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c_j40p5icwf9xv | In mathematics, specifically in order theory and functional analysis, if C {\displaystyle C} is a cone at the origin in a topological vector space X {\displaystyle X} such that 0 ∈ C {\displaystyle 0\in C} and if U {\displaystyle {\mathcal {U}}} is the neighborhood filter at the origin, then C {\displaystyle C} is called normal if U = C , {\displaystyle {\mathcal {U}}=\left_{C},} where C := { C: U ∈ U } {\displaystyle \left_{C}:=\left\{_{C}:U\in {\mathcal {U}}\right\}} and where for any subset S ⊆ X , {\displaystyle S\subseteq X,} C := ( S + C ) ∩ ( S − C ) {\displaystyle _{C}:=(S+C)\cap (S-C)} is the C {\displaystyle C} -saturatation of S . {\displaystyle S.} Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices. | Normal cone (functional analysis) |
c_wv60gjyumgv6 | In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space X {\displaystyle X} is the set of all linear functionals on X {\displaystyle X} that map order intervals, which are sets of the form := { x ∈ X: a ≤ x and x ≤ b } , {\displaystyle :=\{x\in X:a\leq x{\text{ and }}x\leq b\},} to bounded sets. The order bound dual of X {\displaystyle X} is denoted by X b . {\displaystyle X^{\operatorname {b} }.} This space plays an important role in the theory of ordered topological vector spaces. | Order bound dual |
c_mseksucbdq22 | In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space X {\displaystyle X} is the set Pos ( X ∗ ) − Pos ( X ∗ ) {\displaystyle \operatorname {Pos} \left(X^{*}\right)-\operatorname {Pos} \left(X^{*}\right)} where Pos ( X ∗ ) {\displaystyle \operatorname {Pos} \left(X^{*}\right)} denotes the set of all positive linear functionals on X {\displaystyle X} , where a linear function f {\displaystyle f} on X {\displaystyle X} is called positive if for all x ∈ X , {\displaystyle x\in X,} x ≥ 0 {\displaystyle x\geq 0} implies f ( x ) ≥ 0. {\displaystyle f(x)\geq 0.} The order dual of X {\displaystyle X} is denoted by X + {\displaystyle X^{+}} . Along with the related concept of the order bound dual, this space plays an important role in the theory of ordered topological vector spaces. | Order dual (functional analysis) |
c_evnq32k4pjz0 | In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} is the finest locally convex topological vector space (TVS) topology on X {\displaystyle X} for which every order interval is bounded, where an order interval in X {\displaystyle X} is a set of the form := { z ∈ X: a ≤ z and z ≤ b } {\displaystyle :=\left\{z\in X:a\leq z{\text{ and }}z\leq b\right\}} where a {\displaystyle a} and b {\displaystyle b} belong to X . {\displaystyle X.} The order topology is an important topology that is used frequently in the theory of ordered topological vector spaces because the topology stems directly from the algebraic and order theoretic properties of ( X , ≤ ) , {\displaystyle (X,\leq ),} rather than from some topology that X {\displaystyle X} starts out having. | Order topology (functional analysis) |
c_wm98fxu41mg6 | This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of ( X , ≤ ) . {\displaystyle (X,\leq ).} For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology. | Order topology (functional analysis) |
c_3lpus2ptyc95 | In mathematics, specifically in order theory and functional analysis, two elements x and y of a vector lattice X are lattice disjoint or simply disjoint if inf { | x | , | y | } = 0 {\displaystyle \inf \left\{|x|,|y|\right\}=0} , in which case we write x ⊥ y {\displaystyle x\perp y} , where the absolute value of x is defined to be | x | := sup { x , − x } {\displaystyle |x|:=\sup \left\{x,-x\right\}} . We say that two sets A and B are lattice disjoint or disjoint if a and b are disjoint for all a in A and all b in B, in which case we write A ⊥ B {\displaystyle A\perp B} . If A is the singleton set { a } {\displaystyle \{a\}} then we will write a ⊥ B {\displaystyle a\perp B} in place of { a } ⊥ B {\displaystyle \{a\}\perp B} . For any set A, we define the disjoint complement to be the set A ⊥ := { x ∈ X: x ⊥ A } {\displaystyle A^{\perp }:=\left\{x\in X:x\perp A\right\}} . | Lattice disjoint |
c_tendtq4yef1e | In mathematics, specifically in order theory, a binary relation ≤ {\displaystyle \,\leq \,} on a vector space X {\displaystyle X} over the real or complex numbers is called Archimedean if for all x ∈ X , {\displaystyle x\in X,} whenever there exists some y ∈ X {\displaystyle y\in X} such that n x ≤ y {\displaystyle nx\leq y} for all positive integers n , {\displaystyle n,} then necessarily x ≤ 0. {\displaystyle x\leq 0.} An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean. A preordered vector space X {\displaystyle X} is called almost Archimedean if for all x ∈ X , {\displaystyle x\in X,} whenever there exists a y ∈ X {\displaystyle y\in X} such that − n − 1 y ≤ x ≤ n − 1 y {\displaystyle -n^{-1}y\leq x\leq n^{-1}y} for all positive integers n , {\displaystyle n,} then x = 0. {\displaystyle x=0.} | Archimedean ordered vector space |
c_0jy8ehmq8dnn | In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . The theorem states that each infinite bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence. An equivalent formulation is that a subset of R n {\displaystyle \mathbb {R} ^{n}} is sequentially compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem. | Bolzano-Weierstrass theorem |
c_8d42jzjubxnx | In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra g {\displaystyle {\mathfrak {g}}} is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra g {\displaystyle {\mathfrak {g}}} is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup. | Borel subalgebra |
c_i7lmvg110dlk | In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematical notion of semisimplicity. Many representations that appear in applications of representation theory are semisimple or can be approximated by semisimple representations. A semisimple module over an algebra over a field is an example of a semisimple representation. Conversely, a semisimple representation of a group G over a field k is a semisimple module over the group ring k. | Isotypic decomposition |
c_ys6pl817p76o | In mathematics, specifically in representation theory, the Frobenius formula, introduced by G. Frobenius, computes the characters of irreducible representations of the symmetric group Sn. Among the other applications, the formula can be used to derive the hook length formula. | Frobenius formula |
c_wgi0gm2e9nqk | In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon the Lie bracket. (There is no Lie bracket for many algebras over commutative rings; a Lie algebra involves its Lie bracket, whereas, there is no Lie bracket defined in the general case of an algebra over a commutative ring.) Another possible source of confusion in terminology is the quantum nilpotent algebra, a concept related to quantum groups and Hopf algebras. | Nilpotent algebra |
c_saez4yg8sdef | In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is torsion-free if its torsion submodule comprises only the zero element. | Torsion point |
c_rffm1xe44lxv | This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements. This terminology applies to abelian groups (with "module" and "submodule" replaced by "group" and "subgroup"). This is allowed by the fact that the abelian groups are the modules over the ring of integers (in fact, this is the origin of the terminology, that has been introduced for abelian groups before being generalized to modules). In the case of groups that are noncommutative, a torsion element is an element of finite order. Contrary to the commutative case, the torsion elements do not form a subgroup, in general. | Torsion point |
c_p6d27rqobcwm | In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring R. | Simple module |
c_6tpq6h9oz4ij | In mathematics, specifically in spectral theory, an eigenvalue of a closed linear operator is called normal if the space admits a decomposition into a direct sum of a finite-dimensional generalized eigenspace and an invariant subspace where A − λ I {\displaystyle A-\lambda I} has a bounded inverse. The set of normal eigenvalues coincides with the discrete spectrum. | Normal eigenvalue |
c_mieiomah2za5 | In mathematics, specifically in surgery theory, the surgery obstructions define a map θ: N ( X ) → L n ( π 1 ( X ) ) {\displaystyle \theta \colon {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))} from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when n ≥ 5 {\displaystyle n\geq 5}: A degree-one normal map ( f , b ): M → X {\displaystyle (f,b)\colon M\to X} is normally cobordant to a homotopy equivalence if and only if the image θ ( f , b ) = 0 {\displaystyle \theta (f,b)=0} in L n ( Z ) {\displaystyle L_{n}(\mathbb {Z} )} . | Surgery obstruction |
c_adrsx3218m04 | In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums. | Hamiltonian action |
c_vamne3dr0w6n | In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that glues two manifolds together into one. The symplectic cut can also be viewed as a generalization of symplectic blow up. The cut was introduced in 1995 by Eugene Lerman, who used it to study the symplectic quotient and other operations on manifolds. | Symplectic cut |
c_wh3dhvcuc2nk | In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a submanifold, often called a fiber sum. The symplectic sum is the inverse of the symplectic cut, which decomposes a given manifold into two pieces. Together the symplectic sum and cut may be viewed as a deformation of symplectic manifolds, analogous for example to deformation to the normal cone in algebraic geometry. The symplectic sum has been used to construct previously unknown families of symplectic manifolds, and to derive relationships among the Gromov–Witten invariants of symplectic manifolds. | Symplectic sum |
c_kn69bw3jpzjx | In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. | Gromov-Witten theory |
c_b5d977arw7n1 | They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten. The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of what the invariants mean, how they are computed, and why they are important. | Gromov-Witten theory |
c_hpj6nfm1idkb | In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more information than the former. In each, the choice of coefficient ring (typically a Novikov ring, described below) significantly affects its structure, as well. While the cup product of ordinary cohomology describes how submanifolds of the manifold intersect each other, the quantum cup product of quantum cohomology describes how subspaces intersect in a "fuzzy", "quantum" way. | Quantum cohomology ring |
c_g5kbaeqe662z | More precisely, they intersect if they are connected via one or more pseudoholomorphic curves. Gromov–Witten invariants, which count these curves, appear as coefficients in expansions of the quantum cup product. | Quantum cohomology ring |
c_1u7l55ij56ea | Because it expresses a structure or pattern for Gromov–Witten invariants, quantum cohomology has important implications for enumerative geometry. It also connects to many ideas in mathematical physics and mirror symmetry. In particular, it is ring-isomorphic to symplectic Floer homology. Throughout this article, X is a closed symplectic manifold with symplectic form ω. | Quantum cohomology ring |
c_uuyi8d86vkam | In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable map was proposed by Maxim Kontsevich around 1992 and published in Kontsevich (1995). Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov–Witten invariants article itself. | Stable map |
c_84nr7q0pcowp | In mathematics, specifically in the area of hyperbolic geometry, Hilbert's arithmetic of ends is a method for endowing a geometric set, the set of ideal points or "ends" of a hyperbolic plane, with an algebraic structure as a field. It was introduced by German mathematician David Hilbert. | Hilbert's arithmetic of ends |
c_11thurr79bbj | In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf. The fundamental lemma of the calculus of variations is typically used to transform this weak formulation into the strong formulation (differential equation), free of the integration with arbitrary function. The proof usually exploits the possibility to choose δf concentrated on an interval on which f keeps sign (positive or negative). | Fundamental lemma of calculus of variations |
c_8msqxm1w8ov5 | Several versions of the lemma are in use. Basic versions are easy to formulate and prove. More powerful versions are used when needed. | Fundamental lemma of calculus of variations |
c_xqmfkpvhh6qo | In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the terminal object is the monoidal unit. Dually, a monoidal finite coproduct category with the monoidal structure given by the coproduct and unit the initial object is called a cocartesian monoidal category, and any finite coproduct category can be thought of as a cocartesian monoidal category. Cartesian categories with an internal Hom functor that is an adjoint functor to the product are called Cartesian closed categories. | Cartesian monoidal category |
c_aqhfz6aznn2i | In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology. Morse homology also serves as a model for the various infinite-dimensional generalizations known as Floer homology theories. | Morse homology |
c_wtmrc4dgr924 | In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number p {\displaystyle p} , a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G {\displaystyle G} is a maximal p {\displaystyle p} -subgroup of G {\displaystyle G} , i.e., a subgroup of G {\displaystyle G} that is a p-group (meaning its cardinality is a power of p , {\displaystyle p,} or equivalently, the order of every group element is a power of p {\displaystyle p} ) that is not a proper subgroup of any other p {\displaystyle p} -subgroup of G {\displaystyle G} . The set of all Sylow p {\displaystyle p} -subgroups for a given prime p {\displaystyle p} is sometimes written Syl p ( G ) {\displaystyle {\text{Syl}}_{p}(G)} . | Sylow p-subgroup |
c_od3lxoqz4z9g | The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group G {\displaystyle G} the order (number of elements) of every subgroup of G {\displaystyle G} divides the order of G {\displaystyle G} . The Sylow theorems state that for every prime factor p {\displaystyle p} of the order of a finite group G {\displaystyle G} , there exists a Sylow p {\displaystyle p} -subgroup of G {\displaystyle G} of order p n {\displaystyle p^{n}} , the highest power of p {\displaystyle p} that divides the order of G {\displaystyle G} . Moreover, every subgroup of order p n {\displaystyle p^{n}} is a Sylow p {\displaystyle p} -subgroup of G {\displaystyle G} , and the Sylow p {\displaystyle p} -subgroups of a group (for a given prime p {\displaystyle p} ) are conjugate to each other. Furthermore, the number of Sylow p {\displaystyle p} -subgroups of a group for a given prime p {\displaystyle p} is congruent to 1 (mod p {\displaystyle p} ). | Sylow p-subgroup |
c_3wgta0h6zjns | In mathematics, specifically in the field of group theory, the McKay conjecture is a conjecture of equality between the number of irreducible complex characters of degree not divisible by a prime number p {\displaystyle p} to that of the normalizer of a Sylow p {\displaystyle p} -subgroup. It is named after Canadian mathematician John McKay. | McKay conjecture |
c_sfk9ebv72nid | In mathematics, specifically in the field of numerical analysis, Kummer's transformation of series is a method used to accelerate the convergence of an infinite series. The method was first suggested by Ernst Kummer in 1837. | Kummer's transformation of series |
c_8logvj72nwrb | In mathematics, specifically in the field of topology, a monotonically normal space is a particular kind of normal space, defined in terms of a monotone normality operator. It satisfies some interesting properties; for example metric spaces and linearly ordered spaces are monotonically normal, and every monotonically normal space is hereditarily normal. | Monotonically normal space |
c_awv2a9qee179 | In mathematics, specifically in the field of topology, a topological space is said to be a door space if every subset is open or closed (or both). The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither". | Door space |
c_rsjwkafsruw4 | In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation ( ρ , V ) {\displaystyle (\rho ,V)} or irrep of an algebraic structure A {\displaystyle A} is a nonzero representation that has no proper nontrivial subrepresentation ( ρ | W , W ) {\displaystyle (\rho |_{W},W)} , with W ⊂ V {\displaystyle W\subset V} closed under the action of { ρ ( a ): a ∈ A } {\displaystyle \{\rho (a):a\in A\}} . Every finite-dimensional unitary representation on a Hilbert space V {\displaystyle V} is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible. | Irreducible representation |
c_9hb4l2uc35gl | In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system evolves. As a phase space trajectory is uniquely determined for any given set of phase space coordinates, it is not possible for different orbits to intersect in phase space, therefore the set of all orbits of a dynamical system is a partition of the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems. For discrete-time dynamical systems, the orbits are sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces. | Positive semi-orbit |
c_mqm88ck17nkn | In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems. | Cauchy–Peano theorem |
c_9czwk8jag8g6 | In mathematics, specifically in the study of topology and open covers of a topological space X, a star refinement is a particular kind of refinement of an open cover of X. A related concept is the notion of barycentric refinement. Star refinements are used in the definition of fully normal space and in one definition of uniform space. It is also useful for stating a characterization of paracompactness. | Star refinement |
c_9o9tcqpjwhm3 | In mathematics, specifically in the study of vector bundles over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the Kodaira vanishing theorem. Given a compact complex manifold M with a holomorphic line bundle F over M, the Nakano vanishing theorem provides a condition on when the cohomology groups H q ( M ; Ω p ( F ) ) {\textstyle H^{q}(M;\Omega ^{p}(F))} equal zero. Here, Ω p ( F ) {\textstyle \Omega ^{p}(F)} denotes the sheaf of holomorphic (p,0)-forms taking values on F. The theorem states that, if the first Chern class of F is negative, Alternatively, if the first Chern class of F is positive, | Nakano vanishing theorem |
c_tda0gqbwe7h2 | In mathematics, specifically in the theory of Markovian stochastic processes in probability theory, the Chapman–Kolmogorov equation(CKE) is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process. The equation was derived independently by both the British mathematician Sydney Chapman and the Russian mathematician Andrey Kolmogorov. CKE prominently used in recent Variational Bayesian methods. | Chapman–Kolmogorov equation |
c_tit5130p276u | In mathematics, specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches. The distance, suitably quantified, to the limiting distribution can be made arbitrarily small by selecting a distribution sufficiently far along the sequence. This notion generalizes a limit of a sequence of functions; a limit as a distribution may exist when a limit of functions does not. The notion is a part of distributional calculus, a generalized form of calculus that is based on the notion of distributions, as opposed to classical calculus, which is based on the narrower concept of functions. | Limit of a distribution |
c_scpa4h6d1j5k | In mathematics, specifically in topology and functional analysis, a subspace S of a uniform space X is said to be sequentially complete or semi-complete if every Cauchy sequence in S converges to an element in S. X is called sequentially complete if it is a sequentially complete subset of itself. | Sequentially complete space |
c_vm8d9tz0dk1v | In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. In particular, they lead to the Gromov–Witten invariants and Floer homology, and play a prominent role in string theory. | Pseudoholomorphic curve |
c_eqvtht4bulj0 | In mathematics, specifically in topology of manifolds, a compact codimension-one submanifold F {\displaystyle F} of a manifold M {\displaystyle M} is said to be 2-sided in M {\displaystyle M} when there is an embedding h: F × → M {\displaystyle h\colon F\times \to M} with h ( x , 0 ) = x {\displaystyle h(x,0)=x} for each x ∈ F {\displaystyle x\in F} and h ( F × ) ∩ ∂ M = h ( ∂ F × ) {\displaystyle h(F\times )\cap \partial M=h(\partial F\times )} .In other words, if its normal bundle is trivial.This means, for example that a curve in a surface is 2-sided if it has a tubular neighborhood which is a cartesian product of the curve times an interval. A submanifold which is not 2-sided is called 1-sided. | 2-sided |
c_cu4xjnilv1k6 | In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its definition relies on the notion of a measured foliation introduced by William Thurston, who also coined the term "pseudo-Anosov diffeomorphism" when he proved his classification of diffeomorphisms of a surface. | Measured foliation |
c_vncrbyrub0qr | In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions. The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty). | Interior points |
c_84eeho7aqtpu | In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces. More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots. | Knot sum |
c_72hm3dav58ki | In mathematics, specifically in transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials was proven by Axel Thue; Thue's proof used Dirichlet's box principle. Carl Ludwig Siegel published his lemma in 1929. It is a pure existence theorem for a system of linear equations. Siegel's lemma has been refined in recent years to produce sharper bounds on the estimates given by the lemma. | Siegel's lemma |
c_yapxpy1fime5 | In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension. In linear algebra, given a quotient map V → Q {\displaystyle V\to Q} , the difference dim V − dim Q is the relative dimension; this equals the dimension of the kernel. In fiber bundles, the relative dimension of the map is the dimension of the fiber. More abstractly, the codimension of a map is the dimension of the cokernel, while the relative dimension of a map is the dimension of the kernel. | Relative dimension |
c_pwlj3yinmngo | These are dual in that the inclusion of a subspace V → W {\displaystyle V\to W} of codimension k dualizes to yield a quotient map W ∗ → V ∗ {\displaystyle W^{*}\to V^{*}} of relative dimension k, and conversely. The additivity of codimension under intersection corresponds to the additivity of relative dimension in a fiber product. Just as codimension is mostly used for injective maps, relative dimension is mostly used for surjective maps. == References == | Relative dimension |
c_0qx1gvmti8ln | In mathematics, specifically linear algebra, a degenerate bilinear form f (x, y ) on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V ) given by v ↦ (x ↦ f (x, v )) is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero x in V such that f ( x , y ) = 0 {\displaystyle f(x,y)=0\,} for all y ∈ V . {\displaystyle \,y\in V.} | Nondegenerate bilinear form |
c_bxlpub1ygoeo | In mathematics, specifically linear algebra, a real matrix A is copositive if x T A x ≥ 0 {\displaystyle x^{T}Ax\geq 0} for every nonnegative vector x ≥ 0 {\displaystyle x\geq 0} . The collection of all copositive matrices is a proper cone; it includes as a subset the collection of real positive-definite matrices. Copositive matrices find applications in economics, operations research, and statistics. | Copositive matrix |
c_jrzvi7apfffs | In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a product of square matrices is equal to the product of their determinants. The formula is valid for matrices with the entries from any commutative ring. | Cauchy-Binet formula |
c_lqdwls9c8v6z | In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number. | Complex measure |
c_auo6fptfx6lo | In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ∞ {\displaystyle \infty } if the subset is infinite.The counting measure can be defined on any measurable space (that is, any set X {\displaystyle X} along with a sigma-algebra) but is mostly used on countable sets.In formal notation, we can turn any set X {\displaystyle X} into a measurable space by taking the power set of X {\displaystyle X} as the sigma-algebra Σ ; {\displaystyle \Sigma ;} that is, all subsets of X {\displaystyle X} are measurable sets. Then the counting measure μ {\displaystyle \mu } on this measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} is the positive measure Σ → {\displaystyle \Sigma \to } defined by for all A ∈ Σ , {\displaystyle A\in \Sigma ,} where | A | {\displaystyle \vert A\vert } denotes the cardinality of the set A . {\displaystyle A.} The counting measure on ( X , Σ ) {\displaystyle (X,\Sigma )} is σ-finite if and only if the space X {\displaystyle X} is countable. | Counting measure |
c_gnxjmgwc1xov | In mathematics, specifically module theory, given a ring R and an R-module M with a submodule N, the module M is said to be an essential extension of N (or N is said to be an essential submodule or large submodule of M) if for every submodule H of M, H ∩ N = { 0 } {\displaystyle H\cap N=\{0\}\,} implies that H = { 0 } {\displaystyle H=\{0\}\,} As a special case, an essential left ideal of R is a left ideal that is essential as a submodule of the left module RR. The left ideal has non-zero intersection with any non-zero left ideal of R. Analogously, an essential right ideal is exactly an essential submodule of the right R module RR. The usual notations for essential extensions include the following two expressions: N ⊆ e M {\displaystyle N\subseteq _{e}M\,} (Lam 1999), and N ⊴ M {\displaystyle N\trianglelefteq M} (Anderson & Fuller 1992)The dual notion of an essential submodule is that of superfluous submodule (or small submodule). A submodule N is superfluous if for any other submodule H, N + H = M {\displaystyle N+H=M\,} implies that H = M {\displaystyle H=M\,} .The usual notations for superfluous submodules include: N ⊆ s M {\displaystyle N\subseteq _{s}M\,} (Lam 1999), and N ≪ M {\displaystyle N\ll M} (Anderson & Fuller 1992) | Essential extension |
c_tsrjw2tgafsx | In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. | Dyadics |
c_8g7wdieol6ms | The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context. A dyadic can be used to contain physical or geometric information, although in general there is no direct way of geometrically interpreting it. The dyadic product is distributive over vector addition, and associative with scalar multiplication. | Dyadics |
c_3j8v9is96qte | Therefore, the dyadic product is linear in both of its operands. In general, two dyadics can be added to get another dyadic, and multiplied by numbers to scale the dyadic. However, the product is not commutative; changing the order of the vectors results in a different dyadic. | Dyadics |
c_u31agasonuku | The formalism of dyadic algebra is an extension of vector algebra to include the dyadic product of vectors. The dyadic product is also associative with the dot and cross products with other vectors, which allows the dot, cross, and dyadic products to be combined to obtain other scalars, vectors, or dyadics. It also has some aspects of matrix algebra, as the numerical components of vectors can be arranged into row and column vectors, and those of second order tensors in square matrices. | Dyadics |
c_3rogg27pk31s | Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic. | Dyadics |
c_qy796kxzbl32 | The effect that a given dyadic has on other vectors can provide indirect physical or geometric interpretations. Dyadic notation was first established by Josiah Willard Gibbs in 1884. | Dyadics |
c_386kwg0cckn9 | The notation and terminology are relatively obsolete today. Its uses in physics include continuum mechanics and electromagnetism. In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over- or underbars. | Dyadics |
c_3z9nxyrhjeff | In mathematics, specifically number theory, Granville numbers, also known as S {\displaystyle {\mathcal {S}}} -perfect numbers, are an extension of the perfect numbers. | Granville number |
c_xvq5cetywbyq | In mathematics, specifically order theory, a partially ordered set is chain-complete if every chain in it has a least upper bound. It is ω-complete when every increasing sequence of elements (a type of countable chain) has a least upper bound; the same notion can be extended to other cardinalities of chains. | Chain-complete partial order |
c_erzojrh40vro | In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set X {\displaystyle X} is a quasi-ordering of X {\displaystyle X} for which every infinite sequence of elements x 0 , x 1 , x 2 , … {\displaystyle x_{0},x_{1},x_{2},\ldots } from X {\displaystyle X} contains an increasing pair x i ≤ x j {\displaystyle x_{i}\leq x_{j}} with i < j . {\displaystyle i | Well-quasi-ordering |
c_p0tjo7aueh4i | In mathematics, specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and constructed it, and after Richard Dedekind because its construction generalizes the Dedekind cuts used by Dedekind to construct the real numbers from the rational numbers. It is also called the completion by cuts or normal completion. | Dedekind–MacNeille completion |
c_m31g9889dw7j | In mathematics, specifically order theory, the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least upper bound) of S , {\displaystyle S,} denoted ⋁ S , {\textstyle \bigvee S,} and similarly, the meet of S {\displaystyle S} is the infimum (greatest lower bound), denoted ⋀ S . {\textstyle \bigwedge S.} In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion. | Join and meet |
c_cw994vmfhucg | A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. | Join and meet |
c_zelgkjw6x287 | A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms.The join/meet of a subset of a totally ordered set is simply the maximal/minimal element of that subset, if such an element exists. If a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is also an (upward) directed set, then its join (if it exists) is called a directed join or directed supremum. Dually, if S {\displaystyle S} is a downward directed set, then its meet (if it exists) is a directed meet or directed infimum. | Join and meet |
c_e2r0dotfa32a | In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book Geometrie der Lage, in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book Anschauliche Geometrie, reprinted in English as Hilbert & Cohn-Vossen (1952). Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be realizable in that geometry), or as a type of abstract incidence geometry. In the latter case they are closely related to regular hypergraphs and biregular bipartite graphs, but with some additional restrictions: every two points of the incidence structure can be associated with at most one line, and every two lines can be associated with at most one point. That is, the girth of the corresponding bipartite graph (the Levi graph of the configuration) must be at least six. | Geometric configuration |
c_3oi0x1wtpsbd | In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space H1, and H2 is another Hilbert space, and f: U → H 2 {\displaystyle f:U\rightarrow H_{2}} is a Lipschitz-continuous map, then there is a Lipschitz-continuous map F: H 1 → H 2 {\displaystyle F:H_{1}\rightarrow H_{2}} that extends f and has the same Lipschitz constant as f. Note that this result in particular applies to Euclidean spaces En and Em, and it was in this form that Kirszbraun originally formulated and proved the theorem. The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). If H1 is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient.The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true in general, not even for finite-dimensional Banach spaces. | Kirszbraun theorem |
c_akdwrqenefw5 | It is for instance possible to construct counterexamples where the domain is a subset of R n {\displaystyle \mathbb {R} ^{n}} with the maximum norm and R m {\displaystyle \mathbb {R} ^{m}} carries the Euclidean norm. More generally, the theorem fails for R m {\displaystyle \mathbb {R} ^{m}} equipped with any ℓ p {\displaystyle \ell _{p}} norm ( p ≠ 2 {\displaystyle p\neq 2} ) (Schwartz 1969, p. 20). | Kirszbraun theorem |
c_i6zrapc4sm1s | In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra. Tilting theory was motivated by the introduction of reflection functors by Joseph Bernšteĭn, Israel Gelfand, and V. A. Ponomarev (1973); these functors were used to relate representations of two quivers. These functors were reformulated by Maurice Auslander, María Inés Platzeck, and Idun Reiten (1979), and generalized by Sheila Brenner and Michael C. R. Butler (1980) who introduced tilting functors. Dieter Happel and Claus Michael Ringel (1982) defined tilted algebras and tilting modules as further generalizations of this. | Coxeter functor |
c_p35cs2h7zy9o | In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. | Primitive spectrum |
c_pl6s22df2eu9 | Primitive ideals are prime. The quotient of a ring by a left primitive ideal is a left primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields. | Primitive spectrum |
c_nm60tomlshzd | In mathematics, specifically ring theory, a principal ideal is an ideal I {\displaystyle I} in a ring R {\displaystyle R} that is generated by a single element a {\displaystyle a} of R {\displaystyle R} through multiplication by every element of R . {\displaystyle R.} The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P {\displaystyle P} generated by a single element x ∈ P , {\displaystyle x\in P,} which is to say the set of all elements less than or equal to x {\displaystyle x} in P . {\displaystyle P.} The remainder of this article addresses the ring-theoretic concept. | Principal ideal |
c_gzlaxomnoe86 | In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset. | Stationary set |
c_1yn0skocbxim | In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W α {\displaystyle W_{\alpha }} indexed by ordinals α {\displaystyle \alpha } such that W α ⊆ W α + 1 {\displaystyle W_{\alpha }\subseteq W_{\alpha +1}} If λ {\displaystyle \lambda } is a limit ordinal, then W λ = ⋃ α < λ W α {\textstyle W_{\lambda }=\bigcup _{\alpha <\lambda }W_{\alpha }} Some authors additionally require that W α + 1 ⊆ P ( W α ) {\displaystyle W_{\alpha +1}\subseteq {\mathcal {P}}(W_{\alpha })} or that W 0 ≠ ∅ {\displaystyle W_{0}\neq \emptyset } .The union W = ⋃ α ∈ O n W α {\textstyle W=\bigcup _{\alpha \in \mathrm {On} }W_{\alpha }} of the sets of a cumulative hierarchy is often used as a model of set theory.The phrase "the cumulative hierarchy" usually refers to the standard cumulative hierarchy V α {\displaystyle \mathrm {V} _{\alpha }} of the von Neumann universe with V α + 1 = P ( W α ) {\displaystyle \mathrm {V} _{\alpha +1}={\mathcal {P}}(W_{\alpha })} introduced by Zermelo (1930). | Cumulative hierarchy |
c_5tsejen0zl8o | In mathematics, specifically set theory, a dimensional operator on a set E is a function from the subsets of E to the subsets of E. | Dimensional operator |
c_qkuqrwpd1yw8 | In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. In terms of set-builder notation, that is A × B = { ( a , b ) ∣ a ∈ A and b ∈ B } . {\displaystyle A\times B=\{(a,b)\mid a\in A\ {\mbox{ and}}\ b\in B\}.} A table can be created by taking the Cartesian product of a set of rows and a set of columns. | Cylinder (algebra) |
c_4kioyx8d99kz | If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. | Cylinder (algebra) |
c_ffzu6n3tay96 | In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that there is no set whose cardinality is strictly between that of the integers and the real numbers, or equivalently, that any subset of the real numbers is finite, is countably infinite, or has the same cardinality as the real numbers. In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2 ℵ 0 = ℵ 1 {\displaystyle 2^{\aleph _{0}}=\aleph _{1}} , or even shorter with beth numbers: ℶ 1 = ℵ 1 {\displaystyle \beth _{1}=\aleph _{1}} . The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940.The name of the hypothesis comes from the term the continuum for the real numbers. | Aleph hypothesis |
c_uft42bk67bnt | In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted as probability distributions – notably as either values of the parameter of a parametric model or as a data set of observed values – the resulting distance is a statistical distance. The most basic Bregman divergence is the squared Euclidean distance. Bregman divergences are similar to metrics, but satisfy neither the triangle inequality (ever) nor symmetry (in general). | Bregman distance |
c_qowbwdz8lee9 | However, they satisfy a generalization of the Pythagorean theorem, and in information geometry the corresponding statistical manifold is interpreted as a (dually) flat manifold. This allows many techniques of optimization theory to be generalized to Bregman divergences, geometrically as generalizations of least squares. Bregman divergences are named after Russian mathematician Lev M. Bregman, who introduced the concept in 1967. | Bregman distance |
c_43cd2m6nnqoy | In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number theory, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory. | Normal basis |
c_49tjydp65ohb | In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using either quadratic irrationals or simply rational numbers. It is named after Harold Davenport and Wolfgang M. Schmidt. | Davenport–Schmidt theorem |
c_4sz4o9e85tvr | In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three. | Cyclic cubic field |
c_uiv90abjcrnd | In mathematics, specifically the field of abstract algebra, Bergman's Diamond Lemma (after George Bergman) is a method for confirming whether a given set of monomials of an algebra forms a k {\displaystyle k} -basis. It is an extension of Gröbner bases to non-commutative rings. The proof of the lemma gives rise to an algorithm for obtaining a non-commutative Gröbner basis of the algebra from its defining relations. However, in contrast to Buchberger's algorithm, in the non-commutative case, this algorithm may not terminate. | Bergman's diamond lemma |
c_0hipw8bvsrv0 | In mathematics, specifically the field of algebra, Sklyanin algebras are a class of noncommutative algebra named after Evgeny Sklyanin. This class of algebras was first studied in the classification of Artin-Schelter regular algebras of global dimension 3 in the 1980s. Sklyanin algebras can be grouped into two different types, the non-degenerate Sklyanin algebras and the degenerate Sklyanin algebras, which have very different properties. A need to understand the non-degenerate Sklyanin algebras better has led to the development of the study of point modules in noncommutative geometry. | Sklyanin algebra |
c_8e767w3zqu2y | In mathematics, specifically the field of algebraic number theory, a Minkowski space is a Euclidean space associated with an algebraic number field.If K is a number field of degree d then there are d distinct embeddings of K into C. We let KC be the image of K in the product Cd, considered as equipped with the usual Hermitian inner product. If c denotes complex conjugation, let KR denote the subspace of KC fixed by c, equipped with a scalar product. This is the Minkowski space of K. | Minkowski space (number field) |
c_16xk7foto4xj | In mathematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials. The conjecture, along with two related, stronger conjectures, is at the top of a hierarchy of conjectures and theorems concerning the arithmetic nature of a certain number of values of the exponential function. | Four exponentials conjecture |
c_f6cgpb71ui8c | In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. | Cauchy-Lipschitz theorem |
c_fvjc38f2ecmw | In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if π: g → g l ( V ) {\displaystyle \pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)} is a finite-dimensional representation of a solvable Lie algebra, then there's a flag V = V 0 ⊃ V 1 ⊃ ⋯ ⊃ V n = 0 {\displaystyle V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0} of invariant subspaces of π ( g ) {\displaystyle \pi ({\mathfrak {g}})} with codim V i = i {\displaystyle \operatorname {codim} V_{i}=i} , meaning that π ( X ) ( V i ) ⊆ V i {\displaystyle \pi (X)(V_{i})\subseteq V_{i}} for each X ∈ g {\displaystyle X\in {\mathfrak {g}}} and i. Put in another way, the theorem says there is a basis for V such that all linear transformations in π ( g ) {\displaystyle \pi ({\mathfrak {g}})} are represented by upper triangular matrices. This is a generalization of the result of Frobenius that commuting matrices are simultaneously upper triangularizable, as commuting matrices generate an abelian Lie algebra, which is a fortiori solvable. A consequence of Lie's theorem is that any finite dimensional solvable Lie algebra over a field of characteristic 0 has a nilpotent derived algebra (see #Consequences). Also, to each flag in a finite-dimensional vector space V, there correspond a Borel subalgebra (that consist of linear transformations stabilizing the flag); thus, the theorem says that π ( g ) {\displaystyle \pi ({\mathfrak {g}})} is contained in some Borel subalgebra of g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} . | Lie's theorem |
c_oyc0igivy715 | In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; ε = ±1, accordingly for symmetric or skew-symmetric. They are also called ( − ) n {\displaystyle (-)^{n}} -quadratic forms, particularly in the context of surgery theory. There is the related notion of ε-symmetric forms, which generalizes symmetric forms, skew-symmetric forms (= symplectic forms), Hermitian forms, and skew-Hermitian forms. More briefly, one may refer to quadratic, skew-quadratic, symmetric, and skew-symmetric forms, where "skew" means (−) and the * (involution) is implied. The theory is 2-local: away from 2, ε-quadratic forms are equivalent to ε-symmetric forms: half the symmetrization map (below) gives an explicit isomorphism. | Quadratic refinement |
c_xcg5sof6o6vd | In mathematics, specifically topology, a sequence covering map is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include sequentially quotient maps, sequence coverings, 1-sequence coverings, and 2-sequence coverings. These classes of maps are closely related to sequential spaces. If the domain and/or codomain have certain additional topological properties (often, the spaces being Hausdorff and first-countable is more than enough) then these definitions become equivalent to other well-known classes of maps, such as open maps or quotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms of sequential continuity or the characterization of compactness in terms of sequential compactness (whenever such characterizations hold). | Sequence covering map |
c_wun7e48joja6 | In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers. | Schanuel's conjecture |
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