full functor 全函子
A functor inducing a surjective map on each hom-set for a fixed pair of source objects.
Terminology & Translations
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function between sets 映射
For specified sets A and B, an assignment giving each member of A exactly one member of B.
functor 函子
An assignment of objects and arrows between categories that respects source, target, identities and composition.
functor category 函子范畴
For small C and locally small D, the category whose objects are functors C→D and whose morphisms are natural transformations.
fundamental group 基本群
The group of loops at a chosen point modulo homotopy fixing the endpoints, with multiplication given here by traversing the first loop then the second.
fundamental theorem of arithmetic 算术基本定理
Every integer greater than one is a finite product of positive primes, unique up to the order of its factors.
fundamental theorem of calculus 微积分基本定理
For continuous real f on [a,b], the function G(x)=integral_a^x f(t) dt has derivative f(x) in (a,b); if F is any antiderivative of f on that interval, continuous at the endpoints, then integral_a^b f=F(b)-F(a).
fundamental unit of a real quadratic field 实二次域的基本单位
For a real quadratic field $K$, a unit $\varepsilon$ generating $\mathcal O_K^\times/\{\pm1\}$, so every unit is $\pm\varepsilon^n$ for some $n\in\mathbb Z$.
Galois extension Galois 扩张
An algebraic field extension that is both normal and separable.
Gauss sum of a Dirichlet character 狄利克雷特征的高斯和
For a Dirichlet character $\chi$ modulo $N$, $\tau(\chi)=\sum_{a\bmod N}\chi(a)e^{2\pi i a/N}$.
Gaussian process 高斯过程
A real stochastic process for which every finite vector of observations has a multivariate normal distribution, allowing degenerate Gaussian vectors.
genus of a closed orientable surface 亏格
For a connected closed orientable surface, the nonnegative integer g determined by its Euler characteristic through chi=2-2g; equivalently, the number of torus summands in its classification, with the sphere having…
gradient of a real function 梯度
For a differentiable real scalar function f on an open subset of Euclidean R^n, its gradient at a is the vector of its coordinate partial derivatives, satisfying Df(a)h=gradient f(a) dot h.
graph isomorphism 图的同构
An isomorphism of simple graphs G=(V,E) and H=(W,F) is a bijection f:V→W such that uv is in E if and only if f(u)f(v) is in F for each distinct u,v.
greatest common divisor of integers 整数的最大公因数
For integers $a,b$ not both zero, the positive common divisor divisible by every common divisor; set $\gcd(0,0)=0$ by convention.
group 群
A set with an associative binary operation, an identity, and a two-sided inverse for every element.
group action 群作用
A map $G\times X\to X$ with $ex=x$ and $(gh)x=g(hx)$; this is a left action.
group homomorphism 群同态
A map $f:G\to H$ between groups satisfying $f(ab)=f(a)f(b)$ for all $a,b$.
groupoid 广群
A category in which every morphism is invertible.
Hahn–Banach extension theorem Hahn-Banach定理
A bounded linear scalar functional on any linear subspace of a real or complex normed space extends to the whole space with the same norm.
Hall’s marriage theorem Hall定理
A finite bipartite graph with parts L and R has a matching saturating L if and only if every S ⊆ L has at least |S| distinct neighbors in R.
Hamiltonian Floer homology Hamilton 系统周期解的 Floer 同调
In the closed symplectic setting where the symplectic form and first Chern class vanish on sphere classes, the homology over the field with two elements of the complex generated by nondegenerate contractible…
Hamiltonian vector field Hamilton 矢量场
For a smooth real function H on a symplectic manifold with form omega, the unique smooth vector field X_H satisfying contraction of omega with X_H equals dH; this entry uses the positive-dH sign convention.
harmonic function 调和函数
A real or complex C^2 function u on an open subset of R^n is harmonic when its Laplacian, the sum of the coordinate second derivatives, vanishes throughout the domain.
Hausdorff space Hausdorff 空间
A topological space in which any two distinct points have disjoint open neighborhoods.