Additive Combinatorics 加法组合学
Additive combinatorics studies arithmetic patterns and additive structure in finite sets, including inverse questions about small sumsets.
Mathematics Encyclopedia
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Adjunction 伴随对
An adjunction identifies maps out of one functor’s value with maps into another’s value, naturally in both variables.
Algebraic Space
Algebraic spaces extend schemes by allowing scheme charts in the étale topology. Use the representable diagonal and the surjective étale chart in the definition as a checklist, then check how schemes fit into the resulting category.
Algebraic Stack
Algebraic stacks describe families of geometric objects while retaining their automorphisms. Start with the three conditions in the definition and identify the separate roles of descent, the diagonal, and a smooth scheme atlas.
Almost Sure Convergence 几乎必然收敛
Almost sure convergence means pointwise convergence on one event of probability one and differs from other probabilistic limits.
Arzelà–Ascoli Theorem
Uniform boundedness and equicontinuity provide convergent subsequences in spaces of continuous functions on compact domains. Compare this criterion with mere pointwise boundedness to see what controls convergence uniformly.
Banach Space 巴拿赫空间
A Banach space is a normed vector space complete for the distance induced by its norm.
Banach–Alaoglu Theorem
The closed dual unit ball is compact for pointwise evaluation on the original normed space. The separate sequential theorem identifies where separability enters arguments that extract convergent subsequences.
binomial random graph 随机图
Choosing each edge independently gives a graph model in which subgraph counts can be estimated by expectation. The deletion argument converts those estimates into deterministic graphs avoiding a prescribed subgraph.
Borel–Cantelli Lemmas
Summable event probabilities force only finitely many occurrences almost surely. With independence, a divergent sum gives infinitely many occurrences; the repeated-event example explains why that hypothesis cannot simply be omitted.
Brownian Motion 布朗运动
Standard Brownian motion has continuous paths and independent Gaussian increments with variance equal to elapsed time.
Cardinality of a Set 集合的基数
Cardinality measures set size by bijections, with injections providing a comparison of sizes.
Categorical Limit 极限
A categorical limit is a universal cone through which every compatible cone factors uniquely.
Categorical Product 积
A product turns any compatible family of component maps into a unique map to one object.
Categorical Pullback 拉回
A pullback combines two maps to the same target while requiring their images to agree.
Category 范畴
A category specifies objects, morphisms and associative composition, including an identity at each object.
cellular homology 胞腔同调
Cellular homology computes singular homology using generators indexed by cells. Track how attaching-map degrees determine boundary maps, making a finite CW decomposition into an explicit chain computation.
Central Limit Theorem 中心极限定理
For iid observations with finite positive variance, the centered sum divided by its standard deviation converges in distribution to a standard normal law.
Chain Complex of Modules 模的链复形
A chain complex is a sequence of modules with degree-lowering differentials whose consecutive composites vanish, allowing homology to measure failure of exactness.
Characteristic Function of a Probability Distribution 特征函数
The characteristic function is the expected complex exponential of a random variable and uniquely determines its probability law.
Chebotarev Density Theorem
Frobenius conjugacy classes control how often unramified primes exhibit each splitting behavior in a finite Galois extension. Polynomial examples translate the density statement into factorization patterns modulo primes.
Chinese Remainder Theorem 中国剩余定理
Congruences with pairwise coprime moduli can be solved independently and recombined into one residue class.
Chow Group
Chow groups turn algebraic cycles into invariants by identifying cycles related by rational functions on larger subvarieties. Start with the precise rational-equivalence relation, then use the localization sequence to see how an open subset and its complement contribute.
Chromatic Number 色数
The chromatic number of a finite simple graph is the minimum number of colors needed to give adjacent vertices different colors.
Commutative Ring 交换环
A commutative ring supports addition, subtraction, and commutative multiplication with identity, without requiring division or cancellation.