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In particular, if F {\displaystyle F} is grade 1 (vector-valued function), then we can write ∇ F = ∇ ⋅ F + ∇ ∧ F {\displaystyle \nabla F=\nabla \cdot F+\nabla \wedge F} and identify the divergence and curl as ∇ ⋅ F = div F , {\displaystyle \nabla \cdot F=\operatorname {div} F,} ∇ ∧ F = I curl F . {\displaystyle \na... | Wikipedia - Geometric calculus | null | null | null |
Let F {\displaystyle \mathbb {F} } be a real closed field. (For example F {\displaystyle \mathbb {F} } could be the field of real numbers R {\displaystyle \mathbb {R} } .) A subset S {\displaystyle S} of F n {\displaystyle \mathbb {F} ^{n}} is a semialgebraic set if it is a finite union of sets defined by polynomial eq... | Wikipedia - Semialgebraic set | null | null | null |
. , x n ) ∈ F n ∣ P ( x 1 , . . | Wikipedia - Semialgebraic set | null | null | null |
. , x n ) = 0 } {\displaystyle \{(x_{1},...,x_{n})\in \mathbb {F} ^{n}\mid P(x_{1},...,x_{n})=0\}} and of sets defined by polynomial inequalities of the form { ( x 1 , . . | Wikipedia - Semialgebraic set | null | null | null |
. , x n ) ∈ F n ∣ Q ( x 1 , . . | Wikipedia - Semialgebraic set | null | null | null |
. , x n ) > 0 } . {\displaystyle \{(x_{1},...,x_{n})\in \mathbb {F} ^{n}\mid Q(x_{1},...,x_{n})>0\}.} | Wikipedia - Semialgebraic set | null | null | null |
Let F: C ⟶ C {\displaystyle F:{\mathcal {C}}\longrightarrow {\mathcal {C}}} be an endofunctor on a category C {\displaystyle {\mathcal {C}}} . An F {\displaystyle F} -coalgebra is an object A {\displaystyle A} of C {\displaystyle {\mathcal {C}}} together with a morphism α: A ⟶ F A {\displaystyle \alpha :A\longrightarro... | Wikipedia - F-coalgebra | null | null | null |
Let F: R n → R n {\displaystyle {\boldsymbol {F}}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} be a (time-independent) vector field and x: R → R n {\displaystyle {\boldsymbol {x}}:\mathbb {R} \to \mathbb {R} ^{n}} the solution of the initial value problem x ˙ ( t ) = F ( x ( t ) ) , x ( 0 ) = x 0 . {\displaystyle {\dot {\bold... | Wikipedia - Flow (geometry) | null | null | null |
Let F: X → Y {\displaystyle F:X\to Y} be a linear operator between topological vector spaces (not necessarily Hausdorff). The following are equivalent: F {\displaystyle F} is (locally) bounded; (Definition): F {\displaystyle F} maps bounded subsets of its domain to bounded subsets of its codomain; F {\displaystyle F} m... | Wikipedia - Bounded linear operator | null | null | null |
{\displaystyle Y.} A sequence x ∙ = ( x i ) i = 1 ∞ {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }} is said to be Mackey convergent to the origin in X {\displaystyle X} if there exists a divergent sequence r ∙ = ( r i ) i = 1 ∞ → ∞ {\displaystyle r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty }\to \infty... | Wikipedia - Bounded linear operator | null | null | null |
if X {\displaystyle X} and Y {\displaystyle Y} are locally convex then the following may be add to this list: F {\displaystyle F} maps bounded disks into bounded disks. F − 1 {\displaystyle F^{-1}} maps bornivorous disks in Y {\displaystyle Y} into bornivorous disks in X . {\displaystyle X.} | Wikipedia - Bounded linear operator | null | null | null |
if X {\displaystyle X} is a bornological space and Y {\displaystyle Y} is locally convex then the following may be added to this list: F {\displaystyle F} is sequentially continuous at some (or equivalently, at every) point of its domain.A sequentially continuous linear map between two TVSs is always bounded, but the c... | Wikipedia - Bounded linear operator | null | null | null |
Let Fk be a free group of finite rank k ≥ 2. Fix a free basis A of Fk and an identification of Fk with the fundamental group of the rose Rk which is a wedge of k circles corresponding to the basis elements of A. Let φ ∈ Out(Fk) be an outer automorphism of Fk. A topological representative of φ is a triple (τ, Γ, f) wher... | Wikipedia - Train track map | null | null | null |
f: Γ → Γ is a combinatorial map which is also a homotopy equivalence. If σ: Γ → Rk is a homotopy inverse of τ then the compositionσfτ: Rk → Rk induces an automorphism of Fk = π1(Rk) whose outer automorphism class is equal to φ.The map τ in the above definition is called a marking and is typically suppressed when topolo... | Wikipedia - Train track map | null | null | null |
Let G ( V , E ) {\displaystyle G(V,E)} be a graph, and for each edge from u to v, let c ( u , v ) {\displaystyle c(u,v)} be the capacity and f ( u , v ) {\displaystyle f(u,v)} be the flow. We want to find the maximum flow from the source s to the sink t. After every step in the algorithm the following is maintained: Th... | Wikipedia - Ford-Fulkerson algorithm | null | null | null |
The path in step 2 can be found with, for example, a breadth-first search (BFS) or a depth-first search in G f ( V , E f ) {\displaystyle G_{f}(V,E_{f})} . If you use the former, the algorithm is called Edmonds–Karp. When no more paths in step 2 can be found, s will not be able to reach t in the residual network. | Wikipedia - Ford-Fulkerson algorithm | null | null | null |
If S is the set of nodes reachable by s in the residual network, then the total capacity in the original network of edges from S to the remainder of V is on the one hand equal to the total flow we found from s to t, and on the other hand serves as an upper bound for all such flows. This proves that the flow we found is... | Wikipedia - Ford-Fulkerson algorithm | null | null | null |
If the graph G ( V , E ) {\displaystyle G(V,E)} has multiple sources and sinks, we act as follows: Suppose that T = { t ∣ t is a sink } {\displaystyle T=\{t\mid t{\text{ is a sink}}\}} and S = { s ∣ s is a source } {\displaystyle S=\{s\mid s{\text{ is a source}}\}} . Add a new source s ∗ {\displaystyle s^{*}} with an e... | Wikipedia - Ford-Fulkerson algorithm | null | null | null |
Then apply the Ford–Fulkerson algorithm. Also, if a node u has capacity constraint d u {\displaystyle d_{u}} , we replace this node with two nodes u i n , u o u t {\displaystyle u_{\mathrm {in} },u_{\mathrm {out} }} , and an edge ( u i n , u o u t ) {\displaystyle (u_{\mathrm {in} },u_{\mathrm {out} })} , with capacity... | Wikipedia - Ford-Fulkerson algorithm | null | null | null |
Let G = ( V , E ) {\displaystyle G=(V,E)} be a directed graph with | V | = n {\displaystyle |V|=n} nodes and | E | = m {\displaystyle |E|=m} edges. Let s {\displaystyle s} be a distinguished vertex (called "source") and c {\displaystyle c} be a function assigning a non-negative real-valued weight to each edge. The goal... | Wikipedia - Parallel single-source shortest path algorithm | null | null | null |
The weight of a path is the sum of the weights of its edges. We set dist ( u , v ) := ∞ {\displaystyle \operatorname {dist} (u,v):=\infty } if v {\displaystyle v} is unreachable from u {\displaystyle u} .Sequential shortest path algorithms commonly apply iterative labeling methods based on maintaining a tentative dis... | Wikipedia - Parallel single-source shortest path algorithm | null | null | null |
Let G = ( V , E , w ) {\displaystyle G=(V,E,w)} be a directed Graph with the set of nodes V {\displaystyle V} and the set of edges E ⊆ V × V {\displaystyle E\subseteq V\times V} . Each edge e ∈ E {\displaystyle e\in E} has a weight w ( e ) {\displaystyle w(e)} assigned. The goal of the all-pair-shortest-paths problem i... | Wikipedia - Parallel all-pairs shortest path algorithm | null | null | null |
For this path to be unique it is required that the graph does not contain cycles with a negative weight. In the remainder of the article it is assumed that the graph is represented using an adjacency matrix. We expect the output of the algorithm to be a distancematrix D {\displaystyle D} . In D {\displaystyle D} , ever... | Wikipedia - Parallel all-pairs shortest path algorithm | null | null | null |
Let G = ( V , E , w ) {\displaystyle G=(V,E,w)} be a finite weighted graph and let f ∈ H ( V ) {\displaystyle f\in {\mathcal {H}}(V)} be a vertex function. Then the weighted difference (or weighted graph derivative) of f {\displaystyle f} along a directed edge ( x i , x j ) ∈ E {\displaystyle (x_{i},x_{j})\in E} is ∂ x... | Wikipedia - Calculus on finite weighted graphs | null | null | null |
Let G = ( V , E , w ) {\displaystyle G=(V,E,w)} be a weighted undirected graph. Suppose that s , t ∈ V {\displaystyle s,t\in V} . The cut is called an s {\displaystyle s} - t {\displaystyle t} cut if exactly one of s {\displaystyle s} or t {\displaystyle t} is in S {\displaystyle S} . The minimal cut of G {\displaystyl... | Wikipedia - Stoer–Wagner algorithm | null | null | null |
For any pair { s , t } {\displaystyle \left\{s,t\right\}} , there are two possible situations: either ( S , T ) {\displaystyle (S,T)} is a global min-cut of G {\displaystyle G} , or s {\displaystyle s} and t {\displaystyle t} belong to the same side of the global min-cut of G {\displaystyle G} . Therefore, the global m... | Wikipedia - Stoer–Wagner algorithm | null | null | null |
If s {\displaystyle s} and t {\displaystyle t} both have edges to some vertex v {\displaystyle v} , then the weight of the edge from the new vertex s t {\displaystyle st} to v {\displaystyle v} is w ( s , v ) + w ( t , v ) {\displaystyle w(s,v)+w(t,v)} . The algorithm is described as: MinimumCutPhase ( G , w , a ) {\di... | Wikipedia - Stoer–Wagner algorithm | null | null | null |
In the MinimumCutPhase, the subset A {\displaystyle A} of the graphs vertices grows starting with an arbitrary single vertex until A {\displaystyle A} is equal to V {\displaystyle V} . In each step, the vertex which is outside of A {\displaystyle A} , but most tightly connected with A {\displaystyle A} is added to the ... | Wikipedia - Stoer–Wagner algorithm | null | null | null |
So, in a single phase, a pair of vertices s {\displaystyle s} and t {\displaystyle t} , and a min s - t {\displaystyle s{\text{-}}t} cut C {\displaystyle C} is determined. After one phase of the MinimumCutPhase, the two vertices are merged as a new vertex, and edges from the two vertices to a remaining vertex are repla... | Wikipedia - Stoer–Wagner algorithm | null | null | null |
If there is a minimum cut of G {\displaystyle G} separating s {\displaystyle s} and t {\displaystyle t} , the C {\displaystyle C} is a minimum cut of G {\displaystyle G} . If not, then the minimum cut of G {\displaystyle G} must have s {\displaystyle s} and t {\displaystyle t} on a same side. Therefore, the algorithm w... | Wikipedia - Stoer–Wagner algorithm | null | null | null |
Let G = (V,w) be an instance of the travelling salesman problem. That is, G is a complete graph on the set V of vertices, and the function w assigns a nonnegative real weight to every edge of G. According to the triangle inequality, for every three vertices u, v, and x, it should be the case that w(uv) + w(vx) ≥ w(ux).... | Wikipedia - Christofides algorithm | null | null | null |
Find a minimum-weight perfect matching M in the induced subgraph given by the vertices from O. Combine the edges of M and T to form a connected multigraph H in which each vertex has even degree. Form an Eulerian circuit in H. Make the circuit found in previous step into a Hamiltonian circuit by skipping repeated vertic... | Wikipedia - Christofides algorithm | null | null | null |
Let G = G 1 × ⋯ × G r {\displaystyle G=G_{1}\times \cdots \times G_{r}} be a direct product of Lie groups and p i: G → G i {\displaystyle p_{i}:G\to G_{i}} projections. Then the differentials d p i: Lie ( G ) → Lie ( G i ) {\displaystyle dp_{i}:\operatorname {Lie} (G)\to \operatorname {Lie} (G_{i})} give the canoni... | Wikipedia - Lie group–Lie algebra correspondence | null | null | null |
{\displaystyle \operatorname {Lie} (H\cap H')=\operatorname {Lie} (H)\cap \operatorname {Lie} (H').} Let G be a connected Lie group. If H is a Lie group, then any Lie group homomorphism f: G → H {\displaystyle f:G\to H} is uniquely determined by its differential d f {\displaystyle df} . | Wikipedia - Lie group–Lie algebra correspondence | null | null | null |
Precisely, there is the exponential map exp: Lie ( G ) → G {\displaystyle \exp :\operatorname {Lie} (G)\to G} (and one for H) such that f ( exp ( X ) ) = exp ( d f ( X ) ) {\displaystyle f(\exp(X))=\exp(df(X))} and, since G is connected, this determines f uniquely. In general, if U is a neighborhood of the identi... | Wikipedia - Lie group–Lie algebra correspondence | null | null | null |
For example, if G is the Lie group of invertible real square matrices of size n (general linear group), then Lie ( G ) {\displaystyle \operatorname {Lie} (G)} is the Lie algebra of real square matrices of size n and exp ( X ) = e X = ∑ 0 ∞ X j / j ! {\textstyle \exp(X)=e^{X}=\sum _{0}^{\infty }{X^{j}/j!}} . | Wikipedia - Lie group–Lie algebra correspondence | null | null | null |
Let G = G a {\displaystyle G=\mathbb {G} _{a}} be the additive group; i.e., G(R) = R for any k-algebra R. As a variety G is the affine line; i.e., the coordinate ring is k and In0 = (tn). | Wikipedia - Distribution on a linear algebraic group | null | null | null |
Let G = G m {\displaystyle G=\mathbb {G} _{m}} be the multiplicative group; i.e., G(R) = R* for any k-algebra R. The coordinate ring of G is k (since G is really GL1(k).) | Wikipedia - Distribution on a linear algebraic group | null | null | null |
Let G and H be groups and let f be a group homomorphism from G to H. If eH is the identity element of H, then the kernel of f is the preimage of the singleton set {eH}; that is, the subset of G consisting of all those elements of G that are mapped by f to the element eH. The kernel is usually denoted ker f (or a variat... | Wikipedia - Kernel of a homomorphism | null | null | null |
{\displaystyle \ker f=\{g\in G:f(g)=e_{H}\}.} Since a group homomorphism preserves identity elements, the identity element eG of G must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {eG}. | Wikipedia - Kernel of a homomorphism | null | null | null |
If f were not injective, then the non-injective elements can form a distinct element of its kernel: there would exist a , b ∈ G {\displaystyle a,b\in G} such that a ≠ b {\displaystyle a\neq b} and f ( a ) = f ( b ) {\displaystyle f(a)=f(b)} . Thus f ( a ) f ( b ) − 1 = e H {\displaystyle f(a)f(b)^{-1}=e_{H}} . f is a g... | Wikipedia - Kernel of a homomorphism | null | null | null |
Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element g ≠ e G ∈ ker f {\displaystyle g\neq e_{G}\in \ker f} , then f ( g ) = f ( e G ) = e H {\displaystyle f(g)=f(e_{G})=e_{H}} , thus f would not be injective. ker f is a subgroup of G and further it is a normal subg... | Wikipedia - Kernel of a homomorphism | null | null | null |
Let G and H be groups, let P = G × H, and consider the following two subsets of P: G′ = { (g, 1): g ∈ G } and H′ = { (1, h): h ∈ H }.Both of these are in fact subgroups of P, the first being isomorphic to G, and the second being isomorphic to H. If we identify these with G and H, respectively, then we can think of the ... | Wikipedia - Direct product (group theory) | null | null | null |
Let G be a Lie group. The associated Lie algebra Lie ( G ) {\displaystyle \operatorname {Lie} (G)} of G may be alternatively defined as follows. Let A ( G ) {\displaystyle A(G)} be the algebra of distributions on G with support at the identity element with the multiplication given by convolution. A ( G ) {\displaysty... | Wikipedia - Lie correspondence | null | null | null |
Let G be a complex semisimple Lie group. Then G admits a natural structure of a linear algebraic group as follows: let A {\displaystyle A} be the ring of holomorphic functions f on G such that G ⋅ f {\displaystyle G\cdot f} spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G ac... | Wikipedia - Complex Lie group | null | null | null |
Let G be a connected Lie group. Then a symmetric space for G is a homogeneous space G/H where the stabilizer H of a typical point is an open subgroup of the fixed point set of an involution σ in Aut(G). Thus σ is an automorphism of G with σ2 = idG and H is an open subgroup of the invariant set G σ = { g ∈ G: σ ( g ) = ... | Wikipedia - Riemannian symmetric space | null | null | null |
Because H is open, it is a union of components of Gσ (including, of course, the identity component). As an automorphism of G, σ fixes the identity element, and hence, by differentiating at the identity, it induces an automorphism of the Lie algebra g {\displaystyle {\mathfrak {g}}} of G, also denoted by σ, whose square... | Wikipedia - Riemannian symmetric space | null | null | null |
The +1 eigenspace is the Lie algebra h {\displaystyle {\mathfrak {h}}} of H (since this is the Lie algebra of Gσ), and the −1 eigenspace will be denoted m {\displaystyle {\mathfrak {m}}} . Since σ is an automorphism of g {\displaystyle {\mathfrak {g}}} , this gives a direct sum decomposition g = h ⊕ m {\displaystyle {\... | Wikipedia - Riemannian symmetric space | null | null | null |
The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer h {\displaystyle {\mathfrak {h}}} is a Lie subalgebra of g {\displaystyle {\mathfrak {g}}} . The second condition means that m {\displaystyle {\mathfrak {m}}} is an h {\displaystyle {\mathfrak {h}}} -invariant compleme... | Wikipedia - Riemannian symmetric space | null | null | null |
Let G be a finite group and H any subgroup of G. Furthermore let (π, V) be a representation of H. Let n = be the index of H in G and let g1, ..., gn be a full set of representatives in G of the left cosets in G/H. The induced representation IndGH π can be thought of as acting on the following space: W = ⨁ i = 1 n g i ... | Wikipedia - Method of induced representation | null | null | null |
(This is just another way of saying that g1, ..., gn is a full set of representatives.) Via the induced representation G acts on W as follows: g ⋅ ∑ i = 1 n g i v i = ∑ i = 1 n g j ( i ) π ( h i ) v i {\displaystyle g\cdot \sum _{i=1}^{n}g_{i}v_{i}=\sum _{i=1}^{n}g_{j(i)}\pi (h_{i})v_{i}} where v i ∈ V {\displaystyle v... | Wikipedia - Method of induced representation | null | null | null |
Let G be a graph. There is a 1-to-1 correspondence between the vertex covers of G and the independent sets of G. For any set S of vertices of G, S is a vertex cover in G if and only if the complement of S is an independent set in G. Thus, the number of vertex covers in G is exactly the same as the number of independent... | Wikipedia - Holographic algorithm | null | null | null |
Let G be a group and L a field on which G acts as automorphisms. A cocycle or (Noether) factor system: 31 is a map c: G × G → L* satisfying c ( h , k ) g c ( h k , g ) = c ( h , k g ) c ( k , g ) . {\displaystyle c(h,k)^{g}c(hk,g)=c(h,kg)c(k,g).} Cocycles are equivalent if there exists some system of elements a: G → L*... | Wikipedia - Factor system | null | null | null |
{\displaystyle c'(g,h)=c(g,h)(a_{g}^{h}a_{h}a_{gh}^{-1}).} Cocycles of the form c ( g , h ) = a g h a h a g h − 1 {\displaystyle c(g,h)=a_{g}^{h}a_{h}a_{gh}^{-1}} are called split. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group H2(G,L*). | Wikipedia - Factor system | null | null | null |
Let G be a group and k a field. The group Hopf algebra of G over k, denoted kG (or k), is as a set (and a vector space) the free vector space on G over k. As an algebra, its product is defined by linear extension of the group composition in G, with multiplicative unit the identity in G; this product is also known as co... | Wikipedia - Group Hopf algebra | null | null | null |
Let G be a locally compact abelian group, and Ĝ the dual group of G. Then L 1 ( G ^ ) {\displaystyle L_{1}({\hat {\mathit {G}}})} is the space of all functions on Ĝ which are integrable with respect to the Haar measure on Ĝ, and it has a Banach algebra structure where the product of two functions is convolution. We def... | Wikipedia - Fourier algebra | null | null | null |
It is a closed sub-algebra of C B ( G ) {\displaystyle CB(G)} , the space of bounded continuous complex-valued functions on G with pointwise multiplication. We call B ( G ) {\displaystyle B(G)} the Fourier-Stieltjes algebra of G. Equivalently, B ( G ) {\displaystyle B(G)} can be defined as the linear span of the set P ... | Wikipedia - Fourier algebra | null | null | null |
Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows. The objects are the elements of G, and for any two elements x and y of G, there exists a unique morphism from x to y if and only if x ∼ y . {\displaystyle x\sim y.} The advant... | Wikipedia - Equivalence relations | null | null | null |
Let G be a split graph, partitioned into a clique C and an independent set i. Then every maximal clique in a split graph is either C itself, or the neighborhood of a vertex in i. Thus, it is easy to identify the maximum clique, and complementarily the maximum independent set in a split graph. In any split graph, one of... | Wikipedia - Split graph | null | null | null |
In this case, i ∪ {x} is a maximum independent set and C is a maximum clique. C is a maximal clique and i is a maximal independent set. In this case, G has a unique partition (C, i) into a clique and an independent set, C is the maximum clique, and i is the maximum independent set.Some other optimization problems that ... | Wikipedia - Split graph | null | null | null |
Let G be a weighted directed graph with vertex set V and edge set E (figure A); let s be a designated source vertex in G, and let t be a designated destination vertex. Let each edge (u,v) in E, from vertex u to vertex v, have a non-negative cost w(u,v). Define d(s,u) to be the cost of the shortest path to vertex u from... | Wikipedia - Suurballe's algorithm | null | null | null |
Let G be an algebraic group acting from the right on a scheme X of finite type over a field k. Then the group action of G on X determines a prestack (but not a stack) over the category C of k-schemes, as follows. Let F be the category where an object is a pair ( U , x ) {\displaystyle (U,x)} consisting of a scheme U in... | Wikipedia - Prestack | null | null | null |
When X is a point ∗ = Spec ( k ) {\displaystyle *=\operatorname {Spec} (k)} and G is affine, the quotient p r e = B G p r e {\displaystyle ^{pre}=BG^{pre}} is the classifying prestack of G and its stackification is the classifying stack of G. One viewing X as a prestack (in fact a stack), there is the obvious canoni... | Wikipedia - Prestack | null | null | null |
Let G be an algebraic group that is "finite" as a group scheme; for example, any finite group may be viewed as a finite algebraic group. There is an equivalence of categories between the category of finite algebraic groups and the category of finite-dimensional cocommutative Hopf algebras given by mapping G to k*, the ... | Wikipedia - Distribution on a linear algebraic group | null | null | null |
Let G be the cyclic group on 6 elements {0, 1, 2, 3, 4, 5} with modular addition, H be the cyclic on 2 elements {0, 1} with modular addition, and f the homomorphism that maps each element g in G to the element g modulo 2 in H. Then ker f = {0, 2, 4} , since all these elements are mapped to 0H. The quotient group G/(ker... | Wikipedia - Kernel (group theory) | null | null | null |
Let G be the gravitational constant and let M be the mass of the earth (or other gravitating body) and m be the mass of the escaping body or projectile. At a distance r from the centre of gravitation the body feels an attractive force F = G M m r 2 . {\displaystyle F=G{\frac {Mm}{r^{2}}}.} The work needed to move the b... | Wikipedia - Escape velocity | null | null | null |
{\displaystyle dW=F\,dr=G{\frac {Mm}{r^{2}}}\,dr.} The total work needed to move the body from the surface r0 of the gravitating body to infinity is then W = ∫ r 0 ∞ G M m r 2 d r = G M m r 0 = m g r 0 . {\displaystyle W=\int _{r_{0}}^{\infty }G{\frac {Mm}{r^{2}}}\,dr=G{\frac {Mm}{r_{0}}}=mgr_{0}.} In order to do this ... | Wikipedia - Escape velocity | null | null | null |
Let G denote the absolute Galois group of K. Then the category of étale K-algebras is equivalent to the category of finite G-sets with continuous G-action. In particular, étale algebras of dimension n are classified by conjugacy classes of continuous homomorphisms from G to the symmetric group Sn. These globalize to e.... | Wikipedia - Étale algebra | null | null | null |
Let G i ( R ) {\displaystyle G_{i}(R)} be the algebraic K-theory of the category of finitely generated modules over a noetherian ring R; explicitly, we can take G i ( R ) = π i ( B + f-gen-Mod R ) {\displaystyle G_{i}(R)=\pi _{i}(B^{+}{\text{f-gen-Mod}}_{R})} , where B + = Ω B Q {\displaystyle B^{+}=\Omega BQ} is given... | Wikipedia - Fundamental theorem of algebraic K-theory | null | null | null |
Let G n ( R n + k ) {\displaystyle G_{n}(\mathbb {R} ^{n+k})} be the Grassmannian of n-dimensional vector subspaces in R n + k ; {\displaystyle \mathbb {R} ^{n+k};} as a set it is the set of all n-dimensional vector subspaces of R n + k . {\displaystyle \mathbb {R} ^{n+k}.} For example, if n = 1, it is the real project... | Wikipedia - Tautological subbundle | null | null | null |
We define the tautological bundle γn, k over G n ( R n + k ) {\displaystyle G_{n}(\mathbb {R} ^{n+k})} as follows. The total space of the bundle is the set of all pairs (V, v) consisting of a point V of the Grassmannian and a vector v in V; it is given the subspace topology of the Cartesian product G n ( R n + k ) × R ... | Wikipedia - Tautological subbundle | null | null | null |
The projection map π is given by π(V, v) = V. If F is the pre-image of V under π, it is given a structure of a vector space by a(V, v) + b(V, w) = (V, av + bw). Finally, to see local triviality, given a point X in the Grassmannian, let U be the set of all V such that the orthogonal projection p onto X maps V isomorphic... | Wikipedia - Tautological subbundle | null | null | null |
{\displaystyle \mathbb {C} .} By definition, the infinite Grassmannian G n {\displaystyle G_{n}} is the direct limit of G n ( R n + k ) {\displaystyle G_{n}(\mathbb {R} ^{n+k})} as k → ∞ . {\displaystyle k\to \infty .} | Wikipedia - Tautological subbundle | null | null | null |
Taking the direct limit of the bundles γn, k gives the tautological bundle γn of G n . {\displaystyle G_{n}.} It is a universal bundle in the sense: for each compact space X, there is a natural bijection { → Vect n R ( X ) f ↦ f ∗ ( γ n ) {\displaystyle {\begin{cases}\to \operatorname {Vect} _{n}^{\mathbb {R} }(X)\\... | Wikipedia - Tautological subbundle | null | null | null |
Remark: In turn, one can define a tautological bundle as a universal bundle; suppose there is a natural bijection = Vect n R ( X ) {\displaystyle =\operatorname {Vect} _{n}^{\mathbb {R} }(X)} for any paracompact space X. Since G n {\displaystyle G_{n}} is the direct limit of compact spaces, it is paracompact and so ... | Wikipedia - Tautological subbundle | null | null | null |
Let G now be a connected reductive group over an algebraically closed field. Then any two Borel subgroups are conjugate by an inner automorphism, so to study outer automorphisms it suffices to consider automorphisms that fix a given Borel subgroup. Associated to the Borel subgroup is a set of simple roots, and the oute... | Wikipedia - Outer automorphism | null | null | null |
Let G {\displaystyle G} be a connected reductive group over the complex plane C {\displaystyle \mathbb {C} } , with a distinguished Borel subgroup B = B G ⊂ G {\displaystyle B=B_{G}\subset G} . Set N = {\displaystyle N=} , so that H = B / N {\displaystyle H=B/N} is the Cartan group. Denote by n < b < g {\displaystyle ... | Wikipedia - Oper (mathematics) | null | null | null |
Let G {\displaystyle G} be a connected, weighted graph and let Y {\displaystyle Y} be the subgraph of G {\displaystyle G} produced by the algorithm. Y {\displaystyle Y} cannot have a cycle, as by definition an edge is not added if it results in a cycle. Y {\displaystyle Y} cannot be disconnected, since the first encoun... | Wikipedia - Kruskal's algorithm | null | null | null |
Let G {\displaystyle G} be a finite group, let K {\displaystyle K} be a commutative ring and let K {\displaystyle K} be the group algebra of G {\displaystyle G} over K . {\displaystyle K.} This algebra is free and a basis can be indexed by the elements of G . {\displaystyle G.} | Wikipedia - Complex representations of finite groups | null | null | null |
Most often the basis is identified with G {\displaystyle G} . Every element f ∈ K {\displaystyle f\in K} can then be uniquely expressed as f = ∑ s ∈ G a s s {\displaystyle f=\sum _{s\in G}a_{s}s} with a s ∈ K {\displaystyle a_{s}\in K} .The multiplication in K {\displaystyle K} extends that in G {\displaystyle G} dis... | Wikipedia - Complex representations of finite groups | null | null | null |
{\displaystyle V.} We define s v = ρ ( s ) v {\displaystyle sv=\rho (s)v} for all s ∈ G {\displaystyle s\in G} and v ∈ V {\displaystyle v\in V} . By linear extension V {\displaystyle V} is endowed with the structure of a left- K {\displaystyle K} –module. | Wikipedia - Complex representations of finite groups | null | null | null |
Vice versa we obtain a linear representation of G {\displaystyle G} starting from a K {\displaystyle K} –module V {\displaystyle V} . Additionally, homomorphisms of representations are in bijective correspondence with group algebra homomorphisms. | Wikipedia - Complex representations of finite groups | null | null | null |
Therefore, these terms may be used interchangeably. This is an example of an isomorphism of categories. | Wikipedia - Complex representations of finite groups | null | null | null |
Suppose K = C . {\displaystyle K=\mathbb {C} .} In this case the left C {\displaystyle \mathbb {C} } –module given by C {\displaystyle \mathbb {C} } itself corresponds to the left-regular representation. | Wikipedia - Complex representations of finite groups | null | null | null |
In the same way C {\displaystyle \mathbb {C} } as a right C {\displaystyle \mathbb {C} } –module corresponds to the right-regular representation. In the following we will define the convolution algebra: Let G {\displaystyle G} be a group, the set L 1 ( G ) := { f: G → C } {\displaystyle L^{1}(G):=\{f:G\to \mathbb {C}... | Wikipedia - Complex representations of finite groups | null | null | null |
The convolution of two elements f , h ∈ L 1 ( G ) {\displaystyle f,h\in L^{1}(G)} defined by f ∗ h ( s ) := ∑ t ∈ G f ( t ) h ( t − 1 s ) {\displaystyle f*h(s):=\sum _{t\in G}f(t)h(t^{-1}s)} makes L 1 ( G ) {\displaystyle L^{1}(G)} an algebra. The algebra L 1 ( G ) {\displaystyle L^{1}(G)} is called the convolution alg... | Wikipedia - Complex representations of finite groups | null | null | null |
{\displaystyle \delta _{s}(t)={\begin{cases}1&t=s\\0&{\text{otherwise. }}\end{cases}}} Using the properties of the convolution we obtain: δ s ∗ δ t = δ s t . {\displaystyle \delta _{s}*\delta _{t}=\delta _{st}.} | Wikipedia - Complex representations of finite groups | null | null | null |
We define a map between L 1 ( G ) {\displaystyle L^{1}(G)} and C , {\displaystyle \mathbb {C} ,} by defining δ s ↦ e s {\displaystyle \delta _{s}\mapsto e_{s}} on the basis ( δ s ) s ∈ G {\displaystyle (\delta _{s})_{s\in G}} and extending it linearly. Obviously the prior map is bijective. A closer inspection of the c... | Wikipedia - Complex representations of finite groups | null | null | null |
{\displaystyle \mathbb {C} .} Thus, the convolution algebra and the group algebra are isomorphic as algebras. | Wikipedia - Complex representations of finite groups | null | null | null |
The involution f ∗ ( s ) = f ( s − 1 ) ¯ {\displaystyle f^{*}(s)={\overline {f(s^{-1})}}} turns L 1 ( G ) {\displaystyle L^{1}(G)} into a ∗ {\displaystyle ^{*}} –algebra. We have δ s ∗ = δ s − 1 . {\displaystyle \delta _{s}^{*}=\delta _{s^{-1}}.} | Wikipedia - Complex representations of finite groups | null | null | null |
A representation ( π , V π ) {\displaystyle (\pi ,V_{\pi })} of a group G {\displaystyle G} extends to a ∗ {\displaystyle ^{*}} –algebra homomorphism π: L 1 ( G ) → End ( V π ) {\displaystyle \pi :L^{1}(G)\to {\text{End}}(V_{\pi })} by π ( δ s ) = π ( s ) . {\displaystyle \pi (\delta _{s})=\pi (s).} Since multiplicity ... | Wikipedia - Complex representations of finite groups | null | null | null |
{\displaystyle \pi (f*h)=\pi (f)\pi (h).} If π {\displaystyle \pi } is unitary, we also obtain π ( f ) ∗ = π ( f ∗ ) . {\displaystyle \pi (f)^{*}=\pi (f^{*}).} | Wikipedia - Complex representations of finite groups | null | null | null |
For the definition of a unitary representation, please refer to the chapter on properties. In that chapter we will see that (without loss of generality) every linear representation can be assumed to be unitary. Using the convolution algebra we can implement a Fourier transformation on a group G . | Wikipedia - Complex representations of finite groups | null | null | null |
{\displaystyle G.} In the area of harmonic analysis it is shown that the following definition is consistent with the definition of the Fourier transformation on R . {\displaystyle \mathbb {R} .} | Wikipedia - Complex representations of finite groups | null | null | null |
Let ρ: G → GL ( V ρ ) {\displaystyle \rho :G\to {\text{GL}}(V_{\rho })} be a representation and let f ∈ L 1 ( G ) {\displaystyle f\in L^{1}(G)} be a C {\displaystyle \mathbb {C} } -valued function on G {\displaystyle G} . The Fourier transform f ^ ( ρ ) ∈ End ( V ρ ) {\displaystyle {\hat {f}}(\rho )\in {\text{End}}(V_{... | Wikipedia - Complex representations of finite groups | null | null | null |
Let G {\displaystyle G} be a finite p-group with d {\displaystyle d} generators. Our goal is to compile a complete list of pairwise non-isomorphic immediate descendants of G {\displaystyle G} . It turns out that all immediate descendants can be obtained as quotients of a certain extension G ∗ {\displaystyle G^{\ast }} ... | Wikipedia - P-group generation algorithm | null | null | null |
We can certainly find a presentation of G {\displaystyle G} in the form of an exact sequence ( 9 ) 1 ⟶ R ⟶ F ⟶ G ⟶ 1 {\displaystyle (9)\qquad 1\longrightarrow R\longrightarrow F\longrightarrow G\longrightarrow 1} , where F {\displaystyle F} denotes the free group with d {\displaystyle d} generators and ϑ: F ⟶ G {\displ... | Wikipedia - P-group generation algorithm | null | null | null |
Consequently, R ∗ := ⋅ R p {\displaystyle R^{\ast }:=\lbrack R,F\rbrack \cdot R^{p}} is a characteristic subgroup of R {\displaystyle R} , and the p-multiplicator R / R ∗ {\displaystyle R/R^{\ast }} of G {\displaystyle G} is an elementary abelian p-group, since ( 10 ) ⋅ R p ≤ ⋅ R p = R ∗ {\displaystyle (10)\qquad \l... | Wikipedia - P-group generation algorithm | null | null | null |
Let us assume now that the assigned finite p-group G ≃ F / R {\displaystyle G\simeq F/R} is of p-class c l p ( G ) = c {\displaystyle \mathrm {cl} _{p}(G)=c} . Then the conditions R ◃ F {\displaystyle R\triangleleft F} and c l p ( F / R ) = c {\displaystyle \mathrm {cl} _{p}(F/R)=c} imply P c ( F ) ≤ R {\displaystyle P... | Wikipedia - P-group generation algorithm | null | null | null |
Let G {\displaystyle G} be a linear algebraic group over a perfect field. Then, essentially by definition, there is a closed embedding G ↪ G L n {\displaystyle G\hookrightarrow \mathbf {GL} _{n}} . Now, to each element g ∈ G {\displaystyle g\in G} , by the multiplicative Jordan decomposition, there are a pair of a semi... | Wikipedia - Jordan-Chevalley decomposition | null | null | null |
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