binomial random graph 随机图
For n ≥ 0 and 0 ≤ p ≤ 1, G(n,p) is the random simple graph on a fixed labeled n-vertex set in which each possible edge is included independently with probability p.
Terminology & Translations
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bipartite graph 二部图
A simple graph is bipartite if its vertex set admits a partition into two disjoint parts such that every edge has one endpoint in each part.
Birkhoff pointwise ergodic theorem Birkhoff遍历定理
For a probability-preserving $T$ and $f\in L^1(\mu)$, the averages $n^{-1}\sum_{j=0}^{n-1}f(T^jx)$ converge almost everywhere to $E[f\mid\mathcal I]$, with $\mathcal I$ the invariant sigma-algebra; for ergodic $T$ the…
bound variable occurrence 约束出现
An occurrence of a variable within the scope of a quantifier for that variable, bound by the nearest such quantifier.
bounded linear operator 有界线性算子
A linear map T between normed spaces is bounded when some finite C satisfies norm(Tx)<=C norm(x) for every domain vector x.
cardinality of a set 基数
In ZFC, the least ordinal bijective with a set; equality of cardinalities is equivalently existence of a bijection.
Cartesian product 笛卡尔积
For sets A and B, the set of ordered pairs with first coordinate in A and second in B.
categorical colimit 余极限
For a small diagram, a cocone from which every compatible cocone factors by a unique map of vertices respecting all legs.
categorical coproduct 余积
An object with maps from a specified family such that each family of maps into any Y extends uniquely to a map from that object to Y.
categorical equalizer 等化子
For parallel f,g:A→B, a map e:E→A with fe=ge universal among maps with that equality.
categorical limit 极限
For a small diagram D:J→C, a cone through which every other cone factors by a unique vertex map respecting all legs.
categorical product 积
Objects P with projections to a specified family such that every matching family of maps from any X arises uniquely from a map X→P.
categorical pullback 拉回
For A→C←B, an object P with maps to A and B making the square commute and universal among such commuting pairs.
category 范畴
Objects and directed morphisms with specified sources, targets, associative composition and a two-sided identity at each object.
Cauchy integral formula 柯西积分公式
If f is holomorphic near the closed disk of radius r centered at a, then for every z in that disk's interior, f(z)=(2 pi i)^(-1) integral_{|w-a|=r} f(w)/(w-z) dw with counterclockwise orientation.
Cauchy sequence 基本列
A sequence x_n in a metric space is Cauchy if for every epsilon > 0 some N makes d(x_m,x_n) = N.
Cauchy–Riemann equations 柯西-黎曼方程
For f=u+iv on a plane domain, the Cauchy–Riemann equations are u_x=v_y and u_y=-v_x. Under C^1 regularity they characterize holomorphicity.
cellular homology 胞腔同调
For a CW complex, the homology of the chain complex with C_n=H_n(X^n,X^{n-1};Z), whose boundary is formed from the relative connecting map followed by projection to the next skeletal pair.
central limit theorem 中心极限定理
The classical iid central limit theorem states that for real iid $X_j$ with mean $\mu$ and variance $0<\sigma^2<\infty$, $(\sum_{j=1}^n X_j-n\mu)/(\sigma\sqrt n)$ converges in distribution to $N(0,1)$.
chain complex of modules 模的链复形
Modules $C_n$ and maps $d_n:C_n\to C_{n-1}$ with $d_{n-1}d_n=0$ for every integer $n$.
characteristic function of a probability distribution 特征函数
For a real random variable $X$, the function $\varphi_X(t)=E[e^{itX}]$ for $t\in\mathbb R$, using this sign convention.
characteristic polynomial of a linear map 线性映射的特征多项式
For an endomorphism of an $n$-dimensional $F$-space, the basis-independent monic polynomial $\det(xI-A)$.
Chinese remainder theorem for integers 整数中国剩余定理
For pairwise coprime positive integers $m_1,\ldots,m_r$, any prescribed residues have a simultaneous integer solution unique modulo $\prod_i m_i$.
chromatic number 色数
The chromatic number χ(G) of a finite simple graph is the least nonnegative integer k admitting a vertex coloring with k colors in which adjacent vertices receive different colors.
class number of a number field 数域的类数
The finite cardinality $h_K=|\operatorname{Cl}(K)|$ of the ideal class group of a number field $K$.