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c_pwfxqzj868nx | {\displaystyle \left,z\right]+\left,x\right]+\left,y\right]=\left+\left+\left-\left-\left-\left.} An Akivis algebra with = 0 {\displaystyle \left=0} is a Lie algebra, for the Akivis identity reduces to the Jacobi identity. Note that the terms on the right hand side have positive sign for even permutations and negative... | Akivis algebra |
c_k6aoudrqsbrl | In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of t... | Stable manifold |
c_fj3byxwtmju5 | In mathematics, and in particular the study of game theory, a function is graph continuous if it exhibits the following properties. The concept was originally defined by Partha Dasgupta and Eric Maskin in 1986 and is a version of continuity that finds application in the study of continuous games. | Graph continuous function |
c_3wq1f7vieqgg | In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation. One distinguishes the left regular representation λ given by left translation and the right regular representation ρ giv... | Regular module |
c_lusn3un8x5pe | In mathematics, and in particular universal algebra, the concept of an n-ary group (also called n-group or multiary group) is a generalization of the concept of a group to a set G with an n-ary operation instead of a binary operation. By an n-ary operation is meant any map f: Gn → G from the n-th Cartesian power of G t... | N-ary group |
c_rgr2dy1x1649 | In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense... | Pseudo inverse |
c_br25bsoc3f3r | A matrix A g ∈ R n × m {\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{n\times m}} is a generalized inverse of a matrix A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} if A A g A = A . {\displaystyle AA^{\mathrm {g} }A=A.} A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular ... | Pseudo inverse |
c_1el0cphqxlnp | In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform. | Perron formula |
c_nchw4duiyjv3 | In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubne... | Primorial |
c_xq665pyvmaje | In mathematics, and more particularly in polyhedral combinatorics, Eberhard's theorem partially characterizes the multisets of polygons that can form the faces of simple convex polyhedra. It states that, for given numbers of triangles, quadrilaterals, pentagons, heptagons, and other polygons other than hexagons, there ... | Eberhard's theorem |
c_lwtbyfka8hab | In mathematics, and more particularly in set theory, a cover (or covering) of a set X {\displaystyle X} is a family of subsets of X {\displaystyle X} whose union is all of X {\displaystyle X} . More formally, if C = { U α: α ∈ A } {\displaystyle C=\lbrace U_{\alpha }:\alpha \in A\rbrace } is an indexed family of subset... | Refinement (topology) |
c_b9cnakkr5q89 | In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction x is said to be restricted, or composed of restricted partial quotients, if the sequence of denominators of its partial quotients is bounded; that is x = = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1... | Restricted partial quotients |
c_ygpeowxpie9l | In mathematics, and more precisely in analysis, the Wallis integrals constitute a family of integrals introduced by John Wallis. | Wallis' integrals |
c_u6csdpjdf0hw | In mathematics, and more precisely in semigroup theory, a nilsemigroup or nilpotent semigroup is a semigroup whose every element is nilpotent. | Nilpotent semigroup |
c_asoe2baeiozb | In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups having some nice algebraic properties. Those classes can be defined in two distinct ways, using either algebraic notions or topological notions. Varieties of finite monoids, varieties of finite ordered semigr... | Variety of finite semigroups |
c_9fshjd8h96t6 | In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is of fundamental importance for the study of 3-mani... | Dehn-Nielsen theorem |
c_x5iag2vdqbkz | In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with... | Involutive algebra |
c_v63iq59rdyau | In mathematics, and more specifically in abstract algebra, a pseudo-ring is one of the following variants of a ring: A rng, i.e., a structure satisfying all the axioms of a ring except for the existence of a multiplicative identity. A set R with two binary operations + and ⋅ such that (R, +) is an abelian group with id... | Pseudo-ring |
c_8v9o535is9uv | In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term rng (IPA: ) is meant to suggest that it is a ring without i, that is, without... | Rng (algebra) |
c_q16p49i0oxtw | In mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if x ∗ = x {\displaystyle x^{*}=x} . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoint if it is closed under the involu... | Self adjoint |
c_hzwmosk6pj80 | For example, if x ∗ = y {\displaystyle x^{*}=y} then since y ∗ = x ∗ ∗ = x {\displaystyle y^{*}=x^{**}=x} in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements. In functional analysis, a linear operator A: H → H {\displaystyle A:H\to H} on a Hilbert space is called... | Self adjoint |
c_d2e25jb5kevm | If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is self-adjoint if and only if the matrix describing A with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint. In a dagger catego... | Self adjoint |
c_67viqbt92k5r | In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by ... | Euler–Poincaré characteristic |
c_v64ebdh8hn3s | It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and... | Euler–Poincaré characteristic |
c_zsqcarr918jl | In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomi... | Holonomic function |
c_gqmuieqbtmlj | When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called holonomic. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence... | Holonomic function |
c_u6abq7vfudrp | In mathematics, and more specifically in combinatorial commutative algebra, a zero-divisor graph is an undirected graph representing the zero divisors of a commutative ring. It has elements of the ring as its vertices, and pairs of elements whose product is zero as its edges. | Zero-divisor graph |
c_fme4bj9fyjsa | In mathematics, and more specifically in computer algebra and elimination theory, a regular chain is a particular kind of triangular set of multivariate polynomials over a field, where a triangular set is a finite sequence of polynomials such that each one contains at least one more indeterminate than the preceding one... | Regular chain |
c_zz2nzlow54wg | In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K over a field K. A Gröbner basis allows many important properties of the ideal and the associated a... | Multivariate division algorithm |
c_skna5dutjgop | He named them after his advisor Wolfgang Gröbner. In 2007, Buchberger received the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work. However, the Russian mathematician Nikolai Günther had introduced a similar notion in 1913, published in various Russian mathematical journal... | Multivariate division algorithm |
c_ac84x7pryj8v | These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. An analogous concept for multivariate power series was developed independently by Heisuke Hironaka in 1964, who named them standard bases. This term has been used by some authors to also denote Gröbn... | Multivariate division algorithm |
c_0csg5rr7s1zm | In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold ... | Hermitian manifold |
c_pqxkucn3p46e | By dropping this condition, we get an almost Hermitian manifold. On any almost Hermitian manifold, we can introduce a fundamental 2-form (or cosymplectic structure) that depends only on the chosen metric and the almost complex structure. This form is always non-degenerate. With the extra integrability condition that it... | Hermitian manifold |
c_0lex2nqzxr1m | In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitizati... | Parametrization invariance |
c_uvl6wymm6zu0 | The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system. For example, the position of a point that moves on a curve in t... | Parametrization invariance |
c_pw2ddzk7nguo | If x, y, z are the coordinates of the point, the movement is thus described by a parametric equation x = f ( t ) y = g ( t ) z = h ( t ) , {\displaystyle {\begin{aligned}x&=f(t)\\y&=g(t)\\z&=h(t),\end{aligned}}} where t is the parameter and denotes the time. Such a parametric equation completely determines the curve, w... | Parametrization invariance |
c_u9ef7bb7un7t | In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. | Weighted digraph |
c_wh316bddculd | In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edges), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge. There are 2 distinct notions of multiple edges: Edges without own ... | Multigraph |
c_2mr0mm917zkz | Edges with own identity: Edges are primitive entities just like nodes. When multiple edges connect two nodes, these are different edges.A multigraph is different from a hypergraph, which is a graph in which an edge can connect any number of nodes, not just two. For some authors, the terms pseudograph and multigraph are... | Multigraph |
c_7ldknn34flhm | In mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree or singly connected network) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is ... | Oriented tree |
c_0ru9tgptfs1q | In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic. A polytree is an example of an oriented graph. The term polytree was coined in 1987 by Rebane and Pearl. | Oriented tree |
c_5o3fkethnm4a | In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants characterizing the structure of a specific module or obje... | Minimal resolution (algebra) |
c_xd6y9e0pz4d5 | Thus one speaks of a P resolution. In particular, every module has free resolutions, projective resolutions and flat resolutions, which are left resolutions consisting, respectively of free modules, projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consist... | Minimal resolution (algebra) |
c_hctrtn50p93z | In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence 0 ⟶ A ⟶ q B ⟶ r C ⟶ 0. {\displaystyle 0\longrightarrow A\mathrel {\overset {q}{\longrightarrow }} B\mathrel {\overset {r}{\longrightar... | Splitting lemma |
c_uueneufaojm0 | In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V → W {\displaystyle V\to W} between two vector spaces that preserves the operations of vector addition and scalar mult... | Linear endomorphism |
c_fuekdxtj1zs5 | In the case where V = W {\displaystyle V=W} , a linear map is called a linear endomorphism. Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V {\displaystyle V} and W {\displaystyle... | Linear endomorphism |
c_mjyx09yf3979 | In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces. | Vector subspace |
c_gkzcrtivkm9n | In mathematics, and more specifically in numerical analysis, Householder's methods are a class of root-finding algorithms that are used for functions of one real variable with continuous derivatives up to some order d + 1. Each of these methods is characterized by the number d, which is known as the order of the method... | Householder's method |
c_p39hmwws63v6 | In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upp... | List of order structures in mathematics |
c_z14tbjsam0x1 | In mathematics, and more specifically in partial differential equations, Duhamel's principle is a general method for obtaining solutions to inhomogeneous linear evolution equations like the heat equation, wave equation, and vibrating plate equation. It is named after Jean-Marie Duhamel who first applied the principle t... | Duhamel's principle |
c_wmewvyw5vg2m | The philosophy underlying Duhamel's principle is that it is possible to go from solutions of the Cauchy problem (or initial value problem) to solutions of the inhomogeneous problem. Consider, for instance, the example of the heat equation modeling the distribution of heat energy u in Rn. Indicating by ut (x, t) the tim... | Duhamel's principle |
c_lwt0ibsxxwrs | By contrast, the inhomogeneous problem for the heat equation, corresponds to adding an external heat energy f (x, t) dt at each point. Intuitively, one can think of the inhomogeneous problem as a set of homogeneous problems each starting afresh at a different time slice t = t0. By linearity, one can add up (integrate) ... | Duhamel's principle |
c_mrvpbzchfo1f | In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet a... | Goldberg polyhedron |
c_oloa3plxvuz3 | A Goldberg polyhedron is a dual polyhedron of a geodesic sphere. A consequence of Euler's polyhedron formula is that a Goldberg polyhedron always has exactly twelve pentagonal faces. Icosahedral symmetry ensures that the pentagons are always regular and that there are always 12 of them. | Goldberg polyhedron |
c_02vs7d6l9fml | If the vertices are not constrained to a sphere, the polyhedron can be constructed with planar equilateral (but not in general equiangular) faces. Simple examples of Goldberg polyhedra include the dodecahedron and truncated icosahedron. Other forms can be described by taking a chess knight move from one pentagon to the... | Goldberg polyhedron |
c_vngoqp7giik7 | Such a polyhedron is denoted GP(m,n). A dodecahedron is GP(1,0) and a truncated icosahedron is GP(1,1). | Goldberg polyhedron |
c_7emirimkt0h5 | A similar technique can be applied to construct polyhedra with tetrahedral symmetry and octahedral symmetry. These polyhedra will have triangles or squares rather than pentagons. These variations are given Roman numeral subscripts denoting the number of sides on the non-hexagon faces: GPIII(n,m), GPIV(n,m), and GPV(n,m... | Goldberg polyhedron |
c_usonc2nkcn27 | In mathematics, and more specifically in projective geometry, a projective frame or projective basis is a tuple of points in a projective space that can be used for defining homogeneous coordinates in this space. More precisely, in a projective space of dimension n, a projective frame is a n + 2-tuple of points such th... | Projective frame |
c_cona4pq49o6c | Let P(V) be a projective space of dimension n, where V is a K-vector space of dimension n + 1. Let p: V ∖ { 0 } → P ( V ) {\displaystyle p:V\setminus \{0\}\to \mathbf {P} (V)} be the canonical projection that maps a nonzero vector v to the corresponding point of P(V), which is the vector line that contains v. Every fra... | Projective frame |
c_a2hyrx4qastm | In particular, the only homography fixing the points of a frame is the identity map. This result is much more difficult in synthetic geometry (where projective spaces are defined through axioms). | Projective frame |
c_0033wg2wa5to | It is sometimes called the first fundamental theorem of projective geometry.Every frame can be written as ( p ( e 0 ) , … , p ( e n ) , p ( e 0 + ⋯ + e n ) ) , {\displaystyle (p(e_{0}),\ldots ,p(e_{n}),p(e_{0}+\cdots +e_{n})),} where ( e 0 , … , e n ) {\displaystyle (e_{0},\dots ,e_{n})} is basis of V. The projective c... | Projective frame |
c_6jp0bhfi25ju | Commonly, the projective space Pn(K) = P(Kn+1) is considered. It has a canonical frame consisting of the image by p of the canonical basis of Kn+1 (consisting of the elements having only one nonzero entry, which is equal to 1), and (1, 1, ..., 1). On this basis, the homogeneous coordinates of p(v) are simply the entrie... | Projective frame |
c_ev61j7lr95t7 | The projective coordinates of a point a on the frame F are the homogeneous coordinates of h(a) on the canonical frame of Pn(K). In the case of a projective line, a frame consists of three distinct points. If P1(K) is identified with K with a point at infinity ∞ added, then its canonical frame is (∞, 0, 1). Given any fr... | Projective frame |
c_6r99lnm4zg1m | In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact is equivalent to Zorn'... | Krull's theorem |
c_6al7qpwr8mr1 | In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (eve... | Algebraic ideal |
c_vg5erjg9bx8r | Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when... | Algebraic ideal |
c_p3lch937o1qm | There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory). The related, but distinct, concept of an ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals ar... | Algebraic ideal |
c_d7mahnc07b5e | In mathematics, and more specifically in the theory of C*-algebras, the noncommutative tori Aθ, also known as irrational rotation algebras for irrational values of θ, form a family of noncommutative C*-algebras which generalize the algebra of continuous functions on the 2-torus. Many topological and geometric propertie... | Noncommutative torus |
c_f3fpp7ixu0kb | In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. It is related to the semidirect product construction for groups. (Roughly speaking, crossed product is the expected s... | Crossed product |
c_bzpfg0tlf4aw | In mathematics, and more specifically matrix theory, the spread of a matrix is the largest distance in the complex plane between any two eigenvalues of the matrix. | Spread of a matrix |
c_lg1cf1gi68z6 | In mathematics, and more specifically number theory, the hyperfactorial of a positive integer n {\displaystyle n} is the product of the numbers of the form x x {\displaystyle x^{x}} from 1 1 {\displaystyle 1^{1}} to n n {\displaystyle n^{n}} . | Hyperfactorial |
c_rdqcb73ckxw1 | In mathematics, and more specifically number theory, the superfactorial of a positive integer n {\displaystyle n} is the product of the first n {\displaystyle n} factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials. | Superfactorial |
c_t0ahaakwmsjd | In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets. Lemma: Let A be a Borel subset of Rn, and let s > 0. Then the following are equivalent: Hs(A) > 0, where Hs denotes the s-dimensional Hausdorff measure.... | Frostman lemma |
c_agvf1p2kztnu | The generalization to Borel sets is more involved, and requires the theory of Suslin sets. A useful corollary of Frostman's lemma requires the notions of the s-capacity of a Borel set A ⊂ Rn, which is defined by C s ( A ) := sup { ( ∫ A × A d μ ( x ) d μ ( y ) | x − y | s ) − 1: μ is a Borel measure and μ ( A ) = 1 } .... | Frostman lemma |
c_ofdqpjljp2xr | (Here, we take inf ∅ = ∞ and 1⁄∞ = 0. As before, the measure μ {\displaystyle \mu } is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rn d i m H ( A ) = sup { s ≥ 0: C s ( A ) > 0 } . {\displaystyle \mathrm {dim} _{H}(A)=\sup\{s\geq 0:C_{s}(A)>0\}.} | Frostman lemma |
c_039a1bimyv6k | In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equ... | Coherence axiom |
c_31a2yx8zufka | In mathematics, and particularly complex dynamics, the escaping set of an entire function ƒ consists of all points that tend to infinity under the repeated application of ƒ. That is, a complex number z 0 ∈ C {\displaystyle z_{0}\in \mathbb {C} } belongs to the escaping set if and only if the sequence defined by z n + 1... | Escaping set |
c_m5okxw93y4xp | In mathematics, and particularly functional analysis, the Helly space, named after Eduard Helly, consists of all monotonically increasing functions ƒ: → , where denotes the closed interval given by the set of all x such that 0 ≤ x ≤ 1. In other words, for all 0 ≤ x ≤ 1 we have 0 ≤ ƒ(x) ≤ 1 and also if x ≤ y then ƒ(x)... | Helly space |
c_xpjspoadlurv | In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set X {\displaystyle X} , given by all points ( x , y ) {\displaystyle (x,y)} in the plane such that y ≥ 0 {\displaystyle y\geq 0} . The set X {\displaystyle X} can be termed the closed upper half plane. T... | Half-disk topology |
c_5r0n9snciuyj | In mathematics, and particularly homology theory, Steenrod's Problem (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds. | Steenrod problem |
c_84f6j4sn2pts | In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe (L) and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the axiom of cons... | Diamond principle |
c_szou8bubjcwk | In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principles that are a weaker version of the corresponding ◊S; it was introduced in 1975 by Adam Ostaszewski. | Clubsuit |
c_vtj2e71qz67c | In mathematics, and particularly in category theory, a polygraph is a generalisation of a directed graph. It is also known as a computad. They were introduced as "polygraphs" by Albert Burroni and as "computads" by Ross Street.In the same way that a directed multigraph can freely generate a category, an n-computad is t... | Polygraph (mathematics) |
c_amfue7jds7nu | In mathematics, and particularly in functional analysis, Fichera's existence principle is an existence and uniqueness theorem for solution of functional equations, proved by Gaetano Fichera in 1954. More precisely, given a general vector space V and two linear maps from it onto two Banach spaces, the principle states n... | Fichera's existence principle |
c_3y0o9r8kqjvr | In mathematics, and particularly in graph theory, the dimension of a graph is the least integer n such that there exists a "classical representation" of the graph in the Euclidean space of dimension n with all the edges having unit length. In a classical representation, the vertices must be distinct points, but the edg... | Dimension (graph theory) |
c_patfs12tzlrw | In mathematics, and particularly in number theory, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation 1 N + ∑ p | N 1 p = 1 , {\displaystyle {\frac {1}{N}}+\sum _{p\,|\;\!N}{\frac {1}{p}}=1,} where the sum is over only the prime divisors of N. | Primary pseudoperfect number |
c_lvlhlcg6mvtq | In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation. | Dirichlet principle |
c_tsr7ktl3rm96 | In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. In set theory, universes are often classes that contain (as elements) all sets for which one hopes to p... | Universe (mathematics) |
c_urby3qiy377a | Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations. For instance, the canonical motivating example of a category is Set, the category of all sets, which cannot be formalized in a set theory without some notion of a universe. In type theory, a universe is a... | Universe (mathematics) |
c_5s4dag5fttpd | In mathematics, and particularly in the field of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving its zeroes and an exponential of a polynomial. It is named for Jacques Hadamard. The theorem may be viewed as an extension ... | Hadamard factorization theorem |
c_hvogk63p9lff | In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every ... | Weierstrass factorization theorem |
c_a5ye8xnsyhsr | In mathematics, and particularly in the theory of formal languages, shortlex is a total ordering for finite sequences of objects that can themselves be totally ordered. In the shortlex ordering, sequences are primarily sorted by cardinality (length) with the shortest sequences first, and sequences of the same length ar... | Shortlex order |
c_wc36ptx42n06 | In mathematics, and particularly ordinary differential equations (ODEs), a monodromy matrix is the fundamental matrix of a system of ODEs evaluated at the period of the coefficients of the system. It is used for the analysis of periodic solutions of ODEs in Floquet theory. | Monodromy operator |
c_lt52h0jv8lko | In mathematics, and particularly ordinary differential equations, a characteristic multiplier is an eigenvalue of a monodromy matrix. The logarithm of a characteristic multiplier is also known as characteristic exponent. They appear in Floquet theory of periodic differential operators and in the Frobenius method. | Characteristic multiplier |
c_26if7inhtpnn | In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If f is a complex-valued holomorphic function germ then the Milnor number of f, denoted μ(f), is either a nonnegative integer, or is infinite. It can be considered both a geometric invari... | Milnor number |
c_o8dinrry7yo3 | In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space E {\displaystyle E} and a product space B × F {\displaystyle B\times F} ... | Trivialization (mathematics) |
c_clwtc9dszurx | The space E {\displaystyle E} is known as the total space of the fiber bundle, B {\displaystyle B} as the base space, and F {\displaystyle F} the fiber. In the trivial case, E {\displaystyle E} is just B × F , {\displaystyle B\times F,} and the map π {\displaystyle \pi } is just the projection from the product space to... | Trivialization (mathematics) |
c_giamo5tfxr42 | Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles. | Trivialization (mathematics) |
c_gf0gbna6mxsx | Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, and the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself (with the identity mapping as projection) to E {\displaystyle E} is called a section of ... | Trivialization (mathematics) |
c_w7mqyjimu6le | In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principa... | Connection one-form |
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