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c_p1qomjude1jp | In mathematics, an invariant subspace of a linear mapping T: V → V i.e. from some vector space V to itself, is a subspace W of V that is preserved by T; that is, T(W) ⊆ W. | Invariant vector |
c_r1fng4fl9ob9 | In mathematics, an invertible sheaf is a sheaf on a ringed space which has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of alg... | Invertible sheaf |
c_capyx8850ukg | In mathematics, an involution, involutory function, or self-inverse function is a function f that is its own inverse, f(f(x)) = xfor all x in the domain of f. Equivalently, applying f twice produces the original value. | Ring with involution |
c_2p1t5z158l3g | In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A is an involution if and only if A2 = I, where I is the n × n identity matrix. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any invert... | Involutory matrix |
c_288k798eo21z | In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field to which the coefficients of the ... | Reducible polynomial |
c_23fcpjcecjon | One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals. Polynomial irreducibility can be considered for polynomials with coefficients in an integral domain, and there are two common definitions. Most often, a polynomial over an integral domain R is said to be irreducible if it is no... | Reducible polynomial |
c_11106x9p37fm | For the second definition, a polynomial is irreducible if it cannot be factored into polynomials with coefficients in the same domain that both have a positive degree. Equivalently, a polynomial is irreducible if it is irreducible over the field of fractions of the integral domain. For example, the polynomial 2 ( x 2 −... | Reducible polynomial |
c_t2h87yzdhur8 | On the other hand, x 2 − 2 {\displaystyle x^{2}-2} is irreducible in Z {\displaystyle \mathbb {Z} } for the two definitions, while it is reducible in R . {\displaystyle \mathbb {R} .} A polynomial that is irreducible over any field containing the coefficients is absolutely irreducible. | Reducible polynomial |
c_ylappsxdrnxo | By the fundamental theorem of algebra, a univariate polynomial is absolutely irreducible if and only if its degree is one. On the other hand, with several indeterminates, there are absolutely irreducible polynomials of any degree, such as x 2 + y n − 1 , {\displaystyle x^{2}+y^{n}-1,} for any positive integer n. A poly... | Reducible polynomial |
c_x93urgvehrqi | In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure". | Linear isometry |
c_5x1bovwc78o5 | In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other.Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the structure of the mathematical object are studied. Examples of this are ordinals... | Isomorphism class |
c_syrgn3q0345x | In homotopy theory, the fundamental group of a space X {\displaystyle X} at a point p {\displaystyle p} , though technically denoted π 1 ( X , p ) {\displaystyle \pi _{1}(X,p)} to emphasize the dependence on the base point, is often written lazily as simply π 1 ( X ) {\displaystyle \pi _{1}(X)} if X {\displaystyle X} i... | Isomorphism class |
c_6884hru5h9jm | In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος isos "equal", and μορφή morp... | Isomorphism (algebra) |
c_wflfjtc1ew7t | Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism.An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomo... | Isomorphism (algebra) |
c_1frt7j5zy5ph | For example, for every prime number p, all fields with p elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique. | Isomorphism (algebra) |
c_234rc313zzy4 | The term isomorphism is mainly used for algebraic structures. In this case, mappings are called homomorphisms, and a homomorphism is an isomorphism if and only if it is bijective. In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. | Isomorphism (algebra) |
c_3171t0n6ofk3 | For example: An isometry is an isomorphism of metric spaces. A homeomorphism is an isomorphism of topological spaces. A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically differentiable manifolds. | Isomorphism (algebra) |
c_h8zjn2zo50jw | A symplectomorphism is an isomorphism of symplectic manifolds. A permutation is an automorphism of a set. In geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations.Category theory, which can be viewed as a formaliz... | Isomorphism (algebra) |
c_r1khsxkf00ps | In mathematics, an isotopy from a possibly non-associative algebra A to another is a triple of bijective linear maps (a, b, c) such that if xy = z then a(x)b(y) = c(z). This is similar to the definition of an isotopy of loops, except that it must also preserve the linear structure of the algebra. For a = b = c this is ... | Isotopy of an algebra |
c_0nom40k368i1 | Isotopy of algebras was introduced by Albert (1942), who was inspired by work of Steenrod. Some authors use a slightly different definition that an isotopy is a triple of bijective linear maps a, b, c such that if xyz = 1 then a(x)b(y)c(z) = 1. For alternative division algebras such as the octonions the two definitions... | Isotopy of an algebra |
c_6b5febsw7wnr | In mathematics, an isotropic manifold is a manifold in which the geometry does not depend on directions. Formally, we say that a Riemannian manifold ( M , g ) {\displaystyle (M,g)} is isotropic if for any point p ∈ M {\displaystyle p\in M} and unit vectors v , w ∈ T p M {\displaystyle v,w\in T_{p}M} , there is an isome... | Isotropic manifold |
c_q1w5q0aongy8 | In mathematics, an iterable cardinal is a type of large cardinal introduced by Gitman (2011), and Sharpe and Welch (2011), and further studied by Gitman and Welch (2011). Sharpe and Welch defined a cardinal κ to be iterable if every subset of κ is contained in a weak κ-model M for which there exists an M-ultrafilter on... | Iterable cardinal |
c_v671bdcrbbcr | In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the multiplication operation to ... | Iterated binary operation |
c_ns1z5gs3j70d | In print, summation and product are represented by special symbols; but other iterated operators often are denoted by larger variants of the symbol for the ordinary binary operator. Thus, the iterations of the four operations mentioned above are denoted ∑ , ∏ , ⋃ , {\displaystyle \sum ,\ \prod ,\ \bigcup ,} and ⋂ {\dis... | Iterated binary operation |
c_8t90e3ykci5r | In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f: X → X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial o... | Iterated map |
c_aindd1u1eoxb | In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If M is already a topological manifold, it is requir... | Differential structure |
c_9dqq866iyh9o | In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra. Here, n {\displaystyle n} may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of 2-group... | Higher group |
c_nqvg7x0bv01l | In mathematics, an n-sphere or a hypersphere is a topological space that is homeomorphic to a standard n-sphere, which is the set of points in (n + 1)-dimensional Euclidean space that are situated at a constant distance r from a fixed point, called the center. It is the generalization of an ordinary sphere in the ordin... | Octahedral sphere |
c_1j6qy5nwi81v | In terms of the standard norm, the n-sphere is defined as S n = { x ∈ R n + 1: ‖ x ‖ = 1 } , {\displaystyle S^{n}=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=1\right\},} and an n-sphere of radius r can be defined as S n ( r ) = { x ∈ R n + 1: ‖ x ‖ = r } . {\displaystyle S^{n}(r)=\left\{x\in \mathbb {R} ^{n+1}:\lef... | Octahedral sphere |
c_ee53tvq453xq | An n-sphere is the surface or boundary of an (n + 1)-dimensional ball. In particular: the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, a circle, which is the one-dimensional circumference of a (two-dimensional) disk, is a 1-sphere, the two-dimensional surface of a three-dimensional ball... | Octahedral sphere |
c_cvykuz53ayec | In mathematics, an nth-order Argand system (named after French mathematician Jean-Robert Argand) is a coordinate system constructed around the nth roots of unity. From the origin, n axes extend such that the angle between each axis and the axes immediately before and after it is 360/n degrees. For example, the number l... | Argand system |
c_9etzn7gfwt6z | In mathematics, an octonion algebra or Cayley algebra over a field F is a composition algebra over F that has dimension 8 over F. In other words, it is a unital non-associative algebra A over F with a non-degenerate quadratic form N (called the norm form) such that N ( x y ) = N ( x ) N ( y ) {\displaystyle N(xy)=N(x)N... | Cayley algebra |
c_7wil91js3ncj | Up to F-algebra isomorphism, there is a unique split octonion algebra over any field F. When F is algebraically closed or a finite field, these are the only octonion algebras over F. Octonion algebras are always non-associative. They are, however, alternative algebras, alternativity being a weaker form of associativity... | Cayley algebra |
c_ijlb1tvd59ir | It follows that the invertible elements in any octonion algebra form a Moufang loop, as do the elements of unit norm. The construction of general octonion algebras over an arbitrary field k was described by Leonard Dickson in his book Algebren und ihre Zahlentheorie (1927) (Seite 264) and repeated by Max Zorn. The prod... | Cayley algebra |
c_vj1wgxc97byw | Another octonion may be written r + Re. Then with * denoting the conjugation in the quaternion algebra, their product is ( q + Q e ) ( r + R e ) = ( q r + γ R ∗ Q ) + ( R q + Q r ∗ ) e . | Cayley algebra |
c_oojahzwibe93 | {\displaystyle (q+Qe)(r+Re)=(qr+\gamma R^{*}Q)+(Rq+Qr^{*})e.} Zorn’s German language description of this Cayley–Dickson construction contributed to the persistent use of this eponym describing the construction of composition algebras. Cohl Furey has proposed that octonion algebras can be utilized in an attempt to recon... | Cayley algebra |
c_gbg6zozc8fr7 | In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3-manifold M into a union of surfaces (necessarily with boundary) and solid tori. Open books have relevance to contact geometry, with a famous theorem of Emmanuel Giroux (given below) that shows that contact geom... | Open book decomposition |
c_lz78nbqxpr45 | In mathematics, an open cover of a topological space X {\displaystyle X} is a family of open subsets such that X {\displaystyle X} is the union of all of the open sets. A good cover is an open cover in which all sets and all non-empty intersections of finitely-many sets are contractible (Petersen 2006). The concept was... | Good cover (algebraic topology) |
c_sd1mu7ryprex | In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set along with a distance defined between any two points), an open set is a set that, along with every point P, contains all points that are sufficiently near to P (that is, all points whose distance to P is less ... | Open region |
c_yuihnkhb8r0x | These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, every subset can be open (the discrete topology), or no subset can be open except the space itself and the empty set (the indiscrete topology).In practice, however, open sets are usually chosen to provide a notion o... | Open region |
c_40x162ekt0s2 | In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O {\displaystyle O} , one defines an algebra over O {\displaystyle O} to be a set toge... | Operad |
c_8weyjvepwhnp | In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. | Operand |
c_0yaou73umsyz | In mathematics, an operation is a function which takes zero or more input values (also called "operands" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and mu... | Mathematical operations |
c_dpl1z3ozjiw5 | The mixed product is an example of an operation of arity 3, also called ternary operation. Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, in which case the "usual" operations of finite arity are called finitary operations. A partial operation is defined similarly to... | Mathematical operations |
c_vn5x4mmdra6i | In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an operator, but the term is often used in place of function when the domain is a set of function... | Mathematical operator |
c_8kji74smjw4q | The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example R n {\displaystyle \mathbb {R} ^{n}} to R n {\displaystyle \mathbb {R} ^{n}} . Such operators often preserve properties, such as continuity. | Mathematical operator |
c_2hxui2ywf9bq | For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators. Operator is also used for denoting the symbol of a mathematical operation. This is related with the meaning of "operat... | Mathematical operator |
c_1v55z3vqwu3g | In mathematics, an operator or transform is a function from one space of functions to another. Operators occur commonly in engineering, physics and mathematics. Many are integral operators and differential operators. In the following L is an operator L: F → G {\displaystyle L:{\mathcal {F}}\to {\mathcal {G}}} which tak... | List of mathematic operators |
c_0y4mtluwcfq4 | In mathematics, an orbit portrait is a combinatorial tool used in complex dynamics for understanding the behavior of one-complex dimensional quadratic maps. In simple words one can say that it is: a list of external angles for which rays land on points of that orbit graph showing above list | Orbit portrait |
c_mzlh5i6n4285 | In mathematics, an orbit trap is a method of colouring fractal images based upon how close an iterative function, used to create the fractal, approaches a geometric shape, called a "trap". Typical traps are points, lines, circles, flower shapes and even raster images. Orbit traps are typically used to colour two dimens... | Orbit trap |
c_6225blu0ukcj | In mathematics, an orbital integral is an integral transform that generalizes the spherical mean operator to homogeneous spaces. Instead of integrating over spheres, one integrates over generalized spheres: for a homogeneous space X = G/H, a generalized sphere centered at a point x0 is an orbit of the isotropy group of... | Orbital integral |
c_9ux80iktzm9k | In mathematics, an order in the sense of ring theory is a subring O {\displaystyle {\mathcal {O}}} of a ring A {\displaystyle A} , such that A {\displaystyle A} is a finite-dimensional algebra over the field Q {\displaystyle \mathbb {Q} } of rational numbers O {\displaystyle {\mathcal {O}}} spans A {\displaystyle A} ov... | Order (ring theory) |
c_3ktkq4oiagm1 | In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered set, the order topology on X is generated by the subbase of "open rays" { x ∣ a < x } {\... | Right order topology |
c_zsojfzi20z8h | In mathematics, an ordered algebra is an algebra over the real numbers R {\displaystyle \mathbb {R} } with unit e together with an associated order such that e is positive (i.e. e ≥ 0), the product of any two positive elements is again positive, and when A is considered as a vector space over R {\displaystyle \mathbb {... | Ordered algebra |
c_snea8s0mg5hf | In mathematics, an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of n scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector v on one basis is, in g... | Coordinate change |
c_pf7wu6n86iy7 | Using matrices, this formula can be written x o l d = A x n e w , {\displaystyle \mathbf {x} _{\mathrm {old} }=A\,\mathbf {x} _{\mathrm {new} },} where "old" and "new" refer respectively to the firstly defined basis and the other basis, x o l d {\displaystyle \mathbf {x} _{\mathrm {old} }} and x n e w {\displaystyle \m... | Coordinate change |
c_0z8fuhocse7u | In mathematics, an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of real numbers. | Ordered exponential field |
c_ak7watrvvfi3 | In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field is isomorphic to the reals. Every subfield of an ordered field is also an... | Ordered field |
c_dc9vmxalvv51 | Squares are necessarily non-negative in an ordered field. This implies that the complex numbers cannot be ordered since the square of the imaginary unit i is −1 (which is negative in any ordered field). | Ordered field |
c_agj6z0y9sho5 | Finite fields cannot be ordered. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn. This grew eventually into the Artin–Schreier theory of ordered fields and formally real fields. | Ordered field |
c_nojaq70ie6qp | In mathematics, an ordered pair (a, b) is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. (In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.) | Ordered pairs |
c_ou4lsme0zfja | Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) | Ordered pairs |
c_rkl254p0tfue | The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects). For example, the ordered triple (a,b,c) can be defined as (a, (b,c)), i.e., as one pair nested in another. In the ordered pair (a, b), the object a is called the first entry, an... | Ordered pairs |
c_0i53srvyx4uy | In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S. An ordered monoid and an ordered group are, respectively, a monoid or a group that are endowed with a partia... | Ordered monoid |
c_m1sfhexmlnak | In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. | Positive cone |
c_omay9co9wm2b | In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differenti... | System of ordinary differential equation |
c_wnam7mxb2o7i | In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form y ′ + P ( x ) y = Q ( x ) y n , {\displaystyle y'+P(x)y=Q(x)y^{n},} where n {\displaystyle n} is a real number. Some authors allow any real n {\displaystyle n} , whereas others require that n {\displaystyl... | Bernoulli differential equation |
c_xj6x6xoph1y6 | In mathematics, an ordinary singularity of an algebraic curve is a singular point of multiplicity r where the r tangents at the point are distinct (Walker 1950, p. 54). In higher dimensions the literature on algebraic geometry contains many inequivalent definitions of ordinary singular points. | Ordinary singularity |
c_ovyzcvvvzbmj | In mathematics, an orientation of a curve is the choice of one of the two possible directions for travelling on the curve. For example, for Cartesian coordinates, the x-axis is traditionally oriented toward the right, and the y-axis is upward oriented. In the case of a planar simple closed curve (that is, a curve in th... | Curve orientation |
c_sjfsfwq9g36b | This definition relies on the fact that every simple closed curve admits a well-defined interior, which follows from the Jordan curve theorem. The inner loop of a beltway road in a country where people drive on the right side of the road is an example of a negatively oriented (clockwise) curve. | Curve orientation |
c_628plz551uqs | In trigonometry, the unit circle is traditionally oriented counterclockwise. The concept of orientation of a curve is just a particular case of the notion of orientation of a manifold (that is, besides orientation of a curve one may also speak of orientation of a surface, hypersurface, etc.). Orientation of a curve is ... | Curve orientation |
c_8apstay6ywug | In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map (which is a bundle map) ϕ U:... | Orientation of a vector bundle |
c_tdq1oqpyp0io | A vector bundle together with an orientation is called an oriented bundle. A vector bundle that can be given an orientation is called an orientable vector bundle. The basic invariant of an oriented bundle is the Euler class. The multiplication (that is, cup product) by the Euler class of an oriented bundle gives rise t... | Orientation of a vector bundle |
c_v76s9sivjz9v | In mathematics, an orthodox semigroup is a regular semigroup whose set of idempotents forms a subsemigroup. In more recent terminology, an orthodox semigroup is a regular E-semigroup. The term orthodox semigroup was coined by T. E. Hall and presented in a paper published in 1969. Certain special classes of orthodox sem... | Orthodox semigroup |
c_qualmt32at1r | In mathematics, an orthogonal array (more specifically, a fixed-level orthogonal array) is a "table" (array) whose entries come from a fixed finite set of symbols (for example, {1,2,...,v}), arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t-tuples of ... | Orthogonal array |
c_fzc1g51g9323 | The second and third columns would give, (1,1), (2,1), (2,2) and (1,2); again, all possible ordered pairs each appearing once. The same statement would hold had the first and second columns been used. This is thus an orthogonal array of strength two. | Orthogonal array |
c_kf2dpbu0i6sh | In the example on the right, the rows restricted to the first three columns contain the 8 possible ordered triples consisting of 0's and 1's, each appearing once. The same holds for any other choice of three columns. Thus this is an orthogonal array of strength 3. | Orthogonal array |
c_imlcwhwaw81s | A mixed-level orthogonal array is one in which each column may have a different number of symbols. An example is given below. Orthogonal arrays generalize, in a tabular form, the idea of mutually orthogonal Latin squares. These arrays have many connections to other combinatorial designs and have applications in the sta... | Orthogonal array |
c_t11njvsrpbkb | In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Lag... | Orthogonal polynomials |
c_y80g5kzysab8 | The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A. A. Markov and T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (quadrature rules), probability theory, representation theory (of Lie groups, qua... | Orthogonal polynomials |
c_ks7m0fxgdaak | In mathematics, an orthogonal symmetric Lie algebra is a pair ( g , s ) {\displaystyle ({\mathfrak {g}},s)} consisting of a real Lie algebra g {\displaystyle {\mathfrak {g}}} and an automorphism s {\displaystyle s} of g {\displaystyle {\mathfrak {g}}} of order 2 {\displaystyle 2} such that the eigenspace u {\displaysty... | Symmetric Lie algebra |
c_x3u7x8ufrwqo | In practice, effectiveness is often assumed; we do this in this article as well. The canonical example is the Lie algebra of a symmetric space, s {\displaystyle s} being the differential of a symmetry. Let ( g , s ) {\displaystyle ({\mathfrak {g}},s)} be effective orthogonal symmetric Lie algebra, and let p {\displayst... | Symmetric Lie algebra |
c_58ky42j0a8fv | In mathematics, an orthostochastic matrix is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix. The detailed definition is as follows. A square matrix B of size n is doubly stochastic (or bistochastic) if all its rows and columns sum to 1 and all it... | Orthostochastic matrix |
c_xigi8vlb1v1y | It is orthostochastic if there exists an orthogonal matrix O such that B i j = O i j 2 for i , j = 1 , … , n . {\displaystyle B_{ij}=O_{ij}^{2}{\text{ for }}i,j=1,\dots ,n.\,} All 2-by-2 doubly stochastic matrices are orthostochastic (and also unistochastic) since for any B = {\displaystyle B={\begin{bmatrix}a&1-a\\1-... | Orthostochastic matrix |
c_7fr8yg2y19mk | In mathematics, an outer measure μ on n-dimensional Euclidean space Rn is called a Borel regular measure if the following two conditions hold: Every Borel set B ⊆ Rn is μ-measurable in the sense of Carathéodory's criterion: for every A ⊆ Rn, μ ( A ) = μ ( A ∩ B ) + μ ( A ∖ B ) . {\displaystyle \mu (A)=\mu (A\cap B)+\mu... | Borel regular measure |
c_qg5cmym8lfsn | An outer measure satisfying only the first of these two requirements is called a Borel measure, while an outer measure satisfying only the second requirement (with the Borel set B replaced by a measurable set B) is called a regular measure. The Lebesgue outer measure on Rn is an example of a Borel regular measure. It c... | Borel regular measure |
c_dp5mrffsskiu | In mathematics, an overline can be used as a vinculum. The vinculum can indicate a line segment:The vinculum can indicate a repeating decimal value: When it is not possible to format the number so that the overline is over the digit(s) that repeat, one overline character is placed to the left of the digit(s) that repea... | Overhead bar |
c_nu01uu8947sm | In mathematics, an overring of an integral domain contains the integral domain, and the integral domain's field of fractions contains the overring. Overrings provide an improved understanding of different types of rings and domains. | Overring |
c_rgkhifhugnfr | In mathematics, an ovoid O of a (finite) polar space of rank r is a set of points, such that every subspace of rank r − 1 {\displaystyle r-1} intersects O in exactly one point. | Ovoid (polar space) |
c_wtj34g5yr4ck | In mathematics, an ultra limit is a geometric construction that assigns a limit metric space to a sequence of metric spaces X n {\displaystyle X_{n}} . The concept of such captures the limiting behavior of finite configurations in the X n {\displaystyle X_{n}} spaces and employs an ultrafilter to bypass the need for re... | Asymptotic cone |
c_nqv9g2ejrwzx | In mathematics, an ultragraph C*-algebra is a universal C*-algebra generated by partial isometries on a collection of Hilbert spaces constructed from ultragraphs.pp. 6-7. These C*-algebras were created in order to simultaneously generalize the classes of graph C*-algebras and Exel–Laca algebras, giving a unified framew... | Ultragraph C*-algebra |
c_6d2is4r52er7 | In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d ( x , z ) ≤ max { d ( x , y ) , d ( y , z ) } {\displaystyle d(x,z)\leq \max \left\{d(x,y),d(y,z)\right\}} . Sometimes the associated metric is also called a non-Archimedean metric or super-metric. | Ultrametric space |
c_3v40enufnjbz | In mathematics, an ultrapolynomial is a power series in several variables whose coefficients are bounded in some specific sense. | Ultrapolynomial |
c_fntliut7ienn | In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers. | Uncountable set |
c_bg4f6q2neih6 | In mathematics, an unfoldable cardinal is a certain kind of large cardinal number. Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementa... | Unfoldable cardinal |
c_f51a7zoi8lsb | In mathematics, an unfolding of a smooth real-valued function ƒ on a smooth manifold, is a certain family of functions that includes ƒ. | Unfolding (functions) |
c_i2or7kxlo6tb | In mathematics, an unordered pair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b with no particular relation between them, where {a, b} = {b, a}. In contrast, an ordered pair (a, b) has a as its first element and b as its second element, which means (a, b) ≠ (b, a). While the two elemen... | Unordered pair |
c_smgzj7cds8ae | A set with precisely two elements is also called a 2-set or (rarely) a binary set. An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1. In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing. More gener... | Unordered pair |
c_0qtbg3q7jogn | In mathematics, an untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 ... | Untouchable number |
c_jrrg34eh1zke | In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set ( X , ≤ ) {\displaystyle (X,\leq )} is a subset S ⊆ X {\displaystyle S\subseteq X} with the following property: if s is in S and if x in X is larger than s (that is, if s < x {\displaystyle s | Principal down-set |
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