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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.
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https://en.wikipedia.org/wiki/Invariant_vector
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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 algebraic varieties.
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https://en.wikipedia.org/wiki/Invertible_sheaf
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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.
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https://en.wikipedia.org/wiki/Ring_with_involution
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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 invertible matrix multiplied by its inverse is the identity.
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https://en.wikipedia.org/wiki/Involutory_matrix
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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 polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as ( x − 2 ) ( x + 2 ) {\displaystyle \left(x-{\sqrt {2}}\right)\left(x+{\sqrt {2}}\right)} if it is considered as a polynomial with real coefficients.
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https://en.wikipedia.org/wiki/Reducible_polynomial
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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 not the product of two polynomials that have their coefficients in R, and that are not unit in R. Equivalently, for this definition, an irreducible polynomial is an irreducible element in the rings of polynomials over R. If R is a field, the two definitions of irreducibility are equivalent.
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https://en.wikipedia.org/wiki/Reducible_polynomial
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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 − 2 ) ∈ Z {\displaystyle 2(x^{2}-2)\in \mathbb {Z} } is irreducible for the second definition, and not for the first one.
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https://en.wikipedia.org/wiki/Reducible_polynomial
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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.
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https://en.wikipedia.org/wiki/Reducible_polynomial
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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 polynomial that is not irreducible is sometimes said to be a reducible polynomial.Irreducible polynomials appear naturally in the study of polynomial factorization and algebraic field extensions. It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal magnitude) are the irreducible integers. They exhibit many of the general properties of the concept of "irreducibility" that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors. When the coefficient ring is a field or other unique factorization domain, an irreducible polynomial is also called a prime polynomial, because it generates a prime ideal.
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https://en.wikipedia.org/wiki/Reducible_polynomial
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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".
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https://en.wikipedia.org/wiki/Linear_isometry
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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 and graphs. However, there are circumstances in which the isomorphism class of an object conceals vital internal information about it; consider these examples: The associative algebras consisting of coquaternions and 2 × 2 real matrices are isomorphic as rings. Yet they appear in different contexts for application (plane mapping and kinematics) so the isomorphism is insufficient to merge the concepts.
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https://en.wikipedia.org/wiki/Isomorphism_class
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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} is path connected. The reason for this is that the existence of a path between two points allows one to identify loops at one with loops at the other; however, unless π 1 ( X , p ) {\displaystyle \pi _{1}(X,p)} is abelian this isomorphism is non-unique. Furthermore, the classification of covering spaces makes strict reference to particular subgroups of π 1 ( X , p ) {\displaystyle \pi _{1}(X,p)} , specifically distinguishing between isomorphic but conjugate subgroups, and therefore amalgamating the elements of an isomorphism class into a single featureless object seriously decreases the level of detail provided by the theory. == References ==
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https://en.wikipedia.org/wiki/Isomorphism_class
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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 morphe "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects).
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https://en.wikipedia.org/wiki/Isomorphism_(algebra)
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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 isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as is the case for solutions of a universal property), or if the isomorphism is much more natural (in some sense) than other isomorphisms.
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https://en.wikipedia.org/wiki/Isomorphism_(algebra)
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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.
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https://en.wikipedia.org/wiki/Isomorphism_(algebra)
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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.
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https://en.wikipedia.org/wiki/Isomorphism_(algebra)
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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.
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https://en.wikipedia.org/wiki/Isomorphism_(algebra)
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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 formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea.
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https://en.wikipedia.org/wiki/Isomorphism_(algebra)
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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 the same as an isomorphism. The autotopy group of an algebra is the group of all isotopies to itself (sometimes called autotopies), which contains the group of automorphisms as a subgroup.
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https://en.wikipedia.org/wiki/Isotopy_of_an_algebra
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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 of isotopy are equivalent, but in general they are not.
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https://en.wikipedia.org/wiki/Isotopy_of_an_algebra
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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 isometry φ {\displaystyle \varphi } of M {\displaystyle M} with φ ( p ) = p {\displaystyle \varphi (p)=p} and φ ∗ ( v ) = w {\displaystyle \varphi _{\ast }(v)=w} . Every connected isotropic manifold is homogeneous, i.e. for any p , q ∈ M {\displaystyle p,q\in M} there is an isometry φ {\displaystyle \varphi } of M {\displaystyle M} with φ ( p ) = q . {\displaystyle \varphi (p)=q.} This can be seen by considering a geodesic γ: → M {\displaystyle \gamma :\to M} from p {\displaystyle p} to q {\displaystyle q} and taking the isometry which fixes γ ( 1 ) {\displaystyle \gamma (1)} and maps γ ′ ( 1 ) {\displaystyle \gamma '(1)} to − γ ′ ( 1 ) . {\displaystyle -\gamma '(1).}
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https://en.wikipedia.org/wiki/Isotropic_manifold
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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 κ which allows for wellfounded iterations by ultrapowers of arbitrary length. Gitman gave a finer notion, where a cardinal κ is defined to be α-iterable if ultrapower iterations only of length α are required to wellfounded. (By standard arguments iterability is equivalent to ω1-iterability.)
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https://en.wikipedia.org/wiki/Iterable_cardinal
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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 the product operation. Other operations, e.g., the set-theoretic operations union and intersection, are also often iterated, but the iterations are not given separate names.
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https://en.wikipedia.org/wiki/Iterated_binary_operation
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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 ⋂ {\displaystyle \bigcap } , respectively.More generally, iteration of a binary function is generally denoted by a slash: iteration of f {\displaystyle f} over the sequence ( a 1 , a 2 … , a n ) {\displaystyle (a_{1},a_{2}\ldots ,a_{n})} is denoted by f / ( a 1 , a 2 … , a n ) {\displaystyle f/(a_{1},a_{2}\ldots ,a_{n})} , following the notation for reduce in Bird–Meertens formalism. In general, there is more than one way to extend a binary operation to operate on finite sequences, depending on whether the operator is associative, and whether the operator has identity elements.
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https://en.wikipedia.org/wiki/Iterated_binary_operation
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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 object, the result of applying a given function is fed again in the function as input, and this process is repeated. For example on the image on the right: L = F {\displaystyle {\mathit {F}}\,} ( K ), M = F ∘ F {\displaystyle {\mathit {F}}\,\circ {\mathit {F}}\,} ( K ) = F 2 {\displaystyle {\mathit {F}}\;^{2}\,} ( K ),with the circle‑shaped symbol of function composition.Iterated functions are objects of study in computer science, fractals, dynamical systems, mathematics and renormalization group physics.
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https://en.wikipedia.org/wiki/Iterated_map
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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 required that the new topology be identical to the existing one.
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https://en.wikipedia.org/wiki/Differential_structure
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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-groups under the moniker 'gr-category'. The general definition of n {\displaystyle n} -group is a matter of ongoing research. However, it is expected that every topological space will have a homotopy n {\displaystyle n} -group at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group π n {\displaystyle \pi _{n}} , or the entire Postnikov tower for n = ∞ {\displaystyle n=\infty } .
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https://en.wikipedia.org/wiki/Higher_group
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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 ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit n-sphere or simply the n-sphere for brevity.
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https://en.wikipedia.org/wiki/Octahedral_sphere
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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}:\left\|x\right\|=r\right\}.} The dimension of n-sphere is n, and must not be confused with the dimension (n + 1) of the Euclidean space in which it is naturally embedded.
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https://en.wikipedia.org/wiki/Octahedral_sphere
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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 is a 2-sphere, often simply called a sphere, the three-dimensional boundary of a (four-dimensional) 4-ball is a 3-sphere, the (n – 1)-dimensional boundary of a (n-dimensional) n-ball is an (n – 1)-sphere.For n ≥ 2, the n-spheres that are differential manifolds can be characterized (up to a diffeomorphism) as the simply connected n-dimensional manifolds of constant, positive curvature. The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional Euclidean spaces together, by identifying the boundary of an n-cube with a point, or (inductively) by forming the suspension of an (n − 1)-sphere. The 1-sphere is the 1-manifold that is a circle, which is not simply connected. The 0-sphere is the 0-manifold, which is not even connected, consisting of two points.
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https://en.wikipedia.org/wiki/Octahedral_sphere
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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 line is the 2nd-order Argand system because the two axes extending from the origin represent 1 and −1, the 2nd roots of unity. The complex plane (sometimes called the Argand plane, also named after Argand) is the 4th-order Argand system because the 4 axes extending from the origin represent 1, i, −1, and −i, the 4th roots of unity.
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https://en.wikipedia.org/wiki/Argand_system
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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(y)} for all x and y in A. The most well-known example of an octonion algebra is the classical octonions, which are an octonion algebra over R, the field of real numbers. The split-octonions also form an octonion algebra over R. Up to R-algebra isomorphism, these are the only octonion algebras over the reals. The algebra of bioctonions is the octonion algebra over the complex numbers C. The octonion algebra for N is a division algebra if and only if the form N is anisotropic. A split octonion algebra is one for which the quadratic form N is isotropic (i.e., there exists a non-zero vector x with N(x) = 0).
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https://en.wikipedia.org/wiki/Cayley_algebra
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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. Moreover, the Moufang identities hold in any octonion algebra.
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https://en.wikipedia.org/wiki/Cayley_algebra
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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 product depends on selection of a γ from k. Given q and Q from a quaternion algebra over k, the octonion is written q + Qe.
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https://en.wikipedia.org/wiki/Cayley_algebra
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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 .
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https://en.wikipedia.org/wiki/Cayley_algebra
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{\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 reconcile components of the standard model.
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https://en.wikipedia.org/wiki/Cayley_algebra
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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 geometry can be studied from an entirely topological viewpoint.
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https://en.wikipedia.org/wiki/Open_book_decomposition
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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 introduced by André Weil in 1952 for differentiable manifolds, demanding the U α 1 … α n {\displaystyle U_{\alpha _{1}\ldots \alpha _{n}}} to be differentiably contractible. A modern version of this definition appears in Bott & Tu (1982).
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https://en.wikipedia.org/wiki/Good_cover_(algebraic_topology)
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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 than some value depending on P). More generally, an open set is a member of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology.
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https://en.wikipedia.org/wiki/Open_region
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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 of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such as continuity, connectedness, and compactness, which were originally defined by means of a distance. The most common case of a topology without any distance is given by manifolds, which are topological spaces that, near each point, resemble an open set of a Euclidean space, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the Zariski topology, which is fundamental in algebraic geometry and scheme theory.
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https://en.wikipedia.org/wiki/Open_region
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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 together with concrete operations on this set which behave just like the abstract operations of O {\displaystyle O} . For instance, there is a Lie operad L {\displaystyle L} such that the algebras over L {\displaystyle L} are precisely the Lie algebras; in a sense L {\displaystyle L} abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations.
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https://en.wikipedia.org/wiki/Operad
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In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on.
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https://en.wikipedia.org/wiki/Operand
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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 multiplication, and unary operations (i.e., operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant.
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https://en.wikipedia.org/wiki/Mathematical_operations
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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 an operation, but with a partial function in place of a function.
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https://en.wikipedia.org/wiki/Mathematical_operations
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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 functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an integral operator), and may be extended so as to act on related objects (an operator that acts on functions may act also on differential equations whose solutions are functions that satisfy the equation). See Operator (physics) for other examples.
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https://en.wikipedia.org/wiki/Mathematical_operator
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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.
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https://en.wikipedia.org/wiki/Mathematical_operator
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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 "operator" in computer programming; see Operator (computer programming).
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https://en.wikipedia.org/wiki/Mathematical_operator
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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 takes a function y ∈ F {\displaystyle y\in {\mathcal {F}}} to another function L ∈ G {\displaystyle L\in {\mathcal {G}}} . Here, F {\displaystyle {\mathcal {F}}} and G {\displaystyle {\mathcal {G}}} are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions.
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https://en.wikipedia.org/wiki/List_of_mathematic_operators
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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
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https://en.wikipedia.org/wiki/Orbit_portrait
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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 dimensional fractals representing the complex plane.
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https://en.wikipedia.org/wiki/Orbit_trap
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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 x0.
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https://en.wikipedia.org/wiki/Orbital_integral
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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} over Q {\displaystyle \mathbb {Q} } , and O {\displaystyle {\mathcal {O}}} is a Z {\displaystyle \mathbb {Z} } -lattice in A {\displaystyle A} .The last two conditions can be stated in less formal terms: Additively, O {\displaystyle {\mathcal {O}}} is a free abelian group generated by a basis for A {\displaystyle A} over Q {\displaystyle \mathbb {Q} } . More generally for R {\displaystyle R} an integral domain contained in a field K {\displaystyle K} , we define O {\displaystyle {\mathcal {O}}} to be an R {\displaystyle R} -order in a K {\displaystyle K} -algebra A {\displaystyle A} if it is a subring of A {\displaystyle A} which is a full R {\displaystyle R} -lattice.When A {\displaystyle A} is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal orders exist in general, but need not be unique: there is in general no largest order, but a number of maximal orders. An important class of examples is that of integral group rings.
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https://en.wikipedia.org/wiki/Order_(ring_theory)
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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 } {\displaystyle \{x\mid a
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https://en.wikipedia.org/wiki/Right_order_topology
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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 {R} } then it is an Archimedean ordered vector space.
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https://en.wikipedia.org/wiki/Ordered_algebra
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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 general, different from the coordinate vector that represents v on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis.Such a conversion results from the change-of-basis formula which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis.
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https://en.wikipedia.org/wiki/Coordinate_change
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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 \mathbf {x} _{\mathrm {new} }} are the column vectors of the coordinates of the same vector on the two bases, and A {\displaystyle A} is the change-of-basis matrix (also called transition matrix), which is the matrix whose columns are the coordinate vectors of the new basis vectors on the old basis. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.
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https://en.wikipedia.org/wiki/Coordinate_change
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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.
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https://en.wikipedia.org/wiki/Ordered_exponential_field
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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 in the inherited order. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers.
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https://en.wikipedia.org/wiki/Ordered_field
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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).
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https://en.wikipedia.org/wiki/Ordered_field
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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.
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https://en.wikipedia.org/wiki/Ordered_field
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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}.)
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https://en.wikipedia.org/wiki/Ordered_pairs
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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.)
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https://en.wikipedia.org/wiki/Ordered_pairs
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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, and the object b the second entry of the pair. Alternatively, the objects are called the first and second components, the first and second coordinates, or the left and right projections of the ordered pair. Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs, cf. picture.
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https://en.wikipedia.org/wiki/Ordered_pairs
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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 partial order that makes them ordered semigroups. The terms posemigroup, pogroup and pomonoid are sometimes used, where "po" is an abbreviation for "partially ordered". The positive integers, the nonnegative integers and the integers form respectively a posemigroup, a pomonoid, and a pogroup under addition and the natural ordering. Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=". A morphism or homomorphism of posemigroups is a semigroup homomorphism that preserves the order (equivalently, that is monotonically increasing).
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https://en.wikipedia.org/wiki/Ordered_monoid
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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.
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https://en.wikipedia.org/wiki/Positive_cone
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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 differential equations which may be with respect to more than one independent variable.
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https://en.wikipedia.org/wiki/System_of_ordinary_differential_equation
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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 {\displaystyle n} not be 0 or 1. The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz, who published his result in the same year and whose method is the one still used today.Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A notable special case of the Bernoulli equation is the logistic differential equation.
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https://en.wikipedia.org/wiki/Bernoulli_differential_equation
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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.
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https://en.wikipedia.org/wiki/Ordinary_singularity
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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 the plane whose starting point is also the end point and which has no other self-intersections), the curve is said to be positively oriented or counterclockwise oriented, if one always has the curve interior to the left (and consequently, the curve exterior to the right), when traveling on it. Otherwise, that is if left and right are exchanged, the curve is negatively oriented or clockwise oriented.
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https://en.wikipedia.org/wiki/Curve_orientation
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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.
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https://en.wikipedia.org/wiki/Curve_orientation
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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 associated with parametrization of its points by a real variable. A curve may have equivalent parametrizations when there is a continuous increasing monotonic function relating the parameter of one curve to the parameter of the other. When there is a decreasing continuous function relating the parameters, then the parametric representations are opposite and the orientation of the curve is reversed.
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https://en.wikipedia.org/wiki/Curve_orientation
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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: π − 1 ( U ) → U × R n {\displaystyle \phi _{U}:\pi ^{-1}(U)\to U\times \mathbf {R} ^{n}} is fiberwise orientation-preserving, where Rn is given the standard orientation. In more concise terms, this says that the structure group of the frame bundle of E, which is the real general linear group GLn(R), can be reduced to the subgroup consisting of those with positive determinant. If E is a real vector bundle of rank n, then a choice of metric on E amounts to a reduction of the structure group to the orthogonal group O(n). In that situation, an orientation of E amounts to a reduction from O(n) to the special orthogonal group SO(n).
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https://en.wikipedia.org/wiki/Orientation_of_a_vector_bundle
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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 to a Gysin sequence.
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https://en.wikipedia.org/wiki/Orientation_of_a_vector_bundle
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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 semigroups had been studied earlier. For example, semigroups that are also unions of groups, in which the sets of idempotents form subsemigroups were studied by P. H. H. Fantham in 1960.
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https://en.wikipedia.org/wiki/Orthodox_semigroup
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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 the symbols, formed by taking the entries in each row restricted to these columns, appear the same number of times. The number t is called the strength of the orthogonal array. Here are two examples: The example at left is that of an orthogonal array with symbol set {1,2} and strength 2. Notice that the four ordered pairs (2-tuples) formed by the rows restricted to the first and third columns, namely (1,1), (2,1), (1,2) and (2,2), are all the possible ordered pairs of the two element set and each appears exactly once.
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https://en.wikipedia.org/wiki/Orthogonal_array
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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.
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https://en.wikipedia.org/wiki/Orthogonal_array
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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.
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https://en.wikipedia.org/wiki/Orthogonal_array
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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 statistical design of experiments, coding theory, cryptography and various types of software testing.
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https://en.wikipedia.org/wiki/Orthogonal_array
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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 Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases.
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https://en.wikipedia.org/wiki/Orthogonal_polynomials
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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, quantum groups, and related objects), enumerative combinatorics, algebraic combinatorics, mathematical physics (the theory of random matrices, integrable systems, etc.), and number theory. Some of the mathematicians who have worked on orthogonal polynomials include Gábor Szegő, Sergei Bernstein, Naum Akhiezer, Arthur Erdélyi, Yakov Geronimus, Wolfgang Hahn, Theodore Seio Chihara, Mourad Ismail, Waleed Al-Salam, Richard Askey, and Rehuel Lobatto.
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https://en.wikipedia.org/wiki/Orthogonal_polynomials
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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 {\displaystyle {\mathfrak {u}}} of s corresponding to 1 (i.e., the set u {\displaystyle {\mathfrak {u}}} of fixed points) is a compact subalgebra. If "compactness" is omitted, it is called a symmetric Lie algebra. An orthogonal symmetric Lie algebra is said to be effective if u {\displaystyle {\mathfrak {u}}} intersects the center of g {\displaystyle {\mathfrak {g}}} trivially.
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https://en.wikipedia.org/wiki/Symmetric_Lie_algebra
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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 {\displaystyle {\mathfrak {p}}} denotes the -1 eigenspace of s {\displaystyle s} .
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https://en.wikipedia.org/wiki/Symmetric_Lie_algebra
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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 its entries are nonnegative real numbers.
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https://en.wikipedia.org/wiki/Orthostochastic_matrix
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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-a&a\end{bmatrix}}} we find the corresponding orthogonal matrix O = , {\displaystyle O={\begin{bmatrix}\cos \phi &\sin \phi \\-\sin \phi &\cos \phi \end{bmatrix}},} with cos 2 ϕ = a , {\displaystyle \cos ^{2}\phi =a,} such that B i j = O i j 2 . {\displaystyle B_{ij}=O_{ij}^{2}.} For larger n the sets of bistochastic matrices includes the set of unistochastic matrices, which includes the set of orthostochastic matrices and these inclusion relations are proper.
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https://en.wikipedia.org/wiki/Orthostochastic_matrix
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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 (A\setminus B).} For every set A ⊆ Rn there exists a Borel set B ⊆ Rn such that A ⊆ B and μ(A) = μ(B).Notice that the set A need not be μ-measurable: μ(A) is however well defined as μ is an outer measure.
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https://en.wikipedia.org/wiki/Borel_regular_measure
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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 can be proved that a Borel regular measure, although introduced here as an outer measure (only countably subadditive), becomes a full measure (countably additive) if restricted to the Borel sets.
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https://en.wikipedia.org/wiki/Borel_regular_measure
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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 repeat: Historically, the vinculum was used to group together symbols so that they could be treated as a unit. Today, parentheses are more commonly used for this purpose.
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https://en.wikipedia.org/wiki/Overhead_bar
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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.
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https://en.wikipedia.org/wiki/Overring
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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.
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https://en.wikipedia.org/wiki/Ovoid_(polar_space)
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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 repeatedly considering subsequences to ensure convergence. Ultra limits generalize the idea of Gromov Hausdorff convergence in metric spaces.
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https://en.wikipedia.org/wiki/Asymptotic_cone
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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 framework for studying these objects. This is because every graph can be encoded as an ultragraph, and similarly, every infinite graph giving an Exel-Laca algebras can also be encoded as an ultragraph.
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https://en.wikipedia.org/wiki/Ultragraph_C*-algebra
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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.
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https://en.wikipedia.org/wiki/Ultrametric_space
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In mathematics, an ultrapolynomial is a power series in several variables whose coefficients are bounded in some specific sense.
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https://en.wikipedia.org/wiki/Ultrapolynomial
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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.
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https://en.wikipedia.org/wiki/Uncountable_set
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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 elementary embedding j of M into a transitive model with the critical point of j being κ and j(κ) ≥ λ. A cardinal is unfoldable if and only if it is an λ-unfoldable for all ordinals λ. A cardinal number κ is strongly 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 elementary embedding j of M into a transitive model "N" with the critical point of j being κ, j(κ) ≥ λ, and V(λ) is a subset of N. Without loss of generality, we can demand also that N contains all its sequences of length λ. Likewise, a cardinal is strongly unfoldable if and only if it is strongly λ-unfoldable for all λ. These properties are essentially weaker versions of strong and supercompact cardinals, consistent with V = L. Many theorems related to these cardinals have generalizations to their unfoldable or strongly unfoldable counterparts. For example, the existence of a strongly unfoldable implies the consistency of a slightly weaker version of the proper forcing axiom.
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https://en.wikipedia.org/wiki/Unfoldable_cardinal
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In mathematics, an unfolding of a smooth real-valued function ƒ on a smooth manifold, is a certain family of functions that includes ƒ.
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https://en.wikipedia.org/wiki/Unfolding_(functions)
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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 elements of an ordered pair (a, b) need not be distinct, modern authors only call {a, b} an unordered pair if a ≠ b. But for a few authors a singleton is also considered an unordered pair, although today, most would say that {a, a} is a multiset. It is typical to use the term unordered pair even in the situation where the elements a and b could be equal, as long as this equality has not yet been established.
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https://en.wikipedia.org/wiki/Unordered_pair
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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 generally, an unordered n-tuple is a set of the form {a1, a2,... an}.
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https://en.wikipedia.org/wiki/Unordered_pair
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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 and 5 are untouchable.
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https://en.wikipedia.org/wiki/Untouchable_number
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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
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https://en.wikipedia.org/wiki/Principal_down-set
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