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wiki_3900_chunk_0 | Inner product spaces | In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in ⟨ a , b ⟩ {\displaystyle \langle a,b\rang... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3901_chunk_0 | Inner product spaces | Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.An inner product naturally induc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3902_chunk_0 | Integer matrix | In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3903_chunk_0 | Integer sequences | In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3904_chunk_0 | Integer sequences | The sequence 0, 3, 8, 15, ... is formed according to the formula n2 − 1 for the nth term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3905_chunk_0 | Integer-valued function | In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain. The floor and ceiling functions are examples of integer-valued functions of a real variable, but on real numbers and, generally, on (non-disconnected... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3906_chunk_0 | Numerical polynomial | In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P ( t ) {\displaystyle P(t)} is a polynomial whose value P ( n ) {\displaystyle P(n)} is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polyno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3907_chunk_0 | Integral curve | In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3908_chunk_0 | Definite Integrals | In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3909_chunk_0 | Definite Integrals | Today integration is used in a wide variety of scientific fields. The integrals enumerated here are called definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the hori... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3910_chunk_0 | Definite Integrals | Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3911_chunk_0 | Definite Integrals | Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3912_chunk_0 | Ehrhart polynomial | In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3913_chunk_0 | Kernel (integral operator) | In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3914_chunk_0 | Integrating factor technique | In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact diffe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3915_chunk_0 | Integrodifferential equation | In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3916_chunk_0 | Integrodifference equation | In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form: n t + 1 ( x ) = ∫ Ω k ( x , y ) f ( n t ( y ) ) d y , {\displaystyle n_{t+1}(x)=\int _{\Omega }k(x,y)\,f(n_{t}(y))\,dy,} where { n t } {\displaystyle \{n_{t}\}\,} is a sequence in the function space and Ω ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3917_chunk_0 | Integrodifference equation | In this case, n t ( x ) {\displaystyle n_{t}(x)} is the population size or density at location x {\displaystyle x} at time t {\displaystyle t} , f ( n t ( x ) ) {\displaystyle f(n_{t}(x))} describes the local population growth at location x {\displaystyle x} and k ( x , y ) {\displaystyle k(x,y)} , is the probability o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3918_chunk_0 | Interval contractor | In mathematics, an interval contractor (or contractor for short) associated to a set X {\displaystyle X} is an operator C {\displaystyle C} which associates to a hyperrectangle {\displaystyle } in R n {\displaystyle {\mathbf {R}}^{n}} another box C ( ) {\displaystyle C()} of R n {\displaystyle {\mathbf {R}}^{n}} such... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3919_chunk_0 | Invariant convex cone | In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant. For a simple Lie algebra, the existence of an invariant convex cone forces the Lie algebra ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3920_chunk_0 | Invariant convex cone | The dual cone with respect to the Killing form is the maximal invariant convex cone. Any intermediate cone is uniquely determined by its intersection with the Lie algebra of a maximal torus in a maximal compact subgroup. The intersection is invariant under the Weyl group of the maximal torus and the orbit of every poin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3921_chunk_0 | Invariant convex cone | They are naturally associated with Hermitian symmetric spaces and their associated holomorphic discrete series. The semigroup is made up of those elements in the complexification which, when acting on the Hermitian symmetric space of compact type, leave invariant the bounded domain corresponding to the noncompact dual. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3922_chunk_0 | Invariant convex cone | The semigroup acts by contraction operators on the holomorphic discrete series; its interior acts by Hilbert–Schmidt operators. The unitary part of their polar decomposition is the operator corresponding to an element in the original real Lie group, while the positive part is the exponential of an imaginary multiple of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3923_chunk_0 | Invariant set | More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3924_chunk_0 | Invariant measure | In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference of slopes is invariant under shear mapping.Ergodic theory ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3925_chunk_0 | Invariant polynomial | In mathematics, an invariant polynomial is a polynomial P {\displaystyle P} that is invariant under a group Γ {\displaystyle \Gamma } acting on a vector space V {\displaystyle V} . Therefore, P {\displaystyle P} is a Γ {\displaystyle \Gamma } -invariant polynomial if P ( γ x ) = P ( x ) {\displaystyle P(\gamma x)=P(x)}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3926_chunk_0 | Invariant vector | 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 |
wiki_3927_chunk_0 | 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, they play a central role in the study of alg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3928_chunk_0 | 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 |
wiki_3929_chunk_0 | 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 simply a consequence of the fact that any invert... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3930_chunk_0 | 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 integral domain R is said to be irreducible if it is no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3931_chunk_0 | 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 integral domain. For example, the polynomial 2 ( x 2 −... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3932_chunk_0 | 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 x^{2}+y^{n}-1,} for any positive integer n. A poly... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3933_chunk_0 | 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 object are studied. Examples of this are ordinals... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3934_chunk_0 | 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 ) {\displaystyle \pi _{1}(X)} if X {\displaystyle X} i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3935_chunk_0 | 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 the Ancient Greek: ἴσος isos "equal", and μορφή morp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3936_chunk_0 | 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 isomorphism between two structures is a canonical isomo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3937_chunk_0 | 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 |
wiki_3938_chunk_0 | 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 on the type of structure under consideration. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3939_chunk_0 | 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 |
wiki_3940_chunk_0 | 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 transformations.Category theory, which can be viewed as a formaliz... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3941_chunk_0 | 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 linear structure of the algebra. For a = b = c this is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3942_chunk_0 | 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 division algebras such as the octonions the two definitions... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3943_chunk_0 | 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 {\displaystyle v,w\in T_{p}M} , there is an isome... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3944_chunk_0 | 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 operation, and the extension of the multiplication operation to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3945_chunk_0 | 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 ∑ , ∏ , ⋃ , {\displaystyle \sum ,\ \prod ,\ \bigcup ,} and ⋂ {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3946_chunk_0 | Iterated map | In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f: X → X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3947_chunk_0 | 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 Sính was an in-depth study of 2-group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3948_chunk_0 | 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 ( x ) N ( y ) {\displaystyle N(xy)=N(x)N... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3949_chunk_0 | 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, alternativity being a weaker form of associativity... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3950_chunk_0 | 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 264) and repeated by Max Zorn. The prod... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3951_chunk_0 | 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 |
wiki_3952_chunk_0 | 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 can be utilized in an attempt to recon... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3953_chunk_0 | 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 Emmanuel Giroux (given below) that shows that contact geom... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3954_chunk_0 | Open region | These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, every subset can be open (the discrete topology), or no subset can be open except the space itself and the empty set (the indiscrete topology).In practice, however, open sets are usually chosen to provide a notion o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3955_chunk_0 | Operad | In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O {\displaystyle O} , one defines an algebra over O {\displaystyle O} to be a set toge... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3956_chunk_0 | Mathematical operations | In mathematics, an operation is a function which takes zero or more input values (also called "operands" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and mu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3957_chunk_0 | Mathematical operations | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3958_chunk_0 | 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 place of function when the domain is a set of function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3959_chunk_0 | 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 preserve properties, such as continuity. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3960_chunk_0 | 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 operation. This is related with the meaning of "operat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3961_chunk_0 | 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 typically used to colour two dimens... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3962_chunk_0 | 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 point x0 is an orbit of the isotropy group of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3963_chunk_0 | Order (ring theory) | In mathematics, an order in the sense of ring theory is a subring O {\displaystyle {\mathcal {O}}} of a ring A {\displaystyle A} , such that A {\displaystyle A} is a finite-dimensional algebra over the field Q {\displaystyle \mathbb {Q} } of rational numbers O {\displaystyle {\mathcal {O}}} spans A {\displaystyle A} ov... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3964_chunk_0 | 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 vector space over R {\displaystyle \mathbb {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3965_chunk_0 | 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 represents a vector v on one basis is, in g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3966_chunk_0 | 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} _{\mathrm {old} }} and x n e w {\displaystyle \m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3967_chunk_0 | 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 |
wiki_3968_chunk_0 | Ordered pairs | 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.) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3969_chunk_0 | 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 |
wiki_3970_chunk_0 | System of 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 those functions. The term "ordinary" is used in contrast with partial differenti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3971_chunk_0 | 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 any real n {\displaystyle n} , whereas others require that n {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3972_chunk_0 | 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 ordinary singular points. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3973_chunk_0 | 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 (clockwise) curve. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3974_chunk_0 | 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, hypersurface, etc.). Orientation of a curve is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3975_chunk_0 | 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 one demands that each trivialization map (which is a bundle map) ϕ U:... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3976_chunk_0 | 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 is, cup product) by the Euler class of an oriented bundle gives rise t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3977_chunk_0 | 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), probability theory, representation theory (of Lie groups, qua... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3978_chunk_0 | 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 {\displaystyle 2} such that the eigenspace u {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3979_chunk_0 | 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 orthogonal symmetric Lie algebra, and let p {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3980_chunk_0 | 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 bistochastic) if all its rows and columns sum to 1 and all it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3981_chunk_0 | 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 for any B = {\displaystyle B={\begin{bmatrix}a&1-a\\1-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3982_chunk_0 | 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 |
wiki_3983_chunk_0 | 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 an ultrafilter to bypass the need for re... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3984_chunk_0 | 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*-algebras and Exel–Laca algebras, giving a unified framew... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3985_chunk_0 | Ultrametric space | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3986_chunk_0 | 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 |
wiki_3987_chunk_0 | 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 |
wiki_3988_chunk_0 | 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 axiom, the axiom of pairing. More gener... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3989_chunk_0 | 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-Baghdadi (circa 1000 AD), who observed that both 2 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3990_chunk_0 | Ω-bounded space | In mathematics, an ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if P is some property of subspaces, then a P-bounded space is one in which every subspace with property P has compact closure. Every compact space is ω-bounded, and every ω-bounded space ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3991_chunk_0 | ∞-topos | In mathematics, an ∞-topos is, roughly, an ∞-category such that its objects behave like sheaves of spaces with some choice of Grothendieck topology; in other words, it gives an intrinsic notion of sheaves without reference to an external space. The prototypical example of an ∞-topos is the ∞-category of sheaves of spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3992_chunk_0 | Three dimensional space | In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z. Relative to these ax... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3993_chunk_0 | Plane coordinates | In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin. They are usually labeled x and y. Relative to these axes, the position of any point in two-dime... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3994_chunk_0 | Coordinate geometry | In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3995_chunk_0 | Coordinate geometry | It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3996_chunk_0 | Analytic Number Theory | In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithme... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3997_chunk_0 | Egyptian multiplication | It is still used in some areas.The second Egyptian multiplication and division technique was known from the hieratic Moscow and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe Ahmes.Although in ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3998_chunk_0 | Hand-waving | In mathematics, and disciplines in which mathematics plays a major role, hand-waving refers to either absence of formal proof or methods that do not meet mathematical rigor. In practice, it often involves the use of unrepresentative examples, unjustified assumptions, key omissions and faulty logic, and while these may ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3999_chunk_0 | Hand-waving | The objector in such a case might receive some measure credit for the theorem the hand-waver presented. The opposite of hand-waving in mathematics (and related fields) is sometimes called nose-following, which refers to the unimaginative development of a narrow line of reasoning that—while correct—can also end up makin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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