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Invariant vector Summary Invariant_subspace 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invertible sheaf Summary Invertible_sheaf 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, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ring with involution Summary Ring_with_involution 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Involutory matrix Summary Involutory_matrix 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 simp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reducible polynomial Summary Reducible_polynomial 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, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reducible polynomial Summary Reducible_polynomial 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 integ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reducible polynomial Summary Reducible_polynomial 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 inte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reducible polynomial Summary Reducible_polynomial 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 absol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reducible polynomial Summary Reducible_polynomial 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Linear isometry Summary Isometric_isomorphism 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isomorphism class Summary Isomorphism_class 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 objec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isomorphism class Summary Isomorphism_class 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 ) {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isomorphism (algebra) Summary Isomorphism_(algebra) 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isomorphism (algebra) Summary Isomorphism_(algebra) 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 iso... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isomorphism (algebra) Summary Isomorphism_(algebra) 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isomorphism (algebra) Summary Isomorphism_(algebra) 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isomorphism (algebra) Summary Isomorphism_(algebra) 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isomorphism (algebra) Summary Isomorphism_(algebra) 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 transformation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isotopy of an algebra Summary Isotopy_of_an_algebra 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 lin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isotopy of an algebra Summary Isotopy_of_an_algebra 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 divisio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Isotropic manifold Summary Isotropic_manifold 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 {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Iterable cardinal Summary Iterable_cardinal 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Iterated binary operation Summary Iterated_binary_operation 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 oper... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Iterated binary operation Summary Iterated_binary_operation 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 ∑ , ∏ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Iterated map Summary Function_iteration 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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differential structure Summary Differentiable_structure 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 manifo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Higher group Summary Higher_group 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 Sín... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Octahedral sphere Summary Hyperspherical_coordinates 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Octahedral sphere Summary Hyperspherical_coordinates 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 } . {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Octahedral sphere Summary Hyperspherical_coordinates 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, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Argand system Summary Argand_system 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/... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cayley algebra Summary Cayley_algebra 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 ( ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cayley algebra Summary Cayley_algebra 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, alternativit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cayley algebra Summary Cayley_algebra 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 2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cayley algebra Summary Cayley_algebra 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 . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cayley algebra Summary Cayley_algebra {\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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Open book decomposition Summary Open_book_decomposition 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 Em... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Good cover (algebraic topology) Summary Good_cover_(algebraic_topology) 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Open region Summary Open_subset 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 poi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Open region Summary Open_subset 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 usua... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Operad Summary Operad_theory 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 {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Operand Summary Operand In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical operations Summary Mathematical_operation 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical operations Summary Mathematical_operation 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical operator Summary Mathematical_operator 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 pl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical operator Summary Mathematical_operator 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 prese... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical operator Summary Mathematical_operator 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 op... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of mathematic operators Summary List_of_mathematic_operators 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: ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orbit portrait Summary Orbit_portrait 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orbit trap Summary Orbit_trap 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 typic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orbital integral Summary Orbital_integral 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 poi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Order (ring theory) Summary Maximal_order 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 {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Right order topology Summary Order_topology 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordered algebra Summary Ordered_algebra 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 vect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coordinate change Summary Coordinate_change 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coordinate change Summary Coordinate_change 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} _{\m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordered exponential field Summary Ordered_exponential_field 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordered field Summary Preordered_field 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordered field Summary Preordered_field 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). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordered field Summary Preordered_field 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 for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordered pairs Summary Ordered_pair 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}.) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordered pairs Summary Ordered_pair 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 vecto... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordered pairs Summary Ordered_pair 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 obj... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordered monoid Summary Ordered_monoid 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Positive cone Summary Positive_cone 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
System of ordinary differential equation Summary Ordinary_differential_equation 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bernoulli differential equation Summary Bernoulli_differential_equation 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 an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ordinary singularity Summary Ordinary_singularity 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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Curve orientation Summary Curve_orientation 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Curve orientation Summary Curve_orientation 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Curve orientation Summary Curve_orientation 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, hyp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orientation of a vector bundle Summary Orientation_of_a_vector_bundle 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orientation of a vector bundle Summary Orientation_of_a_vector_bundle 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthodox semigroup Summary Orthodox_semigroup 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal array Summary Orthogonal_array 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 colu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal array Summary Orthogonal_array 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal array Summary Orthogonal_array 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal array Summary Orthogonal_array 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 combinatoria... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal polynomials Summary Orthogonal_polynomials 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 polyn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthogonal polynomials Summary Orthogonal_polynomials 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), probabi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric Lie algebra Summary Symmetric_Lie_algebra 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 {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric Lie algebra Summary Symmetric_Lie_algebra 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 ort... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthostochastic matrix Summary Orthostochastic_matrix 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 bistoch... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Orthostochastic matrix Summary Orthostochastic_matrix 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 f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Borel regular measure Summary Borel_regular_measure 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Borel regular measure Summary Borel_regular_measure 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Overhead bar Vinculum Overhead_bar > Uses > Math and science > Vinculum 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Overring Summary Overring 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ovoid (polar space) Summary Ovoid_(polar_space) 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Asymptotic cone Summary Asymptotic_cone 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ultragraph C*-algebra Summary Ultragraph_C*-algebra 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*-alge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ultrametric space Summary Strong_triangle_inequality 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ultrapolynomial Summary Ultrapolynomial In mathematics, an ultrapolynomial is a power series in several variables whose coefficients are bounded in some specific sense. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Uncountable set Summary Denumerably_infinite 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unfoldable cardinal Summary Unfoldable_cardinal 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 len... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unfolding (functions) Summary Unfolding_(functions) In mathematics, an unfolding of a smooth real-valued function ƒ on a smooth manifold, is a certain family of functions that includes ƒ. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unordered pair Summary Unordered_pair 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unordered pair Summary Unordered_pair 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Untouchable number Summary Untouchable_number 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-Baghd... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal down-set Summary Downward_closure 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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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