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c_x3k73ldrsrws | 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, every directed set. | Linked set |
c_9fbyh8qbovtw | 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. | Eta set |
c_x5qajzyxmgn8 | 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 ... | Ω-bounded space |
c_g4hncvd5hqul | 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... | ∞-topos |
c_htz4jiasqn4y | 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... | Three dimensional space |
c_2vxa4f27g9zq | 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... | Plane coordinates |
c_rqt8sfd12mnp | 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. | Coordinate geometry |
c_bim7v7s1j0cb | 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 ... | Coordinate geometry |
c_z3ry9ggjktb1 | 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... | Analytic Number Theory |
c_z5c17fwzjh45 | 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 two numbers that does not require the multiplication table, only... | Egyptian multiplication |
c_qhysa8owm54n | 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... | Egyptian multiplication |
c_v55kwj6t6kc9 | 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 ... | Hand-waving |
c_kg2uymlqqygr | 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... | Hand-waving |
c_jqnqgjriokci | 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 ... | Affine focal set |
c_sxtoy83fymfo | 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... | Affine focal set |
c_g4fe3np7zn0z | 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... | Bridgeland stability condition |
c_3ibnkstxsnpc | 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... | Stability (algebraic geometry) |
c_zr5vg02gqv69 | 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... | K-energy functional |
c_xsqd69v8n1tq | 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... | Holomorphic tangent bundle |
c_at93la0dqewc | 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... | K-stability |
c_1ay5yfcpc1ki | 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... | Stable principal bundle |
c_qg7vr1s7b6gr | 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... | Connection (principal bundle) |
c_l26b2pt4s8e6 | 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... | Connection on a vector bundle |
c_33rtyi508zs4 | 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... | Connection on a vector bundle |
c_l5tcyglm9vem | 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... | Gauge theory (mathematics) |
c_u94cu8j1gwpn | 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 ... | Gauge theory (mathematics) |
c_5lb00btcf0n4 | 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... | Affine sphere |
c_mzzukkxxkjw7 | 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... | Quillen metric |
c_13yfl2g7foow | 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 ... | Donaldson theorem |
c_uwvuxrwcspal | 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 mathematician Hassler Whitney. | Whitney topology |
c_njm60jq4dwlc | 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 ... | Donaldson invariant |
c_ahdjxv9ozvtk | 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 investigations of Seiberg–Witten gauge theory. Seiberg–Witten ... | Seiberg–Witten equations |
c_uvazqovoprwp | 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 history see (Jackson 1995). | Seiberg–Witten equations |
c_4u8f17cmmvn2 | 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... | Bogomolny equation |
c_6dte7epp46nb | 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. | Euclidean topology |
c_amrghseq8lh3 | 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 to say which subsets of S are "open", and to do so in a way that the ... | Interlocking interval topology |
c_mi2ybpaff86w | 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... | Intersection multiplicity |
c_9d9k1vtg4dtb | 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... | Intersection multiplicity |
c_n1uik9b3u8hn | 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... | Mathematical diagram |
c_8m5rt43xhj1o | 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. | Commutative diagrams |
c_68ti93to9lt2 | 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... | Icosahedral group |
c_yzgxlfnqi7os | 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)\mapsto g\cdot h} , and a homomorphism of gr... | Crossed module |
c_j27neim7mhww | 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)}... | Nucleus (order theory) |
c_0100o1cvsklu | 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... | Thomas–Yau conjecture |
c_nxu41gfpba9g | 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. ... | Thomas–Yau conjecture |
c_5am9l8kytqz2 | 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... | Schur indicator |
c_fsddynv044ze | 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. | Pinched torus |
c_2l9iyz8s06vd | 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 by Poincaré duality. Poincaré duality is a... | Poincaré complex |
c_9nvomgkmxca0 | 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. | Pytkeev space |
c_bqkajmanzlis | 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... | Bound variable |
c_zrvxpb0fq9mp | 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... | Bound variable |
c_p65328hrylls | 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... | Bound variable |
c_ev09aqn5h5vn | 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... | Bound variable |
c_0ocsbl8plspe | 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 ... | Bound variable |
c_9hk8hlxzivq5 | 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... | K-stability of Fano varieties |
c_rwd0c75ion07 | 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 and independently Gang Tian proved that a smooth Fano manifold is K-... | K-stability of Fano varieties |
c_ldf8a27jtc6m | 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 ... | K-stability of Fano varieties |
c_69pccz7t5dhf | 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... | Analytic variety |
c_qq3hmbn5gcuv | 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... | Hitchin's equations |
c_98icifeprdnc | 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 ... | Hitchin's equations |
c_tol1dw0apdao | 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... | Unilateral shift |
c_ydxf7ptghas6 | 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 ... | Unilateral shift |
c_69iz55oet7ae | 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... | Tensor product of Hilbert spaces |
c_n10p1xzbt2ih | 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... | Sion's minimax theorem |
c_fup3wbilfwhd | 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... | Hermitian Yang–Mills connection |
c_7m7pvfonb0p5 | 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... | Hypercovering |
c_ziw7jbtmwuzq | 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... | Salem–Spencer set |
c_nt4b78h3pvoq | 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... | Combinatorial number system |
c_fjjtalfzuhj8 | 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 from N to the k-combinations taken from N; in this view the corresp... | Combinatorial number system |
c_wqh3ywe6wdz1 | 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... | Combinatorial number system |
c_6odis627tj9o | 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 widen this definition to include permutations with fixed p... | Adjacent transposition |
c_50gjzyn5kqeb | 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 sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cycli... | Adjacent transposition |
c_u0uxxrwylsvr | 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... | Gagliardo–Nirenberg interpolation inequality |
c_o8xe1gljofyc | 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... | Hilbert–Poincaré series |
c_51kk7utycx3h | 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... | No-ghost theorem |
c_cd3trnhcv69j | 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... | Dym equation |
c_7ri87znkfcnt | 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. | Dym equation |
c_kem6ijcwww2q | C.S. Gardner, J.M. | Dym equation |
c_krvz9uo5pj01 | 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. | Dym equation |
c_w6y1x89y1as0 | 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... | Dym equation |
c_r4a8655vuuxa | 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... | Moore-Penrose inverse |
c_en3g1t66ep6t | 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 to find the minimum (Euclidean) norm solution to a s... | Moore-Penrose inverse |
c_o6wpn072ql67 | 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... | Moore-Penrose inverse |
c_48ooc15mynm9 | 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 model M {\displaystyle M} satisfyin... | Prime model |
c_65f0nk79m78c | 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... | Decomposition matrix |
c_u1qumdjvl68g | 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... | Green's matrix |
c_avzjo88mfmz4 | {\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... | Green's matrix |
c_gaw9m9tlp2ye | 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... | Green's matrix |
c_ps8kt2wkpfyy | {\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... | Green's matrix |
c_zgzaz8teohos | 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... | Frobenius reciprocity |
c_2qf3mxjmfboo | 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... | Ak singularity |
c_v7mho31tvrpo | 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... | Ak singularity |
c_45xhhicecptg | 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 ) } . {\displaystyle {\mbox{orb}}(f)=\{\psi \circ ... | Ak singularity |
c_k7nnbwo2ztn0 | 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... | Ak singularity |
c_pltnlegzvthp | 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. | Hobby–Rice theorem |
c_s61cd8nrg20j | 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 ⟨ ⋅ , ⋅ ⟩ . {\displaystyle \langle ... | Crinkled arc |
c_g4981uuurq1b | 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... | Equianharmonic |
c_du6geh9vgdtu | 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. | Equianharmonic |
c_9h3zta2ahw2k | 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 scaling transformation. | Equianharmonic |
c_w6ucgyd10q8w | 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... | Akivis algebra |
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