quadratic number field 二次数域
A degree-two extension of $\mathbb Q$, equivalently $\mathbb Q(\sqrt d)$ for a squarefree integer $d\ne0,1$.
Terminology & Translations
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quotient group 商群
For a normal subgroup $N$ of $G$, the group of cosets with $(gN)(hN)=ghN$.
quotient map 商映射
A surjective map q of topological spaces such that a subset of the target is open exactly when its inverse image under q is open.
quotient module 商模
For a submodule $N\le M$, the additive quotient $M/N$ with scalar action $r(m+N)=rm+N$.
quotient ring 商环
For an ideal $I$ of a commutative ring $R$, the cosets $R/I$ with addition and multiplication induced by representatives.
quotient set 商集
The set of equivalence classes of a specified equivalence relation on a set X.
quotient topology 商拓扑
For a surjection q from a topological space X onto a set Y, the topology in which U is open in Y exactly when its inverse image under q is open in X.
ramification index of a local number-field extension 局部数域扩张的分歧指数
For a finite extension $L/K$ of finite extensions of $\mathbb Q_p$, the integer $e$ determined by $\mathfrak m_K\mathcal O_L=\mathfrak m_L^e$.
random variable 随机变量
A real-valued measurable map $X:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ on a probability space.
rank of a linear map 线性映射的秩
The dimension of the image of a linear map, measured over its scalar field.
rank–nullity theorem 维数公式
For a linear map $T:V\to W$ with $V$ finite-dimensional, $\dim V=\dim\ker T+\dim\operatorname{im}T$.
real projective space 实射影空间
For a nonnegative integer n, the space of one-dimensional real linear subspaces of real (n+1)-space, topologized as the quotient of nonzero vectors by multiplication by nonzero real scalars.
recurrence relation 递推关系
A recurrence relation specifies each term of a sequence after a stated starting index in terms of earlier terms; an associated recursive specification also gives enough initial data to determine the sequence.
reduced residue system 简化剩余系
For $m\ge2$, a set of integers representing each invertible congruence class modulo $m$ exactly once.
regular covering space 正则复迭空间
A covering map for which the deck-transformation group acts transitively on each fiber, so any two points over the same base point are related by a deck transformation.
regular space T3 分离公理
A space in which a point and a disjoint closed subset can be enclosed in disjoint open sets; T1 is not included in this record.
removable complex singularity 可去奇点
An isolated singularity a of a holomorphic f is removable if a value can be assigned at a so that f extends holomorphically to the whole disk.
removable discontinuity 可去间断点
At an accumulation point a of a real domain, a finite punctured limit exists but the function is absent or has a different value at a.
representable functor 可表函子
A set-valued functor naturally isomorphic to Hom(-,A) in the contravariant case, or Hom(A,-) in the covariant case.
residue at an isolated singularity 留数
If f is holomorphic on a punctured disk centered at a, its residue at a is the coefficient of (z-a)^(-1) in the Laurent expansion on that disk.
residue degree of a local number-field extension 局部数域扩张的惯性指数
For a finite extension $L/K$ of $p$-adic local fields, $f(L/K)=[\mathcal O_L/\mathfrak m_L:\mathcal O_K/\mathfrak m_K]$.
retraction onto a subspace 收缩
For a subspace A of a topological space X, a continuous map from X to A whose restriction to A is the identity.
Riemann curvature tensor 曲率张量
For a smooth Riemannian metric with Levi–Civita connection, the tensor R(X,Y)Z = ∇X∇Y Z − ∇Y∇X Z − ∇[X,Y]Z; pairing its output with the metric gives its covariant four-tensor form.
Riemann integral 黎曼积分
For a bounded real function on [a,b], the Riemann integral is the common value of the supremum of its lower partition sums and the infimum of its upper partition sums, when those values agree.
Riemann surface 黎曼曲面
A connected Hausdorff, second countable topological surface without boundary equipped with an atlas to open subsets of the complex plane whose transition maps are holomorphic; equivalently, a connected complex manifold…