abelian group 交换群
A group whose operation satisfies $ab=ba$ for every pair of elements.
Terminology & Translations
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absolute convergence of a numerical series 绝对收敛
A real or complex numerical series sum a_n is absolutely convergent when the nonnegative series sum |a_n| has a finite sum.
adapted stochastic process 适应过程
Relative to a filtration $(\mathcal F_t)$, a process $(X_t)$ is adapted when $X_t$ is $\mathcal F_t$-measurable at every time $t$.
addition principle 加法原理
For finitely many pairwise disjoint finite sets, the cardinality of their union equals the sum of their cardinalities.
additive combinatorics 加法组合学
Additive combinatorics studies arithmetic configurations, additive growth, and structural inverse questions for finite sets, especially in integers and abelian groups.
adele ring of a number field 数域的阿代尔环
For a number field $K$, the ring of tuples $(x_v)\in\prod_v K_v$ such that $x_v\in\mathcal O_v$ at all but finitely many finite places, with the restricted product topology.
adjunction 伴随对
Functors F:C→D and G:D→C together with bijections Hom_D(FX,Y)≅Hom_C(X,GY) natural in X and Y.
algebraic element over a field 代数元
An element $\alpha$ of an extension $L/K$ annihilated by a nonzero polynomial in $K[x]$.
algebraic integer 代数整数
A complex number satisfying a monic polynomial with coefficients in $\mathbb Z$.
almost complex structure 近复结构
A smooth real bundle endomorphism of the tangent bundle of a smooth manifold whose square is minus the identity on every tangent space.
almost everywhere 几乎处处
A property holds mu-almost everywhere on X when there is a measurable null set N such that the property holds at every point of X minus N.
almost sure convergence 几乎必然收敛
On one probability space, $X_n\to X$ almost surely if $P(\{\omega:\lim_nX_n(\omega)=X(\omega)\})=1$.
alpha-limit set α极限点集
For a homeomorphism $T$, $\alpha(x)$ is the set of limits of $T^{-n_k}x$ along sequences $n_k\to\infty$.
annihilator of a module 模的零化子
For a module $M$ over a commutative ring $R$, the ideal $\{r:rM=0\}$.
antiderivative of a real function 原函数
An antiderivative of f on an open real interval I is a differentiable function F on I with F'(x)=f(x) at every point of I.
argument principle 辐角原理
For a meromorphic f near the closure of a bounded domain with positively oriented piecewise smooth boundary, if f has neither zeros nor poles on the boundary, (2 pi i)^(-1) integral f'/f equals the number of interior…
atomic formula 原子公式
A first-order formula obtained by applying a relation symbol to terms, or by equating two terms when equality is present.
augmenting path for a matching 增广路
For a matching M in a finite bipartite graph, an augmenting path is a path of positive length whose edges alternate outside and inside M and whose two endpoints are unmatched; switching membership along it increases…
Banach space Banach空间
A Banach space is a real or complex normed vector space in which every Cauchy sequence for the norm distance converges to an element of the space.
basis for a topology 拓扑基
A family of open subsets such that every open subset is a union of members of that family.
basis of a vector space 向量空间的基
A linearly independent family spanning the vector space by finite linear combinations.
Bernoulli distribution Bernoulli分布
A distribution supported on $\{0,1\}$ with $P(X=1)=p$ and $P(X=0)=1-p$, where $0\le p\le1$.
bijective function 双射
A function that is both injective and surjective, equivalently one admitting a two-sided inverse.
binomial coefficient 二项式系数
For integers n ≥ 0 and 0 ≤ k ≤ n, the binomial coefficient counts the k-element subsets of an n-element set and equals n!/(k!(n−k)!). It is defined as zero for integer k outside that range.
binomial distribution 二项分布
For an integer $n\ge0$ and $0\le p\le1$, the law of the sum of $n$ independent Bernoulli$(p)$ variables; its mass is $\binom nk p^k(1-p)^{n-k}$ for $0\le k\le n$.