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Rather than defining ⋀ ( V ) {\textstyle \bigwedge (V)} first and then identifying the exterior powers ⋀ k ( V ) {\textstyle \bigwedge \nolimits ^{k}(V)} as certain subspaces, one may alternatively define the spaces ⋀ k ( V ) {\textstyle \bigwedge \nolimits ^{k}(V)} first and then combine them to form the algebra ⋀ ( V... | Wikipedia - Alternating form | null | null | null |
Let V be the three-dimensional vector space defined over the field F. The projective plane P(V) = PG(2, F) consists of the one-dimensional vector subspaces of V, called points, and the two-dimensional vector subspaces of V, called lines. Incidence of a point and a line is given by containment of the one-dimensional sub... | Wikipedia - Incidence (geometry) | null | null | null |
The non-zero scalar multiples, written as coordinate triples, are the homogeneous coordinates of the given point, called point coordinates. With respect to this basis, the solution space of a single linear equation {(x, y, z) | ax + by + cz = 0} is a two-dimensional subspace of V, and hence a line of P(V). This line ma... | Wikipedia - Incidence (geometry) | null | null | null |
Let V h ⊂ H 0 1 ( Ω ) {\displaystyle V_{h}\subset H_{0}^{1}(\Omega )} be spanned by the finite basis ( ψ i ) i ∈ I {\displaystyle (\psi _{i})_{i\in I}} . The Galerkin method in V h {\displaystyle V_{h}} is identical to the GDM where one defines X D , 0 = { u = ( u i ) i ∈ I } = R I , {\displaystyle X_{D,0}=\{u=(u_{i})_... | Wikipedia - Gradient discretization method | null | null | null |
Then (4) and (5) are implied by Céa's lemma. The "mass-lumped" P 1 {\displaystyle P^{1}} finite element case enters the framework of the GDM, replacing Π D u {\displaystyle \Pi _{D}u} by Π ~ D u = ∑ i ∈ I u i χ Ω i {\textstyle {\widetilde {\Pi }}_{D}u=\sum _{i\in I}u_{i}\chi _{\Omega _{i}}} , where Ω i {\displaystyle \... | Wikipedia - Gradient discretization method | null | null | null |
Let V {\displaystyle V} be a Euclidean space of a finite dimension and Σ {\displaystyle \Sigma } an affine root system on V {\displaystyle V} . An affine Hecke algebra is a certain associative algebra that deforms the group algebra C {\displaystyle \mathbb {C} } of the Weyl group W {\displaystyle W} of Σ {\displaystyl... | Wikipedia - Affine Hecke algebra | null | null | null |
Let V {\displaystyle V} be a braided vector space, this means there is an action of the braid group B n {\displaystyle \mathbb {B} _{n}} on V ⊗ n {\displaystyle V^{\otimes n}} for any n ∈ N {\displaystyle n\in \mathbb {N} } , where the transposition ( i , i + 1 ) {\displaystyle (i,i+1)} acts as i d ⊗ ⋯ ⊗ τ ⊗ i d ⋯ i d ... | Wikipedia - Nichols algebra | null | null | null |
This is not a group homomorphism, but Matsumoto's theorem (group theory) tells us that the action of any s ( σ ) {\displaystyle s(\sigma )} on V ⊗ n {\displaystyle V^{\otimes n}} is well-defined independently of the choice of a reduced expression. Finally the Nichols algebra is then W n := ∑ σ ∈ S n s ( σ ): V ⊗ n → V ... | Wikipedia - Nichols algebra | null | null | null |
Let V {\displaystyle V} be a finite-dimensional vector space over an algebraically closed field of characteristic zero, K {\displaystyle K} , carrying a representation of a group G {\displaystyle G} , and consider the polynomial algebra on V {\displaystyle V} , K {\displaystyle K} . The algebra K {\displaystyle K} ca... | Wikipedia - Miracle flatness | null | null | null |
The zero set of the { θ i } {\displaystyle \{\theta _{i}\}} , { v ∈ V | θ i = 0 } {\displaystyle \{v\in V|\theta _{i}=0\}} , coincides with the nullcone (link) of R {\displaystyle R} .Importantly, this implies that the algebra can then be expressed as a finitely-generated module over the subalgebra generated by the HSO... | Wikipedia - Miracle flatness | null | null | null |
Let V {\displaystyle V} be a real Z 2 {\displaystyle \mathbb {Z} _{2}} -graded vector space equipped with a nonsingular antisymmetric bilinear superform ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} (i.e. ( g , f ) = − ( − 1 ) | f | | g | ( f , g ) {\displaystyle (g,f)=-(-1)^{|f||g|}(f,g)} ) such that ( f , g ) {\displayst... | Wikipedia - Canonical anticommutation relation | null | null | null |
Let V {\displaystyle V} be a real vector space equipped with a nonsingular real antisymmetric bilinear form ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} (i.e. a symplectic vector space). The unital *-algebra generated by elements of V {\displaystyle V} subject to the relations f g − g f = i ( f , g ) {\displaystyle fg-gf=... | Wikipedia - CAR algebra | null | null | null |
Let V {\displaystyle V} be a vector space over a field F {\displaystyle \mathbb {F} } and let ( e i ) i ∈ I {\displaystyle \left(e_{i}\right)_{i\in I}} be an I {\displaystyle I} -indexed basis of vectors for V . {\displaystyle V.} By the definition of a basis, every vector v ∈ V {\displaystyle v\in V} can be expressed ... | Wikipedia - Primordial element (algebra) | null | null | null |
Given a subspace W {\displaystyle W} of V , {\displaystyle V,} a nonzero vector p ∈ W {\displaystyle p\in W} is said to be primordial if it has both of the following two properties: I ( p ) {\displaystyle I(p)} is minimal among the sets I ( w ) , {\displaystyle I(w),} where 0 ≠ w ∈ W , {\displaystyle 0\neq w\in W,} and... | Wikipedia - Primordial element (algebra) | null | null | null |
Let V {\displaystyle V} be an n-dimensional vector space over field K {\displaystyle K} with basis { e 1 , … , e n } . {\displaystyle \{e_{1},\ldots ,e_{n}\}.} For A ∈ End ( V ) , {\displaystyle A\in \operatorname {End} (V),} define ⋀ k A ∈ End ( ⋀ k V ) {\textstyle \bigwedge ^{k}A\in \operatorname {End} {\bigl (}\... | Wikipedia - Exterior power | null | null | null |
For k = p = n {\displaystyle k=p=n} , this recovers the determinant of A {\displaystyle A} . For k = 1 , p = n {\displaystyle k=1,p=n} , this recovers the trace of A {\displaystyle A} . If p < k , {\displaystyle p | Wikipedia - Exterior power | null | null | null |
Let V ∈ H H Y D {\displaystyle V\in {}_{H}^{H}{\mathcal {YD}}} . There exists a largest ideal I ⊂ T V {\displaystyle {\mathfrak {I}}\subset TV} with the following properties: I ⊂ ⨁ n = 2 ∞ T n V , {\displaystyle {\mathfrak {I}}\subset \bigoplus _{n=2}^{\infty }T^{n}V,} Δ ( I ) ⊂ I ⊗ T V + T V ⊗ I {\displaystyle \Delta ... | Wikipedia - Nichols algebra | null | null | null |
Let V1, V2 be algebraic varieties. We say V1 and V2 are isomorphic, and write V1 ≅ V2, if there are regular maps φ: V1 → V2 and ψ: V2 → V1 such that the compositions ψ ∘ φ and φ ∘ ψ are the identity maps on V1 and V2 respectively. | Wikipedia - Projective algebraic set | null | null | null |
Let W {\displaystyle W} be a finite-dimensional vector space and P {\displaystyle P} be a projection on W {\displaystyle W} . Suppose the subspaces U {\displaystyle U} and V {\displaystyle V} are the image and kernel of P {\displaystyle P} respectively. Then P {\displaystyle P} has the following properties: P {\display... | Wikipedia - Projector operator | null | null | null |
{\displaystyle \mathbf {u} \in U,\mathbf {v} \in V.} The image and kernel of a projection are complementary, as are P {\displaystyle P} and Q = I − P {\displaystyle Q=I-P} . The operator Q {\displaystyle Q} is also a projection as the image and kernel of P {\displaystyle P} become the kernel and image of Q {\displaysty... | Wikipedia - Projector operator | null | null | null |
Let X # {\displaystyle X^{\#}} denote the algebraic dual space of a vector space X . {\displaystyle X.} Let X {\displaystyle X} and Y {\displaystyle Y} be vector spaces over the same field K . | Wikipedia - Algebraic adjoint | null | null | null |
{\displaystyle {\mathcal {K}}.} If u: X → Y {\displaystyle u:X\to Y} is a linear map, then its algebraic adjoint or dual, is the map # u: Y # → X # {\displaystyle {}^{\#}u:Y^{\#}\to X^{\#}} defined by f ↦ f ∘ u . {\displaystyle f\mapsto f\circ u.} | Wikipedia - Algebraic adjoint | null | null | null |
The resulting functional # u ( f ) := f ∘ u {\displaystyle {}^{\#}u(f):=f\circ u} is called the pullback of f {\displaystyle f} by u . {\displaystyle u.} The continuous dual space of a topological vector space (TVS) X {\displaystyle X} is denoted by X ′ . | Wikipedia - Algebraic adjoint | null | null | null |
{\displaystyle X^{\prime }.} If X {\displaystyle X} and Y {\displaystyle Y} are TVSs then a linear map u: X → Y {\displaystyle u:X\to Y} is weakly continuous if and only if # u ( Y ′ ) ⊆ X ′ , {\displaystyle {}^{\#}u\left(Y^{\prime }\right)\subseteq X^{\prime },} in which case we let t u: Y ′ → X ′ {\displaystyle {}^{t... | Wikipedia - Algebraic adjoint | null | null | null |
The map t u {\displaystyle {}^{t}u} is called the transpose or algebraic adjoint of u . {\displaystyle u.} The following identity characterizes the transpose of u {\displaystyle u} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \left\langle \cdot ,\cdot \right\rangle } is the natural pairing defined by ⟨ z , h ⟩ := z ( h ) . {\display... | Wikipedia - Algebraic adjoint | null | null | null |
Let X , C ∈ R m × n {\displaystyle X,C\in \mathbb {R} ^{m\times n}} , and assume that the eigenvalues of A {\displaystyle A} are distinct from the eigenvalues of B {\displaystyle B} . Then, the matrix equation A X − X B = C {\displaystyle AX-XB=C} has a unique solution. The Bartels–Stewart algorithm computes X {\displa... | Wikipedia - Bartels–Stewart algorithm | null | null | null |
The matrices R {\displaystyle R} and S {\displaystyle S} are block-upper triangular matrices, with diagonal blocks of size 1 × 1 {\displaystyle 1\times 1} or 2 × 2 {\displaystyle 2\times 2} . 2. | Wikipedia - Bartels–Stewart algorithm | null | null | null |
Set F = U T C V . {\displaystyle F=U^{T}CV.} 3. | Wikipedia - Bartels–Stewart algorithm | null | null | null |
Solve the simplified system R Y − Y S T = F {\displaystyle RY-YS^{T}=F} , where Y = U T X V {\displaystyle Y=U^{T}XV} . This can be done using forward substitution on the blocks. Specifically, if s k − 1 , k = 0 {\displaystyle s_{k-1,k}=0} , then ( R − s k k I ) y k = f k + ∑ j = k + 1 n s k j y j , {\displaystyle (R-s... | Wikipedia - Bartels–Stewart algorithm | null | null | null |
When s k − 1 , k ≠ 0 {\displaystyle s_{k-1,k}\neq 0} , columns {\displaystyle } should be concatenated and solved for simultaneously. 4. Set X = U Y V T . {\displaystyle X=UYV^{T}.} | Wikipedia - Bartels–Stewart algorithm | null | null | null |
Let X 1 , . . . , X n {\displaystyle X_{1},...,X_{n}} be an i.i.d. | Wikipedia - Uniform measure | null | null | null |
sample from U ( 0 , 1 ) , {\displaystyle U(0,1),} and let X ( k ) {\displaystyle X_{(k)}} be the k {\displaystyle k} -th order statistic from this sample. X ( k ) {\displaystyle X_{(k)}} has a beta distribution, with parameters k {\displaystyle k} and n − k + 1. {\displaystyle n-k+1.} | Wikipedia - Uniform measure | null | null | null |
The expected value is: E ( X ( k ) ) = k n + 1 . {\displaystyle \operatorname {E} (X_{(k)})={k \over n+1}.} This fact is useful when making Q–Q plots. The variance is: V ( X ( k ) ) = k ( n − k + 1 ) ( n + 1 ) 2 ( n + 2 ) . {\displaystyle \operatorname {V} (X_{(k)})={k(n-k+1) \over (n+1)^{2}(n+2)}.} | Wikipedia - Uniform measure | null | null | null |
Let X 1 , . . X n {\displaystyle X_{1},..X_{n}} be a set of random variables. | Wikipedia - Rank statistics | null | null | null |
By sorting them into order, we have defined their order statistics X n , ( 1 ) ≤ . . . | Wikipedia - Rank statistics | null | null | null |
≤ X n , ( n ) {\displaystyle X_{n,(1)}\leq ...\leq X_{n,(n)}} If all the values are unique, the rank of variable number i {\displaystyle i} is the unique solution R n , i {\displaystyle R_{n,i}} to the equation X i = X N , ( R n , i ) {\displaystyle X_{i}=X_{N,(R_{n,i})}} . In the presence of ties, we may either use a ... | Wikipedia - Rank statistics | null | null | null |
Let X 1 , X 2 , ⋯ {\displaystyle X_{1},X_{2},\cdots } be independent random variables with mean 0 and variance 1, then let S n := 1 n ∑ i = 1 n X i {\displaystyle S_{n}:={\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}X_{i}} . We can compute the moments of S n {\displaystyle S_{n}} asExplicit expansion shows thatwhere the numerat... | Wikipedia - Method of moments (statistics) | null | null | null |
Let X 1 , … , X n {\displaystyle X_{1},\dotsc ,X_{n}} be n {\displaystyle n} independent and identically distributed exponential random variables with rate parameter λ. Let X ( 1 ) , … , X ( n ) {\displaystyle X_{(1)},\dotsc ,X_{(n)}} denote the corresponding order statistics. For i < j {\displaystyle i | Wikipedia - Exponential distribution | null | null | null |
Let X = ( x i j ) ∈ R m × n {\displaystyle \mathbf {X} =(x_{ij})\in \mathbb {R} ^{m\times n}} denote an observed data matrix whose n {\displaystyle n} columns correspond to observations of m {\displaystyle m} -variate mixed vectors. It is assumed that X {\displaystyle \mathbf {X} } is prewhitened, that is, its rows hav... | Wikipedia - Joint Approximation Diagonalization of Eigen-matrices | null | null | null |
Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Y can be identified with the space of all lines in X which are parallel to Y. That is to say that, the elements of the set X/Y are lines in X parallel to Y. Note that the points along any one such line wil... | Wikipedia - Quotient space (linear algebra) | null | null | null |
Let X = Spec A be an affine scheme over a field k and let Ix be the kernel of the restriction map A → k ( x ) {\displaystyle A\to k(x)} , the residue field of x. By definition, a distribution f supported at x'' is a k-linear functional on A such that f ( I x n ) = 0 {\displaystyle f(I_{x}^{n})=0} for some n. (Note: the... | Wikipedia - Distribution on a linear algebraic group | null | null | null |
The multiplication turns out to be associative (use 1 ⊗ Δ ∘ Δ = Δ ⊗ 1 ∘ Δ {\displaystyle 1\otimes \Delta \circ \Delta =\Delta \otimes 1\circ \Delta } ) and thus Dist(G) is an associative algebra, as the set is closed under the muplication by the formula: (*) Δ ( I 1 n ) ⊂ ∑ r = 0 n I 1 r ⊗ I 1 n − r . {\displaystyle \D... | Wikipedia - Distribution on a linear algebraic group | null | null | null |
The Lie algebra Lie(G) sits inside Dist(G). Indeed, by definition, Lie(G) is the tangent space to G at the identity element 1; i.e., the dual space of I 1 / I 1 2 {\displaystyle I_{1}/I_{1}^{2}} . Thus, a tangent vector amounts to a linear functional on I1 that has no constant term and kills the square of I1 and the fo... | Wikipedia - Distribution on a linear algebraic group | null | null | null |
Let g = Lie ( G ) {\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)} be the Lie algebra of G. Then, by the universal property, the inclusion g ↪ Dist ( G ) {\displaystyle {\mathfrak {g}}\hookrightarrow \operatorname {Dist} (G)} induces the algebra homomorphism: U ( g ) → Dist ( G ) . {\displaystyle U({\mathf... | Wikipedia - Distribution on a linear algebraic group | null | null | null |
Let X = {a,b,c} be a set with 3 elements. There are 29 distinct topologies on X but only 9 inequivalent topologies: {∅, {a,b,c}} {∅, {c}, {a,b,c}} {∅, {a,b}, {a,b,c}} {∅, {c}, {a,b}, {a,b,c}} {∅, {c}, {b,c}, {a,b,c}} (T0) {∅, {c}, {a,c}, {b,c}, {a,b,c}} (T0) {∅, {a}, {b}, {a,b}, {a,b,c}} (T0) {∅, {b}, {c}, {a,b}, {b,c}... | Wikipedia - Finite topological space | null | null | null |
Let X = {x1, x2, ...., xn} be any finite set. Suppose Ar = {x1, x2, ..., xr}. Then the collection τ1 = {φ, A1, A2, ..., An = X} will be a topology on X. If τ1, τ2, ..., τm be m such topologies (chain topologies) defined on X, then the structure (X, τ1, τ2, ..., τm) is an m-topological space. == References == | Wikipedia - N-topological space | null | null | null |
Let X and Y be closed connected oriented m-dimensional manifolds. Orientability of a manifold implies that its top homology group is isomorphic to Z. Choosing an orientation means choosing a generator of the top homology group. A continuous map f: X →Y induces a homomorphism f∗ from Hm(X) to Hm(Y). Let , resp. | Wikipedia - Degree of a mapping | null | null | null |
be the chosen generator of Hm(X), resp. Hm(Y) (or the fundamental class of X, Y). | Wikipedia - Degree of a mapping | null | null | null |
Then the degree of f is defined to be f*(). In other words, f ∗ ( ) = deg ( f ) . {\displaystyle f_{*}()=\deg(f)\,.} If y in Y and f −1(y) is a finite set, the degree of f can be computed by considering the m-th local homology groups of X at each point in f −1(y). | Wikipedia - Degree of a mapping | null | null | null |
Let X be a Banach space, L(X) be the bounded operators on X, and σ(T) denote the spectrum of T ∈ L(X). The holomorphic functional calculus is defined as follows: Fix a bounded operator T. Consider the family Hol(T) of complex functions that is holomorphic on some open set G containing σ(T). Let Γ = {γi} be a finite col... | Wikipedia - Jordan canonical form | null | null | null |
The open set G could vary with f and need not be connected. The integral is defined as the limit of the Riemann sums, as in the scalar case. Although the integral makes sense for continuous f, we restrict to holomorphic functions to apply the machinery from classical function theory (for example, the Cauchy integral fo... | Wikipedia - Jordan canonical form | null | null | null |
The assumption that σ(T) lie in the inside of Γ ensures f(T) is well defined; it does not depend on the choice of Γ. The functional calculus is the mapping Φ from Hol(T) to L(X) given by Φ ( f ) = f ( T ) . {\displaystyle \;\Phi (f)=f(T).} We will require the following properties of this functional calculus: Φ extends ... | Wikipedia - Jordan canonical form | null | null | null |
Let X be a CW complex and C n ( X ) {\displaystyle C^{n}(X)} be the singular cochains with coboundary map d n: C n − 1 ( X ) → C n ( X ) {\displaystyle d^{n}:C^{n-1}(X)\to C^{n}(X)} . Then elements of ker d {\displaystyle {\text{ker }}d} are cocycles. Elements of im d {\displaystyle {\text{im }}d} are coboundaries. If ... | Wikipedia - Cocycle | null | null | null |
Let X be a Riemann surface. Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function c: S 1 → X {\displaystyle c:S^{1}\to X} ), we can associate a differential form η c {\displaystyle \eta _{c}} of compact support, the P... | Wikipedia - Intersection number (algebraic geometry) | null | null | null |
Let X be a compact Hausdorff topological space. For any finite open cover C of X, let H(C) be the logarithm (usually to base 2) of the smallest number of elements of C that cover X. For two covers C and D, let C ∨ D {\displaystyle C\vee D} be their (minimal) common refinement, which consists of all the non-empty inters... | Wikipedia - Topological entropy | null | null | null |
Let X be a compact complex manifold with a fixed Hermitian metric, viewed as a positive (1,1)-form ω {\displaystyle \omega } . Following Jean-Pierre Demailly, Thomas Peternell and Michael Schneider, a holomorphic line bundle L on X is said to be nef if for every ϵ > 0 {\displaystyle \epsilon >0} there is a smooth Hermi... | Wikipedia - Nef divisor | null | null | null |
Let X be a complex Banach space, and L(X) denote the family of bounded operators on X. Recall the Cauchy integral formula from classical function theory. Let f: C → C be holomorphic on some open set D ⊂ C, and Γ be a rectifiable Jordan curve in D, that is, a closed curve of finite length without self-intersections. Ass... | Wikipedia - Polynomial functional calculus | null | null | null |
As the resolvent mapping ζ → (ζ−T)−1 is undefined on the spectrum of T, σ(T), the Jordan curve Γ should not intersect σ(T). Now, the resolvent mapping will be holomorphic on the complement of σ(T). So to obtain a non-trivial functional calculus, Γ must enclose (at least part of) σ(T). | Wikipedia - Polynomial functional calculus | null | null | null |
The functional calculus should be well-defined in the sense that f(T) has to be independent of Γ.The full definition of the functional calculus is as follows: For T ∈ L(X), define f ( T ) = 1 2 π i ∫ Γ f ( ζ ) ζ − T d ζ , {\displaystyle f(T)={\frac {1}{2\pi i}}\int \nolimits _{\Gamma }{\frac {f(\zeta )}{\zeta -T}}\,d\z... | Wikipedia - Polynomial functional calculus | null | null | null |
Let X be a connected space and x be a cut point in X such that X\{x}=A|B. Then {x} is either open or closed. if {x} is open, A and B are closed. | Wikipedia - Cut point | null | null | null |
If {x} is closed, A and B are open. Let X be a cut-point space. The set of closed points of X is infinite. | Wikipedia - Cut point | null | null | null |
Let X be a locally compact Hausdorff space, A = C0(X), the commutative C*-algebra of continuous functions that vanish at infinity. Then M(A) is Cb(X), the continuous bounded functions on X. By the Gelfand–Naimark theorem, one has the isomorphism of C*-algebras C b ( X ) ≃ C ( Y ) {\displaystyle C_{b}(X)\simeq C(Y)} whe... | Wikipedia - Multiplier algebra | null | null | null |
Let X be a locally compact Hausdorff space. The space C 0 ( X ) {\displaystyle C_{0}(X)} of complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness) form a commutative C*-algebra C 0 ( X ) {\displaystyle C_{0}(X)} under pointwise multiplication and addition. The inv... | Wikipedia - Commutative C*-algebra | null | null | null |
As does any C*-algebra, C 0 ( X ) {\displaystyle C_{0}(X)} has an approximate identity. In the case of C 0 ( X ) {\displaystyle C_{0}(X)} this is immediate: consider the directed set of compact subsets of X {\displaystyle X} , and for each compact K {\displaystyle K} let f K {\displaystyle f_{K}} be a function of compa... | Wikipedia - Commutative C*-algebra | null | null | null |
Any such sequence of functions { f K } {\displaystyle \{f_{K}\}} is an approximate identity. The Gelfand representation states that every commutative C*-algebra is *-isomorphic to the algebra C 0 ( X ) {\displaystyle C_{0}(X)} , where X {\displaystyle X} is the space of characters equipped with the weak* topology. Furt... | Wikipedia - Commutative C*-algebra | null | null | null |
Let X be a projective variety over a number field K. Let L be a line bundle on X. One defines the Weil height on X with respect to L as follows. First, suppose that L is very ample. A choice of basis of the space Γ ( X , L ) {\displaystyle \Gamma (X,L)} of global sections defines a morphism ϕ from X to projective space... | Wikipedia - Height of a polynomial | null | null | null |
Let X be a random sample from a probability distribution with a real non-negative parameter θ ∈ [ 0 , ∞ ) {\displaystyle \theta \in [0,\infty )} . A CLs upper limit for the parameter θ, with confidence level 1 − α ′ {\displaystyle 1-\alpha '} , is a statistic (i.e., observable random variable) θ u p ( X ) {\displaystyl... | Wikipedia - CLs upper limits | null | null | null |
An equivalent definition can be made by considering a hypothesis test of the null hypothesis H 0: θ = θ 0 {\displaystyle H_{0}:\theta =\theta _{0}} against the alternative H 1: θ = 0 {\displaystyle H_{1}:\theta =0} . Then the numerator in (1), when evaluated at θ 0 {\displaystyle \theta _{0}} , correspond to the type-I... | Wikipedia - CLs upper limits | null | null | null |
This can be interpreted intuitively as saying that θ 0 {\displaystyle \theta _{0}} is excluded because it is α ′ {\displaystyle \alpha '} less likely to observe such an extreme outcome as X when θ 0 {\displaystyle \theta _{0}} is true than it is when the alternative θ = 0 {\displaystyle \theta =0} is true. The calculat... | Wikipedia - CLs upper limits | null | null | null |
Let X be a scheme which is finite type over a field k. An algebraic r-cycle on X is a formal linear combination ∑ n i {\displaystyle \sum n_{i}} of r-dimensional closed integral k-subschemes of X. The coefficient ni is the multiplicity of Vi. The set of all r-cycles is the free abelian group Z r X = ⨁ V ⊆ X Z ⋅ , {\d... | Wikipedia - Algebraic cycles | null | null | null |
A cycle is effective or positive if all its coefficients are non-negative. Closed integral subschemes of X are in one-to-one correspondence with the scheme-theoretic points of X under the map that, in one direction, takes each subscheme to its generic point, and in the other direction, takes each point to the unique re... | Wikipedia - Algebraic cycles | null | null | null |
The cycles rationally equivalent to zero are a subgroup Z r ( X ) rat ⊆ Z r ( X ) {\displaystyle Z_{r}(X)_{\text{rat}}\subseteq Z_{r}(X)} , and the group of r-cycles modulo rational equivalence is the quotient A r ( X ) = Z r ( X ) / Z r ( X ) rat . {\displaystyle A_{r}(X)=Z_{r}(X)/Z_{r}(X)_{\text{rat}}.} This group is... | Wikipedia - Algebraic cycles | null | null | null |
Elements of the group A ∗ ( X ) = ⨁ r A r ( X ) {\displaystyle A_{*}(X)=\bigoplus _{r}A_{r}(X)} are called cycle classes on X. Cycle classes are said to be effective or positive if they can be represented by an effective cycle. If X is smooth, projective, and of pure dimension N, the above groups are sometimes reindexe... | Wikipedia - Algebraic cycles | null | null | null |
In this case, A ∗ X {\displaystyle A^{*}X} is called the Chow ring of X because it has a multiplication operation given by the intersection product. There are several variants of the above definition. We may substitute another ring for integers as our coefficient ring. | Wikipedia - Algebraic cycles | null | null | null |
The case of rational coefficients is widely used. Working with families of cycles over a base, or using cycles in arithmetic situations, requires a relative setup. Let ϕ: X → S {\displaystyle \phi \colon X\to S} , where S is a regular Noetherian scheme. | Wikipedia - Algebraic cycles | null | null | null |
An r-cycle is a formal sum of closed integral subschemes of X whose relative dimension is r; here the relative dimension of Y ⊆ X {\displaystyle Y\subseteq X} is the transcendence degree of k ( Y ) {\displaystyle k(Y)} over k ( ϕ ( Y ) ¯ ) {\displaystyle k({\overline {\phi (Y)}})} minus the codimension of ϕ ( Y ) ¯ {\d... | Wikipedia - Algebraic cycles | null | null | null |
Let X be a scheme. A pre-log structure on X consists of a sheaf of (commutative) monoids M {\displaystyle {\mathcal {M}}} on X together with a homomorphism of monoids α: M → O X {\displaystyle \alpha \colon {\mathcal {M}}\to {\mathcal {O}}_{X}} , where O X {\displaystyle {\mathcal {O}}_{X}} is considered as a monoid un... | Wikipedia - Log structure | null | null | null |
Let X be a set and let R be a ring. Since addition and multiplication are defined in R, we can construct an algebraic structure known as an algebra out of the functions from X to R by defining addition, multiplication, and scalar multiplication of functions to be done pointwise. If RX denotes the set of functions from ... | Wikipedia - Pointwise product | null | null | null |
Let X be a set of v elements. Consider a partition of the 2-element subsets of X into n non-empty subsets, R1, ..., Rn such that: given an x ∈ X {\displaystyle x\in X} , the number of y ∈ X {\displaystyle y\in X} such that { x , y } ∈ R i {\displaystyle \{x,y\}\in R_{i}} depends only on i (and not on x). This number wi... | Wikipedia - Bose–Mesner algebra | null | null | null |
This enhancement permits the parameters i, j, and k to take on the value of zero, and lets some of x,y or z be equal. A set with such an enhanced partition is called an association scheme. One may view an association scheme as a partition of the edges of a complete graph (with vertex set X) into n classes, often though... | Wikipedia - Bose–Mesner algebra | null | null | null |
In this representation, there is a loop at each vertex and all the loops receive the same 0th color. The association scheme can also be represented algebraically. | Wikipedia - Bose–Mesner algebra | null | null | null |
Consider the matrices Di defined by: ( D i ) x , y = { 1 , if ( x , y ) ∈ R i , 0 , otherwise. ( 1 ) {\displaystyle (D_{i})_{x,y}={\begin{cases}1,&{\text{if }}\left(x,y\right)\in R_{i},\\0,&{\text{otherwise. }}\end{cases}}\qquad (1)} Let A {\displaystyle {\mathcal {A}}} be the vector space consisting of all matrices ∑ ... | Wikipedia - Bose–Mesner algebra | null | null | null |
{\displaystyle D_{i}D_{j}=\sum _{k=0}^{n}p_{ij}^{k}D_{k}=D_{j}D_{i},\qquad i,j=0,\ldots ,n.} The (x,y)-th entry of the left side of 4. is the number of two colored paths of length two joining x and y (using "colors" i and j) in the graph. Note that the rows and columns of D i {\displaystyle D_{i}} contain v i {\display... | Wikipedia - Bose–Mesner algebra | null | null | null |
( 2 ) {\displaystyle D_{i}J=JD_{i}=v_{i}J.\qquad (2)} From 1., these matrices are symmetric. From 2., D 0 , … , D n {\displaystyle D_{0},\ldots ,D_{n}} are linearly independent, and the dimension of A {\displaystyle {\mathcal {A}}} is n + 1 {\displaystyle n+1} . From 4., A {\displaystyle {\mathcal {A}}} is closed under... | Wikipedia - Bose–Mesner algebra | null | null | null |
This associative commutative algebra A {\displaystyle {\mathcal {A}}} is called the Bose–Mesner algebra of the association scheme. Since the matrices in A {\displaystyle {\mathcal {A}}} are symmetric and commute with each other, they can be simultaneously diagonalized. This means that there is a matrix S {\displaystyle... | Wikipedia - Bose–Mesner algebra | null | null | null |
This means that A {\displaystyle {\mathcal {A}}} is semi-simple and has a unique basis of primitive idempotents J 0 , … , J n {\displaystyle J_{0},\ldots ,J_{n}} . These are complex n × n matrices satisfying J i 2 = J i , i = 0 , … , n , ( 3 ) {\displaystyle J_{i}^{2}=J_{i},i=0,\ldots ,n,\qquad (3)} J i J k = 0 , i ≠ k... | Wikipedia - Bose–Mesner algebra | null | null | null |
By definition, there exist well-defined complex numbers such that D i = ∑ k = 0 n p i ( k ) E k , ( 6 ) {\displaystyle D_{i}=\sum _{k=0}^{n}p_{i}(k)E_{k},\qquad (6)} and | X | E k = ∑ i = 0 n q k ( i ) D i . ( 7 ) {\displaystyle |X|E_{k}=\sum _{i=0}^{n}q_{k}\left(i\right)D_{i}.\qquad (7)} The p-numbers p i ( k ) {\disp... | Wikipedia - Bose–Mesner algebra | null | null | null |
Let X be a smooth complex projective variety. A complex subvariety Y in X of codimension p defines an element of the cohomology group H 2 p ( X , Z ) {\displaystyle H^{2p}(X,\mathbb {Z} )} . Moreover, the resulting class has a special property: its image in the complex cohomology H 2 p ( X , C ) {\displaystyle H^{2p}(X... | Wikipedia - Hodge decomposition | null | null | null |
(Such a linear combination is called an algebraic cycle on X.) A crucial point is that the Hodge decomposition is a decomposition of cohomology with complex coefficients that usually does not come from a decomposition of cohomology with integral (or rational) coefficients. As a result, the intersection ( H 2 p ( X , Z ... | Wikipedia - Hodge decomposition | null | null | null |
In short, the Hodge conjecture predicts that the possible "shapes" of complex subvarieties of X (as described by cohomology) are determined by the Hodge structure of X (the combination of integral cohomology with the Hodge decomposition of complex cohomology). The Lefschetz (1,1)-theorem says that the Hodge conjecture ... | Wikipedia - Hodge decomposition | null | null | null |
In particular, definite integrals of algebraic functions, known as periods, can be transcendental numbers. The difficulty of the Hodge conjecture reflects the lack of understanding of such integrals in general. Example: For a smooth complex projective K3 surface X, the group H2(X, Z) is isomorphic to Z22, and H1,1(X) i... | Wikipedia - Hodge decomposition | null | null | null |
Their intersection can have rank anywhere between 1 and 20; this rank is called the Picard number of X. The moduli space of all projective K3 surfaces has a countably infinite set of components, each of complex dimension 19. The subspace of K3 surfaces with Picard number a has dimension 20−a. (Thus, for most projective... | Wikipedia - Hodge decomposition | null | null | null |
This example suggests several different roles played by Hodge theory in complex algebraic geometry. First, Hodge theory gives restrictions on which topological spaces can have the structure of a smooth complex projective variety. Second, Hodge theory gives information about the moduli space of smooth complex projective... | Wikipedia - Hodge decomposition | null | null | null |
The best case is when the Torelli theorem holds, meaning that the variety is determined up to isomorphism by its Hodge structure. Finally, Hodge theory gives information about the Chow group of algebraic cycles on a given variety. The Hodge conjecture is about the image of the cycle map from Chow groups to ordinary coh... | Wikipedia - Hodge decomposition | null | null | null |
Let X be a smooth projective variety over a number field. The Bloch-Kato conjecture on values of L-functions predicts that the order of vanishing of an L-function of X at an integer point is equal to the rank of a suitable motivic cohomology group. This is one of the central problems of number theory, incorporating ear... | Wikipedia - Motivic cohomology | null | null | null |
Let X be a smooth projective variety where all of its irreducible components have dimension n. In this situation, the canonical sheaf ωX, defined as the sheaf of Kähler differentials of top degree (i.e., algebraic n-forms), is a line bundle. | Wikipedia - Projective algebraic variety | null | null | null |
Let X be a smooth variety of dimension n over a field k. Define the canonical line bundle K X {\displaystyle K_{X}} to be the bundle of n-forms on X, the top exterior power of the cotangent bundle: K X = Ω X n = ⋀ n ( T ∗ X ) . {\displaystyle K_{X}=\Omega _{X}^{n}={\bigwedge }^{n}(T^{*}X).} Suppose in addition that X i... | Wikipedia - Serre duality | null | null | null |
It follows that the dimensions of the two cohomology groups are equal: h i ( X , E ) = h n − i ( X , K X ⊗ E ∗ ) . {\displaystyle h^{i}(X,E)=h^{n-i}(X,K_{X}\otimes E^{\ast }).} As in Poincaré duality, the isomorphism in Serre duality comes from the cup product in sheaf cohomology. | Wikipedia - Serre duality | null | null | null |
Namely, the composition of the cup product with a natural trace map on H n ( X , K X ) {\displaystyle H^{n}(X,K_{X})} is a perfect pairing: H i ( X , E ) × H n − i ( X , K X ⊗ E ∗ ) → H n ( X , K X ) → k . {\displaystyle H^{i}(X,E)\times H^{n-i}(X,K_{X}\otimes E^{\ast })\to H^{n}(X,K_{X})\to k.} The trace map is the an... | Wikipedia - Serre duality | null | null | null |
Let X be a split quadric over a field k. (In particular, X can be any smooth quadric over an algebraically closed field.) In low dimensions, X and the linear spaces it contains can be described as follows. | Wikipedia - Quadric (algebraic geometry) | null | null | null |
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