multiplication principle 乘法原理
If a finite construction has k stages and every valid prefix has exactly n_i continuations at stage i, its number of completed outcomes is the product of the n_i, provided construction histories uniquely specify…
Terminology & Translations
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natural isomorphism 自然同构
A natural transformation all of whose components are isomorphisms, equivalently an invertible morphism in the functor category.
natural transformation 自然变换
For $F,G:C\to D$, components $\eta_X:F(X)\to G(X)$ satisfying $G(f)\eta_X=\eta_YF(f)$ for every $f:X\to Y$.
negation 否定
In classical semantics, the connective reversing a statement’s truth value.
neighborhood basis 邻域基
At a point of a topological space, a family of neighborhoods such that every neighborhood of that point contains a member of the family; here a neighborhood may be any set containing an open set that contains the point.
newform in the classical cuspidal sense 新形式
A normalized Hecke eigenform in $S_k(\Gamma_1(N))$ lying in the Petersson orthogonal complement of the old subspace generated by degeneracy images from proper divisor levels.
Noetherian commutative ring Noether 环
A commutative ring in which every ascending chain of ideals stabilizes, equivalently every ideal is finitely generated.
nonwandering point 非游荡点
For a continuous self-map $T$, a point $x$ such that every neighborhood $U$ of $x$ has $T^n(U)\cap U\ne\varnothing$ for some integer $n\ge1$.
norm of a number-field element 数域元素的范数
For $\alpha\in K$ with $K$ a number field, $N_{K/\mathbb Q}(\alpha)$ is the determinant of multiplication by $\alpha$ on the $\mathbb Q$-vector space $K$.
norm on a vector space 范数
A norm on a real or complex vector space is a nonnegative real function N with N(x)=0 exactly for x=0, N(alpha x)=|alpha|N(x), and N(x+y)<=N(x)+N(y).
normal algebraic extension 正规扩张
An algebraic extension $L/K$ such that every irreducible polynomial over $K$ having a root in $L$ splits completely in $L$.
normal distribution 正态分布
For $\mu\in\mathbb R$ and $\sigma>0$, the law with density $(\sigma\sqrt{2\pi})^{-1}\exp(-(x-\mu)^2/(2\sigma^2))$ on $\mathbb R$.
normal space T4 分离公理
A space in which any two disjoint closed subsets have disjoint open neighborhoods; T1 is not included here.
normal subgroup 正规子群
A subgroup $N\le G$ satisfying $gNg^{-1}=N$ for every $g\in G$.
normalized full-level holomorphic Eisenstein series 全模群的正规化艾森斯坦级数
For even $k\ge4$ and $\operatorname{Im}\tau>0$, $E_k(\tau)=\tfrac12\sum_{(c,d)\in\mathbb Z^2,\ \gcd(c,d)=1}(c\tau+d)^{-k}$, an absolutely convergent weight-$k$ modular form with constant term $1$.
normalized Hecke eigenform 正规化 Hecke 特征形式
A nonzero modular form on $\Gamma_1(N)$ that is a simultaneous eigenvector for its Hecke algebra and has Fourier coefficient $a_1=1$ at infinity.
normed vector space 赋范线性空间
A normed vector space is a real or complex vector space equipped with a specified norm, inducing the distance d(x,y)=norm(x-y).
null set 零测集
A null set for a specified measure mu is a measurable set N with mu(N)=0.
null-homotopic map 零伦
A continuous map homotopic to a constant map, with any required based or relative condition stated separately.
number field 数域
A field $K$ containing $\mathbb Q$ with finite degree $[K:\mathbb Q]$.
omega-limit set ω极限点集
For a continuous map $T$ on a metric space, $\omega(x)$ is the set of limits of $T^{n_k}x$ along sequences $n_k\to\infty$.
one-point compactification 一点紧化
For a noncompact locally compact Hausdorff space X, the compact Hausdorff space formed by adding infinity, whose neighborhoods have compact closed complements in X.
open map 开映射
A map between topological spaces that sends every open subset of its domain to an open subset of its codomain; continuity and surjectivity are not included in the definition.
open mapping theorem for Banach spaces 开映射定理
A bounded surjective linear map between Banach spaces sends every open set in its domain to an open set in its codomain.
open neighborhood 邻域
An open subset of a topological space containing the specified point.