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wiki_4000_chunk_0 | Affine focal set | In mathematics, and especially affine differential geometry, the affine focal set of a smooth submanifold M embedded in a smooth manifold N is the caustic generated by the affine normal lines. It can be realised as the bifurcation set of a certain family of functions. The bifurcation set is the set of parameter values ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4001_chunk_0 | Affine focal set | Assume that M is an n-dimensional smooth hypersurface in real (n+1)-space. Assume that M has no points where the second fundamental form is degenerate. From the article affine differential geometry, there exists a unique transverse vector field over M. This is the affine normal vector field, or the Blaschke normal fiel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4002_chunk_0 | Bridgeland stability condition | In mathematics, and especially algebraic geometry, a Bridgeland stability condition, defined by Tom Bridgeland, is an algebro-geometric stability condition defined on elements of a triangulated category. The case of original interest and particular importance is when this triangulated category is the derived category o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4003_chunk_0 | Stability (algebraic geometry) | In mathematics, and especially algebraic geometry, stability is a notion which characterises when a geometric object, for example a point, an algebraic variety, a vector bundle, or a sheaf, has some desirable properties for the purpose of classifying them. The exact characterisation of what it means to be stable depend... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4004_chunk_0 | K-energy functional | In mathematics, and especially complex geometry, the Mabuchi functional or K-energy functional is a functional on the space of Kähler potentials of a compact Kähler manifold whose critical points are constant scalar curvature Kähler metrics. The Mabuchi functional was introduced by Toshiki Mabuchi in 1985 as a function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4005_chunk_0 | Holomorphic tangent bundle | In mathematics, and especially complex geometry, the holomorphic tangent bundle of a complex manifold M {\displaystyle M} is the holomorphic analogue of the tangent bundle of a smooth manifold. The fibre of the holomorphic tangent bundle over a point is the holomorphic tangent space, which is the tangent space of the u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4006_chunk_0 | K-stability | In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian and reformulated more algebraically later by Simon Donaldson. The definition was... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4007_chunk_0 | Stable principal bundle | In mathematics, and especially differential geometry and algebraic geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles. The concept of stability for principal bundles was introduced by Annamalai Ramanathan for the purpose of defining the mo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4008_chunk_0 | Connection (principal bundle) | In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4009_chunk_0 | Connection on a vector bundle | In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for whic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4010_chunk_0 | Connection on a vector bundle | Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear. Linear connections are also called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them (Koszul 1950). This article defines the connection on a vector bundle using a common mathem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4011_chunk_0 | Gauge theory (mathematics) | In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4012_chunk_0 | Gauge theory (mathematics) | These are differential equations involving connections on vector bundles or principal bundles, or involving sections of vector bundles, and so there are strong links between gauge theory and geometric analysis. These equations are often physically meaningful, corresponding to important concepts in quantum field theory ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4013_chunk_0 | Affine sphere | In mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point. The term affine sphere is used because they play an analogous role in affine differential geometry to that of ordinary spheres in Euclidean differential geometry. An aff... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4014_chunk_0 | Quillen metric | In mathematics, and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel Quillen for certain elliptic operators over a Riemann surface, and generalized to higher-dimensional manifolds by Jean-Michel Bismut and Dan Freed.The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4015_chunk_0 | Donaldson theorem | In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4016_chunk_0 | Donaldson invariant | In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4017_chunk_0 | Bogomolny equation | In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation F A = ⋆ d A Φ , {\displaystyle F_{A}=\star d_{A}\Phi ,} where F A {\displaystyle F_{A}} is the curvature of a connection A {\displaystyle A} on a principal G {\displaystyle G} -bundle over a 3-manifold M {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4018_chunk_0 | Intersection multiplicity | In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4019_chunk_0 | Intersection multiplicity | The complexity enters when calculating intersections at points of tangency, and intersections which are not just points, but have higher dimension. For example, if a plane is tangent to a surface along a line, the intersection number along the line should be at least two. These questions are discussed systematically in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4020_chunk_0 | Mathematical diagram | In mathematics, and especially in category theory, a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition. Commutative diagrams play the role i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4021_chunk_0 | Commutative diagrams | In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4022_chunk_0 | Icosahedral group | In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4023_chunk_0 | Nucleus (order theory) | In mathematics, and especially in order theory, a nucleus is a function F {\displaystyle F} on a meet-semilattice A {\displaystyle {\mathfrak {A}}} such that (for every p {\displaystyle p} in A {\displaystyle {\mathfrak {A}}} ): p ≤ F ( p ) {\displaystyle p\leq F(p)} F ( F ( p ) ) = F ( p ) {\displaystyle F(F(p))=F(p)}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4024_chunk_0 | Thomas–Yau conjecture | In mathematics, and especially symplectic geometry, the Thomas–Yau conjecture asks for the existence of a stability condition, similar to those which appear in algebraic geometry, which guarantees the existence of a solution to the special Lagrangian equation inside a Hamiltonian isotopy class of Lagrangian submanifold... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4025_chunk_0 | Thomas–Yau conjecture | The conjecture is intimately related to mirror symmetry, a conjecture in string theory and mathematical physics which predicts that mirror to a symplectic manifold (which is a Calabi–Yau manifold) there should be another Calabi–Yau manifold for which the symplectic structure is interchanged with the complex structure. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4026_chunk_0 | Schur indicator | In mathematics, and especially the discipline of representation theory, the Schur indicator, named after Issai Schur, or Frobenius–Schur indicator describes what invariant bilinear forms a given irreducible representation of a compact group on a complex vector space has. It can be used to classify the irreducible repre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4027_chunk_0 | Pinched torus | In mathematics, and especially topology and differential geometry, a pinched torus (or croissant surface) is a kind of two-dimensional surface. It gets its name from its resemblance to a torus that has been pinched at a single point. A pinched torus is an example of an orientable, compact 2-dimensional pseudomanifold. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4028_chunk_0 | Bound variable | In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. The terms are opposites. A free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4029_chunk_0 | Bound variable | Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol. In computer programming, the term free variable... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4030_chunk_0 | Bound variable | The term non-local variable is often a synonym in this context. An instance of a variable symbol is bound, in contrast, if the value of that variable symbol has been bound to a specific value or range of values in the domain of discourse or universe. This may be achieved through the use of logical quantifiers, variable... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4031_chunk_0 | Bound variable | A variable symbol overall is bound if at least one occurrence of it is bound.pp.142--143 Since the same variable symbol may appear in multiple places in an expression, some occurrences of the variable symbol may be free while others are bound,p.78 hence "free" and "bound" are at first defined for occurrences and then g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4032_chunk_0 | Bound variable | For example, consider the following expression in which both variables are bound by logical quantifiers: ∀ y ∃ x ( x = y ) . {\displaystyle \forall y\,\exists x\,\left(x={\sqrt {y}}\right).} This expression evaluates to false if the domain of x {\displaystyle x} and y {\displaystyle y} is the real numbers, but true if ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4033_chunk_0 | K-stability of Fano varieties | In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular importance for the case of Fano varieties, where it is the correct stability condition to allow the formation of moduli spa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4034_chunk_0 | K-stability of Fano varieties | K-stability is important in constructing moduli spaces of Fano varieties, where observations going back to the original development of geometric invariant theory show that it is necessary to restrict to a class of stable objects to form good moduli. It is now known through the work of Chenyang Xu and others that there ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4035_chunk_0 | Analytic variety | In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties are locally ringed spaces which are locally isomorphic to local model spaces... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4036_chunk_0 | Hitchin's equations | In mathematics, and in particular differential geometry and gauge theory, Hitchin's equations are a system of partial differential equations for a connection and Higgs field on a vector bundle or principal bundle over a Riemann surface, written down by Nigel Hitchin in 1987. Hitchin's equations are locally equivalent t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4037_chunk_0 | Hitchin's equations | The moduli space of solutions to Hitchin's equations was constructed by Hitchin in the rank two case on a compact Riemann surface and was one of the first examples of a hyperkähler manifold constructed. The nonabelian Hodge correspondence shows it is isomorphic to the Higgs bundle moduli space, and to the moduli space ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4038_chunk_0 | Unilateral shift | In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function x ↦ f(x) to its translation x ↦ f(x + a). In time series analysis, the shift operator is called the lag operator. Shift operators are examples of linear operators, impo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4039_chunk_0 | Unilateral shift | The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4040_chunk_0 | Tensor product of Hilbert spaces | In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4041_chunk_0 | Sion's minimax theorem | In mathematics, and in particular game theory, Sion's minimax theorem is a generalization of John von Neumann's minimax theorem, named after Maurice Sion. It states: Let X {\displaystyle X} be a compact convex subset of a linear topological space and Y {\displaystyle Y} a convex subset of a linear topological space. If... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4042_chunk_0 | Hermitian Yang–Mills connection | In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations: namely, the contraction o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4043_chunk_0 | Hypercovering | In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover U → X {\displaystyle {\mathcal {U}}\to X} , one can show that if the space X {\displaystyle X} is compact and if every intersection of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4044_chunk_0 | Salem–Spencer set | In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also called 3-AP-free sequences or progression-free sets. They have also been called non-averaging sets, but this term has also been used to den... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4045_chunk_0 | Combinatorial number system | In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) N and k-combinations. The combinations are repres... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4046_chunk_0 | Combinatorial number system | Indeed, a greedy algorithm finds the k-combination corresponding to N: take ck maximal with ( c k k ) ≤ N {\displaystyle {\tbinom {c_{k}}{k}}\leq N} , then take ck−1 maximal with ( c k − 1 k − 1 ) ≤ N − ( c k k ) {\displaystyle {\tbinom {c_{k-1}}{k-1}}\leq N-{\tbinom {c_{k}}{k}}} , and so forth. Finding the number N, u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4047_chunk_0 | Gagliardo–Nirenberg interpolation inequality | In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the L p {\displaystyle L^{p}} -norms of different weak derivatives of a function through an interpolation inequality. The theorem is of particular importa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4048_chunk_0 | Hilbert–Poincaré series | In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4049_chunk_0 | No-ghost theorem | In mathematics, and in particular in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem describing properties of a functor that quantizes bosonic strings. It is named after Peter Goddard and Charles Thorn. The name "no-ghost theorem" stems from the fa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4050_chunk_0 | Dym equation | In mathematics, and in particular in the theory of solitons, the Dym equation (HD) is the third-order partial differential equation u t = u 3 u x x x . {\displaystyle u_{t}=u^{3}u_{xxx}.\,} It is often written in the equivalent form for some function v of one space variable and time v t = ( v − 1 / 2 ) x x x . {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4051_chunk_0 | Dym equation | HD is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It obeys an infinite number of conservation laws; it does not possess the Painlevé property. The Dym equation has strong links to the Korteweg–de Vries equation. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4052_chunk_0 | Dym equation | C.S. Gardner, J.M. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4053_chunk_0 | Dym equation | Greene, Kruskal and R.M. Miura applied to the solution of corresponding problem in Korteweg–de Vries equation. The Lax pair of the Harry Dym equation is associated with the Sturm–Liouville operator. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4054_chunk_0 | Dym equation | The Liouville transformation transforms this operator isospectrally into the Schrödinger operator. Thus by the inverse Liouville transformation solutions of the Korteweg–de Vries equation are transformed into solutions of the Dym equation. An explicit solution of the Dym equation, valid in a finite interval, is found b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4055_chunk_0 | Moore-Penrose inverse | In mathematics, and in particular linear algebra, the Moore–Penrose inverse A + {\displaystyle A^{+}} of a matrix A {\displaystyle A} is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roger Penrose in 1955. Earlier, Erik Iv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4056_chunk_0 | Moore-Penrose inverse | The pseudoinverse facilitates the statement and proof of results in linear algebra. The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition. In the special case where A {\displaystyle A} is a normal matrix (for example... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4057_chunk_0 | Decomposition matrix | In mathematics, and in particular modular representation theory, a decomposition matrix is a matrix that results from writing the irreducible ordinary characters in terms of the irreducible modular characters, where the entries of the two sets of characters are taken to be over all conjugacy classes of elements of orde... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4058_chunk_0 | Green's matrix | In mathematics, and in particular ordinary differential equations, a Green's matrix helps to determine a particular solution to a first-order inhomogeneous linear system of ODEs. The concept is named after George Green. For instance, consider x ′ = A ( t ) x + g ( t ) {\displaystyle x'=A(t)x+g(t)\,} where x {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4059_chunk_0 | Green's matrix | {\displaystyle X(t)=\left.\,} Now X ( t ) {\displaystyle X(t)\,} is an n × n {\displaystyle n\times n\,} matrix solution of X ′ = A X {\displaystyle X'=AX\,} . This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogeneous equatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4060_chunk_0 | Green's matrix | Now, x ′ = X ′ y + X y ′ = A X y + X y ′ = A x + X y ′ . {\displaystyle {\begin{aligned}x'&=X'y+Xy'\\&=AXy+Xy'\\&=Ax+Xy'.\end{aligned}}} This implies X y ′ = g {\displaystyle Xy'=g\,} or y = c + ∫ a t X − 1 ( s ) g ( s ) d s {\displaystyle y=c+\int _{a}^{t}X^{-1}(s)g(s)\,ds\,} where c {\displaystyle c\,} is an arbitrar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4061_chunk_0 | Green's matrix | {\displaystyle x=X(t)c+X(t)\int _{a}^{t}X^{-1}(s)g(s)\,ds.\,} The first term is the homogeneous solution and the second term is the particular solution. Now define the Green's matrix G 0 ( t , s ) = { 0 t ≤ s ≤ b X ( t ) X − 1 ( s ) a ≤ s < t . {\displaystyle G_{0}(t,s)={\begin{cases}0&t\leq s\leq b\\X(t)X^{-1}(s)&a\le... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4062_chunk_0 | Frobenius reciprocity | In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting. It can be used to leverage knowledge about representations of a subgroup to find and classify representations of "large" groups that contain them. It is name... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4063_chunk_0 | Ak singularity | In mathematics, and in particular singularity theory, an Ak singularity, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold. Let f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be a smooth function. We denote by Ω ( R n , R ) {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4064_chunk_0 | Ak singularity | Let diff ( R n ) {\displaystyle \operatorname {diff} (\mathbb {R} ^{n})} denote the infinite-dimensional Lie group of diffeomorphisms R n → R n , {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{n},} and diff ( R ) {\displaystyle \operatorname {diff} (\mathbb {R} )} the infinite-dimensional Lie group of diffeomorph... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4065_chunk_0 | Ak singularity | A function f is said to have a type Ak-singularity if it lies in the orbit of f ( x 1 , … , x n ) = 1 + ε 1 x 1 2 + ⋯ + ε n − 1 x n − 1 2 ± x n k + 1 {\displaystyle f(x_{1},\ldots ,x_{n})=1+\varepsilon _{1}x_{1}^{2}+\cdots +\varepsilon _{n-1}x_{n-1}^{2}\pm x_{n}^{k+1}} where ε i = ± 1 {\displaystyle \varepsilon _{i}=\p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4066_chunk_0 | Hobby–Rice theorem | In mathematics, and in particular the necklace splitting problem, the Hobby–Rice theorem is a result that is useful in establishing the existence of certain solutions. It was proved in 1965 by Charles R. Hobby and John R. Rice; a simplified proof was given in 1976 by A. Pinkus. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4067_chunk_0 | Equianharmonic | In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g2 = 0 and g3 = 1. This page follows the terminology of Abramowitz and Stegun; see also the lemniscatic case. (These are special examples of complex multiplication.) In t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4068_chunk_0 | Akivis algebra | In mathematics, and in particular the study of algebra, an Akivis algebra is a nonassociative algebra equipped with a binary operator, the commutator {\displaystyle } and a ternary operator, the associator {\displaystyle } that satisfy a particular relationship known as the Akivis identity. They are named in honour o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4069_chunk_0 | Akivis algebra | {\displaystyle \left,z\right]+\left,x\right]+\left,y\right]=\left+\left+\left-\left-\left-\left.} An Akivis algebra with = 0 {\displaystyle \left=0} is a Lie algebra, for the Akivis identity reduces to the Jacobi identity. Note that the terms on the right hand side have positive sign for even permutations and negative... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4070_chunk_0 | Graph continuous function | In mathematics, and in particular the study of game theory, a function is graph continuous if it exhibits the following properties. The concept was originally defined by Partha Dasgupta and Eric Maskin in 1986 and is a version of continuity that finds application in the study of continuous games. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4071_chunk_0 | N-ary group | In mathematics, and in particular universal algebra, the concept of an n-ary group (also called n-group or multiary group) is a generalization of the concept of a group to a set G with an n-ary operation instead of a binary operation. By an n-ary operation is meant any map f: Gn → G from the n-th Cartesian power of G t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4072_chunk_0 | Pseudo inverse | In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4073_chunk_0 | Pseudo inverse | A matrix A g ∈ R n × m {\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{n\times m}} is a generalized inverse of a matrix A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} if A A g A = A . {\displaystyle AA^{\mathrm {g} }A=A.} A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4074_chunk_0 | Perron formula | In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4075_chunk_0 | Primorial | In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubne... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4076_chunk_0 | Eberhard's theorem | In mathematics, and more particularly in polyhedral combinatorics, Eberhard's theorem partially characterizes the multisets of polygons that can form the faces of simple convex polyhedra. It states that, for given numbers of triangles, quadrilaterals, pentagons, heptagons, and other polygons other than hexagons, there ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4077_chunk_0 | Refinement (topology) | In mathematics, and more particularly in set theory, a cover (or covering) of a set X {\displaystyle X} is a family of subsets of X {\displaystyle X} whose union is all of X {\displaystyle X} . More formally, if C = { U α: α ∈ A } {\displaystyle C=\lbrace U_{\alpha }:\alpha \in A\rbrace } is an indexed family of subset... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4078_chunk_0 | Restricted partial quotients | In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction x is said to be restricted, or composed of restricted partial quotients, if the sequence of denominators of its partial quotients is bounded; that is x = = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4079_chunk_0 | Variety of finite semigroups | In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups having some nice algebraic properties. Those classes can be defined in two distinct ways, using either algebraic notions or topological notions. Varieties of finite monoids, varieties of finite ordered semigr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4080_chunk_0 | Dehn-Nielsen theorem | In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is of fundamental importance for the study of 3-mani... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4081_chunk_0 | Involutive algebra | In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4082_chunk_0 | Pseudo-ring | In mathematics, and more specifically in abstract algebra, a pseudo-ring is one of the following variants of a ring: A rng, i.e., a structure satisfying all the axioms of a ring except for the existence of a multiplicative identity. A set R with two binary operations + and ⋅ such that (R, +) is an abelian group with id... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4083_chunk_0 | Rng (algebra) | In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term rng (IPA: ) is meant to suggest that it is a ring without i, that is, without... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4084_chunk_0 | Self adjoint | In mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if x ∗ = x {\displaystyle x^{*}=x} . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoint if it is closed under the involu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4085_chunk_0 | Self adjoint | For example, if x ∗ = y {\displaystyle x^{*}=y} then since y ∗ = x ∗ ∗ = x {\displaystyle y^{*}=x^{**}=x} in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements. In functional analysis, a linear operator A: H → H {\displaystyle A:H\to H} on a Hilbert space is called... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4086_chunk_0 | Self adjoint | If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is self-adjoint if and only if the matrix describing A with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint. In a dagger catego... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4087_chunk_0 | Euler–Poincaré characteristic | In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4088_chunk_0 | Euler–Poincaré characteristic | It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4089_chunk_0 | Holonomic function | In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4090_chunk_0 | Holonomic function | When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called holonomic. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4091_chunk_0 | Zero-divisor graph | In mathematics, and more specifically in combinatorial commutative algebra, a zero-divisor graph is an undirected graph representing the zero divisors of a commutative ring. It has elements of the ring as its vertices, and pairs of elements whose product is zero as its edges. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4092_chunk_0 | Regular chain | In mathematics, and more specifically in computer algebra and elimination theory, a regular chain is a particular kind of triangular set of multivariate polynomials over a field, where a triangular set is a finite sequence of polynomials such that each one contains at least one more indeterminate than the preceding one... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4093_chunk_0 | Multivariate division algorithm | In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K over a field K. A Gröbner basis allows many important properties of the ideal and the associated a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4094_chunk_0 | Multivariate division algorithm | He named them after his advisor Wolfgang Gröbner. In 2007, Buchberger received the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work. However, the Russian mathematician Nikolai Günther had introduced a similar notion in 1913, published in various Russian mathematical journal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4095_chunk_0 | Multivariate division algorithm | These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. An analogous concept for multivariate power series was developed independently by Heisuke Hironaka in 1964, who named them standard bases. This term has been used by some authors to also denote Gröbn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4096_chunk_0 | Hermitian manifold | In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4097_chunk_0 | Parametrization invariance | In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitizati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4098_chunk_0 | Parametrization invariance | The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system. For example, the position of a point that moves on a curve in t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4099_chunk_0 | Parametrization invariance | If x, y, z are the coordinates of the point, the movement is thus described by a parametric equation x = f ( t ) y = g ( t ) z = h ( t ) , {\displaystyle {\begin{aligned}x&=f(t)\\y&=g(t)\\z&=h(t),\end{aligned}}} where t is the parameter and denotes the time. Such a parametric equation completely determines the curve, w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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