classical modular curve 模曲线
For a congruence subgroup $\Gamma$, let $\mathcal H=\{\tau\in\mathbb C:\operatorname{Im}\tau>0\}$. The modular curves are the Riemann surface $Y(\Gamma)=\Gamma\backslash\mathcal H$ and its compactification $X(\Gamma)$…
Terminology & Translations
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clique 团
A clique in a simple graph is a vertex subset in which every pair of distinct vertices is adjacent, so its induced subgraph is complete.
closed graph theorem 闭图像定理
An everywhere-defined linear map T between Banach spaces is bounded if its graph {(x,Tx)} is closed in their product norm topology.
closed manifold 闭流形
A compact topological manifold without boundary; manifolds here are Hausdorff and second countable.
closed map 闭映射
A map between topological spaces that sends every closed subset of its domain to a closed subset of its codomain; it need not be continuous or surjective merely by this definition.
closed subset 闭集
A subset whose complement in the specified ambient topological space is open.
closure of a subset 闭包
The intersection of all closed subsets of the ambient space containing the given subset.
cocountable topology 余可数拓扑
The topology on a set whose open subsets are the empty set together with subsets having at most countable complement; finite complements are allowed.
cofinite topology 余有限拓扑
The topology on a set whose open subsets are the empty set together with subsets having finite complement.
commutative local ring 交换局部环
A nonzero commutative ring with a unique maximal ideal.
commutative ring 交换环
A unital ring whose multiplication satisfies $ab=ba$ for all elements.
compact linear operator 紧算子
A linear map between Banach spaces is compact if the image of every bounded set has compact closure in the codomain.
compactness 紧致性
The property that every open cover of a topological space admits a finite subcover; Hausdorff separation is not imposed here.
complete graph 完全图
A complete graph K_n, for n ≥ 0, is a simple graph on n vertices with an edge between every pair of distinct vertices.
complete measure space 完备测度空间
A measure space is complete if every subset of each measurable null set is measurable.
complete residue system 完全剩余系
For $m\ge1$, a set of integers containing exactly one representative of every congruence class modulo $m$.
complex contour integral 复积分
For continuous f along a piecewise C^1 path gamma:[a,b] to C, its contour integral is integral_a^b f(gamma(t)) gamma'(t) dt, summed over the smooth pieces.
composite number 合数
An integer $n>1$ having a positive divisor strictly between $1$ and $n$.
conditional expectation 条件期望
For integrable $X$ and a sub-sigma-algebra $\mathcal G$, an integrable $\mathcal G$-measurable variable $Y$ satisfying $\int_A Y\,dP=\int_A X\,dP$ for every $A\in\mathcal G$; $Y$ is unique almost surely.
conditional probability 条件概率
For events $A,B$ with $P(B)>0$, the probability of $A$ given $B$ is $P(A\mid B)=P(A\cap B)/P(B)$.
conditionally independent events 条件独立
For an event $C$ with $P(C)>0$, events $A,B$ are independent given $C$ when $P(A\cap B\mid C)=P(A\mid C)P(B\mid C)$.
conductor of a Dirichlet character 狄利克雷特征的导子
For a character modulo $N$, the least positive divisor $f$ of $N$ from whose character group it is induced by reduction of units.
congruence subgroup of the modular group 模群的同余子群
A subgroup of $\operatorname{SL}_2(\mathbb Z)$ containing $\Gamma(N)=\{\gamma:\gamma\equiv I\pmod N\}$ for some $N\ge1$.
conjunction 合取
The connective producing a statement true exactly when both component statements are true.
connected component 连通分支
A maximal connected subset of a topological space.