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Linked set Summary Linked_set In mathematics, an upwards linked set A is a subset of a partially ordered set, P, in which any two of elements A have a common upper bound in P. Similarly, every pair of elements of a downwards linked set has a lower bound. Every centered set is linked, which includes, in particular, ever... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Eta set Summary Η_set In mathematics, an η set (eta set) is a type of totally ordered set introduced by Hausdorff (1907, p. 126, 1914, chapter 6 section 8) that generalizes the order type η of the rational numbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ω-bounded space Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
∞-topos Summary Lurie_characterization_theorem 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Three dimensional space Coordinate systems 3D_space > In Euclidean geometry > Coordinate systems 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Plane coordinates Coordinate systems Plane_(geometry) > In geometry > Coordinate systems 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coordinate geometry Summary 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, sp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Coordinate geometry Summary 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Analytic Number Theory Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Egyptian multiplication Summary Ancient_Egyptian_multiplication In mathematics, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two multiplication methods used by scribes, is a systematic method for multiplying ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Egyptian multiplication Summary Ancient_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 b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hand-waving In mathematics (and formal logic, philosophy, theoretical science) Hand-waving > In mathematics (and formal logic, philosophy, theoretical science) 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 mathem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hand-waving In mathematics (and formal logic, philosophy, theoretical science) Hand-waving > In mathematics (and formal logic, philosophy, theoretical science) 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Affine focal set Summary 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 bifur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Affine focal set Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bridgeland stability condition Summary 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stability (algebraic geometry) Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
K-energy functional Summary 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 int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Holomorphic tangent bundle Summary Holomorphic_cotangent_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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
K-stability Summary 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 Sim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Stable principal bundle Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Connection (principal bundle) Summary Principal_connection 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Connection on a vector bundle Summary Connection_(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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Connection on a vector bundle Summary Connection_(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 articl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gauge theory (mathematics) Summary 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 relat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gauge theory (mathematics) Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Affine sphere Summary 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 Euc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quillen metric Summary 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 b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Donaldson theorem Summary 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 di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Whitney topology Summary Whitney_topologies In mathematics, and especially differential topology, functional analysis and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mappings between two smooth manifolds. They are named after the American mathema... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Donaldson invariant Summary Donaldson's_polynomial_invariants 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 possibl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Seiberg–Witten equations Summary Seiberg-Witten_equations In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten (1994), using the Seiberg–Witten theory studied by Nathan Seiberg and Witten (1994a, 1994b) during their inve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Seiberg–Witten equations Summary Seiberg-Witten_equations For detailed descriptions of Seiberg–Witten invariants see (Donaldson 1996), (Moore 2001), (Morgan 1996), (Nicolaescu 2000), (Scorpan 2005, Chapter 10). For the relation to symplectic manifolds and Gromov–Witten invariants see (Taubes 2000). For the early histor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bogomolny equation Summary 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 {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Euclidean topology Summary Euclidean_topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} by the Euclidean metric. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Interlocking interval topology Summary Interlocking_interval_topology In mathematics, and especially general topology, the interlocking interval topology is an example of a topology on the set S := R+ \ Z+, i.e. the set of all positive real numbers that are not positive whole numbers. To give the set S a topology means... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Intersection multiplicity Summary Intersection_number_(algebraic_geometry) 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Intersection multiplicity Summary Intersection_number_(algebraic_geometry) 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 l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical diagram Commutative diagram Mathematical_diagram > Specific types of mathematical diagrams > Commutative 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Commutative diagrams Summary Commutative_square 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 equation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Icosahedral group Summary 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 triacont... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Crossed module Summary Crossed_module In mathematics, and especially in homotopy theory, a crossed module consists of groups G {\displaystyle G} and H {\displaystyle H} , where G {\displaystyle G} acts on H {\displaystyle H} by automorphisms (which we will write on the left, ( g , h ) ↦ g ⋅ h {\displaystyle (g,h)\mapst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nucleus (order theory) Summary 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)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Thomas–Yau conjecture Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Thomas–Yau conjecture Summary 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 st... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Schur indicator Summary 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 b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pinched torus Summary 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, co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Poincaré complex Summary Poincaré_complex In mathematics, and especially topology, a Poincaré complex (named after the mathematician Henri Poincaré) is an abstraction of the singular chain complex of a closed, orientable manifold. The singular homology and cohomology groups of a closed, orientable manifold are related ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pytkeev space Summary Pytkeev_space In mathematics, and especially topology, a Pytkeev space is a topological space that satisfies qualities more subtle than a convergence of a sequence. They are named after E. G. Pytkeev, who proved in 1983 that sequential spaces have this property. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bound variable Summary Free_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 sub... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bound variable Summary Free_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 compute... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bound variable Summary Free_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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bound variable Summary Free_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 fir... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Bound variable Summary Free_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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
K-stability of Fano varieties Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
K-stability of Fano varieties Summary K-stability_of_Fano_varieties Tian's definition of K-stability was reformulated by Simon Donaldson in 2001 in a purely algebro-geometric way.K-stability has become an important notion in the study and classification of Fano varieties. In 2012 Xiuxiong Chen, Donaldson, and Song Sun ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
K-stability of Fano varieties Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Analytic variety Summary Analytic_varieties 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hitchin's equations Summary 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 19... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hitchin's equations Summary 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 Hig... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unilateral shift Summary 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 operat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Unilateral shift Summary 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 integ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product of Hilbert spaces Summary 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 s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sion's minimax theorem Summary 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 {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hermitian Yang–Mills connection Summary Hermitian–Einstein_metric 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 satisf... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hypercovering Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Salem–Spencer set Summary 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 s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combinatorial number system Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combinatorial number system Summary Combinatorial_number_system The numbers less than ( n k ) {\displaystyle {\tbinom {n}{k}}} correspond to all k-combinations of {0, 1, ..., n − 1}. The correspondence does not depend on the size n of the set that the k-combinations are taken from, so it can be interpreted as a map fro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Combinatorial number system Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Adjacent transposition Summary Anticyclic_permutation In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has k elements, it may be called a k-cycle. Some authors wide... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Adjacent transposition Summary Anticyclic_permutation For example, the permutation (1 3 2 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a 4-cycle, and the permutation (1 3 2)(4) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 is considered a 3-cycle by some authors. On the other hand, the permutation (1 3)(2 4) that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Gagliardo–Nirenberg interpolation inequality Summary 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 d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hilbert–Poincaré series Summary Hilbert–Serre_theorem 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 (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
No-ghost theorem Summary Goddard–Thorn_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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dym equation Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dym equation Summary 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 |
Dym equation Summary Dym_equation C.S. Gardner, J.M. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dym equation Summary 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 |
Dym equation Summary 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, vali... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Moore-Penrose inverse Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Moore-Penrose inverse Summary Moore–Penrose_inverse The term generalized inverse is sometimes used as a synonym for pseudoinverse. A common use of the pseudoinverse is to compute a "best fit" (least squares) solution to a system of linear equations that lacks a solution (see below under § Applications). Another use is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Moore-Penrose inverse Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime model Summary Prime_model In mathematics, and in particular model theory, a prime model is a model that is as simple as possible. Specifically, a model P {\displaystyle P} is prime if it admits an elementary embedding into any model M {\displaystyle M} to which it is elementarily equivalent (that is, into any mod... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Decomposition matrix Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Green's matrix Summary 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 ) {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Green's matrix Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Green's matrix Summary 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\,} wher... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Green's matrix Summary 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{cas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Frobenius reciprocity Summary 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 representat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ak singularity Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ak singularity Summary 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 infinit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ak singularity Summary Ak_singularity We define the group action as follows: ( φ , ψ ) ⋅ f := ψ ∘ f ∘ φ − 1 {\displaystyle (\varphi ,\psi )\cdot f:=\psi \circ f\circ \varphi ^{-1}} The orbit of f , denoted orb(f), of this group action is given by orb ( f ) = { ψ ∘ f ∘ φ − 1: φ ∈ diff ( R n ) , ψ ∈ diff ( R ) } . {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ak singularity Summary 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 = ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hobby–Rice theorem Summary 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. Pin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Crinkled arc Summary Crinkled_arc In mathematics, and in particular the study of Hilbert spaces, a crinkled arc is a type of continuous curve. The concept is usually credited to Paul Halmos. Specifically, consider f: → X , {\displaystyle f\colon \to X,} where X {\displaystyle X} is a Hilbert space with inner product ⟨... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equianharmonic Summary 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 exa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equianharmonic Summary Equianharmonic The half period is ω 1 = 1 2 ( − 1 + 3 i ) ω 2 . {\displaystyle \omega _{1}={\tfrac {1}{2}}(-1+{\sqrt {3}}i)\omega _{2}.} Here the period lattice is a real multiple of the Eisenstein integers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Equianharmonic Summary Equianharmonic The constants e1, e2 and e3 are given by e 1 = 4 − 1 / 3 e ( 2 / 3 ) π i , e 2 = 4 − 1 / 3 , e 3 = 4 − 1 / 3 e − ( 2 / 3 ) π i . {\displaystyle e_{1}=4^{-1/3}e^{(2/3)\pi i},\qquad e_{2}=4^{-1/3},\qquad e_{3}=4^{-1/3}e^{-(2/3)\pi i}.} The case g2 = 0, g3 = a may be handled by a scal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Akivis algebra Summary 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 Akivi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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