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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. | https://en.wikipedia.org/wiki/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). | https://en.wikipedia.org/wiki/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. | https://en.wikipedia.org/wiki/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 | https://en.wikipedia.org/wiki/Minuscule_representation |
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 e... | https://en.wikipedia.org/wiki/Coherent_set_of_characters |
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 semis... | https://en.wikipedia.org/wiki/Eisenstein_integral |
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. | https://en.wikipedia.org/wiki/Hecke_algebra_of_a_pair |
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. | https://en.wikipedia.org/wiki/Naimark_equivalence |
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 {\displaysty... | https://en.wikipedia.org/wiki/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 t... | https://en.wikipedia.org/wiki/Baumgartner's_axiom |
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 ... | https://en.wikipedia.org/wiki/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 ... | https://en.wikipedia.org/wiki/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 im... | https://en.wikipedia.org/wiki/Cantor's_theorem |
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 c... | https://en.wikipedia.org/wiki/Chang's_model |
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). | https://en.wikipedia.org/wiki/Cohen_algebra |
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 t... | https://en.wikipedia.org/wiki/Permutation_model |
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 natur... | https://en.wikipedia.org/wiki/Pseudo-intersection_number |
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 f... | https://en.wikipedia.org/wiki/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... | https://en.wikipedia.org/wiki/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.... | https://en.wikipedia.org/wiki/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 = O... | https://en.wikipedia.org/wiki/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 ... | https://en.wikipedia.org/wiki/Hereditarily_ordinal_definable |
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 i... | https://en.wikipedia.org/wiki/Square_principle |
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. | https://en.wikipedia.org/wiki/Countable_transitive_model |
In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory. | https://en.wikipedia.org/wiki/Worldly_cardinal |
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 t... | https://en.wikipedia.org/wiki/Aronszajn_line |
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. | https://en.wikipedia.org/wiki/Ulam_matrix |
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} .... | https://en.wikipedia.org/wiki/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 an... | https://en.wikipedia.org/wiki/Jónsson_function |
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). | https://en.wikipedia.org/wiki/Cantor_tree |
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, ... | https://en.wikipedia.org/wiki/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 an... | https://en.wikipedia.org/wiki/Mitchell_order |
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 o... | https://en.wikipedia.org/wiki/Mostowski_model |
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 . | https://en.wikipedia.org/wiki/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 )}.} | https://en.wikipedia.org/wiki/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 op... | https://en.wikipedia.org/wiki/Axiom_of_Adjunction |
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 set... | https://en.wikipedia.org/wiki/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 tha... | https://en.wikipedia.org/wiki/Multiverse_(set_theory) |
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 ingredient... | https://en.wikipedia.org/wiki/Illness_narrative |
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 procedure... | https://en.wikipedia.org/wiki/Haar's_theorem |
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 th... | https://en.wikipedia.org/wiki/Haar's_theorem |
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-\o... | https://en.wikipedia.org/wiki/Standardized_variable |
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 ... | https://en.wikipedia.org/wiki/Asymptotic_estimate |
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 t... | https://en.wikipedia.org/wiki/Asymptotic_estimate |
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 w... | https://en.wikipedia.org/wiki/Growth_curve_(statistics) |
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. | https://en.wikipedia.org/wiki/Darmois–Skitovich_theorem |
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 obse... | https://en.wikipedia.org/wiki/Singular_statistical_model |
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 ... | https://en.wikipedia.org/wiki/Singular_statistical_model |
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 distribut... | https://en.wikipedia.org/wiki/Kullback–Leibler_divergence |
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 m... | https://en.wikipedia.org/wiki/Total_curvature |
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 resea... | https://en.wikipedia.org/wiki/Multidimensional_systems |
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 }}&={... | https://en.wikipedia.org/wiki/Financial_result |
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 ¯ {\d... | https://en.wikipedia.org/wiki/Eddy_covariance |
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 , {\displaystyl... | https://en.wikipedia.org/wiki/Alfvén's_Theorem |
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)} mat... | https://en.wikipedia.org/wiki/Multi-objective_linear_programming |
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 be... | https://en.wikipedia.org/wiki/Statistical_modeling |
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... | https://en.wikipedia.org/wiki/Statistical_modeling |
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 h... | https://en.wikipedia.org/wiki/Statistical_modeling |
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\subset... | https://en.wikipedia.org/wiki/Vector_optimization |
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 ... | https://en.wikipedia.org/wiki/Point_transformation |
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 st... | https://en.wikipedia.org/wiki/Inverse_demand_function |
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 {\di... | https://en.wikipedia.org/wiki/Inverse_demand_function |
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 ) {\disp... | https://en.wikipedia.org/wiki/Quantum_discord |
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 th... | https://en.wikipedia.org/wiki/Quantum_discord |
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 Ga... | https://en.wikipedia.org/wiki/Quantum_discord |
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... | https://en.wikipedia.org/wiki/Jahn–Teller_distortion |
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. | https://en.wikipedia.org/wiki/Jahn–Teller_distortion |
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... | https://en.wikipedia.org/wiki/Jahn–Teller_distortion |
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 e... | https://en.wikipedia.org/wiki/Binary_Golay_code |
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 dodec... | https://en.wikipedia.org/wiki/Binary_Golay_code |
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. | https://en.wikipedia.org/wiki/Binary_Golay_code |
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 ... | https://en.wikipedia.org/wiki/Binary_Golay_code |
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 th... | https://en.wikipedia.org/wiki/Binary_Golay_code |
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. | https://en.wikipedia.org/wiki/Binary_Golay_code |
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... | https://en.wikipedia.org/wiki/Euler_top |
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 ... | https://en.wikipedia.org/wiki/Strict |
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... | https://en.wikipedia.org/wiki/Hylozoism |
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 some... | https://en.wikipedia.org/wiki/Semicircle |
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... | https://en.wikipedia.org/wiki/Major_index |
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 len... | https://en.wikipedia.org/wiki/Major_index |
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 ... | https://en.wikipedia.org/wiki/Major_index |
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 subspac... | https://en.wikipedia.org/wiki/Kronecker_foliation |
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... | https://en.wikipedia.org/wiki/Kronecker_foliation |
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 als... | https://en.wikipedia.org/wiki/Colored_operad |
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... | https://en.wikipedia.org/wiki/Convolution_operation |
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}... | https://en.wikipedia.org/wiki/Convolution_operation |
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... | https://en.wikipedia.org/wiki/Convolution_operation |
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 val... | https://en.wikipedia.org/wiki/Linear_recurrence_with_constant_coefficients |
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 ite... | https://en.wikipedia.org/wiki/Linear_recurrence_with_constant_coefficients |
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 Verrie... | https://en.wikipedia.org/wiki/Faddeev–LeVerrier_algorithm |
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. | https://en.wikipedia.org/wiki/Faddeev–LeVerrier_algorithm |
An elegant shortcut to the proof, bypassing Newton polynomials, was introduced by Hou. The bulk of the presentation here follows Gantmacher, p. 88.) | https://en.wikipedia.org/wiki/Faddeev–LeVerrier_algorithm |
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 eluci... | https://en.wikipedia.org/wiki/Group_cohomology |
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 applic... | https://en.wikipedia.org/wiki/Group_cohomology |
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 ... | https://en.wikipedia.org/wiki/Group_cohomology |
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, inc... | https://en.wikipedia.org/wiki/Group_cohomology |
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 o... | https://en.wikipedia.org/wiki/Argument_of_a_complex_number |
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 reflect... | https://en.wikipedia.org/wiki/Shear_matrix |
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 f... | https://en.wikipedia.org/wiki/Volume_integral |
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 "... | https://en.wikipedia.org/wiki/Radon_metric |
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