn 1 (trivial), 2n (spin). Quasi-minuscule: 2n+1 (vector) Cn 1 (trivial), 2n (vector).
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c_qkohs14auxj9
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Minuscule representation
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Summary
|
Minuscule_representation
|
Quasi-minuscule: 2n2–n–1 if n>1 Dn 1 (trivial), 2n (vector), 2n−1 (half spin), 2n−1 (half spin). Quasi-minuscule: 2n2–n (adjoint) E6 1, 27, 27.
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c_6da7jisogzyb
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Minuscule representation
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Summary
|
Minuscule_representation
|
Quasi-minuscule: 78 (adjoint) E7 1, 56. Quasi-minuscule: 133 (adjoint) E8 1. Quasi-minuscule: 248 (adjoint) F4 1. Quasi-minuscule: 26 G2 1. Quasi-minuscule: 7
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c_p5e6vle0j4ib
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Coherent set of characters
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Summary
|
Coherent_set_of_characters
|
In mathematical representation theory, coherence is a property of sets of characters that allows one to extend an isometry from the degree-zero subspace of a space of characters to the whole space. The general notion of coherence was developed by Feit (1960, 1962), as a generalization of the proof by Frobenius of the existence of a Frobenius kernel of a Frobenius group and of the work of Brauer and Suzuki on exceptional characters. Feit & Thompson (1963, Chapter 3) developed coherence further in the proof of the Feit–Thompson theorem that all groups of odd order are solvable.
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c_qszhzf95w6bh
|
Eisenstein integral
|
Summary
|
Eisenstein_integral
|
In mathematical representation theory, the Eisenstein integral is an integral introduced by Harish-Chandra in the representation theory of semisimple Lie groups, analogous to Eisenstein series in the theory of automorphic forms. Harish-Chandra used Eisenstein integrals to decompose the regular representation of a semisimple Lie group into representations induced from parabolic subgroups. Trombi gave a survey of Harish-Chandra's work on this.
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c_zxfmp2v9it2o
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Hecke algebra of a pair
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Summary
|
Hecke_algebra_of_a_pair
|
In mathematical representation theory, the Hecke algebra of a pair (g,K) is an algebra with an approximate identity, whose approximately unital modules are the same as K-finite representations of the pairs (g,K). Here K is a compact subgroup of a Lie group with Lie algebra g.
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c_mu4qbt3xw633
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Naimark equivalence
|
Summary
|
Naimark_equivalence
|
In mathematical representation theory, two representations of a group on topological vector spaces are called Naimark equivalent (named after Mark Naimark) if there is a closed bijective linear map between dense subspaces preserving the group action.
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c_itzjmumz7gn3
|
Baumgartner's axiom
|
Summary
|
Baumgartner's_axiom
|
In mathematical set theory, Baumgartner's axiom (BA) can be one of three different axioms introduced by James Earl Baumgartner. A subset of the real line is said to be ℵ 1 {\displaystyle \aleph _{1}} -dense if every two points are separated by exactly ℵ 1 {\displaystyle \aleph _{1}} other points, where ℵ 1 {\displaystyle \aleph _{1}} is the smallest uncountable cardinality. This would be true for the real line itself under the continuum hypothesis. An axiom introduced by Baumgartner (1973) states that all ℵ 1 {\displaystyle \aleph _{1}} -dense subsets of the real line are order-isomorphic, providing a higher-cardinality analogue of Cantor's isomorphism theorem that countable dense subsets are isomorphic.
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c_8spi0yt8ccls
|
Baumgartner's axiom
|
Summary
|
Baumgartner's_axiom
|
Baumgartner's axiom is a consequence of the proper forcing axiom. It is consistent with a combination of ZFC, Martin's axiom, and the negation of the continuum hypothesis, but not implied by those hypotheses.Another axiom introduced by Baumgartner (1975) states that Martin's axiom for partially ordered sets MAP(κ) is true for all partially ordered sets P that are countable closed, well met and ℵ1-linked and all cardinals κ less than 2ℵ1. Baumgartner's axiom A is an axiom for partially ordered sets introduced in (Baumgartner 1983, section 7). A partial order (P, ≤) is said to satisfy axiom A if there is a family ≤n of partial orderings on P for n = 0, 1, 2, ... such that ≤0 is the same as ≤ If p ≤n+1q then p ≤nq If there is a sequence pn with pn+1 ≤n pn then there is a q with q ≤n pn for all n. If I is a pairwise incompatible subset of P then for all p and for all natural numbers n there is a q such that q ≤n p and the number of elements of I compatible with q is countable.
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c_wdzz36p85nxj
|
Cantor's theorem
|
Summary
|
Cantor's_theorem
|
In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A {\displaystyle A} , the set of all subsets of A , {\displaystyle A,} the power set of A , {\displaystyle A,} has a strictly greater cardinality than A {\displaystyle A} itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with n {\displaystyle n} elements has a total of 2 n {\displaystyle 2^{n}} subsets, and the theorem holds because 2 n > n {\displaystyle 2^{n}>n} for all non-negative integers. Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also.
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c_549qimbhet2l
|
Cantor's theorem
|
Summary
|
Cantor's_theorem
|
As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for German mathematician Georg Cantor, who first stated and proved it at the end of the 19th century.
|
c_9zybh0cjkhts
|
Cantor's theorem
|
Summary
|
Cantor's_theorem
|
Cantor's theorem had immediate and important consequences for the philosophy of mathematics. For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem, we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it. Consequently, the theorem implies that there is no largest cardinal number (colloquially, "there's no largest infinity").
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c_lfg1kbsmkoje
|
Chang's model
|
Summary
|
Chang's_model
|
In mathematical set theory, Chang's model is the smallest inner model of set theory closed under countable sequences. It was introduced by Chang (1971). More generally Chang introduced the smallest inner model closed under taking sequences of length less than κ for any infinite cardinal κ. For κ countable this is the constructible universe, and for κ the first uncountable cardinal it is Chang's model. Chang's model is a model of ZF. Kenneth Kunen proved in Kunen (1973) that the axiom of choice fails in Chang's model provided there are sufficient large cardinals, such as uncountable many measurable cardinals.
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c_48tnyt94x2kk
|
Cohen algebra
|
Summary
|
Cohen_algebra
|
In mathematical set theory, a Cohen algebra, named after Paul Cohen, is a type of Boolean algebra used in the theory of forcing. A Cohen algebra is a Boolean algebra whose completion is isomorphic to the completion of a free Boolean algebra (Koppelberg 1993).
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c_70ydg6bttpg7
|
Permutation model
|
Summary
|
Hereditarily_symmetric_set
|
In mathematical set theory, a permutation model is a model of set theory with atoms (ZFA) constructed using a group of permutations of the atoms. A symmetric model is similar except that it is a model of ZF (without atoms) and is constructed using a group of permutations of a forcing poset. One application is to show the independence of the axiom of choice from the other axioms of ZFA or ZF. Permutation models were introduced by Fraenkel (1922) and developed further by Mostowski (1938). Symmetric models were introduced by Paul Cohen.
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c_ygm2aatbs18l
|
Pseudo-intersection number
|
Summary
|
Pseudo-intersection_number
|
In mathematical set theory, a pseudo-intersection of a family of sets is an infinite set S such that each element of the family contains all but a finite number of elements of S. The pseudo-intersection number, sometimes denoted by the fraktur letter 𝔭, is the smallest size of a family of infinite subsets of the natural numbers that has the strong finite intersection property but has no pseudo-intersection.
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c_4khy4olj21sq
|
Hereditarily ordinal definable
|
Summary
|
Hereditarily_ordinal_definable
|
In mathematical set theory, a set S is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first-order formula. Ordinal definable sets were introduced by Gödel (1965). A drawback to this informal definition is that it requires quantification over all first-order formulas, which cannot be formalized in the language of set theory.
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c_aw5sfp57u3y6
|
Hereditarily ordinal definable
|
Summary
|
Hereditarily_ordinal_definable
|
However there is a different way of stating the definition that can be so formalized. In this approach, a set S is formally defined to be ordinal definable if there is some collection of ordinals α1, ..., αn such that S ∈ V α 1 {\displaystyle S\in V_{\alpha _{1}}} and S {\displaystyle S} can be defined as an element of V α 1 {\displaystyle V_{\alpha _{1}}} by a first-order formula φ taking α2, ..., αn as parameters. Here V α 1 {\displaystyle V_{{\alpha }_{1}}} denotes the set indexed by the ordinal α1 in the von Neumann hierarchy.
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c_5vckhk5eropf
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Hereditarily ordinal definable
|
Summary
|
Hereditarily_ordinal_definable
|
In other words, S is the unique object such that φ(S, α2...αn) holds with its quantifiers ranging over V α 1 {\displaystyle V_{\alpha _{1}}} . The class of all ordinal definable sets is denoted OD; it is not necessarily transitive, and need not be a model of ZFC because it might not satisfy the axiom of extensionality. A set is hereditarily ordinal definable if it is ordinal definable and all elements of its transitive closure are ordinal definable.
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c_u76wwpjgcdka
|
Hereditarily ordinal definable
|
Summary
|
Hereditarily_ordinal_definable
|
The class of hereditarily ordinal definable sets is denoted by HOD, and is a transitive model of ZFC, with a definable well ordering. It is consistent with the axioms of set theory that all sets are ordinal definable, and so hereditarily ordinal definable. The assertion that this situation holds is referred to as V = OD or V = HOD.
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c_kgrscywxxc1c
|
Hereditarily ordinal definable
|
Summary
|
Hereditarily_ordinal_definable
|
It follows from V = L, and is equivalent to the existence of a (definable) well-ordering of the universe. Note however that the formula expressing V = HOD need not hold true within HOD, as it is not absolute for models of set theory: within HOD, the interpretation of the formula for HOD may yield an even smaller inner model. HOD has been found to be useful in that it is an inner model that can accommodate essentially all known large cardinals. This is in contrast with the situation for core models, as core models have not yet been constructed that can accommodate supercompact cardinals, for example.
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c_f4rxipcd2c7w
|
Square principle
|
Summary
|
Global_square
|
In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of short closed unbounded (club) sets so that no one (long) club set coheres with them all. As such they may be viewed as a kind of incompactness phenomenon. They were introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L.
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c_9wdm7mp5gun3
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Countable transitive model
|
Summary
|
Countable_transitive_model
|
In mathematical set theory, a transitive model is a model of set theory that is standard and transitive. Standard means that the membership relation is the usual one, and transitive means that the model is a transitive set or class.
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c_lghwozjafo9e
|
Worldly cardinal
|
Summary
|
Worldly_cardinal
|
In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory.
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c_4l830fo4n2hm
|
Aronszajn line
|
Summary
|
Aronszajn_line
|
In mathematical set theory, an Aronszajn line (named after Nachman Aronszajn) is a linear ordering of cardinality ℵ 1 {\displaystyle \aleph _{1}} which contains no subset order-isomorphic to ω 1 {\displaystyle \omega _{1}} with the usual ordering the reverse of ω 1 {\displaystyle \omega _{1}} an uncountable subset of the Real numbers with the usual ordering.Unlike Suslin lines, the existence of Aronszajn lines is provable using the standard axioms of set theory. A linear ordering is an Aronszajn line if and only if it is the lexicographical ordering of some Aronszajn tree. == References ==
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c_on1swhhyx1v8
|
Ulam matrix
|
Summary
|
Ulam_matrix
|
In mathematical set theory, an Ulam matrix is an array of subsets of a cardinal number with certain properties. Ulam matrices were introduced by Stanislaw Ulam in his 1930 work on measurable cardinals: they may be used, for example, to show that a real-valued measurable cardinal is weakly inaccessible.
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c_a4n4t7onj1wy
|
Jónsson function
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Summary
|
Jónsson_function
|
In mathematical set theory, an ω-Jónsson function for a set x of ordinals is a function f: ω → x {\displaystyle f:^{\omega }\to x} with the property that, for any subset y of x with the same cardinality as x, the restriction of f {\displaystyle f} to ω {\displaystyle ^{\omega }} is surjective on x {\displaystyle x} . Here ω {\displaystyle ^{\omega }} denotes the set of strictly increasing sequences of members of x {\displaystyle x} , or equivalently the family of subsets of x {\displaystyle x} with order type ω {\displaystyle \omega } , using a standard notation for the family of subsets with a given order type. Jónsson functions are named for Bjarni Jónsson.
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c_dndi8l6vorsq
|
Jónsson function
|
Summary
|
Jónsson_function
|
Erdős and Hajnal (1966) showed that for every ordinal λ there is an ω-Jónsson function for λ. Kunen's proof of Kunen's inconsistency theorem uses a Jónsson function for cardinals λ such that 2λ = λℵ0, and Kunen observed that for this special case there is a simpler proof of the existence of Jónsson functions. Galvin and Prikry (1976) gave a simple proof for the general case. The existence of Jónsson functions shows that for any cardinal there is an algebra with an infinitary operation that has no proper subalgebras of the same cardinality. In particular if infinitary operations are allowed then an analogue of Jónsson algebras exists in any cardinality, so there are no infinitary analogues of Jónsson cardinals.
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c_x81bhvg2x84w
|
Cantor tree
|
Summary
|
Cantor_tree
|
In mathematical set theory, the Cantor tree is either the full binary tree of height ω + 1, or a topological space related to this by joining its points with intervals, that was introduced by Robert Lee Moore in the late 1920s as an example of a non-metrizable Moore space (Jones 1966).
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c_sohz8algwj1n
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Mitchell order
|
Summary
|
Mitchell_order
|
In mathematical set theory, the Mitchell order is a well-founded preorder on the set of normal measures on a measurable cardinal κ. It is named for William Mitchell. We say that M ◅ N (this is a strict order) if M is in the ultrapower model defined by N. Intuitively, this means that M is a weaker measure than N (note, for example, that κ will still be measurable in the ultrapower for N, since M is a measure on it). In fact, the Mitchell order can be defined on the set (or proper class, as the case may be) of extenders for κ; but if it is so defined it may fail to be transitive, or even well-founded, provided κ has sufficiently strong large cardinal properties.
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c_6ltfv6dh0qlj
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Mitchell order
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Summary
|
Mitchell_order
|
Well-foundedness fails specifically for rank-into-rank extenders; but Itay Neeman showed in 2004 that it holds for all weaker types of extender. The Mitchell rank of a measure is the order type of its predecessors under ◅; since ◅ is well-founded this is always an ordinal. Using the method of coherent sequences, for any rank ≤ κ + + {\displaystyle \leq \kappa ^{++}} Mitchell constructed an inner model for a measurable cardinal of rank κ {\displaystyle \kappa } .A cardinal that has measures of Mitchell rank α for each α < β is said to be β-measurable.
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c_45m4j9owzsm6
|
Mostowski model
|
Summary
|
Mostowski_model
|
In mathematical set theory, the Mostowski model is a model of set theory with atoms where the full axiom of choice fails, but every set can be linearly ordered. It was introduced by Mostowski (1939). The Mostowski model can be constructed as the permutation model corresponding to the group of all automorphisms of the ordered set of rational numbers and the ideal of finite subsets of the rational numbers.
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c_vwhe4ok6am2r
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Axiom of Adjunction
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Summary
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Axiom_of_Adjunction
|
In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x ∪ {y} given by "adjoining" the set y to the set x. It is stated as ∀ x . ∀ y . ∃ w . ∀ z .
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c_c8jqm2l0v2io
|
Axiom of Adjunction
|
Summary
|
Axiom_of_Adjunction
|
( z ∈ w ↔ ( z ∈ x ∨ z = y ) ) . {\displaystyle \forall x.\forall y.\exists w.\forall z. {\big (}z\in w\leftrightarrow (z\in x\lor z=y){\big )}.}
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c_0itafuh24h6z
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Axiom of Adjunction
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Summary
|
Axiom_of_Adjunction
|
Bernays (1937, page 68, axiom II (2)) introduced the axiom of adjunction as one of the axioms for a system of set theory that he introduced in about 1929. It is a weak axiom, used in some weak systems of set theory such as general set theory or finitary set theory. The adjunction operation is also used as one of the operations of primitive recursive set functions.
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c_t1skjs2ozqxy
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Multiverse (set theory)
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Summary
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Multiverse_(set_theory)
|
In mathematical set theory, the multiverse view is that there are many models of set theory, but no "absolute", "canonical" or "true" model. The various models are all equally valid or true, though some may be more useful or attractive than others. The opposite view is the "universe" view of set theory in which all sets are contained in some single ultimate model.
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c_ssuq2meq3jki
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Multiverse (set theory)
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Summary
|
Multiverse_(set_theory)
|
The collection of countable transitive models of ZFC (in some universe) is called the hyperverse and is very similar to the "multiverse". A typical difference between the universe and multiverse views is the attitude to the continuum hypothesis. In the universe view the continuum hypothesis is a meaningful question that is either true or false though we have not yet been able to decide which. In the multiverse view it is meaningless to ask whether the continuum hypothesis is true or false before selecting a model of set theory. Another difference is that the statement "For every transitive model of ZFC there is a larger model of ZFC in which it is countable" is true in some versions of the multiverse view of mathematics but is false in the universe view.
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c_2h4jmvkkw1qk
|
Illness narrative
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Mathematical-sociology approach
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Illness_narrative > Social-sciences approaches > Mathematical-sociology approach
|
In mathematical sociology, the theory of comparative narratives was devised in order to describe and compare the structures (expressed as "and" in a directed graph where multiple causal links incident into a node are conjoined) of action-driven sequential events.Narratives so conceived comprise the following ingredients: A finite set of state descriptions of the world S, the components of which are weakly ordered in time; A finite set of actors/agents (individual or collective), P; A finite set of actions A; A mapping of P onto A;The structure (directed graph) is generated by letting the nodes stand for the states and the directed edges represent how the states are changed by specified actions. The action skeleton can then be abstracted, comprising a further digraph where the actions are depicted as nodes and edges take the form "action a co-determined (in context of other actions) action b". Narratives can be both abstracted and generalised by imposing an algebra upon their structures and thence defining homomorphism between the algebras. The insertion of action-driven causal links in a narrative can be achieved using the method of Bayesian narratives.
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c_qh8eh1k0s1ms
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Haar's theorem
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Mathematical statistics
|
Haar's_theorem > Uses > Mathematical statistics
|
In mathematical statistics, Haar measures are used for prior measures, which are prior probabilities for compact groups of transformations. These prior measures are used to construct admissible procedures, by appeal to the characterization of admissible procedures as Bayesian procedures (or limits of Bayesian procedures) by Wald. For example, a right Haar measure for a family of distributions with a location parameter results in the Pitman estimator, which is best equivariant. When left and right Haar measures differ, the right measure is usually preferred as a prior distribution.
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c_yr3ux9w8hu0b
|
Haar's theorem
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Mathematical statistics
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Haar's_theorem > Uses > Mathematical statistics
|
For the group of affine transformations on the parameter space of the normal distribution, the right Haar measure is the Jeffreys prior measure. Unfortunately, even right Haar measures sometimes result in useless priors, which cannot be recommended for practical use, like other methods of constructing prior measures that avoid subjective information.Another use of Haar measure in statistics is in conditional inference, in which the sampling distribution of a statistic is conditioned on another statistic of the data. In invariant-theoretic conditional inference, the sampling distribution is conditioned on an invariant of the group of transformations (with respect to which the Haar measure is defined). The result of conditioning sometimes depends on the order in which invariants are used and on the choice of a maximal invariant, so that by itself a statistical principle of invariance fails to select any unique best conditional statistic (if any exist); at least another principle is needed. For non-compact groups, statisticians have extended Haar-measure results using amenable groups.
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c_lcifs70o20to
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Standardized variable
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Standardizing in mathematical statistics
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Z_score > Standardizing in mathematical statistics
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In mathematical statistics, a random variable X is standardized by subtracting its expected value E {\displaystyle \operatorname {E} } and dividing the difference by its standard deviation σ ( X ) = Var ( X ): {\displaystyle \sigma (X)={\sqrt {\operatorname {Var} (X)}}:} Z = X − E σ ( X ) {\displaystyle Z={X-\operatorname {E} \over \sigma (X)}} If the random variable under consideration is the sample mean of a random sample X 1 , … , X n {\displaystyle \ X_{1},\dots ,X_{n}} of X: X ¯ = 1 n ∑ i = 1 n X i {\displaystyle {\bar {X}}={1 \over n}\sum _{i=1}^{n}X_{i}} then the standardized version is Z = X ¯ − E σ ( X ) / n . {\displaystyle Z={\frac {{\bar {X}}-\operatorname {E} }{\sigma (X)/{\sqrt {n}}}}.}
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c_d3zwpomd5b9o
|
Asymptotic estimate
|
Asymptotic distribution
|
Asymptotic_theory > Asymptotic distribution
|
In mathematical statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables Zi for i = 1, …, n, for some positive integer n. An asymptotic distribution allows i to range without bound, that is, n is infinite. A special case of an asymptotic distribution is when the late entries go to zero—that is, the Zi go to 0 as i goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.
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c_mep6usphm561
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Asymptotic estimate
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Asymptotic distribution
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Asymptotic_theory > Asymptotic distribution
|
This is based on the notion of an asymptotic function which cleanly approaches a constant value (the asymptote) as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon. An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation y = 1 x , {\displaystyle y={\frac {1}{x}},} y becomes arbitrarily small in magnitude as x increases.
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c_afnrmk6l1ofi
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Growth curve (statistics)
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Other uses
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Growth_curve_(statistics) > Other uses
|
In mathematical statistics, growth curves such as those used in biology are often modeled as being continuous stochastic processes, e.g. as being sample paths that almost surely solve stochastic differential equations. Growth curves have been also applied in forecasting market development. When variables are measured with error, a Latent growth modeling SEM can be used.
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c_v66b4ypfpduf
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Darmois–Skitovich theorem
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Summary
|
Darmois–Skitovich_theorem
|
In mathematical statistics, the Darmois–Skitovich theorem characterizes the normal distribution (the Gaussian distribution) by the independence of two linear forms from independent random variables. This theorem was proved independently by G. Darmois and V. P. Skitovich in 1953.
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c_xr3n7vlyw4k7
|
Singular statistical model
|
Summary
|
Fisher_matrix
|
In mathematical statistics, the Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information. The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized by the statistician Ronald Fisher (following some initial results by Francis Ysidro Edgeworth). The Fisher information matrix is used to calculate the covariance matrices associated with maximum-likelihood estimates.
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c_gc1h3bjzrlq3
|
Singular statistical model
|
Summary
|
Fisher_matrix
|
It can also be used in the formulation of test statistics, such as the Wald test. In Bayesian statistics, the Fisher information plays a role in the derivation of non-informative prior distributions according to Jeffreys' rule. It also appears as the large-sample covariance of the posterior distribution, provided that the prior is sufficiently smooth (a result known as Bernstein–von Mises theorem, which was anticipated by Laplace for exponential families). The same result is used when approximating the posterior with Laplace's approximation, where the Fisher information appears as the covariance of the fitted Gaussian.Statistical systems of a scientific nature (physical, biological, etc.) whose likelihood functions obey shift invariance have been shown to obey maximum Fisher information. The level of the maximum depends upon the nature of the system constraints.
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c_9mmk942gu475
|
Kullback–Leibler divergence
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Summary
|
Information_gain
|
In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} , is a type of statistical distance: a measure of how one probability distribution P is different from a second, reference probability distribution Q. A simple interpretation of the KL divergence of P from Q is the expected excess surprise from using Q as a model when the actual distribution is P. While it is a distance, it is not a metric, the most familiar type of distance: it is not symmetric in the two distributions (in contrast to variation of information), and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a type of divergence, a generalization of squared distance, and for certain classes of distributions (notably an exponential family), it satisfies a generalized Pythagorean theorem (which applies to squared distances).In the simple case, a relative entropy of 0 indicates that the two distributions in question have identical quantities of information. Relative entropy is a nonnegative function of two distributions or measures. It has diverse applications, both theoretical, such as characterizing the relative (Shannon) entropy in information systems, randomness in continuous time-series, and information gain when comparing statistical models of inference; and practical, such as applied statistics, fluid mechanics, neuroscience and bioinformatics.
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c_sue8j1o6hlhu
|
Total curvature
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Summary
|
Total_curvature
|
In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length: ∫ a b k ( s ) d s = 2 π N . {\displaystyle \int _{a}^{b}k(s)\,ds=2\pi N.} The total curvature of a closed curve is always an integer multiple of 2π, where N is called the index of the curve or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces.
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c_5zcwcg5w6htx
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Multidimensional systems
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Summary
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Multidimensional_systems
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In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables. Important problems such as factorization and stability of m-D systems (m > 1) have recently attracted the interest of many researchers and practitioners. The reason is that the factorization and stability is not a straightforward extension of the factorization and stability of 1-D systems because, for example, the fundamental theorem of algebra does not exist in the ring of m-D (m > 1) polynomials.
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c_5v10r0gl3s67
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Financial result
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Calculation formula
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Financial_result > Calculation formula
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In mathematical terms financial result is defined as follows: Financial result = Interest income − Interest expense ± Write-downs/write-ups for financial assets ± Write-downs/write-ups for marketable securities + Other financial income and expenses {\displaystyle \textstyle {\begin{aligned}{\mbox{Financial result }}&={\mbox{ Interest income}}\\&-{\mbox{ Interest expense}}\\&\pm {\mbox{ Write-downs/write-ups for financial assets}}\\&\pm {\mbox{ Write-downs/write-ups for marketable securities}}\\&+{\mbox{ Other financial income and expenses}}\end{aligned}}}
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c_l6ks74zomk1c
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Eddy covariance
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Mathematical foundation
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Eddy_covariance > Mathematical foundation
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In mathematical terms, "eddy flux" is computed as a covariance between instantaneous deviation in vertical wind speed ( w ′ {\displaystyle w'} ) from the mean value ( w ¯ {\displaystyle {\bar {w}}} ) and instantaneous deviation in gas concentration, mixing ratio ( s ′ {\displaystyle s'} ), from its mean value ( s ¯ {\displaystyle {\bar {s}}} ), multiplied by mean air density ( ρ a {\displaystyle \rho _{a}} ). Several mathematical operations and assumptions, including Reynolds decomposition, are involved in getting from physically complete equations of the turbulent flow to practical equations for computing "eddy flux," as shown below.
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c_jro1jgzgyms2
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Alfvén's Theorem
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Mathematical statement
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Frozen-in_flux_theorem > Mathematical statement
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In mathematical terms, Alfvén's theorem states that, in an electrically conducting fluid in the limit of a large magnetic Reynolds number, the magnetic flux ΦB through an orientable, open material surface advected by a macroscopic, space- and time-dependent velocity field v is constant, or D Φ B D t = 0 , {\displaystyle {\frac {D\Phi _{B}}{Dt}}=0,} where D/Dt = ∂/∂t + (v ⋅ ∇) is the advective derivative.
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c_4y2kzk9hg8wc
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Multi-objective linear programming
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Problem formulation
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Multi-objective_linear_programming > Problem formulation
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In mathematical terms, a MOLP can be written as: min x P x s.t. a ≤ B x ≤ b , ℓ ≤ x ≤ u {\displaystyle \min _{x}Px\quad {\text{s.t. }}\quad a\leq Bx\leq b,\;\ell \leq x\leq u} where B {\displaystyle B} is an ( m × n ) {\displaystyle (m\times n)} matrix, P {\displaystyle P} is a ( q × n ) {\displaystyle (q\times n)} matrix, a {\displaystyle a} is an m {\displaystyle m} -dimensional vector with components in R ∪ { − ∞ } {\displaystyle \mathbb {R} \cup \{-\infty \}} , b {\displaystyle b} is an m {\displaystyle m} -dimensional vector with components in R ∪ { + ∞ } {\displaystyle \mathbb {R} \cup \{+\infty \}} , ℓ {\displaystyle \ell } is an n {\displaystyle n} -dimensional vector with components in R ∪ { − ∞ } {\displaystyle \mathbb {R} \cup \{-\infty \}} , u {\displaystyle u} is an n {\displaystyle n} -dimensional vector with components in R ∪ { + ∞ } {\displaystyle \mathbb {R} \cup \{+\infty \}}
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c_levqo60mfw62
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Statistical modeling
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Formal definition
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Statistical_Model > Formal definition
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In mathematical terms, a statistical model is usually thought of as a pair ( S , P {\displaystyle S,{\mathcal {P}}} ), where S {\displaystyle S} is the set of possible observations, i.e. the sample space, and P {\displaystyle {\mathcal {P}}} is a set of probability distributions on S {\displaystyle S} .The intuition behind this definition is as follows. It is assumed that there is a "true" probability distribution induced by the process that generates the observed data. We choose P {\displaystyle {\mathcal {P}}} to represent a set (of distributions) which contains a distribution that adequately approximates the true distribution. Note that we do not require that P {\displaystyle {\mathcal {P}}} contains the true distribution, and in practice that is rarely the case.
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c_jax37zyl8q6i
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Statistical modeling
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Formal definition
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Statistical_Model > Formal definition
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Indeed, as Burnham & Anderson state, "A model is a simplification or approximation of reality and hence will not reflect all of reality"—hence the saying "all models are wrong". The set P {\displaystyle {\mathcal {P}}} is almost always parameterized: P = { F θ: θ ∈ Θ } {\displaystyle {\mathcal {P}}=\{F_{\theta }:\theta \in \Theta \}} .
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c_zy5rpb0763ul
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Statistical modeling
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Formal definition
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Statistical_Model > Formal definition
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The set of distributions Θ {\displaystyle \Theta } defines the parameters of the model. A parameterization is generally required to have distinct parameter values give rise to distinct distributions, i.e. F θ 1 = F θ 2 ⇒ θ 1 = θ 2 {\displaystyle F_{\theta _{1}}=F_{\theta _{2}}\Rightarrow \theta _{1}=\theta _{2}} must hold (in other words, it must be injective). A parameterization that meets the requirement is said to be identifiable.
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c_jptzdqz41r69
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Vector optimization
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Problem formulation
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Vector_optimization > Problem formulation
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In mathematical terms, a vector optimization problem can be written as: C - min x ∈ S f ( x ) {\displaystyle C\operatorname {-} \min _{x\in S}f(x)} where f: X → Z {\displaystyle f:X\to Z} for a partially ordered vector space Z {\displaystyle Z} . The partial ordering is induced by a cone C ⊆ Z {\displaystyle C\subseteq Z} . X {\displaystyle X} is an arbitrary set and S ⊆ X {\displaystyle S\subseteq X} is called the feasible set.
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c_3hy6s4atvgau
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Point transformation
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Modern mathematical description
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Canonical_transformation > Modern mathematical description
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In mathematical terms, canonical coordinates are any coordinates on the phase space (cotangent bundle) of the system that allow the canonical one-form to be written as up to a total differential (exact form). The change of variable between one set of canonical coordinates and another is a canonical transformation. The index of the generalized coordinates q is written here as a superscript ( q i {\displaystyle q^{i}} ), not as a subscript as done above ( q i {\displaystyle q_{i}} ). The superscript conveys the contravariant transformation properties of the generalized coordinates, and does not mean that the coordinate is being raised to a power. Further details may be found at the symplectomorphism article.
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c_1q4er33fyy8f
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Inverse demand function
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Definition
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Inverse_demand_function > Definition
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In mathematical terms, if the demand function is d e m a n d = f ( p r i c e ) {\displaystyle {demand}=f({price})} , then the inverse demand function is p r i c e = f − 1 ( d e m a n d ) {\displaystyle {price}=f^{-1}({demand})} . The value of the inverse demand function is the highest price that could be charged and still generate the quantity demanded. This is useful because economists typically place price (P) on the vertical axis and quantity (demand, Q) on the horizontal axis in supply-and-demand diagrams, so it is the inverse demand function that depicts the graphed demand curve in the way the reader expects to see.
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c_0o5g2037c8uz
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Inverse demand function
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Definition
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Inverse_demand_function > Definition
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The inverse demand function is the same as the average revenue function, since P = AR.To compute the inverse demand function, simply solve for P from the demand function. For example, if the demand function has the form Q = 240 − 2 P {\displaystyle Q=240-2P} then the inverse demand function would be P = 120 − .5 Q {\displaystyle P=120-.5Q} . Note that although price is the dependent variable in the inverse demand function, it is still the case that the equation represents how the price determines the quantity demanded, not the reverse.
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c_8ihnn6jnly5y
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Quantum discord
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Definition and mathematical relations
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Quantum_discord > Definition and mathematical relations
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In mathematical terms, quantum discord is defined in terms of the quantum mutual information. More specifically, quantum discord is the difference between two expressions which each, in the classical limit, represent the mutual information. These two expressions are: I ( A ; B ) = H ( A ) + H ( B ) − H ( A , B ) {\displaystyle I(A;B)=H(A)+H(B)-H(A,B)} J ( A ; B ) = H ( A ) − H ( A | B ) {\displaystyle J(A;B)=H(A)-H(A|B)} where, in the classical case, H(A) is the information entropy, H(A, B) the joint entropy and H(A|B) the conditional entropy, and the two expressions yield identical results. In the nonclassical case, the quantum physics analogy for the three terms are used – S(ρA) the von Neumann entropy, S(ρ) the joint quantum entropy and S(ρA|ρB) a quantum generalization of conditional entropy (not to be confused with conditional quantum entropy), respectively, for probability density function ρ; I ( ρ ) = S ( ρ A ) + S ( ρ B ) − S ( ρ ) {\displaystyle I(\rho )=S(\rho _{A})+S(\rho _{B})-S(\rho )} J A ( ρ ) = S ( ρ B ) − S ( ρ B | ρ A ) {\displaystyle J_{A}(\rho )=S(\rho _{B})-S(\rho _{B}|\rho _{A})} The difference between the two expressions defines the basis-dependent quantum discord D A ( ρ ) = I ( ρ ) − J A ( ρ ) , {\displaystyle {\mathcal {D}}_{A}(\rho )=I(\rho )-J_{A}(\rho ),} which is asymmetrical in the sense that D A ( ρ ) {\displaystyle {\mathcal {D}}_{A}(\rho )} can differ from D B ( ρ ) {\displaystyle {\mathcal {D}}_{B}(\rho )} .
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c_7ebl14zyyvth
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Quantum discord
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Definition and mathematical relations
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Quantum_discord > Definition and mathematical relations
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The notation J represents the part of the correlations that can be attributed to classical correlations and varies in dependence on the chosen eigenbasis; therefore, in order for the quantum discord to reflect the purely nonclassical correlations independently of basis, it is necessary that J first be maximized over the set of all possible projective measurements onto the eigenbasis: D A ( ρ ) = I ( ρ ) − max { Π j A } J { Π j A } ( ρ ) = S ( ρ A ) − S ( ρ ) + min { Π j A } S ( ρ B | { Π j A } ) {\displaystyle {\mathcal {D}}_{A}(\rho )=I(\rho )-\max _{\{\Pi _{j}^{A}\}}J_{\{\Pi _{j}^{A}\}}(\rho )=S(\rho _{A})-S(\rho )+\min _{\{\Pi _{j}^{A}\}}S(\rho _{B|\{\Pi _{j}^{A}\}})} Nonzero quantum discord indicates the presence of correlations that are due to noncommutativity of quantum operators. For pure states, the quantum discord becomes a measure of quantum entanglement, more specifically, in that case it equals the entropy of entanglement.Vanishing quantum discord is a criterion for the pointer states, which constitute preferred effectively classical states of a system. It could be shown that quantum discord must be non-negative and that states with vanishing quantum discord can in fact be identified with pointer states.
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c_8jaa09huydrx
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Quantum discord
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Definition and mathematical relations
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Quantum_discord > Definition and mathematical relations
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Other conditions have been identified which can be seen in analogy to the Peres–Horodecki criterion and in relation to the strong subadditivity of the von Neumann entropy.Efforts have been made to extend the definition of quantum discord to continuous variable systems, in particular to bipartite systems described by Gaussian states. A very recent work has demonstrated that the upper-bound of Gaussian discord indeed coincides with the actual quantum discord of a Gaussian state, when the latter belongs to a suitable large family of Gaussian states. Computing quantum discord is NP-complete and hence difficult to compute in the general case. For certain classes of two-qubit states, quantum discord can be calculated analytically.
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c_or2gljygcxph
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Jahn–Teller distortion
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Mexican-hat potential
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Jahn–Teller_effect > Theory > Potential energy surfaces > Mexican-hat potential
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In mathematical terms, the APESs characterising the JT distortion arise as the eigenvalues of the potential energy matrix. Generally, the APESs take the characteristic appearance of a double cone, circular or elliptic, where the point of contact, i.e. degeneracy, denotes the high-symmetry configuration for which the JT theorem applies. For the above case of the linear E ⊗ e JT effect the situation is illustrated by the APES V = μ ω 2 2 ( Q θ 2 + Q ϵ 2 ) ± k Q θ 2 + Q ϵ 2 {\displaystyle V={\frac {\mu \omega ^{2}}{2}}(Q_{\theta }^{2}+Q_{\epsilon }^{2})\pm k{\sqrt {Q_{\theta }^{2}+Q_{\epsilon }^{2}}}} displayed in the figure, with part cut away to reveal its shape, which is known as a Mexican Hat potential. Here, ω {\displaystyle \omega } is the frequency of the vibrational e mode, μ {\displaystyle \mu } is its mass and k {\displaystyle k} is a measure of the strength of the JT coupling.
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c_l8fccovufaf6
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Jahn–Teller distortion
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Mexican-hat potential
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Jahn–Teller_effect > Theory > Potential energy surfaces > Mexican-hat potential
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The conical shape near the degeneracy at the origin makes it immediately clear that this point cannot be stationary, that is, the system is unstable against asymmetric distortions, which leads to a symmetry lowering. In this particular case there are infinitely many isoenergetic JT distortions.
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c_f3y8erc8co8s
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Jahn–Teller distortion
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Mexican-hat potential
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Jahn–Teller_effect > Theory > Potential energy surfaces > Mexican-hat potential
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The Q i {\displaystyle Q_{i}} giving these distortions are arranged in a circle, as shown by the red curve in the figure. Quadratic coupling or cubic elastic terms lead to a warping along this "minimum energy path", replacing this infinite manifold by three equivalent potential minima and three equivalent saddle points. In other JT systems, linear coupling results in discrete minima.
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c_llx04yadacmx
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Binary Golay code
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Mathematical definition
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Extended_binary_Golay_code > Mathematical definition
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In mathematical terms, the extended binary Golay code G24 consists of a 12-dimensional linear subspace W of the space V = F242 of 24-bit words such that any two distinct elements of W differ in at least 8 coordinates. W is called a linear code because it is a vector space. In all, W comprises 4096 = 212 elements. The elements of W are called code words.
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c_de59dmiuafrf
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Binary Golay code
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Mathematical definition
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Extended_binary_Golay_code > Mathematical definition
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They can also be described as subsets of a set of 24 elements, where addition is defined as taking the symmetric difference of the subsets. In the extended binary Golay code, all code words have Hamming weights of 0, 8, 12, 16, or 24. Code words of weight 8 are called octads and code words of weight 12 are called dodecads.
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c_3k633gu7ns1n
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Binary Golay code
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Mathematical definition
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Extended_binary_Golay_code > Mathematical definition
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Octads of the code G24 are elements of the S(5,8,24) Steiner system. There are 759 = 3 × 11 × 23 octads and 759 complements thereof. It follows that there are 2576 = 24 × 7 × 23 dodecads.
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c_21ne8u7gbbu2
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Binary Golay code
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Mathematical definition
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Extended_binary_Golay_code > Mathematical definition
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Two octads intersect (have 1's in common) in 0, 2, or 4 coordinates in the binary vector representation (these are the possible intersection sizes in the subset representation). An octad and a dodecad intersect at 2, 4, or 6 coordinates. Up to relabeling coordinates, W is unique.The binary Golay code, G23 is a perfect code.
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c_iolkpig9ufx3
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Binary Golay code
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Mathematical definition
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Extended_binary_Golay_code > Mathematical definition
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That is, the spheres of radius three around code words form a partition of the vector space. G23 is a 12-dimensional subspace of the space F232. The automorphism group of the perfect binary Golay code G23 (meaning the subgroup of the group S23 of permutations of the coordinates of F232 which leave G23 invariant), is the Mathieu group M 23 {\displaystyle M_{23}} .
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c_1thllxrhxn59
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Binary Golay code
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Mathematical definition
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Extended_binary_Golay_code > Mathematical definition
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The automorphism group of the extended binary Golay code is the Mathieu group M 24 {\displaystyle M_{24}} , of order 210 × 33 × 5 × 7 × 11 × 23. M 24 {\displaystyle M_{24}} is transitive on octads and on dodecads. The other Mathieu groups occur as stabilizers of one or several elements of W.
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c_zt96t6ykdrl7
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Euler top
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Mathematical description of phase space
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Lagrange,_Euler,_and_Kovalevskaya_tops > Hamiltonian formulation of classical tops > Mathematical description of phase space
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In mathematical terms, the spatial configuration of the body is described by a point on the Lie group S O ( 3 ) {\displaystyle SO(3)} , the three-dimensional rotation group, which is the rotation matrix from the lab frame to the body frame. The full configuration space or phase space is the cotangent bundle T ∗ S O ( 3 ) {\displaystyle T^{*}SO(3)} , with the fibers T R ∗ S O ( 3 ) {\displaystyle T_{R}^{*}SO(3)} parametrizing the angular momentum at spatial configuration R {\displaystyle R} . The Hamiltonian is a function on this phase space.
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c_axslppjvdhb7
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Strict
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Summary
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Strict
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In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusive meaning of the term is to be understood. The opposite is non-strict, which is often understood to be the case but can be put explicitly for clarity. In some contexts, the word "proper" can also be used as a mathematical synonym for "strict".
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c_pxf6qwxkg1cs
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Hylozoism
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Literature
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Hylozoism > In popular culture > Literature
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In mathematician and writer Rudy Rucker's novels Postsingular and Hylozoic, the emergent sentience of all material things is described as a property of the technological singularity. The Hylozoist is one of the Culture ships mentioned in Iain M. Banks's novel Surface Detail – appropriately, this ship is a member of the branch of Contact dealing with smart matter outbreaks.
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c_0fxivdv3lotr
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Semicircle
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Summary
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Semicircle
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In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, π radians, or a half-turn). It has only one line of symmetry (reflection symmetry). In non-technical usage, the term "semicircle" is sometimes used to refer to either a closed curve that also includes the diameter segment from one end of the arc to the other or to the half-disk, which is a two-dimensional geometric region that further includes all the interior points. By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a right triangle, with a right angle at the third vertex. All lines intersecting the semicircle perpendicularly are concurrent at the center of the circle containing the given semicircle.
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c_1vbf5izcbxyv
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Major index
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Summary
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Major_index
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In mathematics (and particularly in combinatorics), the major index of a permutation is the sum of the positions of the descents of the permutation. In symbols, the major index of the permutation w is maj ( w ) = ∑ w ( i ) > w ( i + 1 ) i . {\displaystyle \operatorname {maj} (w)=\sum _{w(i)>w(i+1)}i.} For example, if w is given in one-line notation by w = 351624 (that is, w is the permutation of {1, 2, 3, 4, 5, 6} such that w(1) = 3, w(2) = 5, etc.) then w has descents at positions 2 (from 5 to 1) and 4 (from 6 to 2) and so maj(w) = 2 + 4 = 6.
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c_wsyp5zhd92qf
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Major index
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Summary
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Major_index
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This statistic is named after Major Percy Alexander MacMahon who showed in 1913 that the distribution of the major index on all permutations of a fixed length is the same as the distribution of inversions. That is, the number of permutations of length n with k inversions is the same as the number of permutations of length n with major index equal to k. (These numbers are known as Mahonian numbers, also in honor of MacMahon.)
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c_6x1t9946nari
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Major index
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Summary
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Major_index
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In fact, a stronger result is true: the number of permutations of length n with major index k and i inversions is the same as the number of permutations of length n with major index i and k inversions, that is, the two statistics are equidistributed. For example, the number of permutations of length 4 with given major index and number of inversions is given in the table below. 0 1 2 3 4 5 6 0 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 2 0 1 2 1 1 0 0 3 0 1 1 2 1 1 0 4 0 0 1 1 2 1 0 5 0 0 0 1 1 1 0 6 0 0 0 0 0 0 1 {\displaystyle {\begin{array}{c|ccccccc}&0&1&2&3&4&5&6\\\hline 0&1&0&0&0&0&0&0\\1&0&1&1&1&0&0&0\\2&0&1&2&1&1&0&0\\3&0&1&1&2&1&1&0\\4&0&0&1&1&2&1&0\\5&0&0&0&1&1&1&0\\6&0&0&0&0&0&0&1\end{array}}}
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c_xxa3n6ajeri9
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Kronecker foliation
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Summary
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Foliation_theory
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In mathematics (differential geometry), a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class Cr), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1 (otherwise, C0 is a topological foliation).
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c_hdqurpotams7
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Kronecker foliation
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Summary
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Foliation_theory
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The number p (the dimension of the leaves) is called the dimension of the foliation and q = n − p is called its codimension. In some papers on general relativity by mathematical physicists, the term foliation (or slicing) is used to describe a situation where the relevant Lorentz manifold (a (p+1)-dimensional spacetime) has been decomposed into hypersurfaces of dimension p, specified as the level sets of a real-valued smooth function (scalar field) whose gradient is everywhere non-zero; this smooth function is moreover usually assumed to be a time function, meaning that its gradient is everywhere time-like, so that its level-sets are all space-like hypersurfaces. In deference to standard mathematical terminology, these hypersurface are often called the leaves (or sometimes slices) of the foliation. Note that while this situation does constitute a codimension-1 foliation in the standard mathematical sense, examples of this type are actually globally trivial; while the leaves of a (mathematical) codimension-1 foliation are always locally the level sets of a function, they generally cannot be expressed this way globally, as a leaf may pass through a local-trivializing chart infinitely many times, and the holonomy around a leaf may also obstruct the existence of a globally-consistent defining functions for the leaves. For example, while the 3-sphere has a famous codimension-1 foliation discovered by Reeb, a codimension-1 foliation of a closed manifold cannot be given by the level sets of a smooth function, since a smooth function on a closed manifold necessarily has critical points at its maxima and minima.
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c_gvkf9o79jgxt
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Colored operad
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Summary
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Colored_operad
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In mathematics (especially category theory), a multicategory is a generalization of the concept of category that allows morphisms of multiple arity. If morphisms in a category are viewed as analogous to functions, then morphisms in a multicategory are analogous to functions of several variables. Multicategories are also sometimes called operads, or colored operads.
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c_24temcrj4ly9
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Convolution operation
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Summary
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Convolution_kernel
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In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function ( f ∗ g {\displaystyle f*g} ) that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity).
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c_keotmbg726zq
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Convolution operation
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Summary
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Convolution_kernel
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The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution ( f ∗ g {\displaystyle f*g} ) differs from cross-correlation ( f ⋆ g {\displaystyle f\star g} ) only in that either f(x) or g(x) is reflected about the y-axis in convolution; thus it is a cross-correlation of g(−x) and f(x), or f(−x) and g(x). For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator.
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c_nae4hgw6ghrk
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Convolution operation
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Summary
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Convolution_kernel
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Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, geophysics, engineering, physics, computer vision and differential equations.The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.Computing the inverse of the convolution operation is known as deconvolution.
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c_uc3gy6gndz7j
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Linear recurrence with constant coefficients
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Summary
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Linear_recurrence
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In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients: ch. 17: ch. 10 (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence. The polynomial's linearity means that each of its terms has degree 0 or 1.
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c_3yxm4czgvomr
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Linear recurrence with constant coefficients
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Summary
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Linear_recurrence
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A linear recurrence denotes the evolution of some variable over time, with the current time period or discrete moment in time denoted as t, one period earlier denoted as t − 1, one period later as t + 1, etc. The solution of such an equation is a function of t, and not of any iterate values, giving the value of the iterate at any time. To find the solution it is necessary to know the specific values (known as initial conditions) of n of the iterates, and normally these are the n iterates that are oldest. The equation or its variable is said to be stable if from any set of initial conditions the variable's limit as time goes to infinity exists; this limit is called the steady state. Difference equations are used in a variety of contexts, such as in economics to model the evolution through time of variables such as gross domestic product, the inflation rate, the exchange rate, etc. They are used in modeling such time series because values of these variables are only measured at discrete intervals. In econometric applications, linear difference equations are modeled with stochastic terms in the form of autoregressive (AR) models and in models such as vector autoregression (VAR) and autoregressive moving average (ARMA) models that combine AR with other features.
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c_gmtsf0k84ivq
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Faddeev–LeVerrier algorithm
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Summary
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Faddeev–LeVerrier_algorithm
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In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial p A ( λ ) = det ( λ I n − A ) {\displaystyle p_{A}(\lambda )=\det(\lambda I_{n}-A)} of a square matrix, A, named after Dmitry Konstantinovich Faddeev and Urbain Le Verrier. Calculation of this polynomial yields the eigenvalues of A as its roots; as a matrix polynomial in the matrix A itself, it vanishes by the Cayley–Hamilton theorem. Computing the characteristic polynomial directly from the definition of the determinant is computationally cumbersome insofar as it introduces a new symbolic quantity λ {\displaystyle \lambda } ; by contrast, the Faddeev-Le Verrier algorithm works directly with coefficients of matrix A {\displaystyle A} . The algorithm has been independently rediscovered several times in different forms.
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c_oz57781eiswh
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Faddeev–LeVerrier algorithm
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Summary
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Faddeev–LeVerrier_algorithm
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It was first published in 1840 by Urbain Le Verrier, subsequently redeveloped by P. Horst, Jean-Marie Souriau, in its present form here by Faddeev and Sominsky, and further by J. S. Frame, and others. (For historical points, see Householder.
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c_z0hz0n8xtohg
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Faddeev–LeVerrier algorithm
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Summary
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Faddeev–LeVerrier_algorithm
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An elegant shortcut to the proof, bypassing Newton polynomials, was introduced by Hou. The bulk of the presentation here follows Gantmacher, p. 88.)
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c_n52s7gw72bhj
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Group cohomology
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Summary
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Group_homology
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In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of G n {\displaystyle G^{n}} representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups H n ( G , M ) {\displaystyle H^{n}(G,M)} . The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves.
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c_54gdc58bt75x
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Group cohomology
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Summary
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Group_homology
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Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology.
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c_xxterx8wy0mb
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Group cohomology
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Summary
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Group_homology
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The techniques of group cohomology can also be extended to the case that instead of a G-module, G acts on a nonabelian G-group; in effect, a generalization of a module to non-Abelian coefficients. These algebraic ideas are closely related to topological ideas. The group cohomology of a discrete group G is the singular cohomology of a suitable space having G as its fundamental group, namely the corresponding Eilenberg–MacLane space.
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c_iisxw04esoud
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Group cohomology
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Summary
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Group_homology
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Thus, the group cohomology of Z {\displaystyle \mathbb {Z} } can be thought of as the singular cohomology of the circle S1, and similarly for Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } and P ∞ ( R ) . {\displaystyle \mathbb {P} ^{\infty }(\mathbb {R} ).} A great deal is known about the cohomology of groups, including interpretations of low-dimensional cohomology, functoriality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.
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c_2z5oriry1j2r
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Argument of a complex number
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Summary
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Argument_of_a_complex_number
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In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as φ {\displaystyle \varphi } in Figure 1. It is a multivalued function operating on the nonzero complex numbers. To define a single-valued function, the principal value of the argument (sometimes denoted Arg z) is used. It is often chosen to be the unique value of the argument that lies within the interval (−π, π].
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c_uydoznf12w6l
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Shear matrix
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Summary
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Shear_matrix
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In mathematics (particularly linear algebra), a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another. Such a matrix may be derived by taking the identity matrix and replacing one of the zero elements with a non-zero value. The name shear reflects the fact that the matrix represents a shear transformation. Geometrically, such a transformation takes pairs of points in a vector space that are purely axially separated along the axis whose row in the matrix contains the shear element, and effectively replaces those pairs by pairs whose separation is no longer purely axial but has two vector components. Thus, the shear axis is always an eigenvector of S.
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c_xqyd69m79pbk
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Volume integral
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Summary
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Volume_integral
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In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities, or to calculate mass from a corresponding density function.
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c_wyy3tur5lb4y
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Radon metric
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Summary
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Outer_regular_measure
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In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.
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